text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
\section{Introduction}
The affine Coxeter arrangement of an irreducible crystallographic
root system $\Phi\subset V\simeq \mathbb{R}^r$ is
obtained by taking all integer affine translations
$H_{\alpha,k} = \{x\in V\mid (\alpha,x) = k\}$, $\alpha\in\Phi$, $k\in{\mathbb Z}$,
of the hyperplanes perpendicular to the roots. The regions of the affine
Coxeter arrangements are simplices called alcoves. They are in a one-to-one
correspondence with elements of the associated affine Weyl group.
We define an {\it alcoved polytope\/} ${\mathcal P}$ as a convex polytope that
is the union of several alcoves. In other words, an alcoved polytope is the
intersection of some half-spaces bounded by the hyperplanes $H_{\alpha,k}$:
$$
{\mathcal P}= \{x\in V \mid b_\alpha\leq (\alpha,x)\leq c_\alpha,\ \alpha\in\Phi\},
$$
where $b_\alpha$ and $c_\alpha$ are some integer parameters.
These polytopes come naturally equipped with coherent
triangulations into alcoves. Alcoved polytopes include many
interesting classes of polytopes: hypersimplices, order polytopes,
some special matroid polytopes, Fomin-Zelevinsky's generalized associahedra,
and many others.
This is the first of two papers about alcoved polytopes. In this paper,
we concentrate on alcoved polytopes of the Lie type $A$ case
and on related combinatorial objects. In~\cite{AP2}, we will
treat the general case of an arbitrary root system.
Hypersimplices are integer polytopes which appear in algebraic and geometric
contexts. For example, they are moment polytopes for torus actions on
Grassmannians. They are also weight polytopes of the fundamental
representations of the general linear group $GL_n$.
The $(k,n)$-th hypersimplex can be defined as the slice of the hypercube
$[0,1]^{n-1}$ located between the two hyperplanes $\sum x_i = k-1$
and $\sum x_i = k$. It is well-know that the normalized volume of this
hypersimplex equals the Eulerian number $A_{k,n-1}$,
i.e., the number of permutations of size $n-1$ with $k-1$ descents.
Stanley~\cite{Sta1} explained this fact by
constructing a triangulation of the hypersimplex into $A_{k,n-1}$ unit
simplices.
Another construction of a triangulation of the hypersimplex was given by
Sturmfels~\cite{Stu}. It naturally appears in the context of Gr\"obner bases.
These two constructions of triangulations are quite different.
In Section~\ref{sec:hypersimplex},
we compare these triangulations and show that they are actually
identical to each other and that they can be naturally described in terms of
alcoved polytopes.\footnote{C.~Haase~\cite{Haa} reported to us that he also
discovered this equivalence (unpublished). G.~Ziegler~\cite{Zie} reported that
the alcove triangulation of the hypersimplex appeared as an example in his
1997/98 class on triangulations.} In Section~\ref{sec:triang_alcove},
we extend the
descriptions of this triangulation to general alcoved polytopes, and give a
formula for the volume of an alcoved polytope. In
Sections~\ref{sec:matroid}--\ref{sec:multihyper}, we study in detail three
examples of alcoved polytopes: the matroid polytopes, the second hypersimplex
and the multi-hypersimplices.
In the second part~\cite{AP2} of this paper, we will extend the
hypersimplices to all Lie types and calculate their volumes.
We will prove a general theorem on volumes of alcoved polytopes.
We will give uniform generalizations of the descent and major index statistics, appropriate for our geometric
approach.
Some alcoved polytopes have also been studied from a more
algebraic perspective earlier; see~\cite{KKMS,BGT}.
\section{Four triangulations of the hypersimplex}
\label{sec:hypersimplex}
Let us fix integers $0< k < n$. Let $[n]:=\{1,\dots,n\}$ and
$\binom{[n]}{k}$
denote the collection of $k$-element subsets of $[n]$.
To each $k$-subset $I\in\binom{[n]}{k}$
we associate the $01$-vector $\epsilon_I =
(\epsilon_1,\dots,\epsilon_n)$ such that $\epsilon_i=1$, for $i\in I$;
and $\epsilon_i = 0$, for $i\not\in I$.
The {\it hypersimplex\/} $\Delta_{k,n}\subset\mathbb{R}^n$ is the convex polytope
defined as the convex hull of the points $\epsilon_I$,
for $I\in\binom{[n]}{k}$.
All these $\binom{n}{k}$ points are actually vertices
of the hypersimplex because they are obtained from each other
by permutations of the coordinates.
This $(n-1)$-dimensional polytope can also be defined as
$$
\Delta_{k,n} = \{(x_1,\dots,x_n)\mid 0\leq x_1,\dots,x_n\leq 1;\
x_1+\cdots + x_{n} =k\}.
$$
The hypersimplex is linearly equivalent to the
polytope $\tilde\Delta_{k,n}\subset\mathbb{R}^{n-1}$ given by
\[
\tilde\Delta_{k,n}=\{(x_1,\dots,x_{n-1})\mid 0\leq x_1,\dots,x_{n-1}\leq 1;\
k-1 \leq x_1 + \cdots + x_{n-1} \leq k\}.
\]
Indeed, the projection $p:(x_1,\dots,x_n)\mapsto (x_1,\dots,x_{n-1})$
sends $\Delta_{k,n}$ to $\tilde\Delta_{k,n}$.
The hypersimplex $\tilde\Delta_{k,n}$ can be thought of as
the region (slice) of the unit hypercube $[0,1]^{n-1}$
contained between the two hyperplanes $\sum x_i = k-1$ and $\sum x_i = k$.
Recall that a {\it descent\/} in a permutation $w\in S_n$ is an
index $i\in\{1,\dots,n-1\}$ such that $w(i)>w(i+1)$. Let $\mathrm{des}(w)$
denote the number of descents in $w$. The {\it Eulerian number\/}
$A_{k,n}$ is the number of permutations in $S_n$ with $\mathrm{des}(w)=k-1$
descents.
Let us normalize the volume form in $\mathbb{R}^{n-1}$ so that the volume of
a unit simplex is 1 and, thus, the volume of a unit hypercube is
$(n-1)!$. It is a classical result, implicit in the work of
Laplace~\cite[p.~257ff]{Lap},
that the normalized volume of the hypersimplex $\Delta_{k,n}$ equals
the Eulerian number $A_{k,n-1}$. One would like to present a
triangulation of $\Delta_{k,n}$ into $A_{k,n-1}$ unit simplices.
Such a triangulation into unit simplices is called a {\it unimodular
triangulation}.
In this section we define four triangulations of the hypersimplex
$\Delta_{k,n}$. One triangulation is due to Stanley~\cite{Sta1}, one
is due to Sturmfels~\cite{Stu}, one arises from the affine Coxeter
arrangement of type $A$ and the final one, which is new, we call the
circuit triangulation. The main result of this section,
Theorem~\ref{thm:equaltriangulation}, says that these four
triangulations coincide. In addition, we will describe the dual
graphs of these triangulations.
\subsection{Stanley's triangulation} \label{sec:Sta}
The hypercube $[0,1]^{n-1}\subset\mathbb{R}^{n-1}$ can be triangulated into
$(n-1)$-dimensional unit simplices $\nabla_w$ labelled by
permutations $w \in S_{n-1}$ given by
\[
\nabla_w = \set{(y_1,\ldots,y_{n-1}) \in [0,1]^{n-1} \mid 0 <
y_{w(1)} < y_{w(2)} < \cdots < y_{w(n-1)} < 1}.
\]
Stanley~\cite{Sta1} defined a transformation of the hypercube
$\psi:[0,1]^{n-1} \rightarrow [0,1]^{n-1}$ by
$\psi(x_1,\ldots,x_{n-1}) = (y_1,\ldots,y_{n-1})$, where
\[
y_i = (x_1 + x_2 + \cdots + x_i) - \lfloor x_1 + x_2 + \cdots +
x_i \rfloor.
\]
The notation $\lfloor x \rfloor$ denotes the integer part of $x$.
The map $\psi$ is piecewise-linear, bijective on the hypercube
(except for a subset of measure zero), and volume preserving.
Since the inverse map $\psi^{-1}$ is linear and injective when
restricted to the open simplices $\nabla_w$, it transforms the
triangulation of the hypercube given by $\nabla_w$'s into another
triangulation.
\begin{theorem}[Stanley~\cite{Sta1}]
The collection of simplices $\psi^{-1}(\nabla_w)$, $w\in S_{n-1}$,
gives a triangulation of the hypercube $[0,1]^{n-1}$ compatible
with the subdivision of the hypercube into hypersimplices.
The collection of the simplices $\psi^{-1}(\nabla_w)$,
where $w^{-1}$ varies over permutations in $S_{n-1}$
with $k-1$ descents, gives a triangulation of the $k$-th
hypersimplex $\tilde \Delta_{k,n}$.
Thus the normalized volume of $\tilde\Delta_{k,n}$
equals the Eulerian number $A_{k,n-1}$.
\label{thm:Sta}
\end{theorem}
\begin{proof}
Let $\psi(x_1,\dots,x_{n-1}) = (y_1,\dots,y_{n-1})\in\nabla_w$.
For $i=1,\dots,n-2$, we have,
$$
\floor{x_1+\cdots+x_{i+1}}=\left\{
\begin{array}{cl}
\floor{x_1+\cdots+x_i}&\textrm{if } y_i<y_{i+1};\\[.1in]
\floor{x_1+\cdots+x_i}+1&\textrm{if } y_i>y_{i+1}.\\[.1in]
\end{array}
\right.
$$
Thus $\floor{x_1+\cdots+x_{n-1}}=\mathrm{des}(w^{-1})$.
In other words, if $\mathrm{des}(w^{-1})=k-1$, then
$k-1\leq x_1+\cdots+x_{n-1}\leq k$, i.e., $(x_1,\dots,x_{n-1})\in
\tilde\Delta_{k,n}$.
\end{proof}
\subsection{Sturmfels' triangulation} \label{sec:Stu}
Let $S$ be a multiset of elements from $[n]$. We define
$\mathrm{sort}(S)$ to be the unique non-decreasing sequence obtained by ordering
the elements of $S$.
Let $I$ and $J$ be two $k$-element subsets of $[n]$,
and let $\mathrm{sort}(I \cup J) = (a_1,a_2,\dots, a_{2k})$.
Then we set $U(I,J) = \set{a_1,a_3,\dots,a_{2k-1}}$ and
$V(I,J) = \set{a_2,a_4,\dots,a_{2k}}$.
For example, for $I = \set{1,2,3,5}$, $J = \set{2,4,5,6}$, we have
$\mathrm{sort}(I\cup J) = (1,2,2,3,4,5,5,6)$,
$U(I,J) = \set{1,2,4,5}$, and $V(I,J) = \set{2,3,5,6}$.
We say that an ordered pair $(I,J)$ is {\it sorted} if $I = U(I,J)$ and
$J = V(I,J)$. We call an ordered collection ${\mathcal{I}}=(I_1,\ldots,I_r)$
of $k$-subsets of $[n]$ \emph{sorted} if $(I_i,I_j)$ is sorted
for every $1 \leq i < j \leq r$.
Equivalently, if $I_l = \set{I_{l1}<\cdots<I_{lk}}$,
for $l=1,\dots,r$, then ${\mathcal{I}}$ is sorted if and only if
$I_{11} \leq I_{21} \leq \cdots \leq
I_{r1} \leq I_{12} \leq I_{22} \leq \cdots \leq I_{rk}$.
For such a collection ${\mathcal{I}}$, let $\nabla_{\mathcal{I}}$ denote the $(r-1)$-dimensional
simplex with the vertices $\epsilon_{I_1},\dots,\epsilon_{I_r}$.
\begin{theorem}[Sturmfels~\cite{Stu}]
The collection of simplices $\nabla_{\mathcal{I}}$, where ${\mathcal{I}}$ varies over all sorted
collections of $k$-element subsets in $[n]$,
is a simplicial complex that forms a triangulation
of the hypersimplex $\Delta_{k,n}$.
\end{theorem}
It follows that the maximal by inclusion sorted collections,
which correspond to the maximal simplices in the triangulation,
all have the same size $r=n$.
\begin{cor}
\label{cor:sorted_vol}
The normalized volume of the hypersimplex $\Delta_{k,n}$ is
equal to the number of maximal sorted collections of
$k$-subsets in $[n]$.
\end{cor}
This triangulation naturally appears in the context of Gr\"obner
bases.
Let $k[x_I]$ be the polynomial ring in the $\binom{n}{k}$ variables
$x_I$ labelled by $k$-subsets $I\in\binom{[n]}{k}$. Define the map
$\phi: k[x_I] \rightarrow k[t_1,t_2,\ldots,t_n]$ by $x_I \mapsto
t_{i_1}t_{i_2}\cdots t_{i_k}$, for $I = \set{i_1,\ldots,i_k}$. The
kernel of this map is an ideal in $k[x_I]$ that we denote by
${\mathcal{J}}_{k,n}$. Recall that a sufficiently generic height function on
the vertices $\epsilon_I$ of the hypersimplex $\Delta_{k,n}$ induces
a term order on monomials in $k[x_I]$ and defines a Gr\"{o}bner
basis for the ideal ${\mathcal{J}}_{k,n}$. On the other hand, such a height
function gives a coherent triangulation of $\Delta_{k,n}$. This
gives a correspondences between Gr\"obner bases and coherent
triangulations. The initial ideal associated with a Gr\"obner basis
is square-free if and only if the corresponding triangulation is
unimodular. For more details on Gr\"obner bases, see
Appendix~\ref{sec:grobner_intro}.
\begin{theorem}[Sturmfels~\cite{Stu}]
\label{thm:Stu} The marked set of quadratic binomials
\[
{\mathcal G}_{k,n} = \set{\underline{x_I x_J} - x_{U(I,J)}x_{V(I,J)}\mid
I,J\in\binom{[n]}{k} },
\]
is a Gr\"{o}bner basis for ${\mathcal{J}}_{k,n}$ under some term order on
$k[x_I]$ such that the underlined term is the initial monomial.
The simplices of the corresponding triangulation are $\nabla_{\mathcal{I}}$,
where ${\mathcal{I}}$ varies over sorted collections of $k$-subsets of $[n]$.
Moreover, this triangulation is unimodular.
\end{theorem}
In Section~\ref{sec:sort_closed}, we state and prove a more
general statement.
\subsection{Alcove triangulation}
\label{sec:alcove}
The {\it affine Coxeter arrangement\/} of type $A_{n-1}$ is the arrangement
of hyperplanes in $\mathbb{R}^{n-1}$ given by
$$
H_{ij}^l = \{(z_1,\dots,z_{n-1})\in\mathbb{R}^{n-1}\mid z_i-z_j = l\},\qquad\textrm{for }0\leq i<j\leq n-1,\ l\in{\mathbb Z},
$$
where we assume that $z_0=0$. It follows from the general theory
of affine Weyl groups, see~\cite{Hum}, that the hyperplanes
$H_{ij}^l$ subdivide $\mathbb{R}^{n-1}$ into unit simplices, called {\it
alcoves}.
We say that a polytope ${\mathcal P}$ in $\mathbb{R}^{n-1}$ is {\it alcoved\/} if
${\mathcal P}$ is an intersection of some half-spaces bounded by the
hyperplanes $H_{ij}^l$. In other words, an alcoved polytope is a
polytope given by inequalities of the form $b_{ij} \leq z_j - z_i
\leq c_{ij}$, for some collection of integer parameters $b_{ij}$
and $c_{ij}$. We will denote this alcoved polytope by ${\mathcal P}(b_{ij},
c_{ij})$. If the parameters satisfy $b_{ij} = c_{ij} -1$, for all
$i,j$, then the corresponding polytope consists of a single alcove
(or is empty). Each alcoved polytope comes naturally equipped
with the triangulation into alcoves. Conversely, if ${\mathcal P}$ is a
convex polytope which is a union of alcoves, then ${\mathcal P}$ is an
alcoved polytope.
Assume that $z_i = x_1+\cdots + x_i$, for $i=1,\dots,n-1$.
The hypersimplex $\tilde \Delta_{k,n}$ is given by
the following inequalities in the $z$-coordinates:
\begin{equation}
0\leq z_1-z_0,\dots,z_{n-1}-z_{n-2}\leq 1;\ k-1\leq z_{n-1}-z_0\leq k.
\label{eq:hypersimplex-z-coords}
\end{equation}
Thus the hypersimplex is an alcoved polytope.
Let us call its triangulation into alcoves the
{\it alcove triangulation}.
\subsection{Circuit triangulation} \label{sec:graphs}
Let $G_{k,n}$ be the directed graph on the vertices $\epsilon_I$,
$I\in\binom{[n]}{k}$, of the hypersimplex $\Delta_{k,n}$ defined as
follows. Let us regard the indices $i$ of a vector
$\epsilon=(\epsilon_1,\dots, \epsilon_n)$ as elements of ${\mathbb Z}/n{\mathbb Z}$.
Thus we assume that $\epsilon_{n+1}=\epsilon_1$. We connect a vertex
$\epsilon= (\epsilon_1,\dots,\epsilon_n)$ with a vertex $\epsilon'$
by an edge $\epsilon {\stackrel i \longrightarrow} \epsilon'$
labelled by $i\in[n]$ whenever $(\epsilon_i,\epsilon_{i+1})=(1,0)$
and the vector $\epsilon'$ is obtained from $\epsilon$ by switching
$\epsilon_i$ and $\epsilon_{i+1}$. In other words, each edge in the
graph $G_{k,n}$ is given by cyclically shifting a ``1'' in vector
$\epsilon$ one step to the right to the next adjacent place. It is
possible to perform such a shift if and only if the next place is
not occupied by another ``1''.
A circuit in the graph $G_{k,n}$ of minimal possible length is
given by a sequence of shifts of ``1''s so that the first ``1'' in
$\epsilon$ moves to the position of the second ``1'', the second ``1''
moves to the position of the third ``1'', and so on, finally, the last
``1'' cyclically moves to the position of the first ``1''. The length of
such a circuit is $n$. We will call such circuits in $G_{k,n}$ is
{\it minimal}. Here is an example of a minimal circuit in
$G_{26}$:
$$
\begin{array}{ccccc}
(1,0,1,0,0,0) &\stackrel 3 \longrightarrow&
(1,0,0,1,0,0) &\stackrel 1 \longrightarrow&
(0,1,0,1,0,0) \\
\uparrow_6&&&& \downarrow_2\\
(0,0,1,0,0,1)
&\stackrel 5 \longleftarrow&
(0,0,1,0,1,0) &\stackrel 4 \longleftarrow&
(0,0,1,1,0,0)
\end{array}
$$
The sequence of labels of edges in a minimal circuit forms a
permutation $w=w_1\cdots w_n\in S_n$.
For example, the permutation corresponding to the above minimal
circuit is $w=3 1 2 4 5 6$.
If we do not specify the initial vertex in a minimal circuit, then
the permutation $w$ is defined modulo cyclic shifts $w_1\dots
w_n\sim w_{n}w_1\dots w_{n-1}$. By convention, we will pick the
representative $w$ of the class of permutations modulo cyclic
shifts such that $w_n=n$. This corresponds to picking the initial
point in a minimal circuit with lexicographically maximal
01-vector $\epsilon$. Indeed, if $\epsilon\stackrel i
\longrightarrow \epsilon'$ is an edge in $G_{k,n}$, then
$\epsilon>\epsilon'$ in the lexicographic order, for
$i=1,\dots,n-1$; and $\epsilon<\epsilon'$, for $i=n$.
\begin{lemma}
\label{lem:circuit-descents} A minimal circuit in the graph
$G_{k,n}$ is uniquely determined by the permutation $w$ modulo
cyclic shifts. A permutation $w \in S_n$ such that $w_n=n$
corresponds to a minimal circuit in the graph $G_{k,n}$ if and
only if the inverse permutation $w^{-1}$ has exactly $k - 1$
descents.
\end{lemma}
\begin{proof}
The inverse permutation $w^{-1}$ has a descent for each pair $(i <
j)$ such that $b = w_i = w_j + 1$. Thus a ``1'' was moved from the
$b$-th position to the $(b+1)$-th position in $\epsilon$ before a
``1'' was moved from the $(b-1)$-th position to the $b$-th position.
(Here $\epsilon$ denotes the initial vertex of the corresponding
circuit.) This happens if and only if $\epsilon_b = 1$. Since
$\epsilon$ has $k$ ``1''s, this happens exactly $k$ times. But the
occurrence corresponding to $w^{-1}(n) = n > w^{-1}(1)$ is not
counted as a descent, so $w^{-1}$ has exactly $k-1$ descents.
Conversely, if $w^{-1}$ has $k-1$ descents, then we obtain a vector
$\epsilon$ with $k$ ``1''s, so that $w$ corresponds to a minimal
circuit containing $\epsilon$.
\end{proof}
For a permutation, $w=w_1\dots w_n\in S_n$, let $(w)$
denote the long cycle in $S_n$ given by $(w)=(w_1,\dots,w_n)$
in cycle notation. Two permutations $u,w\in S_n$ are equivalent
modulo cyclic shifts if and only if $(u) = (w)$.
The reader should not confuse {\it circuits\/} in the graph $G_{k,n}$
with {\it cycles\/} in the symmetric group $S_n$.
Let $C_{k,n}$ denote the set of long cycles
$(w)=(w_1,\dots,w_{n-1},n)\in S_n$ such that $w^{-1}$ has exactly
$k - 1 $ descents. For $(w)\in C_{k,n}$, let $c_{(w)}$ be the
corresponding minimal circuit in the graph $G_{k,n}$, whose edges
are labelled by $w_1,\dots,w_n$. Lemma~\ref{lem:circuit-descents}
shows that that the map $(w)\mapsto c_{(w)}$ is one-to-one
correspondence between the set long cycles $C_{k,n}$ and the set of
minimal circuits in $G_{k,n}$.
Each minimal circuit $c_{(w)}$ in $G_{k,n}$ determines the simplex
$\Delta_{(w)}$ inside the hypersimplex $\Delta_{k,n}$ with the vertex
set $c_{(w)}$.
\begin{theorem} \label{thm:circuittriangulation}
The collection of simplices
$\Delta_{(w)}$ corresponding to all minimal circuits in $G_{k,n}$
forms a triangulation of the hypersimplex $\Delta_{k,n}$.
\end{theorem}
Let us call this triangulation of the hypersimplex the {\it
circuit triangulation}.
\begin{theorem} \label{thm:equaltriangulation} The following four triangulations of the hypersimplex
are identical:
Stanley's triangulation, Sturmfels' triangulation, the alcove triangulation,
and the circuit triangulation.
\end{theorem}
Let us prove Theorems~\ref{thm:circuittriangulation}
and~\ref{thm:equaltriangulation} together. Let $\Gamma_{k,n}$ be the
collection of (maximal) simplices of the triangulation of
Theorem~\ref{thm:equaltriangulation}.
\begin{proof}
The fact that Stanley's triangulation coincides with the alcove triangulation
follows directly from the definitions. We leave this as an exercise
for the reader.
Let us show that the simplices $\Delta_{(w)}$ are exactly those
in Sturmfels' triangulation. An ordered pair of subsets $I =
\set{i_1 < \cdots < i_k}$ and $J = \set{j_1 < \cdots <
j_k}$ is sorted if and only if the interleaving condition $i_1
\leq j_1 \leq i_2 \leq j_2 \leq \cdots \leq j_k$ is satisfied.
When two vertices $\epsilon_I$ and $\epsilon_J$ belong to
the same minimal circuit, a ``1'' from $\epsilon_I$ is moved towards
the right in $\epsilon_J$ but never past the original position of
another ``1'' in $\epsilon_I$.
Thus the interleaving $i_a \leq j_a \leq i_{a+1}$ condition is
satisfied, and similarly we obtain the other interleaving inequalities.
Conversely, the interleaving condition implies that
each sorted collection belongs to a minimal circuit in $G_{k,n}$.
Let us now show that the circuit triangulation coincides with
Stanley's triangulation. Recall that the latter triangulation
occurs in the space $\mathbb{R}^{n-1}$. To be more precise, in order to
obtain Stanley's triangulation we need to apply the projection
$p:(x_1,\ldots,x_n) \mapsto (x_1,\ldots,x_{n-1})$ to the circuit
triangulation. Let us identify a permutation $w=w_1\cdots
w_{n-1}\in S_{n-1}$ with $k-1$ descents with the permutation
$w_1\cdots w_{n-1} n\in S_n$.
We claim that the projected simplex $p(\Delta_{(w)})$ is exactly
the simplex $\psi^{-1}(\nabla_w)$ in Stanley's triangulation.
Indeed, the map
$\psi^{-1}:(y_1,\dots,y_{n-1})\mapsto(x_1,\dots,x_{n-1})$
restricted to the simplex $\nabla_w = \set{0 < y_{w(1)} < \cdots <
y_{w(n-1)} <1}$, is given by $x_1 = y_1$ and
$$
x_{i+1} = \left\{
\begin{array}{cl}
y_{i+1}-y_i & \text{if } w^{-1}(i+1)>w^{-1}(i),\\
y_{i+1}-y_i+1 & \text{if }w^{-1}(i+1)<w^{-1}(i)
\end{array}
\right.
$$
for $i=1,\dots,n-2$. The vertices of the simplex $\nabla_w$ are
the points $v_0,\dots,v_{n-1}\in\mathbb{R}^{n-1}$ such that $v_r =
(y_1,\dots,y_{n-1})$ is given by $y_{w(1)} = \cdots = y_{w(r)} =
0$ and $y_{w(r+1)} = \cdots = y_{w(n-1)}=1$. The map $\psi^{-1}$
sends the vertex $v_0=(0,\dots,0)$ to the point
$(x_1,\dots,x_{n-1})$ such that $x_1=0$ and $x_{i+1} = 1$ if
$w^{-1}(i+1)<w^{-1}(i)$ and $x_{i+1} = 0$ if
$w^{-1}(i+1)>w^{-1}(i)$, for $i=1,\dots,n-2$. The vertex $v_r$ is
obtained from $v_{r-1}$ by changing $y_{w(n-r)}$ from 0 to 1. Thus
$\psi^{-1}(v_{r})$ differs from $\psi^{-1}(v_{r-1})$ exactly in
the coordinates $x_{w(n-r)}$ and $x_{w(n-r)+1}$. Here $x_n = k -
(x_1 + \ldots + x_{n-1})$. In fact, going from
$\psi^{-1}(v_{r-1})$ to $\psi^{-1}(v_{r})$ we move a ``1'' from
$x_{w(n-r)+1}$ to $x_{w(n-r)}$. Finally, moving from $v_{n-1}$ to
$v_0$ we are changing $x_1$ from $1$ to $0$. Thus as we go from
$\psi^{-1}(v_r)$ to $\psi^{-1}(v_{r+1})$ we are traveling along
the edges of the graph $G_{k,n}$ in the reverse direction. So the
vertices $\psi^{-1}(v_r)$ of the simplex $\psi^{-1}(\nabla_w)$ are
exactly the vertices of $p(\Delta_{(w)})$. This completes the
proof of the theorem.
\end{proof}
\begin{remark}
\label{rem:bijtheta} An explicit bijection $\theta$ between maximal
sorted collections of $k$-subsets of $[n]$ and permutations $w \in
S_n$ with $k-1$ descents satisfying $w_n=n$ can be constructed as
follows. Let ${\mathcal{I}}=(I_1,\ldots,I_n)$ be such a collection. Every
number in $[n]$ must occur in $\bigcup_i I_i$. Set
$(a_1,\ldots,a_{kn}) = \mathrm{sort}(\bigcup_i I_i)$. Let $\alpha_k$ be such
that $a_{\alpha_k} = k$ and $a_{\alpha_k+1} = k+1$. Then
$\theta({\mathcal{I}}) = w_1w_2\cdots w_{n-1}n$, where $w_i \equiv \alpha_i
\pmod {n-1}$ with representatives taken from $[n-1]$. This bijection
is compatible with the correspondences in
Theorem~\ref{thm:equaltriangulation}. For example, $\alpha_i \,\mathrm{mod}\,
n-1$ tells us when a ``1'' is moved from $\epsilon_i$ to
$\epsilon_{i+1}$ where by convention a ``1'' is moved from
$\epsilon_{n}$ to $\epsilon_1$ in the last edge of a circuit.
\end{remark}
\subsection{Adjacency of maximal simplices in the hypersimplex}
\label{sec:adjacency}
Let us say that two simplices in a triangulation are {\it adjacent}
if they share a common facet. Let us describe the adjacent simplices in
the triangulation $\Gamma_{k,n}$, using first the construction of the
circuit triangulation.
\begin{theorem}
\label{thm:adjacency} Two simplices $\Delta_{(u)}$ and
$\Delta_{(w)}$ of $\Gamma_{k,n}$ are adjacent if and only if there
exists $i=1,\dots,n$ such that $u_{i}-u_{i+1}\ne \pm 1\pmod n$ and
the cycle $(w)$ is obtained from $(u)$ by switching
$u_{i}$ with $u_{i+1}$, i.e., $(w) = (u_i,u_{i+1}) (u)
(u_i,u_{i+1})$. Here again we assume that $u_{n+1}=u_1$.
\end{theorem}
\begin{proof}
The two simplices $\Delta_{(u)}$ and $\Delta_{(w)}$ are adjacent
if and only if exactly one pair of their vertices differ. This
means that the corresponding minimal circuits $c_{(u)}$ and
$c_{(w)}$ differ in exactly one place. Let
$\epsilon'\stackrel{u_i} \longrightarrow \epsilon
\stackrel{u_{i+1}} \longrightarrow \epsilon''$ be three vertices
in order along the minimal cycle $c_{(u)}$. Then we can obtain
another cycle $c_{(w)}$ from $c_{(u)}$ by changing only $\epsilon$
if and only if $u_{i}-u_{i+1}\ne \pm 1\pmod n$ so that
$\epsilon'\stackrel{ u_{i+1}} \longrightarrow \epsilon^*
\stackrel{u_{i}} \longrightarrow \epsilon''$ are valid edges. When
$u_{i}-u_{i+1} = \pm 1\pmod n$ we are either moving the same ``1''
twice or moving two adjacent ``1''s one after another. In both
cases, the order of the shifts cannot be reversed, and so
$\epsilon$ cannot be replaced by another vertex.
\end{proof}
Alternatively, let ${\mathcal{I}} = (I_1,\ldots,I_n)$ be a sorted subset
corresponding to the maximal simplex $\nabla_{\mathcal{I}}$ of $\Gamma_{k,n}$. Let
$t \in [n]$ and $I_t = \set{i_1,i_2,\ldots,i_k}$. Then we can
replace $I_t$ in ${\mathcal{I}}$ by another $I'_t \in \binom{[n]}{k}$ to
obtain an adjacent maximal simplex $\nabla_{{\mathcal{I}}'}$ if and only if
the following holds. We must have $I'_t = \set{i_1,\ldots, i'_a,
\ldots, i'_b,\ldots,i_k}$ for some $a \neq b \in [n]$ and $i'_a
\neq i'_b$, satisfying $i_a - i'_a = i'_b - i_b = \pm 1 \pmod n$
and also both $k$-subsets
$\set{i_1,\ldots,i'_a,\ldots,i_b,\ldots,i_k}$ and $\set{i_1 ,
\ldots, i_a, \ldots, i'_b, \ldots i_k}$ must lie in ${\mathcal{I}}$. For
example, we may replace $\set{1,3,5,8}$ by $\set{1,2,6,8}$ if and
only if both $\set{1,2,5,8}$ and $\set{1,3,6,8}$ lie in ${\mathcal{I}}$.
\begin{figure}[ht]
\pspicture(20,-15)(80,115)
\psset{unit=1pt}
\rput(10,80){$\Gamma_{1,4}
= $} \rput(50,80){\ovalnode{A}{1234}} \rput(-25,32){$\Gamma_{2,4} = $}
\rput(10,50){\ovalnode{B}{1324}}
\rput(90,50){\ovalnode{C}{3124}} \rput(90,15){\ovalnode{D}{2314}}
\rput(10,15){\ovalnode{E}{2134}}
\rput(10,-15){$\Gamma_{3,4}=$}\rput(50,-15){\ovalnode{F}{3214}}
\ncline{-}{B}{C} \ncline{-}{B}{E} \ncline{-}{C}{D}
\ncline{-}{D}{E}
\endpspicture
\caption{The graphs of the triangulations of $\Delta_{1,4}, \Delta_{2,4}$
and $\Delta_{3,4}$.} \label{fig:graphk4}
\end{figure}
\begin{figure}[ht]
\pspicture(0 ,-10)(175,190)
\rput(100,20){\ovalnode{A}{34125}}
\rput(-70,90){$\Gamma_{2,5} =$ } \rput(25,-5){\ovalnode{B}{23415}}
\rput(175,-5){\ovalnode{C}{41235}}
\rput(100,80){\ovalnode{D}{31425}}
\rput(25,55){\ovalnode{E}{23145}}
\rput(175,55){\ovalnode{F}{14235}}
\rput(-25,105){\ovalnode{G}{21345}}
\rput(225,105){\ovalnode{H}{12435}}
\rput(50,130){\ovalnode{I}{13425}}
\rput(150,130){\ovalnode{J}{31245}}
\rput(100,180){\ovalnode{K}{13245}} \ncline{-}{A}{B}
\ncline{-}{A}{C} \ncline{-}{A}{D} \ncline{-}{E}{D}
\ncline{-}{B}{E} \ncline{-}{F}{D} \ncline{-}{C}{F}
\ncline{-}{D}{I} \ncline{-}{D}{J} \ncline{-}{E}{G}
\ncline{-}{F}{H} \ncline{-}{G}{I} \ncline{-}{H}{J}
\ncline{-}{I}{K} \ncline{-}{J}{K}
\endpspicture
\caption{Graph of the triangulation of the hypersimplex
$\Delta_{2,5}$.} \label{fig:graph25}
\end{figure}
We can give $\Gamma_{k,n}$ the structure of a graph by letting the
simplices be the vertices and letting an edge join two simplices
if the two simplices are adjacent. Figures~\ref{fig:graphk4} and
\ref{fig:graph25} give examples of these graphs which we will also
denote as $\Gamma_{k,n}$.
We will investigate degrees of vertices in these graphs for $k=2$ in
Section~\ref{sec:adjacency_second}.
\section{Triangulations and volumes of alcoved polytopes}
\label{sec:triang_alcove}
In this section, we generalize (Theorem~\ref{thm:alcove}) the
triangulations of Section~\ref{sec:hypersimplex} to all alcoved
polytopes (see Section~\ref{sec:alcove}). As a consequence of these
descriptions of the alcove triangulation, we obtain a curious
formula (Theorem~\ref{thm:latticepoints}) expressing the volume of
an alcoved polytope as a sum of the number of lattice points in
certain other alcoved polytopes. This formula has a root system
theoretic explanation which we give in~\cite{AP2}. Finally, we
explain a construction of the dual graph of the alcove
triangulation, which we call the {\it alcove lattice}.
\subsection{Triangulations of alcoved polytopes}
\label{sec:alcoved}
Let ${\mathcal P} = {\mathcal P}(b_{ij},c_{ij}) \subset \mathbb{R}^n$ be an alcoved polytope
which we realize in the $x$-coordinates. In other words, ${\mathcal P}$ is a
$(n-1)$-dimensional polytope lying in a hyperplane $x_1 + x_2 +
\cdots + x_n = k$ for some $k \in {\mathbb Z}$; and given by the inequalities
$b_{ij} \leq x_{i+1} + \cdots + x_j \leq c_{ij}$ for each pair
$(i,j)$ satisfying $0 \leq i < j \leq n-1$. By translating ${\mathcal P}$ by
$(m,\ldots,m)$ for some $m \in {\mathbb Z}$ to obtain an affinely equivalent
polytope, we can assume that all the coordinates of the points of
${\mathcal P}$ are non-negative. Let $Z_{\mathcal P} = {\mathcal P} \cap {\mathbb Z}^n \subset {\mathbb N}^n$
denote the set of integer points lying inside ${\mathcal P}$.
Let $G_{\mathcal P}$ be the directed graph defined as follows in analogy with
$G_{k,n}$ in Section~\ref{sec:graphs}. The graph $G_{\mathcal P}$ has
vertices labelled by points $a \in Z_{\mathcal P}$. Two vertices $a, b \in
Z_{\mathcal P}$ are connected by an edge $a \to b$ labelled $i$ if there
exists an index $i \in [1,n]$ such that $a + e_{i+1} - e_i = b$,
where $e_i, e_{i+1}$ are the coordinate vectors and $e_{n+1} :=
e_1$. Let $C_{\mathcal P}$ denote the set of minimal circuits of $G_{\mathcal P}$,
which have length $n$.
For an integer vector $a = (a_1,a_2,\ldots,a_n) \in {\mathbb N}^n$ with
non-negative coordinates lying on $x_1 + x_2 + \cdots + x_n = k$, we
let $I_a$ denote the multiset of size $k$ of $\{1,2,\ldots,n\}$ with
$a_1$ 1's, $a_2$ 2's and so on. If $I, J$ are multisets of size $k$
with elements from $\{1,2,\ldots,n\}$ then we can define $U(I,J)$
and $V(I,J)$ by sorting $I \cup J$ as in Section~\ref{sec:Stu}.
Similarly, we define the notions of sorted and sort-closed for
collections of multisets. In the following theorem, note that $Z_{\mathcal P}$
and $G_{\mathcal P}$ are defined without needing ${\mathcal P}$ to be alcoved.
The proof of the following theorem is exactly analogous to the
arguments of Theorem~\ref{thm:equaltriangulation}.
\begin{theorem}
\label{thm:alcove} Let ${\mathcal P} \subset \mathbb{R}^n$ be a $(n-1)$-dimensional
polytope lying in $x_1 + x_2 + \cdots + x_n = k$ so that all points
of ${\mathcal P}$ have non-negative coordinates. Then the following are
equivalent:
\begin{enumerate}
\item
\label{it:1} ${\mathcal P}$ is an alcoved polytope.
\item
\label{it:3} The set $\mathcal{I} = \set{I_a \mid a \in Z_{\mathcal P}}$ is
sort-closed. A triangulation of ${\mathcal P}$ consists of the maximal
simplices with vertices $\{a_1,a_2,\ldots,a_n\}$ for each sorted
collection $(I_{a_1},I_{a_2},\ldots,I_{a_n})$ where $I_{a_k} \in
\mathcal{I}$.
\item
\label{it:4} The set $C_{\mathcal P}$ of minimal circuits of $G_{\mathcal P}$ gives rise
to a triangulation of ${\mathcal P}$: if $C =(c^{(1)},c^{(2)},\ldots,c^{(n)})
\in C_{\mathcal P}$ then a maximal simplex is given by
$\mathrm{conv}(c^{(1)},c^{(2)},\ldots,c^{(n)})$.
\end{enumerate}
When these conditions hold, all the three above triangulations
agree.
\end{theorem}
An alcoved polytope also gives rise to a Gr\"{o}bner basis ${\mathcal G}_{\mathcal P}$
of the associated toric ideal ${\mathcal{J}}_{\mathcal P}$. The reader will be able to
write it down following Section~\ref{sec:sort_closed}.
Let us identify $w \in S_{n-1}$ with $w_1w_2\cdots w_{n-1} n \in
S_n$ as usual. Recall that $\Delta_{(w)}$ denotes the simplex
$(\psi \circ p)^{-1}(\nabla_w)$ where we view $p$ as a map from $\{x
\in \mathbb{R}^n \mid x_1+\ldots+x_n = k\}$ to $\mathbb{R}^{n-1}$. For an alcoved
polytope ${\mathcal P}$, define the polytopes ${\mathcal P}_{(w)}$ by
\[
{\mathcal P}_{(w)} = \set{x \in \mathbb{R}^n \mid (\Delta_{(w)} + x) \subset {\mathcal P}}.
\]
Denote by $I({\mathcal P})$ the number $|{\mathbb Z}^n \cap {\mathcal P}| = \#Z_{\mathcal P}$ of lattice
points in ${\mathcal P}$.
\begin{theorem}
\label{thm:latticepoints} Each of the polytopes ${\mathcal P}_{(w)}$ is an
alcoved polytope. The normalized volume of ${\mathcal P}$ is given by
\[
\mathrm{Vol}({\mathcal P}) = \sum_w I({\mathcal P}_{(w)})
\]
where the sum is over all permutations $w \in S_{n-1}$.
\end{theorem}
\begin{proof}
The alcoved triangulation is obtained by copying and translating the
triangulations of Theorem~\ref{thm:equaltriangulation} by an integer
vector so that the corresponding simplices cover the polytope ${\mathcal P}$.
For example, if $(c^{(1)},c^{(2)},\ldots,c^{(n)}) \in C_{\mathcal P}$ then
there exists $c \in {\mathbb Z}^n$ so that $c^{(i)} - c$ is a 0-1 vector for
each $i$. Thus $(c^{(1)}-c,c^{(2)}-c,\ldots,c^{(n)}-c)$ is a minimal
circuit in some $G_{k,n}$ corresponding to some simplex
$\Delta_{(w)}$; see Section~\ref{sec:graphs}. The simplex
$\Delta_{(w)} + c = \mathrm{conv}(c^{(1)},c^{(2)},\ldots,c^{(n)})$
will correspond to the circuit $(c^{(1)},c^{(2)},\ldots,c^{(n)})$.
This proves the second statement of the theorem.
Let ${\mathcal P}$ be given by the inequalities $b_{ij} \leq x_{i+1} + \cdots
+ x_j \leq c_{ij}$ within the hyperplane $x_1+ x_2+\ldots+x_n = l$.
We check that ${\mathcal P}_{(w)}$ is an alcoved polytope. In fact this
follows from the fact that $\Delta_{(w)}$ is itself an alcoved
polytope and is given by some inequalities $d_{ij} \leq x_{i+1} +
\cdots + x_j \leq f_{ij}$ and a hyperplane $x_1 + x_2 +\ldots+x_n =
k$, where we pick $d_{ij}$ and $f_{ij}$ so that all equalities are
achieved by some point in $\Delta_{(w)}$. Then ${\mathcal P}_{(w)}$ is the
intersection of the inequalities $b_{ij}-d_{ij} \leq x_{i+1} +
\cdots + x_j \leq c_{ij}-f_{ij}$ with the hyperplane $x_1+
x_2+\ldots+x_n = l-k$, which by definition is an alcoved polytope.
\end{proof}
For the hypersimplex, all the polytopes ${\mathcal P}_{(w)}$ are either empty
or a single point.
To conclude this section, we give one further interpretation of the
volumes of alcoved polytopes in terms of maps of the circle with
marked points. Let $S^1$ be the unit circle and let $S^1_{(n)}$
denote a circle with $n$ distinct marked points
$p_0,p_1,\ldots,p_{n-1}$ arranged in clockwise order. Let ${\mathcal P}$ be an
alcoved polytope with parameters $b_{ij}$ and $c_{ij}$ as in
Section~\ref{sec:alcoved}. Let $M_{\mathcal P}$ denote the set of homotopy
classes of continuous maps $f: S^1_{(n)} \rightarrow S^{1}$
satisfying:
\begin{itemize}
\item
The map $f$ is always locally bijective and locally orientation
preserving. Informally, this means that $f$ traces out $S^{1}$ in
the clockwise direction and never stops.
\item
The images of marked points are distinct.
\item
For each $0 \leq i < j \leq n-1$, The number $d$ of pre-images of
$f(p_i)$ under $f$ in the open interval $(p_i,p_j)$ satisfies
$b_{ij} \leq d < c_{ij}$.
\end{itemize}
Two maps $f$ and $g$ belong to the same homotopy class if and only
if they can be deformed into one another by a homotopy, in the usual
sense, while always satisfying the conditions above. The following
proposition follows from the preceding discussion.
\begin{prop}
Let ${\mathcal P}$ be an alcoved polytope. Then the simplices in the
triangulation $\Gamma_{\mathcal P}$ are in bijection with the elements of $M_{\mathcal P}$.
\end{prop}
\subsection{Alcove lattice and alcoved polytopes} \label{sec:alcove
lattice}
Define the {\it alcove lattice} $\Lambda_n$ as the infinite graph
whose vertices correspond to alcoves (i.e., regions of type
$A_{n-1}$ affine Coxeter arrangement) and edges correspond to pairs
of adjacent alcoves. For example, $\Lambda_3$ is the infinite
hexagonal lattice. For an alcoved polytope ${\mathcal P}$, define its graph
$\Gamma_{\mathcal P}$ as the finite subgraph of $\Lambda_n$ formed by alcoves
in ${\mathcal P}$. For the graphs of Section~\ref{sec:graphs}, we have
$\Gamma_{k,n} = \Gamma_{\Delta_{k,n}}$.
According to~\cite[Sect.~14]{LP}, we have the following
combinatorial construction of the lattice $\Lambda_n$. Let
$[\lambda_0,\dots,\lambda_{n-1}]$ denote an element of
${\mathbb Z}^n/(1,\dots,1){\mathbb Z}$. In other words, we assume that
$[\lambda_0,\dots,\lambda_{n-1}]= [\lambda_0',\dots,\lambda_{n-1}']$
whenever the $\lambda_i'$ are obtained from the $\lambda_i$ by
adding the same integer. The vertices of $\Lambda_n$ can be
identified with the following subset of ${\mathbb Z}^n/(1,\dots,1){\mathbb Z}$,
see~\cite{LP}:
$$
\Lambda_n = \{[\lambda_0,\dots,\lambda_{n-1}]\mid \textrm{the
integers } \lambda_0,\dots,\lambda_{n-1} \textrm{ have different
residues modulo } n\}.
$$
Two vertices $[\lambda_0,\dots,\lambda_{n-1}]$ and
$[\mu_0,\dots,\mu_{n-1}]$ of $\Lambda_n$ are connected by an edge
whenever there exists a pair $(i,j)$, $0\leq i\ne j\leq (n-1)$, such
that $\lambda_i+1\equiv \lambda_j\ \textrm{mod} \ n$ and
$(\mu_0,\dots,\mu_{n-1}) = (\lambda_0,\dots,\lambda_{n-1}) + e_i
-e_j$, where $e_i, e_j$ are the coordinate vectors in ${\mathbb Z}^n$. In
this construction, $\lambda_0,\dots,\lambda_{n-1}$ are the
$z$-coordinates of the central point of the associated alcove scaled
by the factor $n$, see~\cite{LP}. This construction immediately
implies the following description of the graph of the alcoved
polytope ${\mathcal P}(b_{ij},c_{ij})$, defined as in
Section~\ref{sec:alcove}.
\begin{proposition} For an alcoved polytope ${\mathcal P} = {\mathcal P}(b_{ij},c_{ij})$,
its graph $\Gamma_{\mathcal P}$ is the induced subgraph of $\Lambda_n$ given
by the subset of vertices
$$
\left\{[\lambda_0,\dots,\lambda_{n-1}]\in\Lambda_n \mid n\cdot
b_{i,j}\leq \lambda_i - \lambda_j\leq n\cdot c_{i,j} , \textrm{ for
} i,j\in[0,n-1] \right\}.
$$
\end{proposition}
The vertices of the graph $\Gamma_{\mathcal P}$ are in bijection with the
elements of $C_{\mathcal P}$ defined in the previous section. Let $C =
(c^{(1)},c^{(2)},\ldots,c^{(n)}) \in C_{\mathcal P}$ be a minimal circuit in
the graph $\Gamma_{\mathcal P}$. The integer points
$\{c^{(1)},c^{(2)},\ldots,c^{(n)}\}$ are the vertices of an alcove
$A_C$, in the $x$-coordinates. The vertex
$[\lambda_1,\lambda_2,\ldots,\lambda_n]$ of $\Lambda_n$ associated
to $A_C$ is given by $\lambda_i = \alpha_1 + \alpha_2 + \cdots +
\alpha_i$ where $\alpha := (\alpha_1,\ldots,\alpha_n)$ is given by
$\alpha = \sum_{i = 1}^n c^{(i)}$. The point
$\frac{1}{n}[\lambda_1,\lambda_2,\ldots,\lambda_n] \in
\mathbb{R}^n/(1,\dots,1)\mathbb{R}$ is the central point of the alcove $A_C$ in the
$z$-coordinates.
\begin{example}
The $k$-th hypersimplex is given by
inequalities~(\ref{eq:hypersimplex-z-coords}). Thus the vertex set
of the graph $\Gamma_{\Delta_{k,n}}$ is the subset of $\Lambda_n$
given by the inequalities $0\leq
\lambda_1-\lambda_0,\lambda_2-\lambda_1,\dots,\lambda_{n-1}-\lambda_{n-2}
\leq n$, and $(k-1)\cdot n\leq \lambda_{n-1}-\lambda_0\leq k\cdot
n$.
\end{example}
One can give an abstract characterization of subgraphs of
$\Lambda_n$ corresponding to alcoved polytopes. Let us say that an
induced subgraph $H$ of some graph $G$ is {\it convex} if, for any
pair of vertices $u,v$ in $H$, and any path $P$ in $G$ from $u$ to
$v$ of minimal possible length, all vertices of $P$ are in $H$.
In~\cite{AP2}, we will prove, in the more general context of an
arbitrary Weyl group, that an induced subgraph $\Gamma$ in
$\Lambda_n$ is the graph of some alcoved polytope if and only if
$\Gamma$ is convex.
\section{Matroid polytopes}
\label{sec:matroid}
Let ${\mathcal M}$ be a collection of $k$-subsets of $[n]$. The polytope
${\mathcal P}_{\mathcal M}$ is the convex hull in ${\mathbb R}^n$ of the points
$\set{\epsilon_I | \; I \in {\mathcal M}}$ and is a subpolytope of the
hypersimplex $\Delta_{k,n}$. In this section we classify the polytopes
${\mathcal P}_{\mathcal M}$ that are alcoved and in these cases give a combinatorial
interpretation for their volumes.
Our main interest lies in the case when ${\mathcal M}$ is a matroid on the
set $[n]$. In this case the polytopes ${\mathcal P}_{\mathcal M}$ are known as {\it
matroid polytopes} -- we give explicit examples of matroid polytopes
which are alcoved. Matroid polytopes have recently been studied
intensively, motivated by applications to tropical
geometry~\cite{FS,Spe}: the {\it amoeba} of a linear subspace $V
\subset \mathbb{C}^n$ is asymptotically described by its {\it Bergman fan},
and this fan is closely related to the normal fan of the matroid
polytope ${\mathcal P}_{\mathcal M}$ of the matroid of $V$; see also~\cite{AK}.
Some of the results in this section have been obtained
earlier by Blum~\cite{Blu} in the context of Koszul algebras.
\subsection{Sort-closed sets} \label{sec:sort_closed}
\begin{definition}
A collection ${\mathcal M}$ of $k$-subsets of $[n]$ is \emph{sort-closed}
if for every two elements $I$ and $J$ in ${\mathcal M}$, the subsets
$U(I,J)$ and $V(I,J)$ are both in ${\mathcal M}$.
\end{definition}
A sorted subset of ${\mathcal M}$ is a subset of the form
$\{I_1,\ldots,I_r\} \subset {\mathcal M}$ such that $(I_1,\ldots,I_r)$ is a
sorted collection of $k$-subsets of $[n]$.
\begin{theorem}
\label{thm:matroid_tri} The triangulation $\Gamma_{k,n}$ of the
hypersimplex induces a triangulation of the polytope ${\mathcal P}_{\mathcal M}$ if
and only if ${\mathcal M}$ is sort-closed. The normalized volume of
${\mathcal P}_{\mathcal M}$ is equal to the number of sorted subsets of ${\mathcal M}$ of size
${\rm dim}({\mathcal P}_{\mathcal M}) + 1$.
\end{theorem}
The proof is analogous to that of Theorem~\ref{thm:Stu}. We work in
the polynomial ring $k[x_I \mid I \in {\mathcal M}]$. The ideal ${\mathcal{J}}_{\mathcal M}$ is
the kernel of the ring homomorphism $\phi: k[x_I \mid I \in {\mathcal M}]
\rightarrow k[t_1,t_2,\ldots,t_n]$ given by $x_I \mapsto
t_{i_1}t_{i_2}\cdots t_{i_k}$, for $I = \set{i_1,\ldots,i_k}$. The
following result is essentially equivalent to~\cite[Proposition 3.1]
{Blu}.
\begin{prop}
\label{prop:grobner_multi} Suppose that ${\mathcal M}$ is sort-closed. Then
there is a term order, $\prec_{\mathcal M}$, on $k[x_I |I \in {\mathcal M}]$ so that
the reduced Gr\"{o}bner basis of ${\mathcal{J}}_{\mathcal M}$ is given by the nonzero
marked binomials of the form
\begin{equation}
\label{bin} \set{\underline{x_I x_J} - x_{U(I,J)}x_{V(I,J)}}
\end{equation}
with the first monomial being the leading term.
\end{prop}
\begin{proof}[(Sketch of proof.)]
Since this proof is essentially the same as that of Sturmfels
in~\cite[Chapter 14]{Stu}, we will only sketch the argument; see
Appendix~\ref{sec:grobner_intro} for background.
We say a monomial $x_Ax_B\cdots x_V = x_{a_1\cdots a_k}x_{b_1\cdots
b_k} \cdots x_{v_1 \cdots v_k}$ is \emph{sorted} if the ordered
collection of sets $(A,B,\ldots,V)$ is sorted. If a monomial is not
sorted, then there is a pair of adjacent variables $x_I x_J$ which
is unsorted. Using the binomial $x_I x_J - x_{U(I,J)}x_{V(I,J)}$ we
can sort this pair. We can sort a monomial modulo the ideal
generated by the marked binomials of~(\ref{bin}) in a finite number
of steps. Using~\cite[Theorem 3.12]{Stu}, we conclude that there is
a term order $\prec_{\mathcal M}$ which selects the marked term for each
binomial of~(\ref{bin}). Finally, one checks that the sorted
monomials are exactly the $\prec_{\mathcal M}$-standard monomials.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:matroid_tri}]
The `if' direction follows from Proposition~\ref{prop:grobner_multi}
and Theorem~\ref{thm:regtri}. We may
assume that $\prec_{\mathcal M}$ arises from a weight vector since only
finitely many binomials are involved in (\ref{bin}). For the
`only if' direction, suppose ${\mathcal P} = {\mathcal P}_{\mathcal M}$ is a convex polytope
which is a union of simplices in $\Gamma_{k,n}$. Since the
triangulation $\Gamma_{k,n}$ is coherent, this triangulation $\Gamma_{\mathcal P}$ is
also coherent. We know already that all the faces of $\Gamma_{\mathcal P}$ are
sorted collections of $k$-subsets of $[n]$ (if we identify a
simplex with its set of vertices). By the correspondence of
Theorem~\ref{thm:regtri} and Proposition~\ref{prop:sqfree},
$\Gamma_{\mathcal P}$ arises from some term-order $\prec_{\mathcal M}$ which gives rise
to an initial ideal which is the Stanley-Reisner ideal of the
triangulation $\Gamma_{\mathcal P}$. Let $(I,J) \in {\mathcal M} \times {\mathcal M}$ not be
sorted, then $m_1 = x_Ix_J \in {\rm in}_{\prec_{\mathcal M}}({\mathcal{J}}_{\mathcal M})$. As
in the proof of Proposition~\ref{prop:grobner_multi}, this means
there is another monomial $m_2$ so that $m_1 - m_2 \in {\mathcal{J}}_{\mathcal M}$.
After a finite iteration of this argument, we see that $x_Ix_J =
x_Ax_B \ \,\mathrm{mod}\, {\mathcal{J}}_{\mathcal M} $ for some $\prec_{\mathcal M}$-standard monomial
$x_Ax_B$. But this means that $x_Ax_B$ must be a sorted monomial
since it is an edge of an alcove.
Thus $A = U(I,J)$ and $B= V(I,J)$ satisfy
$A, B \in {\mathcal M}$ so that ${\mathcal M}$ is sort-closed.
\end{proof}
\subsection{Sort-closed matroids} \label{sec:matroids} Let $k$ and
$n$ be positive integers satisfying $k \leq n$. A (non-empty)
collection ${\mathcal M}$ of $k$-subsets (called bases) of $[n]$ (the ground
set) is a \emph{matroid} if it satisfies the following axiom
(Exchange Axiom):
\begin{quote}
Let $I$ and $J$ be two bases of ${\mathcal M}$. Then for any $i \in I$
there exists $j \in J$ so that $(I - \set{i}) \cup \set{j}$ is a
base of ${\mathcal M}$.
\end{quote}
The matroid ${\mathcal M}$ is then said to be a rank $k$ matroid on $n$
elements. If $I$ is a base of ${\mathcal M}$ we write $I \in {\mathcal M}$. To a
matroid ${\mathcal M}$ (of rank $k$ on $n$ elements) we associate the
\emph{matroid polytope} ${\mathcal P}_{\mathcal M}$ as in
Section~\ref{sec:sort_closed}. We say that ${\mathcal M}$ is {\it
sort-closed} if it is sort-closed as a collection of $k$-subsets.
Thus by Theorem~\ref{thm:matroid_tri}, the triangulation
$\Gamma_{k,n}$ of the hypersimplex induces a triangulation of the
polytopes ${\mathcal P}_{\mathcal M}$ for sort-closed matroids ${\mathcal M}$. Sort-closed
matroids were introduced by Blum~\cite{Blu}, who called them
base-sortable matroids.
We now describe two classes of matroids which are sort-closed. Let
$\Pi$ be a set partition of $[n]$ with parts
$\set{\pi_i}_{i=1}^{r}$ of sizes $|\pi_i|= a_i$, and $\underline{b} =
(b_1,\ldots,b_r)$, $\underline{c} = (c_1,\ldots,c_r)$ be two sequences of
non-negative integers. We will call the data $(\Pi,\underline{b},\underline{c})$ a
\emph{weighted set partition}. Define ${\mathcal M}_{\Pi,\underline{b},\underline{c},k}$ to be
the collection of $k$-subsets $I$ of $[n]$ such that
\begin{equation} \label{eq:wsp} b_j \leq |I \cap \pi_j| \leq c_j
\end{equation} for all $j$.
\begin{lemma}
The collection of $k$-subsets ${\mathcal M}_{\Pi,\underline{b},\underline{c},k}$ defined above
is a matroid.
\end{lemma}
\begin{proof}
Let $I$ and $J$ be two such subsets and $i \in I$, say $i \in
\pi_s$ for some $s$. If $|I \cap \pi_k| = |J \cap \pi_k|$ for all
$k$ or if $|I \cap \pi_s| \leq |J \cap \pi_s|$, one can again find
some $j \in J \cap \pi_s - (I - \set{i}))$ to add to $I - \set{i}$
form a base. Otherwise there is some $t$ such that $|I \cap \pi_t|
< |J \cap \pi_t| \leq c_t$ in which case one can find some $j \in
(J \cap \pi_t)$ to add to $I - \set{i}$ without violating any of
the inequalities in (\ref{eq:wsp}). This verifies the exchange
axiom.
\end{proof}
This class of matroids is closed under duality. The dual of
${\mathcal M}_{\Pi,\underline{b},\underline{c},k}$ is ${\mathcal M}_{\Pi,\underline{b}',\underline{c}',n-k}$ where $b'_j =
|\pi_j|-c_j$ and $c'_j = |\pi_j| - b_j$. We call the polytope
$\Delta_{\Pi,\underline{b},\underline{c},k}$ associated to the matroid ${\mathcal M}_{\Pi,\underline{b},\underline{c},k}$
a {\it weighted multi-hypersimplex.} When $b_j=0$ and $c_j = 1$ for
all $j$ we will denote the matroid and polytope by ${\mathcal M}_{\Pi,k}$ and
${\Delta}_{\Pi,k}$ respectively, and we call the polytope ${\Delta}_{\Pi,k}$ a
\emph{multi-hypersimplex}. Up to affine equivalence the polytope
${\Delta}_{\Pi,k}$ depends only on the multiset $\set{a_i}_{i=1}^r$. The
polytope ${\Delta}_{\Pi,k}$ is the intersection of the hyperplane $x_1 +
\cdots + x_n = k$ with a product of simplices ${\Delta}_\Pi \simeq
\Delta_{a_1} \times \cdots \times \Delta_{a_r}$ just as the
hypersimplices are slices of cubes. In the $z$-coordinates, this
polytope is determined by intersecting the hypersimplex
$\tilde{\Delta}_{k,n}$ with the inequalities
\begin{align*}
0 \leq z_{a_1} - z_0 \leq 1;\;\; 0 \leq z_{a_1+a_2} - z_{a_1}
\leq 1; \ldots; 0 \leq z_{n} - z_{a_1+a_2+\cdots+a_{r-1}} \leq 1
\end{align*}
where we assume $z_n = k$. A weighted multi-hypersimplex can be
viewed as a slice of a product of unions of hypersimplices. In
particular, when $b_j = c_j -1$ for all $j$, the polytope
$\Delta_{\Pi,\underline{b}, \underline{c},k}$ is the slice $x_1+\cdots+x_n=k$ of
\[
\Delta_{\Pi,\underline{b},\underline{c}} \simeq \tilde{\Delta}_{c_1, a_1+1} \times
\tilde{\Delta}_{c_2,a_2+1} \times \cdots \times \tilde{\Delta}_{c_r,a_r+1}
\]
where the hypersimplex $\tilde{\Delta}_{c_i,a_i}$ (in the notation of
Section~\ref{sec:hypersimplex}) lives in the coordinates
$(x_{a_1+\cdots+a_{i-1}+1},\ldots,x_{a_1+\cdots+a_i})$. Let
$\Pi(a_1,\ldots,a_r) $ denote the set partition
\[\set{\pi_1 = \set{1,\ldots,a_1},\ldots,
\pi_r=\set{a_1+\ldots+a_{r-1}+1,\ldots,a_1+\ldots+a_r = n}}.\]
\begin{prop}
The matroid ${\mathcal M}_{\Pi,\underline{b},\underline{c},k}$ with $\Pi = \Pi(a_1,\ldots,a_r)$
and any $\underline{b}, \underline{c} \in {\mathbb N}^r$ is sort-closed.
\end{prop}
\begin{proof}
Let $I,J \in {\mathcal M}_{\Pi,\underline{b},\underline{c},k}$. Suppose to the contrary that
one of $U(I,J)$ or $V(I,J)$ were not a base. Let
$(q_1,q_2,\ldots,q_{2k}) = \mathrm{sort}(I \cup J)$. If $|U(I,J) \cap
\pi_s| > c_s$ or $|V(I,J) \cap \pi_s|> c_s$ then it must be the
case that for some $i$ the entries $q_i, q_{i+2}, \ldots,
q_{i+2c_s}$ belonged to the same part $\pi_s \in \Pi$. Then
$q_{i+1}, q_{i+3}, \ldots, q_{i+2c_s-1}$ belong to $\pi_s$ as well
since $q_{i+2k} \leq q_{i+2k+1} \leq q_{i+2k+2}$. This is
impossible as $I$ and $J$ were legitimate bases to begin with and
contain at most $c_s$ elements from $\pi_s$ each. A similar
argument guarantees that $|U(I,J) \cap \pi_s| \geq b_s$ and
$|V(I,J) \cap \pi_s| \geq b_s$ for all $s$.
\end{proof}
A matroid ${\mathcal M}$ is \emph{cyclically transversal} if it is a
transversal matroid specified by a set of (not necessarily disjoint)
cyclic intervals $\set{S_1,\ldots,S_k}$ of $[n]$. Recall that the
bases of a transversal matroid are the $k$-element subsets $I =
\{i_1,\ldots,i_k\}$ of $[n]$ such that $i_s \in S_s$.
\begin{prop}[{\cite[Theorem 5.2 (without proof)]{Blu}}]
Let ${\mathcal M}$ be a cyclically transversal matroid defined by the
subsets $\set{S_1,\ldots,S_k}$. Then ${\mathcal M}$ is sort-closed.
\end{prop}
Since the proof is omitted in~\cite{Blu}, we give a simple direct
proof here.
\begin{proof}
Let $I$ be a $k$-element subset of $[n]$. By the Hall marriage
theorem, $I$ is a base of ${\mathcal M}$ if and only if \begin{equation}
\label{eq:hall} |I \cap \bigcup_{r\in R} S_r| \geq |R|
\end{equation} for every subset $R$ of $[k]$. Now let $I$ and $J$
be bases of ${\mathcal M}$ and we now check (\ref{eq:hall}) for $U(I,J)$
and $V(I,J)$. Since each $S_i$ is a cyclic interval of $[n]$ it
suffices to consider the case where $\bigcup_{r\in R} S_r$ is
itself a cyclic interval $[a,b]$. By hypothesis, the multiset $I
\cup J$ intersects $[a,b]$ in at least $2|R|$ elements. Thus each
of $U(I,J)$ and $V(I,J)$ will intersect $[a,b]$ in at least $|R|$
elements.
\end{proof}
Let us describe ${\mathcal P}_{\mathcal M} \subset \mathbb{R}^n$ for a cyclically transversal
matroid explicitly in terms of inequalities. It is given by the
hyperplane $x_1 + x_2 + \cdots + x_n = k$, the inequalities $0 \leq
x_i \leq 1$ together with the inequalities
\[
\sum_{s \in S_R} x_s \geq |R|
\]
with $S_R = \bigcup_{r\in R} S_r$ for every subset $R$ of $[k]$.
We end this section with the question: what other matroids are
sort-closed?
\section{The second hypersimplex}
There is a description of the triangulation of the second
hypersimplex, developed in~\cite{LST} and ~\cite[Chapter~9]{Stu}, in
terms of graphs known as thrackles. In this section, we apply this
description to rank two matroids, and give a precise description of
the dual graph of the triangulation.
Triangulations of the second hypersimplex $\Delta_{2,n}$ arise in
the study of metrics on a finite set of points~\cite{Dre}, and
recently a thorough classification of the triangulations of
$\Delta(2,6)$ was performed in~\cite{SY}. This classification of
triangulations is an important problem in phylogenetic
combinatorics. Our study of the dual graphs of the triangulations
is partly motivated by this connection: the graph $\Gamma_{2,n}$ of
the triangulation is essentially what is known as the {\it tight
span} of the corresponding metric, and generalizes the phylogenetic
trees derived from the metric; see~\cite{Dre}.
\subsection{Thrackles and rank two matroids} \label{sec:ranktwo} When
the rank $k$ is equal to two (which we will assume throughout this
section), every matroid ${\mathcal M}$ arises as ${\mathcal M}_{\Pi,2}$ for some set
partition $\Pi$. Throughout this section we will assume that $\Pi$
has the form $\Pi = \Pi(a_1,\ldots,a_r)$.
Following~\cite[Chapter~9]{Stu} and \cite{LST}, we associate a graph
on $[n]$ to each maximal simplex of the matroid polytope
${\mathcal M}_{\Pi,2}$. The vertices are drawn on a circle so that they are
labelled clockwise in increasing order. Throughout this section, a
``graph on $[n]$'' will refer to such a configuration of the
vertices in the plane. Since the bases are two element subsets of
$[n]$, we may identify them with the edges.
\begin{lemma}
\label{lem:sortedge} Let $A = (a_1,a_2)$ and $B = (b_1,b_2)$ be
two bases. Then the pair $A, B$ is sorted if and only if the edges
$(a_1,a_2)$ and $(b_1,b_2)$ intersect (not necessarily in their
interior) when drawn on the circle.
\end{lemma}
\begin{proof}
Sorted implies that $a_1 \leq b_1 \leq a_2 \leq b_2$ which
immediately gives the lemma.
\end{proof}
Thus sorted subsets of $\binom{[n]}{2}$ correspond to graphs on
$[n]$ drawn on a circle, so that every pair of edges cross. These
graphs are known as {\it thrackles}. Note that two edges sharing
a vertex are considered to cross.
\begin{prop}
\label{prop:volume_graph} Let $\Pi$ be a set partition and
${\Delta}_{\Pi,2}$ have dimension $d$. The maximal simplices in the
alcoved triangulation of ${\Delta}_{\Pi,2}$ are in one-to-one
correspondence with thrackles on $[n]$ with $d+1$ edges such that
all edges are bases.
\end{prop}
\begin{proof}
Follows immediately from Theorem~\ref{thm:matroid_tri} and
Lemma~\ref{lem:sortedge}.
\end{proof}
Without the condition on the number of edges in the thrackles of
Proposition~\ref{prop:volume_graph}, one would obtain graphs
corresponding to all simplices (not just the maximal ones) of the
triangulation.
When the dimension of ${\Delta}_{\Pi,2}$ is $n-1$, each thrackle $G$ is
determined by picking an odd-cycle $C$ such that all the edges cross
pairwise. The remaining edges of $G$ join a vertex not on $C$ to
the unique `opposite' vertex lying on $C$ (so that the edge crosses
every edge of $C$); see Figure~\ref{fig:graphranktwo}. We will call
the resulting thrackle $G(C)$. Let $C$ be a cycle, with pairwise
crossing edges, of length $2k+1$ with vertices $V(C) =
\set{v_1,v_2,\ldots, v_{2k+1}} \subset [n]$ labeled so that $v_1 <
v_2 < \cdots < v_{2k+1}$. Then the edges of $C$ are of the form
$(v_i,v_{k+i+1})$, where the indices are taken modulo $2k+1$. Thus
the condition that all the edges of $C$ are bases is equivalent to
$|V(C) \cap \pi_i| \leq k$ for all $i$. In fact this is enough to
guarantee that $G(C)$ corresponds to a valid maximal simplex of
${\Delta}_{\Pi,2}$ -- that the remaining edges not on the cycle are bases
is implied.
\begin{figure}[ht]
\pspicture(-50,-50)(50,50)
\pscircle[linewidth=0.5pt](0,0){40} \SpecialCoor
\psline[linewidth=1.5pt](40;0)(40;154)(40;308)(40;102)(40;257)(40;0)
\psline(40;51)(40;257) \psline(40;203)(40;0) \rput(50;0){$1$}
\rput(50;51){$7$}\rput(50;102){$6$}\rput(50;154){$5$}\rput(50;203){$4$}\rput(50;257){$3$}\rput(50;308){$2$}\rput(50;0){$1$}
\psdot[dotscale=1.5]
(40;0)\psdot[dotscale=1.5](40;154)\psdot[dotscale=1.5](40;308)\psdot[dotscale=1.5](40;102)
\psdot[dotscale=1.5](40;257)
\endpspicture
\caption{A thrackle $G(C)$. The cycle $C$ has been drawn in
bold.} \label{fig:graphranktwo}
\end{figure}
Suppose $G$ arises from a sorted subset $(I_1,\ldots,I_r)$. Let $w
= \theta(I_1,\ldots,I_r)$ where $\theta$ is the bijection of
Remark~\ref{rem:bijtheta}. The vertices $i$ not on the odd cycle of
$G$ are exactly the positions such that $w_{i} = w_{i-1} + 1$.
\begin{prop}
Let $a_1, \ldots, a_r$ be positive integers and $n = a_1 + \cdots +
a_r$. Then the $(n-1)$-dimensional volume of the second
multi-hypersimplex ${\Delta}_{\Pi(a_1,\ldots,a_r),2}$ is given by
\begin{align*}
\mathrm{Vol}({\mathcal P}_{\Pi(a_1,\ldots,a_r),2}) &= \sum_{k=1}^\infty
\left(\sum_{c_1,..,c_r\leq k; \; c_1+...+c_r=2k+1}
\binom{a_1}{c_1} \cdots \binom{a_r}{c_r} \right) \\
&= 2^{n-1} - \sum_{i=1}^r \sum_{b,d\geq 0}\binom{a_i}{2b+d+1}
\binom{n-a_i}{d}
\end{align*}
\end{prop}
\begin{proof}
The first formula follows from enumerating odd subsets $S \subset
\{1,2,\ldots,n\}$ with size $2k+1$ satisfying $c_i := |S \cap
\pi_i| \leq k$ for all $i$. The second formula comes from
counting the odd subsets $S' \subset \{1,2,\ldots,n\}$, where $|S'
\cap \pi_i| > |S'|/2$ for some $i \in \{1,\ldots,r\}$, and
subtracting them from all odd subsets of $\{1,\ldots,n\}$.
\end{proof}
One can also describe the simplices of the polytopes ${\mathcal P}_{\mathcal M}$ for
higher rank matroids as hypergraphs $G$ satisfying the following
conditions:
\begin{enumerate}
\item
Every hyperedge $A \subset [n]$ of $G$ is a base of ${\mathcal M}$.
\item
Let $A = \set{a_1 < \cdots < a_k}$, $B = \set{b_1 < \cdots < b_k}$
be a pair of hyperedges belonging to $G$. Let $C_A$ be the cycle
on $[n]$ drawn with usual edges $(a_1,a_2), \ldots, (a_k,a_1)$ and
similarly for $C_B$. Then each edge of $C_A$ must touch $C_B$ and
vice versa.
\end{enumerate}
\begin{remark}
Let $\tilde\Gamma_{2,n}$ be the
simplicial complex associated with the triangulation $\Gamma_{2,n}$,
and let $f(\tilde\Gamma_{2,n})= \sum_{i=1}^d
f_i(\tilde\Gamma_{2,n})\, t^i$ be its $f$-polynomial,
where $f_i$ is the number of $i$-dimensional simplices in
this complex.
Using generating function techniques, one can deduce the following
expression for these polynomials:
\[\sum_{n\geq 2} f(\tilde\Gamma_{2,n},t)\, x^n =
\left[\frac{tq^2(1+q)(t^2 q^2 + t^2q -tq
+1)}{(1-tq)^2(1-2tq-tq^2)}\right]_{q \mapsto \frac{x}{1-x}}.\]
\end{remark}
\subsection{Adjacency of alcoves in the second hypersimplex}
\label{sec:adjacency_second} Let $\Delta \in \Gamma_{k,n}$ be a
(maximal) simplex. We say that $\Delta$ has \emph{degree} $d$ if it
is adjacent to $d$ other simplices. We call $\Delta$ an
\emph{internal simplex} if none of its facets lies on the boundary
of $\Delta_{k,n}$. In this case $\Delta$ has maximal degree, namely $n$.
\begin{proposition}
\label{prop:graph_move} The two simplices of $\Gamma_{2,n}$
corresponding to two thrackles $G$ and $G'$ (via the
correspondence of Proposition~\ref{prop:volume_graph}) are
adjacent if and only if there are four distinct vertices labelled
$a, a+1, b, b+1 \ \,\mathrm{mod}\, n$ such that $G$ contains the edges
$(a,b),(a-1,b),(b+1,a)$ and $G'$ is obtained from $G$ by changing
the edge $(a,b)$ to $(a-1,b+1)$; see Figure~\ref{fig:graph_move}.
\end{proposition}
\begin{proof}
The proposition follows immediately from Proposition~\ref{prop:volume_graph}
and Theorem~\ref{thm:adjacency} applied to
the case $k=2$ (more precisely, the comments after the proof of
the theorem).
\end{proof}
\begin{figure}
\pspicture(-50,-50)(150,50)
\pscircle[linewidth=0.5pt](0,0){40}
\pscircle[linewidth=0.5pt](100,0){40} \rput(0,50){$a$}
\rput(-28,48){$a-1$} \rput(-28,-48){$b+1$} \rput(0,-50){$b$}
\rput(100,50){$a$}
\rput(72,48){$a-1$} \rput(72,-48){$b+1$} \rput(100,-50){$b$}
\rput(50,0){$\leftrightarrow$} \SpecialCoor \psline(40;90)(40;270)
\psline(40;90)(40;250) \psline(40;270)(40;110) \psdot(40;90)
\psdot(40;250)\psdot(40;270)\psdot(40;110)
\psset{origin={-100;0}} \psline(40;250)(40;110)
\psline(40;90)(40;250) \psline(40;270)(40;110) \psdot(40;90)
\psdot(40;250)\psdot(40;270)\psdot(40;110)
\endpspicture
\caption{The two simplices in $\Gamma_{2,n}$ corresponding to two
thrackles $G$ and $G'$ are adjacent if $G$ and $G'$ are related by
the above move.}\label{fig:graph_move}
\end{figure}
In fact the proposition is true also for smaller dimensional faces
of the simplices in $\Gamma_{2,n}$. Let $C$ be an odd cycle such that
all edges cross and let $|C|$ denote its length.
\begin{theorem}
\label{thm:degree_of_simplex} Let $\Delta$ be the simplex of
$\Gamma_{2,n}$ corresponding to the thrackle $G(C)$. If $G(C)$ is a
triangle then $\Delta = \Delta_{2,3}$ is the unique simplex in
$\Gamma_{2,3}$ (and has degree 0). If $|C| = 3$ and $G(C)$ has two
vertices of degree two then $\Delta$ has degree two. Otherwise,
$\Delta$ has degree $|C|$.
\end{theorem}
\begin{proof}
The Theorem follows from Proposition~\ref{prop:graph_move}. Indeed
we can perform the move shown in Figure~\ref{fig:graph_move} to an
edge $(a,b)$ if it joins two vertices $a,b$ each of degree at
least two, with the exception of the case where $a$ and $b$ both
have degree exactly two and are joined to the same vertex $c$.
When the edge $(a,b)$ can be replaced, the change is necessarily
unique. The case where $a$ and $b$ both have degree two and are
joined to the same vertex $c$ occurs only when $C$ is a
three-cycle. In all other cases, every edge of $C$ can be
replaced, and so $\Delta$ has degree $|C|$.
\end{proof}
The following corollary is immediate from Theorem~\ref{thm:degree_of_simplex}.
\begin{corollary}
For $d > 1$, the second hypersimplex $\Gamma_{2,n}$ has $\binom{n}{2d+1}$
simplices with degree $2d+1$. No simplex has even degree
greater than or equal to 4. In particular, if $n$ is odd then
$\Gamma_{2,n}$ contains a unique internal simplex. If $n$ is even,
then $\Gamma_{2,n}$ has no internal simplices.
\end{corollary}
\section{Multi-hypersimplices and multi-Eulerian polynomials}
\label{sec:multihyper}
In this section we investigate the volumes of the {\it
multi-hypersimplices}, defined in Section~\ref{sec:matroids}. They
are slices of the product $\Delta_{a_1} \times \Delta_{a_2}\times
\cdots \times \Delta_{a_r}$ of simplices or more generally of
hypersimplices. We define the multi-Eulerian numbers to be the
volumes of these polytopes. They are generalizations of the usual
Eulerian numbers. These numbers, like the usual Eulerian numbers,
satisfy a number of interesting enumerative identities. In the
first non-trivial case (Proposition~\ref{prop:multiEulerian}), we
determine the multi-Eulerian numbers explicitly in terms of Eulerian
numbers.
\subsection{Descent-restricted permutations and alcoved polytopes}
\label{sec:descent}
Let us consider an alcoved polytope ${\mathcal P}$ (as in
Section~\ref{sec:alcove}) which lies within a hypersimplex
$\Delta_{k,n}$. In the $x$-coordinates, ${\mathcal P} \subset \mathbb{R}^n$ is defined by
the hyperplane $x_1 + x_2 + \cdots + x_n = k$, the inequalities $0
\leq x_i \leq 1$ together with inequalities of the form
\[
b_{ij} \leq x_{i+1} + \cdots + x_j \leq c_{ij}
\]
for integer parameters $b_{ij}$ and $c_{ij}$ for each pair $(i,j)$
satisfying $0 \leq i < j \leq n-1$.
Let $W_{\mathcal P} \subset S_{n-1}$ be the set of permutations $w =
w_1w_2\cdots w_{n-1} \in S_{n-1}$ satisfying the following
conditions:
\begin{enumerate}
\item
$w$ has $k-1$ descents.
\item
The sequence $w_{i} \cdots w_{j}$ has at least $b_{ij}$ descents.
Furthermore, if $w_{i} \cdots w_{j}$ has exactly $b_{ij}$ descents,
then $w_{i} < w_{j}$.
\item
The sequence $w_{i} \cdots w_{j}$ has at most $c_{ij}$ descents.
Furthermore, if $w_{i} \cdots w_{j}$ has exactly $c_{ij}$ descents,
then we must have that $w_{i} > w_{j}$.
\end{enumerate}
In the above conditions we assume that $w_0 = 0$.
Let $p: \mathbb{R}^n \rightarrow \mathbb{R}^{n-1}$ denote the projection as in
Section~\ref{sec:hypersimplex}. We may apply Stanley's
piecewise-linear map $\psi$ to $p({\mathcal P})$. Using
Theorem~\ref{thm:equaltriangulation}, we see that a unimodular
triangulation of $p({\mathcal P})$ is given by the set of simplices
$\psi^{-1}(\nabla_{w^{-1}})$ as $w = w_1w_2\cdots w_{n-1}$ varies
over permutations in $W_{\mathcal P}$. As a corollary we obtain the following
result.
\begin{proposition}
\label{prop:descent} The volume of ${\mathcal P}$ is equal to $|W_{\mathcal P}|$.
\end{proposition}
\subsection{Multi-Eulerian polynomials} \label{sec:multi} Let $\Pi$
be a fixed set partition with parts of sizes $\set{a_i}_{i=1}^{r}$
of total size $n = \sum_i a_i$ and let $\underline{b} = (b_1,\ldots,b_r), \underline{c}
= (c_1,\ldots,c_r) \in {\mathbb N}^r$. Define the \emph{weighted
multi-Eulerian number} $A_{\Pi,\underline{b},\underline{c},k} =
\mathrm{Vol}({\Delta}_{\Pi,\underline{b},\underline{c},k})$ as the normalized
$(n-1)$-dimensional volume of the corresponding weighted
multi-hypersimplex. We will consider polytopes of smaller dimension
to have volume 0 for what follows. Now define the \emph{weighted
multi-Eulerian polynomial} $A_{\Pi,\underline{b},\underline{c}}(t)$ as
\[
A_{\Pi,\underline{b},\underline{c}}(t) = \sum_{k=1}^{r} A_{\Pi,\underline{b},\underline{c},k}t^{k}.
\]
Note that when $\Pi$ is the set partition with all $a_i = 1$ and
$b_i = 0$ and $c_i = 1$ for all $i$, then $A_{\Pi,\underline{b},\underline{c}}(t)$
reduces to the usual Eulerian polynomial $A_{n-1}(t) = \sum_k
A_{k,n-1} t^k$. If $\Pi$ is a set partition of $[n]$, we denote by
$\Pi^*$ the set partition of $[n+1]$ with an additional part of
size one containing $n+1$. If $\underline{d} = (d_1,\ldots,d_r)
\in {\mathbb Z}^r$, then we denote by $\underline{d}^* \in {\mathbb Z}^{r+1}$ to be
the integer vector with an additional coordinate $d_{r+1} = 0$ and
$\underline{d}' \in {\mathbb Z}^{r+1}$ similarly with $d_{r+1} = 1$. Then
${\Delta}_{\Pi^*,\underline{b}^*,\underline{c}',k}$ is affinely equivalent to (and has the
same normalized volume as) the intersection of $k-1 \leq
x_1+\cdots + x_n \leq k$ with the product of unions of
hypersimplices
\[
\left(\bigcup_{i = b_1+1}^{c_1}\Delta_{i,a_1+1}\right) \times \cdots \times \left(\bigcup_{i = b_r+1}^{c_r}\Delta_{i,a_r+1}\right).
\]
Thus
\[
A_{\Pi^*,\underline{b}^*,\underline{c}'}(1) =
\binom{n}{a_1,a_2,\ldots,a_r}\left(\sum_{i=b_1+1}^{c_1}A_{i,a_1}\right)\cdots\left(\sum_{i=b_r+1}^{c_r}A_{i,a_r}\right).
\]
In particular if $b_j = 0$ and $c_j = 1$ then this value is simply
a multinomial coefficient. For this special case, we will omit
$\underline{b}$ and $\underline{c}$ in the notation and omit the prefix
\emph{weighted} from the names. We write out the combinatorial
interpretation for $A_{\Pi,k}$ explicitly.
\begin{prop}
\label{prop:multiperm} Let $k$ be a positive integer and $\Pi$ be
a set partition of $[n]$ with parts of sizes $\set{a_i}_{i=1}^{r}$
as before. The following quantities are equal:
\begin{enumerate}
\item the multi-Eulerian number $A_{\Pi,k}$,
\item
\label{it:des} the number of permutations of $w \in S_{n-1}$ with
$k-1$ descents such that the substring $w_{a_1}w_{a_1+a_2} \cdots
w_{a_1+a_2+\cdots+a_{r-1}}$ has $k-1$ descents,
\item the number of sorted subsets $(I_1,\ldots,I_n)$ of ${\mathcal M}_\Pi$ of size $n$.
\end{enumerate}
\end{prop}
\begin{proof}
As described in Section~\ref{sec:matroids} we may consider
multi-hypersimplices subpolytopes of the hypersimplex so the
proposition follows from Theorems~\ref{thm:equaltriangulation},
\ref{thm:matroid_tri} and Proposition~\ref{prop:descent}.
\end{proof}
Note that with $\Pi = \Pi(a_1,a_2,\ldots,a_r)$, then $A_{\Pi,k}$
is a function symmetric in the inputs $\{a_i\}$. Let $\Pi=
\Pi(a,1^{n-a})$. By
Proposition~\ref{prop:multiperm}(\ref{it:des}), $A_{\Pi}(t)$ is
the generating function by descents for permutations satisfying
$w_1 < w_2 < \cdots < w_a$. Thus
\[
A_{\Pi(a,1^{n-a})}(t) = \sum_{w \in S_{n-a}}\binom{a+w_1-2}{a-1}
t^{\mathrm{des}(w)+ 1}.
\]
We give an explicit formula for $a = 2$.
\begin{prop}
\label{prop:multiEulerian} For $n \geq 3$, $A_{\Pi(2,1^{n-2})}(t)
= t\frac{d}{dt}A_{n-1}(t)$, where $A_{n-1}(t)$ is the usual
Eulerian polynomial. In other words,
\begin{equation}
\label{eq:des} \sum_{w \in S_{n-1}} w_1 t^{\mathrm{des}(w) + 1} =
\sum_{w \in S_{n-1}} (\mathrm{des}(w) + 1)t^{\mathrm{des}(w) + 1}.
\end{equation}
\end{prop}
We will prove this statement bijectively. Let $w \in S_n$. A
\emph{circular descent} of $w$ is either a usual descent or the
index $n$ if $w_n
> w_1$. Define the \emph{circular descent number} $\mathrm{cdes}(w)$ as
the number of circular descents. Let $C_n$ denote the subgroup of
$S_n$ generated by the long cycle $c = (12\cdots n)$, written in
cycle notation. We have the following easy lemma.
\begin{lemma}
\label{lem:doublecosets} The statistic $\mathrm{cdes}$ is constant on
double cosets $C_n\backslash S_n/C_n$.
\end{lemma}
\begin{proof}
Left multiplication by the long cycle $c$ maps $w_1 w_2 \cdots
w_n$ to $(w_1+1)(w_2+1)\cdots(w_n+1)$ where ``$n+1$'' is
identified with ``1''. Right multiplication by $c$ maps $w_1 w_2
\cdots w_n$ to $w_2w_3\cdots w_n w_1$. The lemma is immediate
from the definition of circular descent number.
\end{proof}
Actually in the following we will use this Lemma for $S_{n+1}$.
\begin{proof}[Proof of Proposition~\ref{prop:multiEulerian}.]
If $w = w_1w_2\cdots w_{n} \in S_{n}$ and $w' = w_1 w_2 \cdots w_n
(n+1) \in S_{n+1}$ then $\mathrm{cdes}(w') = \mathrm{des}(w) + 1$. By this and
our earlier comments, the left hand side of (\ref{eq:des}) is the
generating function for permutations in $S_{n+1}$ satisfying $w_1
< w_2$ and $w_{n+1} = n+1$, according to their circular descent
number. Alternatively, we may view this as the $\mathrm{cdes}$-generating
function of right cosets $\bar w \in S_{n+1}/C_{n+1}$ satisfying the
property that the two numbers $w_{i},w_{i+1}$ cyclically located
after $w_{i-1} = n+1$ satisfy $w_{i} < w_{i+1}$ for any
representative $w \in \bar w$. Here the indices are taken modulo
$n+1$ and by Lemma~\ref{lem:doublecosets}, $\mathrm{cdes}(\bar w) :=
\mathrm{cdes}(w)$ does not depend on the representative $w$ of $\bar w$ and
so is well defined.
The right hand side of (\ref{eq:des}) is the generating function
for permutations $u = u_1u_2 \cdots u_n \in S_n$ satisfying $u_n =
n$ where one of the circular descents has been marked, again
according to circular descent number. If $u_i
> u_{i+1}$ is the marked circular descent (where $i+1$ is to be taken modulo $n$),
then we can insert the number $n+1$ between $u_i$ and $u_{i+1}$ to
obtain a coset $\bar v \in S_{n+1}/C_{n+1}$ with the same number of
circular descents. If $v \in \bar v$ and $v_i = n+1$ then we
automatically have $v_{i-1} > v_{i+1}$ (in fact this is a
$\mathrm{cdes}$-preserving bijection between cosets $\bar v$ satisfying this
property and permutations of $u \in S_n$ satisfying $u_n = n$ with
a marked circular descent). Let $c = (123\cdots(n+1))$ be the
generator of $C_{n+1}$ and consider $v' = c^{n + 1 -v_{i-1}} v$
for any $v \in \bar v$ where $i$ is determined by $v_i = n+1$. Let
$\bar v' = v'C$. By Lemma~\ref{lem:doublecosets}, $\mathrm{cdes}(\bar v') =
\mathrm{cdes}(\bar v)$. However, it is easy to see that $\bar v'$ is exactly one
of the cosets which are enumerated by the left hand side of
(\ref{eq:des}). Thus we obtain a $\mathrm{cdes}$-preserving bijection
$\bar v \mapsto \bar v'$ between two classes of cosets in
$S_{n+1}/C_{n+1}$, enumerated by the two sides of (\ref{eq:des}).
We illustrate the bijection with an example, where we will pick
representatives of appropriate cosets at our convenience. Let $u
= 53162748 \in S_8$ satisfying $u_8 =8$ with marked circular
descent index $4$ corresponding to $u_4 = 6 > 2 = u_5$. This is an
object enumerated by the right hand side of (\ref{eq:des}), with
$n = 8$. Inserting ``$9$'' between ``$6$'' and ``$2$'' we obtain
$v = 531692748 \in S_9$. Multiplying on the left by $c^3$ (where
$c = (123456789)$) adds 3 to every value, changing ``6'' to ``9'',
giving the permutation $v' = 864935172$. Multiplying on the right
by $c^4$ we move the $9$ to the last position to get $w =
351728649 \in \bar v' = v'C$, which satisfies $w_9 = 9$ and $w_1 <
w_2$. This is exactly a permutation enumerated by the left hand
side of (\ref{eq:des}). Note that $\mathrm{cdes}(u) = 5 = \mathrm{cdes}(w)$ and
that all the steps can be reversed to give a bijection.
\end{proof}
It would be interesting if algebro-geometric proofs of some of our
results concerning (weighted) multi-Eulerian numbers could be
given; see Section~\ref{sec:grassmann}.
\section{Final Remarks} \label{sec:final_remarks}
\subsection{The $h$-vectors of alcoved polytopes}
Let $\tilde{\Gamma}_{\mathcal P}$ denote the simplicial complex associated to the
collection of simplices in the alcoved triangulation of ${\mathcal P}$. The
triangulation $\Gamma_{\mathcal P}$ of an alcoved polytope ${\mathcal P}$ is unimodular and
this implies that the $h$-polynomial $h(\tilde{\Gamma}_{\mathcal P},t)$ is a
non-negative polynomial. We have given many interpretations for the
volume $h(\tilde{\Gamma}_{\mathcal P},1)$ of an alcoved polytope. It would be
interesting to obtain statistics on these interpretations which give
$h(\tilde{\Gamma}_{\mathcal P},t)$.
In fact, the Erhart polynomial of ${\mathcal P}$ equals to the Hilbert
polynomial of the quotient ring $k[x_I]/{\mathcal{J}}_{\mathcal P}$ associated to the
polytope ${\mathcal P}$; see Section~\ref{sec:grobner_intro}. The
$h$-polynomial is the numerator of the associated generating
function.
\subsection{Relation with order polytopes}
Let $P$ be a poset naturally labelled with the numbers $[n]$. The
order polytope ${\mathcal O}_P \subset \mathbb{R}^n$ is defined by the inequalities
$0 \leq x_i \leq 1$ together with $x_i \leq x_j$ for every pair of
elements $i,j \in P$ satisfying $i > j$. It is clear that ${\mathcal O}_P$
is affinely equivalent to an alcoved polytope via the transformation
$y_i = x_i - x_{i-1}$. The triangulation $\Gamma_P$ of ${\mathcal O}_P$ thus
obtained is known as the \emph{canonical triangulation} of ${\mathcal O}_P$.
Denote by ${\mathcal L}_P$ the set of linear extensions of $P$;
see~\cite{EC1}. The simplices of $\Gamma_P$ can be labelled by $w \in
{\mathcal L}_P$ and this is compatible with the labelling used throughout
this paper. It is known~\cite{Sta2} that we have
\[
h(\tilde{\Gamma}_P,t) = \sum_{w \in {\mathcal L}_P} t^{\mathrm{des}(w)}.
\]
The descents counted by this $h$-vector, however, disagree with
the way we have been counting descents in this paper. More
precisely, we have been concerned with the number of descents
$\mathrm{des}(w^{-1})$ of the inverse permutations labelling the
simplices.
\subsection{Weight polytopes for type $A_n$ and alcoved
polytopes} \label{sec:weight} A \emph{weight polytope} is the
convex hull of the weights which occur in some highest weight
representation of a Lie algebra. For example,
the vertices of the hypersimplex $\Delta_{k,n}$ are exactly the weights
which occur in the $k^{th}$ fundamental representation of
$\mathrm{sl_n}$.
We will identify the integral weights $L$ of $\mathrm{sl_n}$ with
the integer vectors $(a_1,\ldots,a_n)$ satisfying $a_1 + \cdots +
a_n = l$, for some fixed $l$. A weight polytope ${\mathcal P}_\lambda$ is
specified completely by giving the highest weight $\lambda =
(\lambda_1, \ldots, \lambda_n)$. We will assume all the
coordinates $\lambda_i$ are non-negative. A weight $\mu$ lies
inside ${\mathcal P}_\lambda$ if $\lambda$ dominates $\mu$ in the usual
sense: $\lambda_1 + \cdots + \lambda_i \geq \mu_1 + \cdots +\mu_i$
for all $i$.
Let $\mu, \nu \in L$ be two weights. Define $U(\mu,\nu),
V(\mu,\nu) \in L$ by requiring that $U(I_\mu,I_\nu) =
I_{U(\mu,\nu)}$ and $V(I_\mu,I_\nu) = I_{V(\mu,\nu)}$, in the
notation of Section~\ref{sec:alcoved}. Alternatively,
$U(\mu,\nu)$ and $V(\mu,\nu)$ are determined by requiring that
$\mu + \nu = U(\mu,\nu) + V(\mu,\nu)$ and $V(\mu,\nu) - U(\mu,\nu)
= \sum_i b_i \alpha_i$ for some $b_i \in \set{0,1}$, where
$\alpha_i = e_{i+1}-e_i$ are the simple roots. We call
${\mathcal P}_\lambda$ \emph{sort-closed} if the set of weights ${\mathcal P}_\lambda
\cap L$ is sort-closed, as in Theorem~\ref{thm:alcove}.
\begin{prop}
\label{prop:sorted_weight} A weight polytope ${\mathcal P}_\lambda$ is
sort-closed if and only if $\lambda$ is equal to $aw_i + bw_{i+1}$ for
non-negative integers $a,b$ and where $w_k =
(1,\ldots,1,0,\ldots,0)$ ($k$ 1's) are the fundamental weights.
\end{prop}
\begin{proof}
The `if' direction is easy as ${\mathcal P}_\lambda$ can be specified by $x_1
+ \cdots + x_n = i \cdot a + (i+1) \cdot b$ and the inequalities $0
\leq x_j \leq a$ for each $j \in [1,n]$ and we can use
Theorem~\ref{thm:alcove}. For the other direction, we may assume the
highest weight $\lambda$ is a $n$-tuple with highest value $a$ and
lowest value $0$. Suppose $\lambda$ is not of the form of the
proposition, then there are two more values $b, c$ not equal to $a$
satisfying $a > b \geq c > 0$ so that $\lambda$ is of the form
$(a,\ldots,a,b,\ldots,b,c,\ldots,c,\ldots,0)$.
Explicitly construct a pair of weights $\delta =
(a,0,b,c,\ldots)$ and $\mu = (a-1,1,b+1,c-1,\ldots)$ where the
tails of the two $n$-tuples are identical, and $\delta$ is just a
permutation of the coordinates of $\lambda$. Both $\delta$ and
$\mu$ are dominated by $\lambda$ and hence lie in ${\mathcal P}_\lambda$.
However, $U(\delta,\mu) = (a,0,b+1,c-1,\ldots)$ does not lie in
${\mathcal P}_\lambda$.
\end{proof}
Sturmfels~\cite[Chapter 14]{Stu} considered exactly this class of
sort-closed weight polytopes.
The following
corollary follows immediately from Theorem~\ref{thm:alcove} and
Proposition~\ref{prop:sorted_weight}.
\begin{cor}
\label{cor:alcoved_weight} A weight polytope in the
$x$-coordinates with highest weight $\lambda$ is alcoved if and
only if $\lambda$ is of the form $a\omega_i + b\omega_{i+1}$ for
$a$ and $b$ non-negative integers. In particular, every weight
polytope for $A_2$ is alcoved.
\end{cor}
\subsection{Geometric motivation: degrees of torus
orbits} \label{sec:grassmann} Let $Gr_{k,n}$ denote the {\it
grassmannian manifold\/} of $k$-dimensional subspaces in the
complex linear space $\mathbb{C}^n$. Elements of $Gr_{k,n}$ can be
represented by $k\times n$-matrices of maximal rank $k$ modulo
left action of $GL_k$. The $\binom{n}{k}$ maximal minors $p_I$ of
such a matrix, where $I$ runs over $k$-element subsets in
$\{1,\dots,n\}$, form projective coordinates on $Gr_{k,n}$, called
the {\it Pl\"ucker coordinates}. The map $Gr_{k,n}\to (p_I)$ gives
the {\it Pl\"ucker embedding\/} of the grassmannian into the
projective space $\mathbb{CP}^{\binom{n}{k} -1}$. Two points
$A,B\in Gr_{k,n}$ are in the same {\it matroid stratum\/} if
$p_I(A)=0$ is equivalent to $p_I(B)=0$, for all $I$. The matroid
${\mathcal M}_A$ of $A$ has as set of bases $\set{I \in \binom{[n]}{k}
\mid p_I(A) \neq 0}$.
The complex torus $T=(\mathbb{C}\setminus\{0\})^n$ acts on $\mathbb{C}^n$ by
stretching the coordinates
$$
(t_1,\dots,t_n):(x_1,\dots,x_n)\mapsto(t_1 x_1,\dots,t_n x_n).
$$
This action lifts to an action of the torus $T$ on the grassmannian
$Gr_{k,n}$. This action was studied in~\cite{GGMS}. The authors
showed that the geometry of the closure of a torus orbit $X_A =
\overline{T\cdot A}$ depends (only) on the matroid stratum of $A$.
The variety $X_A$ is a toric variety and its associated polytope is
exactly the polytope ${\mathcal P}_{{\mathcal M}_A}$ associated to the the matroid
${\mathcal M}_A$ from Section~\ref{sec:matroids}. Our study of the volume of
the polytopes ${\mathcal P}_{{\mathcal M}}$ was motivated by the well known fact
(see~\cite{Ful}) that
\[
\mathrm{deg}(X_A) = \mathrm{Vol}({\mathcal P}_{{\mathcal M}_A})
\]
where $\deg(X_A)$ denotes the degree of $X_A$ as a projective
subvariety of $\mathbb{CP}^{\binom{n}{k}-1}$ and ${\rm Vol}$
denotes the normalized volume with respect to the lattice
generated by the coordinate vectors $e_i$. Note that by
definition only {\it representable} matroids ${\mathcal M}$ arise as
${\mathcal M}_A$ in this manner. Our Theorem~\ref{thm:matroid_tri} gives a
combinatorial description of the degree of a torus orbit closure
corresponding to a stratum of a sort-closed matroid. In fact
Proposition~\ref{prop:grobner_multi} (and
Proposition~\ref{prop:sqfree}) shows that these torus orbit
closures are projectively normal, a fact known for all torus orbit
closures; see~\cite{Whi,Dab}.
It has been conjectured (see~\cite[Conjecture 13.19]{Stu}) that the
ideal of a smooth projectively normal toric variety is always
generated by quadratic binomials. The toric varieties associated to
simple alcoved polytopes give more examples of this.
\section{Appendix: Coherent triangulations and Gr\"{o}bner bases}
\label{sec:grobner_intro} We give a brief introduction to the
relationship between coherent triangulations of integer polytopes
and Gr\"{o}bner bases. See Sturmfels~\cite{Stu} for further
details.
Let $k$ be a field and $k[x] = k[x_1,\ldots,x_n]$ be the
polynomial ring in $n$ variables. A total order $\prec$ on ${\mathbb N}^n$
is a \emph{term order} if it satisfies:
\begin{itemize}
\item
The zero vector is the unique minimal element.
\item
For any ${\underline{a}},\,{\underline{b}},\,\underline{c} \in {\mathbb N}^n$, such that ${\underline{a}} \prec {\underline{b}}$ we have
${\underline{a}}+\underline{c} \prec {\underline{b}}+\underline{c}$.
\end{itemize}
One way to create a term order is by giving a weight vector
${\omega} = ({\omega}_1,\ldots,{\omega}_n) \in \mathbb{R}^n$. Then for sufficiently generic
weight vectors a term order $\prec$ is given by ${\underline{b}} \prec \underline{c}$ if
and only if ${\omega}\cdot{\underline{b}} < {\omega}\cdot\underline{c}$\,. In this situation we will say that
${\omega}$ represents $\prec$.
Given a polynomial $f \in k[x]$ one defines the initial monomial
${\rm in}_{\prec}(f)$ as the monomial $\underline{x}^{\underline{a}}$ with the
largest ${\underline{a}}$ under $\prec$. For an ideal $I$ of $k[x]$ one defines
the initial ideal ${\rm in}_\prec(I)$ as the ideal generated by
the initial monomials of elements of $I$. The monomials which do
not lie in ${\rm in}_\prec(I)$ are called the standard monomials.
A finite subset ${\mathcal G} \subset I$ is a \emph{Gr\"{o}bner basis} for
$I$ with respect to $\prec$ if ${\rm in}_{\prec}({\mathcal G})$ generates
${\rm in}_\prec(I)$. The Gr\"{o}bner basis is called reduced if
for two distinct elements $g, g' \in {\mathcal G}$, no term of $g'$ is
divisible by ${\rm in}_\prec(g)$.
Now let ${\mathcal A} = \set{{\underline{a}}_1,{\underline{a}}_2,\ldots,{\underline{a}}_n}$ be a finite subset of
${\mathbb Z}^d$. We define a ring homomorphism $k[x] \to k[t_1^{\pm
1},\ldots,t_d^{\pm 1}]$ by
\[
x_k \longmapsto \underline{t}^{{\underline{a}}_k}.
\]
The kernel ${\mathcal{J}}_{\mathcal A}$ of this map is an ideal known as a \emph{toric
ideal}.
We now describe the relationship between Gr\"{o}bner bases of
${\mathcal{J}}_{\mathcal A}$ and coherent triangulations of the convex hull of ${\mathcal A}$. For
any term order $\prec$, the initial complex ${\Delta}_\prec({\mathcal{J}}_{\mathcal A})$ of
${\mathcal{J}}_{\mathcal A}$ is the simplicial complex defined as follows. A subset $F
\subset \set{1,2,\ldots,n}$ is a face of ${\Delta}_\prec({\mathcal{J}}_{\mathcal A})$ if there
is no polynomial $f \in {\mathcal{J}}_{\mathcal A}$ such that the support of ${\rm
in}_\prec(f)$ is $F$. Thus the \emph{Stanley-Reisner ideal} of
${\Delta}_\prec({\mathcal{J}}_{\mathcal A})$ is the radical of ${\rm in}_\prec({\mathcal{J}}_{\mathcal A})$.
A triangulation of a set ${\mathcal A} \in {\mathbb Z}^d$ (more specifically, its
convex hull) is \emph{coherent} if one can find a piecewise-linear
convex function $\nu$ on $\mathbb{R}^d$ such that the domains of linearity
are exactly the simplices of the triangulation. Alternatively,
the triangulation is coherent if one can find a `height' vector
${\omega}$ such that the projection of the `lower' faces of the convex
hull of $\set{({\underline{a}}_1,{\omega}_1),({\underline{a}}_2,{\omega}_2),\ldots,({\underline{a}}_n,{\omega}_n)}$ is
exactly the triangulation. We will denote such a triangulation of
${\mathcal A}$ by ${\Delta}_{\omega}({\mathcal A})$. The function $\nu$ and the vector ${\omega}$ can
be related by setting ${\omega}_n = \nu({\underline{a}}_n)$.
The main result we will need is the following~\cite[Chapter
8]{Stu}:
\begin{theorem}
\label{thm:regtri} The coherent triangulations of ${\mathcal A}$ are the
initial complexes of the toric ideal ${\mathcal{J}}_{\mathcal A}$. More precisely, if ${\omega}
\in \mathbb{R}^n$ represents $\prec$ for ${\mathcal{J}}_{\mathcal A}$ then ${\Delta}_\prec({\mathcal{J}}_{\mathcal A}) =
{\Delta}_{\omega}({\mathcal A})$.
\end{theorem}
In the case when ${\mathcal{J}}_{\mathcal A}$ is a homogeneous toric ideal we can say
more.
\begin{prop}
\label{prop:sqfree} Let ${\mathcal A}$ be such that ${\mathcal{J}}_{\mathcal A}$ is a homogeneous
toric ideal. Then the initial ideal ${\rm in}_\prec({\mathcal{J}}_{\mathcal A})$ is
square-free if and only if the corresponding regular triangulation
${\Delta}_\prec$ of ${\mathcal A}$ is unimodular. In that case, let $Y_{{\mathcal A}}$ be the
projective toric variety defined by the ideal ${\mathcal{J}}_{\mathcal A}$. Then
$Y_{{\mathcal A}}$ is projectively normal and the Hilbert polynomial of
$Y_{{\mathcal A}}$ equals to the Erhart polynomial of the convex hull of ${\mathcal A}$.
\end{prop}
In this last case, the $\prec$-standard monomials correspond
exactly to the simplices of the triangulation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,113 |
\section{Introduction}\label{sec:intro}
The TCS community has a long history of applying its perspectives and tools to better understand the processes around us; from learning to multi-agent systems, game theory and mechanism design. By and large, the efforts to understand these areas from a rigorous and algorithmic perspective have been very successful, leading to both rich theories and practical contributions.
Evolution is, perhaps, one of the most blatantly algorithmic processes, yet our computational understanding of it is still in its infancy (see \cite{valiant2009evolvability} for a pioneering study), and we currently lack a computational theory explaining its apparent success.
Algorithmically, how plausible are the origins of evolution and the emergence of self-replication? Is evolution surprisingly efficient or surprisingly inefficient? What are the necessary criteria for evolution-like algorithms to yield rich, interesting, and diverse ecosystems? Why is recombination (i.e., sexual reproduction) more successful than asexual reproduction? Given the reshuffling of genomes that occurs through recombination, how can complex traits that depend on many different genes arise and spread in a population?
In this work, we begin to tackle this last question of why complex traits that may depend on many different genes are able to \textit{efficiently} arise in polynomial populations with recombination. In the standard view of evolution, a variant of a particular gene is more likely to spread across a population if it makes its own contribution to the overall fitness, independent of the contributions of variants of other genes.
How can complex, multi-gene traits spread in a population? This may seem to be especially problematic for multi-gene traits whose contribution to fitness does not decompose into small additive components associated with each gene variant
---traits with the property that
even if one gene variant is different from the one that is needed for the right combination,
there is no benefit, or even a net negative benefit. Here, we provide one rigorous argument for how such complex traits can efficiently spread throughout a population. While we consider this question in a model that makes considerable (but justifiable) simplifications, this model makes a theoretically rigorous contribution to the fundamental problem of how evolution can produce
complexity.
\paragraph{Motivating example: Waddington's experiment.} In 1953 the great experimentalist Conrad Waddington
exposed the pupae of a population of \textit{Drosophilia melanogaster} to a heat shock, and noticed that in some of the adults that developed, the appearance of the wings had changed (they lacked a complete posterior crossvein) \cite{wad}. He then maintained a population of flies where only those with altered wings were allowed to reproduce.
By repeating the procedure of heat shock and selection over the generations, the percentage of flies with altered wings increased over time to values close to one. Even more interestingly, beginning at generation fourteen, some flies exhibited the new trait even without having been treated with heat shock.
At first sight, this surprising phenomenon --- known as genetic assimilation --- recalls Lamarck's now discredited belief that acquired traits can be inherited. However, Boolean functions provide a purely genetic explanation,
which extends the idea originally offered informally by Stern \cite{Stern:1958kx} (see also \cite{Bateman1959b,Falconer1960}):
Suppose that the phenotype ``altered wings'' is a Boolean function of $n$ genes $x_1,\ldots,x_n$ with two alleles (variants) each, thought as $\{-1,1\}$ variables, and of another $\{-1,1\}$ variable $h$ (standing for ``high temperature''). \footnote{Here we assume for simplicity haploid organisms, that is, each individual has only one copy of each gene.}
$$x_1+x_2+\cdots +x_n + \frac{(1+h)}{2}\cdot k\geq n,$$
for some integer $k$ (think of $n \approx 10$ , and $k\approx n/3$).
To see how the percentage of flies with altered wings increases in the population over time, we track the allele frequencies from generation to generation.
Let $\mu_i^t$ be the average value of $x_i$ in the population at time $t$, and assume the genotype frequencies at time $t$ are distributed according to a distribution $\mu^t$ (the reason for denoting the distribution this way will become clear). If mating occurs at random with free recombination\footnote{See the next section for any unfamiliar terms and concepts from evolution.} then, in expectation, the average value of each $x_i$ in the next generation is given by:
\begin{equation}\label{eqn:select}
\mu_i^{t+1} = \frac{{\bf E}_{\mu^t}[ f(x)\cdot x_i ] }{{\bf E}_{\mu^t}[f]},
\end{equation}
where $f(x)=1$ exactly when a fly having genotype $x$ will develop altered wings (i.e., the above inequality is satisfied) and $f(x)=0$ otherwise. We then assume that the next generation will be distributed according to a product distribution $\mu^{t+1}$, where each $x_i$ has expectation $\mu_i^{t+1}$. By approximating the genotype frequencies of the population for each generation in this way,
it can be shown by calculation that a trait with this genotypic specification (a) is very rare in the population under normal temperature $h=1$; (b) it becomes much more common under high temperature $h=1$; (c) jumps to just above $50\%$ after the first breeding under $h= 1$; (d) after successive breedings with $h=1$ it is nearly fixed; and (e) if after this $h$ becomes $-1$, the trait is still quite common.
\paragraph{Note: } Our interpretation of Waddington's experiment is a simplification. First, we consider only the distribution of genotypes in each generation to determine the distribution of the next; instead, we could first take a finite sample according to the present distribution and use that sample to calculate the distribution of the next generation. Such an approximation can only become exact when the population size is infinite, but it is a standard and useful one in population genetics (and we shall eventually consider finite populations for our main result). We also assumed that
each individual of the new generation is produced by sampling each gene independently of the other genes, and with probability equal to the frequencies of the two alleles of this gene in the parent population (the adults of the previous generation with altered wings). This assumption turns out to be justified in the settings that we will consider, as will be discussed in the following section.
\subsection*{Populations of truth assignments}
This way of looking at Waddington's experiment brings about a very natural question: Is this amplification of satisfying truth assignments (outcomes (c) and (d) of experiment described above) a property of threshold functions, or is it more general? Does it hold for all monotone functions, for example? {\em For all Boolean functions?}
Consider any satisfiable Boolean function $f:\fbitset^n\to \bitset$ of $n$ binary genes (in the absence of the environmental variable $h$ which was crucial in Waddington's phenomenon). What if genotypes satisfying this Boolean function had a slight advantage
under natural selection?
(In Waddington's experiment, they had an absolute advantage because of the experimental design.)
For example, imagine that genotypes satisfying $f$ survive to adulthood more than the others, in expectation, by a factor of $(1+\epsilon)$, for some small $\epsilon>0$. Would this trait (that is, satisfaction of the Boolean function $f$) be eventually fixed in the population? And, if so,
could this be
a subtle mechanism for introducing complex adaptation in a population?
To reflect our assumption that satisfaction of $f$ confers only an $\epsilon$-advantage, we may take a function
$f:\fbitset^n \to \{1,1+\epsilon\}$, where we regard the value $1+\epsilon$ as ``satisfied'', and the value $1$ as ``unsatisfied''. We track the allele frequencies from generation to generation as in Waddington's experiment: Equation \eqref{eqn:select} gives us the average value $\mu_i^{t+1}$ of each $x_i$ in the next generation, and we describe the next generation by the product distribution $\mu^{t+1}$.
Suppose that we continue this process, starting from distribution $\mu^0$, and defining \\ $\{\mu^1_i\},\{\mu^2_i\},\ldots, \{\mu^t_i\}\ldots$ as above. Consider the average fitness of the population at time $t$, defined as $\mu^t(f)=\Pr_{\mu^t}[f(x) = 1+\epsilon]$. The question is, when does $\mu^t(f)$ approach one?
Our first result states that, for monotone functions, it does after $O\left(\frac{n}{\epsilon \mu^0(f)}\right)$ steps:
\begin{thm}\label{thm:monotone-main}
If $f$ is monotone, then $\mu^t(f) \geq 1-{n (1+\epsilon)\over \epsilon t \mu^0(f)}.$
\end{thm}
\paragraph{Note:} This nontrivial result also serves to illustrate one point: The work is {\em not} about satisfiability heuristics (monotone functions are not an impressive benchmark in this regard...). Heuristics are about finding good {\em individuals} in a population. In contrast, evolution is about creating good {\em populations}. This is our focus here.
\medskip
Our ambition is to prove the same result for all Boolean functions. Immediately we see that this is impossible if we insist on an infinite population: Consider the function $f=x_1\oplus x_2$: starting with the uniform distribution at time $t=0$, the above dynamics would leave the distribution unchanged, for all time, and hence $\mu^t(f)=1/2$ for all $t$. The parity function is not the only Boolean function with this property: for example the function ``$\sum_{i=1}^n x_i = k$'', if started at $\mu_i = {k\over n}$, will stay at that spot forever, and will always have $\mu^t(f)=O(1/\sqrt{k})$. However, experimentation shows that these ``spurious fixpoints'' are not absorbing, and evolution pulls the distribution away from them and towards satisfaction. That is, this disappointing phenomenon is an artifact of the infinite population simplification.
Indeed, random genetic drift due to sampling effects has been considered to be a significant component of evolution at the molecular level (it is possible for an allele to become fixed in the population even in the absence of selection). Thus, we need to make the model more realistic.
We adopt a model, consisting of the following process: At each generation $t$ we create a large population of $N$ individuals (we call this the ``sampling'' step) by
sampling $N$ times from the product distribution $\mu^t$ to obtain $y^{(1)},\dots,y^{(N)}$ ($N$ is assumed to remain constant from one generation to the next, which is a standard assumption in population genetics \cite{Gillespie2004}).
The empirical allele frequencies of the sample are given by a vector $\nu^t$, where for each $i$ we have:
$$\nu^t_i = \frac{1}{N}\sum_{j=1}^{N} y_i^{(j)}.$$
We write $\nu^t \sim B(\mu^t)$ to denote a draw from this distribution and use $\nu^t$ to denote the implied Bernoulli distribution.
We then enforce the assumed selection advantage of satisfaction to obtain the ``in-expectation'' frequencies of the subsequent generation:
\begin{equation}\label{eqn:select-noise}
\mu_i^{t+1} = \frac{{\bf E}_{\nu^t}[ f(x)\cdot x_i ] }{{\bf E}_{\nu^t}[f]}.
\end{equation}
We show that when selection is weak, \textit{any} satisfiable Boolean function will almost surely be always satisfied after polynomially many time steps.
\begin{thm} \label{thm:main} {\bf (informal statement)}
For any satisfiable Boolean function $f$ of $n$ variables and any sufficiently small $\epsilon>0$, after $T$ generations of $N$ individuals $\mu^T(f)=1$ with probability arbitrarily close to one, where $T$ and $N$ are polynomial functions of $n$, $1\over \epsilon$, and $1\over \mu^0(f)$.
\end{thm}
The proof of Theorem \ref{thm:main} shows why the population does not become stuck at the previously discussed ``spurious fixpoints;''
sampling effects ensure movement over sufficiently many generations, and selection ensures movement is made towards satisfaction.
\subsection*{Outline of the paper}
In the next section we introduce some basic concepts from population genetics, we define and justify our simplified model, and we present a result due to Nagylaki \cite{Nag} implying that, if selection is weak, then one can assume that the genotype distribution is a product distribution. In Section \ref{sec:monotone}, we show Theorem \ref{thm:monotone-main} on monotone functions. Our main result is given in Section \ref{sec:main}, and its proof outlined; the full proof is detailed in the Appendix. In Section \ref{sec:discussion}, we conclude with a discussion of our result, and a number of open problems.
\section{Evolution background and our model}
The genetic makeup of an organism is its {\em genotype}, which specifies one {\em allele} (gene variant) for each genetic site, or ``locus,'' in the haploid case. We shall be focusing on $n$ specific genes of interest (say, a few dozen out of the many thousands of genes of the species). At each locus, we assume that there are two alleles segregating in the population (hence the relevance of Boolean functions). Thus, a genotype will be a vector in $\{\pm 1\}^n$.
We assume the species reproduces sexually (this is crucial, see the discussion in the last section). In a sexual species reproduction proceeds through {\em recombination,} that is, the formation of a new genotype by choosing alleles from two parental genotypes in the previous generation.
To produce each generation, the
individuals
mate at random (we also assume no bipartition into sexes)
and there is no generation overlap (that is, the new generation is produced en masse just before death of the previous one). We assume that
the population size is constant at some large number
$N$ (expressed as a function of $n$, the number of genes of interest, which is the basic parameter). Each genotype $g\in\{\pm 1\}^n$ is assumed to have a {\em fitness value} equal to the expected number of offsprings this genotype will produce. We also assume that the genes {\em recombine freely}, that is, for any two genes $i,j$ of
an offspring,
the probability that the alleles come from the same parent is exactly half (and not larger, as is the case if the two genes are linked).
These assumptions are simplifications of the standard model of population genetics used broadly in the literature, and generally trusted to preserve the essence of selection in sexual populations. The Boolean assumption is of course meant to bring into play mathematical insights from that field, but we believe that it is not restrictive (for example, allele $-1$ could stand for ``any allele other than allele $1$''). In this paper we shall make two more assumptions. The first additional assumption is that the fitness values of our genotypes are either $1$ or $1 + \epsilon$, where $\epsilon>0$ is very small: the organism will reproduce slightly more in expectation if an underlying Boolean function is satisfied
We discuss this restriction in Section \ref{sec:discussion}.
The final assumption is more problematic in general, but justified in the current context: We assume that generating an individual
of the next generation is tantamount to selecting, independently, an allele for each of the $n$ genes, with probability equal to the probability of occurrence of that allele in the parent generation. That is, we assume that the distribution of the genotypes in a generation is a {\em product distribution}. This situation is called in the population genetics literature {\em linkage equilibrium}, or {\em the Wright manifold} \cite{Wright:1931uq,Wright:1932kx}.
In general, genotype frequencies are known to be correlated, and this correlation --- the distance from the product distribution --- is called {\em linkage disequilibrium} \cite{Lewontin:1964ys} and is of importance and interest in the study of evolution. However, in the absence of selection, a standard argument shows that the distribution of a population quickly reaches linkage equilibrium (arguments exist both for finite and infinite populations). Our previous assumption places our experiment in a regime known as {\em weak selection}. Weak selection means that the fitness values are in a small interval $[1-\epsilon, 1+\epsilon]$, where $\epsilon$ is called the {\em selection strength}. An elegant and powerful result due to Thomas Nagylaki \cite{Nag} states that, under weak selection, evolution proceeds to a point very close to linkage equilibrium. In particular, assume that a population evolves as we described above in a regime of weak selection of strength $\epsilon$, and let $m$ be the total number of alleles (this is $2n$ in our case; actually, Nagylaki's Theorem also holds under diploid and partial recombination). By linkage disequilibrium we mean formally the $L_{\infty}$ distance between the genotype distribution and the product distribution:
\begin{thm} (Nagylaki's Theorem, see \cite{Nag}) \label{thm:nagylaki} Under weak selection, and after $O(\log m\cdot \log 1/\epsilon)$ generations, linkage disequilibrium is $O(\epsilon)$.
\end{thm}
In our setting $\epsilon$ is minuscule, so Nagylaki's Theorem motivates our assumption that populations are formed ``by independent sampling of the genetic soup.''
We strongly believe that our theorem is true for large $\epsilon$ as well, but this remains open, as discussed in the last section.
\section{Weak selection on monotone functions}
\label{sec:monotone}
In this section we give a self-contained proof of Theorem \ref{thm:monotone-main}. The proof is simple, once a connection is made to discrete Fourier analysis. In what follows, we assume familiarity with Fourier analysis over the Boolean cube for product distributions. We briefly review some basic facts and describe the notation used in our proofs.
For $\mu = (\mu_1,\dots,\mu_n) \in [-1,1]^n$ and a function $f:\fbitset^n_\mu \to \RR$, where
$\fbitset^n_\mu$ denotes the Boolean cube endowed with the product distribution given by $\mu_i = {\bf E}[x_i]$, we
consider the $\mu$-biased Fourier decomposition of $f$. Let $\sigma_i^2 = 1-\mu_i^2$ be the variance of each bit.
We denote the $\mu$-biased Fourier coefficients by $\hat{f}(S;\mu) = {\bf E}_\mu [f \cdot \phi^\mu_S]$, where $\phi^\mu_S = \prod_{i\in S} \frac{x_i -\mu_i}{\sigma_i}$.
Let $D_i^{(\mu)} f = \frac{\sigma_i}{2} (f_{i=1} - f_{i=-1})$ be the difference operator for Boolean functions over $\fbitset^n_\mu$.
We have that
\begin{equation} \label{eqn:diff} D_i^{(\mu)} f = \sum_{S\ni i} \hat{f}(S;\mu) \phi^\mu_{S\setminus \{i\}},
\end{equation}
and in particular, ${\bf E}_\mu[ D_i^{(\mu)}f ] = \hat{f}(i;\mu)$, which we will use repeatedly throughout our proofs.
Our first step will be to observe that the change in allele frequencies from one generation to the next may be expressed in terms of $f$'s linear Fourier coefficients.
Let $\mu$ be the vector which specifies the allele frequency of the population at time $t$. Then, letting $\mu'$ be the allele frequency vector at time $t+1$ and using the selection specified by Equation \eqref{eqn:select}, we have that
\begin{equation}\label{eqn:select2}
\mu'_i - \mu_i = \sigma_i \frac{\hat{f}(i;\mu)}{{\bf E}_\mu[f]}.
\end{equation}
This follows immediately from the definitions:
\begin{align*}
\sigma_i \cdot \hat{f}(i;\mu) & = \sigma_i\cdot {\bf E}_\mu [ f\cdot \phi_i^\mu] \\
& = {\bf E}_\mu[f\cdot x_i ] - {\bf E}_\mu[ f]\cdot \mu_i \\
& = {\bf E}_\mu[f]\cdot \mu'_i - {\bf E}_\mu[f]\cdot \mu_i.
\end{align*}
Our proof uses the following well-known facts, which are easily derived from the basic notions
(see Chapter 2.3,
\cite{o2014analysis}
). First, we have that the influences of a monotone function
are given by its linear coefficients.
(For a function $f:\fbitset^n_\mu \to \RR$, we denote its influence in direction $i$ by $\sum_{S\ni i} \hat{f}(S;\mu)^2$.) Next, the inequality of Poincar\'{e} lower bounds the total influence of a function by its variance. The versions below have been scaled to our setting and can be obtained by applying the original facts to a Boolean function $g:\fbitset^n\to \fbitset$ and setting $f(x) = 1+\frac{\epsilon}{2} (1+g)$.
\begin{prop}\label{prop:influence} Let $f:\fbitset_\mu^n \to \{1,1+\epsilon\}$ be monotone. Then for all $i\in [n]:$
\begin{eqnarray*}
\sum_{S \ni i} \hat{f}(S;\mu)^2 = \frac{\epsilon \sigma_i}{2} \cdot \hat{f}(i;\mu).
\end{eqnarray*}
\end{prop}
\begin{prop} \label{prop:poinc} Let $f:\fbitset^n_\mu \to \{1,1+\epsilon\}$ and $\operatorname{Var}[f] =
{\bf E}_\mu [f^2] - {\bf E}_\mu[f]^2$. Then
$$\sum_{i \in [n]} \sum_{S\ni i} \hat{f}(S;\mu)^2 = \sum_{S\subseteq [n]} |S|\hat{f}(S;\mu)^2\geq \operatorname{Var}[f].$$
\end{prop}
Equation \eqref{eqn:select2} tells us that the bias of each bit $i$ increases according to the corresponding coefficient $\hat{f}(i)$. Proposition \ref{prop:influence} tells us that for monotone $f$, the linear coefficients correspond to the influences of $f$.
Finally, the inequality of Poincar\'{e}
tells us that the linear coefficients must be large.
We may now prove Theorem \ref{thm:monotone-main}.
\begin{thm_again}{\ref{thm:monotone-main}}
Let $f:\fbitset^n \to \{1,1+\epsilon\}$ be monotone. Then $\mu^t(f)\geq 1-\frac{n(1+\epsilon)}{\epsilon t\mu^0(f)}$.
\end{thm_again}
\begin{proof}
Combining Equation \eqref{eqn:select2} with Propositions \ref{prop:influence} and \ref{prop:poinc} tells us that the sum of the biases increases at each step:
\begin{align*}
\sum_{i\in [n]}(\mu'_i - \mu_i) & = \frac{2}{\epsilon \cdot {\bf E}_\mu[f]} \sum_{i\in [n]}\sum_{S \ni i} \hat{f}(S;\mu)^2 \\
& \geq \frac{2}{\epsilon \cdot {\bf E}_\mu[f ] } \operatorname{Var}[f]\\
& = \frac{2}{\epsilon \cdot {\bf E}_\mu[f]} \epsilon^2 \mu(f)(1-\mu(f))
\end{align*}
Let $\mu^t(f) = 1- \delta.$ Then for all $t'\leq t$, the sum of the biases increases at each step:
\begin{align*}
\sum_{i=1}^n \mu^{t'+1}_i -\sum_{i=1}^n \mu^{t'}_i & \geq \frac{2\epsilon\mu^{t'}(f)(1-\mu^{t'}(f))}{{\bf E}_\mu^{t'}[f]} \\
& \geq \frac{2\epsilon \mu^{t'}(f) \delta }{1+\epsilon} \geq \frac{2\delta \epsilon \mu^{0}(f)}{1+\epsilon}.
\end{align*}
On the other hand, we know that $-n \leq \sum_{i=1}^n \mu^{t'}_i \leq n$ for all $t'$, so $t \leq \frac{n (1+\epsilon)}{\delta \epsilon \mu^0(f)}.$
\end{proof}
We remark that Theorem \ref{thm:monotone-main} (with worse parameters) can also be proven using a generalization of the Russo-Margulis lemma to product distributions, which states that the gradient of ${\bf E}_\mu[f]$ (as a function of $\mu$) corresponds to the influences of $f$ (see Appendix \ref{sec:prelim})
\section{The main result}
\label{sec:main}
For a function $f:\fbitset^n \to \{1,1+\epsilon\}$, consider the multilinear extension $\tilde{f}:[-1,1]^n\to [1,1+\epsilon]$ defined by $\tilde{f}(\mu) = {\bf E}_{x\sim \mu}[f(x)]$. Our goal is to understand when $\tilde{f}(\mu)=1+\epsilon$. We start with the precise statement of the main result (compare with Theorem \ref{thm:main}):
\begin{thm}\label{thm:main-full}
Let \textup{$\beta=\sqrt{\frac{\epsilon}{N\left(1-n\epsilon\right)}}$.} If
\[
\tilde{f}\left(\mu^{0}\right)>1+\sqrt{2 \beta \ln \frac{2}{\beta}}\]
then there is some constant $C$ such that for any $T\geq C\cdot \frac{\epsilon n^{8}\cdot N^{4}}{1-n\epsilon}$:
\[
\Pr\left[\tilde{f}\left(\mu^{\left(T\right)}\right)=1+\epsilon\right]\geq 1-2\beta-2/n.\]
\end{thm}
Note that the conditions in Theorem \ref{thm:main-full} imply restrictions on the initial probability of satisfaction and the strength of selection. In particular, the selection coefficient must be in the range $1/N^{1/3} < \epsilon < 1/n$ (we discuss this restriction in the next section), and the initial probability of satisfaction must be at least $N^{-1/4}$.
The full proof of the theorem is given in the Appendix; in this section we sketch its salient points.
One first difficulty in the proof is this: The convergence proof gauges the improvement in average population fitness obtained during the second of the two steps per generation (the fitness step). However, the first of the two steps (the sampling step) introduces variance, and we must establish that this variance is insignificant in comparison with the increase in fitness. Our first lemma (Lemma \ref{lem:noise-fitness}) establishes that the difference between the average fitness of the sample and the average fitness, squared (that is to say, the variance introduced), is bounded from above by the increase in average fitness obtained in the fitness step:
\begin{align}\label{eqn:variance-lemma}
{\bf E}_{\nu\sim B}[ (\tilde{f}(\nu) -\tilde{f}(\mu))^2 ] \leq
{\bf E}_{\nu\sim B} [ \tilde{f}(\mu') - \tilde{f}(\nu) ] /
[(N-1)\cdot (1-n\epsilon)]
\end{align}
Here we focus on one generation, so $\mu$ denotes the product distribution from which the sampling is made, $\nu$ the empirical product distribution of the sample (note that $\tilde{f}(\nu)$ is a random variable with expectation $\tilde{f}(\mu)$), and $\mu'$ the product distribution resulting from the selection (or fitness) step. Thus, $\mu'$ is the initial product distribution in the next generation.
To establish inequality \eqref{eqn:variance-lemma}, we first show that the right-hand side is lower bounded by the total mass of the singleton Fourier coefficients of the biased transform (Lemma \ref{lem:density}):
\begin{align}\label{eqn:coord}
\tilde{f}(\mu') - \tilde{f}(\nu) \geq (1-n\epsilon) \sum_{i=1}^n \hat{f}(i;\nu)^2.
\end{align}
The intuition in the proof of \eqref{eqn:coord} is that the fitness step is very close to an $\epsilon$-long step of the gradient ascent of the average fitness function (this intuition is very accurate away from the boundary of the hypercube). Gradient ascent in each coordinate is captured by the corresponding singleton coefficient squared. But then there is an analytical complication of approximating the overall ascent by the sum of sequential coordinate-wise ascents; the difficulty is, of course, that the partial derivatives change after each small ascent, and the change must be bounded (Lemma \ref{lem:coord}).
This establishes that the fitness increase in the selection step is larger than the linear Fourier mass, and hence nonnegative when $\epsilon$ is small. However, the linear Fourier mass may be zero, as is the case for the exclusive-or function under the uniform distribution (recall the discussion a few lines after Theorem \ref{thm:monotone-main}). Here, sampling effects will ensure that progress is made in expectation.
We show that, on average, the linear Fourier mass is much larger than the variance (Lemma \ref{lem:variance-noise}):
\begin{align}\label{eqn:lin-coeff}
{\bf E}_{\nu\sim B} \left[ (\tilde{f}(\mu) - \tilde{f}(\nu))^2 \right] \leq
\frac{1}{N-1} &{\bf E}_{\nu\sim B} \left[ \sum_{i=1}^n \hat{f}(i;\nu)^2 \right]
\end{align}
The rather involved proof of \eqref{eqn:lin-coeff} takes place entirely within the biased Fourier domain (see Appendix \ref{subsec:variance-noise}).
Now notice that \eqref{eqn:lin-coeff}, combined with \eqref{eqn:coord}, completes the proof of inequality \eqref{eqn:variance-lemma} and Lemma \ref{lem:noise-fitness}.
Note that the upper bound on the variance in \eqref{eqn:variance-lemma} includes in the denominator a factor of $(1-\epsilon n)\cdot N$. This immediately tells us that our technique is sharpest when the population $N$ is large and the selection strength $\epsilon$ is small --- in particular, it {\em must} be smaller than $1\over n$. This latter point is a rather puzzling limitation of our result: Why does a theorem about the effectiveness of natural selection become harder to prove when selection is stronger? One intuitive explanation is that in this case selection works very much like gradient ascent, and it is well known that the convergence of gradient ascent is harder to establish when the ascent step is large, as a large step can ``skip over'' the stationary point sought. Is this upper limit on $\epsilon$ necessary? This is an intriguing open question discussed in the last section.
Next, we establish that the total effect of the sampling steps is small: For any $\alpha > \sqrt{2\beta\ln 2\beta^{-1}}$,
$$\Pr[ | \sum_{t=1}^T \tilde{f}(\nu^t) -\tilde{f}(\mu^{t-1}) | \geq \alpha ] \leq 2\beta,$$
where $\beta = \left(\frac{\epsilon}{N(1-n\epsilon)}\right)^{1/2}$.
It is not hard to see that the sum $ \sum_{t=1}^T \tilde{f}(\nu^t) -\tilde{f}(\mu^{t-1})$ constitutes a martingale, albeit one with no obvious upper bound on each step. In Lemma \ref{lem:main} we bound the total effect of the sampling step by resorting to a rather exotic martingale inequality derived from a generalization of Bernstein's inequality to martingales with unbounded jumps and proved in \cite{bernstein_generalized} (in fact, a specialization stated in Appendix \ref{sec:martingale} as Lemma \ref{lem:generalized-bernstein}).
Incidentally, notice that this is the place where it is proved, quite indirectly, that the sampling step succeeds in getting the process unstuck from spurious fixpoints such as $({1\over 2},{1\over 2},\ldots,{1\over 2})$ for the exclusive-or function:
Since the total effect of sampling is limited, the increase in average fitness must eventually prevail.
Finally, when the process is near a vertex of the hypercube, fitness increases are too small to help finish the argument, but here we rely on the fact that the process is very likely to drift so close to a vertex that it will eventually get stuck there (Lemma \ref{lem:fast-vertex}), completing the proof of the main result.
\section{Discussion}
\label{sec:discussion}
We proved a novel and highly nontrivial aspect of Boolean satisfiability: By randomly crossing assignments and favoring satisfaction slightly, one can breed a population of pure satisfying truth assignments. We argued that this rather curious property seems important in understanding one intriguing aspect of evolution: how complex traits controlled by many genes can emerge.
There are many roads of mathematical inquiry opened by this theorem. First, can the limitations/restrictions of our model be relaxed so that it better reflects the realities of life?
Some of the assumptions in our model are arguably unrealistic (haploidy, fixed population size, random mating, partly in-expectation fitness calculation), but these follow widely accepted practices in population genetics needed for mathematical simplification.
We also make the assumption of weak selection, but this is also a very defensible one for unlinked loci.
There are, however, a few further restrictions of our model that call for discussion:
\begin{itemize}
\item {\em Two alleles per gene.} The motivation is, of course, that this assumption ushers in the powerful analytical toolbox of Boolean functions. We have no doubt that similar results hold for more alleles, but would require a great number of technical adjustments.
\item {\em Fitness landscape.} We assumed a very specialized fitness landscape with values $1$ and $1+\epsilon$ only. This is a natural simplification that facilitates the connection to Boolean functions, but we do not believe it is an essential one. We believe that this result can be extended to much richer landscapes with a small gap, for example to situations in which fitness values are in $[1-\delta, 1]\cup [1+\epsilon, 1+\epsilon +\delta]$ for some small $\delta>0$.
\end{itemize}
A harder question is, what happens if the fitness gap $\epsilon$ is larger? As we have mentioned, this is an analytical challenge with roots in the difficulty of the analysis of gradient descent. Of course, a constant gap would bring us outside the realm of weak selection, and render our approximation by product distribution baseless. There are two ways we can proceed: One is to prove that the exact recurrence equations of genotype frequencies yield eventual satisfaction. This seems possible but challenging.
Another avenue, which we have followed for some time, is to work with product distributions anyway. In particular, what if the fitness landscape has values $\{0,1\}$ --- that is to say, non-satisfying truth assignments are removed from the population, as in Waddington's experiment? This is a realistic approximation if, for example, this selection does not happen in every generation but every $O(\log n)$ generations (because breeding without selection is known to take you close to the Wright manifold).
In such a setting, our quadratic bound for the in-expectation process of monotone functions no longer requires any dependence on the initial probability of satisfaction $\mu^0(f)$. For the process with sampling, we have the following conjecture.
\paragraph{Conjecture:} {\em If the fitness landscape has values $\{0,1\},$ then the process reaches near universal satisfaction with probability approaching $1$ as the population size goes to infinity.}
\bigskip We now want to point out an obvious and yet surprising aspect of our work:
In the traditional framework of adaptive evolution, each allele spreads in the population mainly either due to an additive contribution to fitness that it makes in and of itself (let us call this ``traditional propagation") or due to random
genetic drift \cite{:1930fk,Wright:1931uq,Wright:1932kx,WadeGoodnight1998}.
In our model, however, alleles at different genes are spreading in the population as governed by the complex interactions between them that are continually subject to selection. Thus, a population can change dramatically through a novel process involving subtle changes in genetic statistics and simultaneous gradual emergence in the whole population \cite{Livnat2013,Mayr1963}, and not by traditional propagation.
Furthermore, notice that since recombination is a crucial ingredient of our analysis, our results inform the question of the role of sex in evolution. In this regard they add to recent works that have begun to examine the role of sex while giving full weight to the importance of genetic interactions \cite{Chastain2014algorithms,Livnat2008mixability}.
Finally, can our bounds be improved?
For the monotone case, it is easy to see that the TRIBES function with appropriate fan-in provides a matching lower bound.
As for the general case, we feel that the very generous bounds of the main result can be improved substantially. For example, the assumed time bound is only necessary in order to finish the last part of the argument (convergence to a vertex) once the vast majority of the population is already satisfying; more analysis is needed to investigate this subtle phenomenon.
Our proof that the population converges to a single satisfying truth assignment may seem a troubling aspect of our result.
Two remarks: First, the loss of genetic diversity should not be surprising in itself. With drift alone, for each locus, one allele will become fixed eventually (where the probability that a particular allele will be the fixed allele is proportional to its current frequency in the population).
Second, in our process many satisfying truth assignments are likely to survive for a very long time before the random walk clears the picture. This fact may be more relevant than the characteristics of eqilibrium; after all, evolution happens in the transient.
\paragraph{Acknowledgments:} We are grateful to Yu Liu of Tsinghua University for some very interesting conversations in the beginning of this research.
\section{Outline of proof}
In the following sections, we prove Theorem \ref{thm:main-full}:
\begin{thm*}
Let \textup{$\beta=\sqrt{\frac{\epsilon}{N\left(1-n\epsilon\right)}}$.} If
\[
\tilde{f}\left(\mu^{0}\right)>1+\sqrt{2 \beta \ln \frac{2}{\beta}}\]
then there is some constant $C$ such that for any $T\geq C\cdot \frac{\epsilon n^{8}\cdot N^{4}}{1-n\epsilon}$:
\[
\Pr\left[\tilde{f}\left(\mu^{\left(T\right)}\right)=1+\epsilon\right]\geq 1-2\beta-2/n.\]
\end{thm*}
Our proof of Theorem \ref{thm:main-full} is structured as follows. In Section \ref{sec:variance-noise}, we consider
the average fitness from one generation to the next. As described in Section \ref{sec:intro}, each generation consists of two steps: the sampling step, which begins with a product distribution $\mu$ and results in an empirical product distribution $\nu$, and the fitness step resulting in a distribution $\mu'$ (which becomes the initial distribution for the next generation). The main lemma (and the most involved step of our proof) of Section \ref{sec:variance-noise} is Lemma \ref{lem:noise-fitness}, which upper bounds the variance of $\tilde{f}(\nu)$,
by a small fraction of ${\bf E}[\tilde{f}(\mu')-\tilde{f}(\nu)]$, the expected increase in average fitness by the fitness step.
In Section \ref{sec:martingale},
we apply Lemma \ref{lem:noise-fitness} with the martingale inequality to prove Lemma \ref{lem:main}, which states that the total fitness decrease will be small with high probability. Finally, we complete the proof of the main theorem in Section \ref{sec:finalproof} by arguing (Lemma \ref{lem:fast-vertex}) that for $T$ as stated in the theorem, $\mu^T$ will reach a vertex of the hypercube (and hence $f(\mu^T) \in \{1,1+\epsilon\}$) with high probability.
\section{Selection vs sampling effects}\label{sec:variance-noise}
In this section we consider just one step of the process.
Let $\mu$ be the initial product distribution of a generation,
$\nu$ be the empirical product distribution from the sampling step, and $\mu'$ be the product distribution after the fitness step.
Our main goal in this section is to show that the variance of $\tilde{f}(\nu)$, the average fitness of the population after the sampling step, is small compared to the expected increase in average fitness from the subsequent selection step, $\tilde{f}(\mu')-\tilde{f}(\nu)$.
The main lemma we will prove is the following:
\begin{lem} \label{lem:noise-fitness}
Let $\nu$ be the vector of expectations of allele frequencies in the population sample of size $N$, drawn according to $B(\mu)$. Then:
$${\bf E}_{\nu\sim B}[ (\tilde{f}(\nu) -\tilde{f}(\mu))^2 ] \leq
{\bf E}_{\nu\sim B} [ \tilde{f}(\mu') - \tilde{f}(\nu) ]/
(N-1)\cdot (1-n\epsilon).$$
\end{lem}
We will prove Lemma \ref{lem:noise-fitness} by proving two intermediate lemmas. First, we show that fitness increase by the selection step $\tilde{f}(\mu') - \tilde{f}(\nu)$ is nearly as large as the $\nu$-biased Fourier weight of the linear coefficients of $f$ (and hence non-negative), provided that $\epsilon$ is sufficiently small.
\begin{lem} \label{lem:density} Let $\mu'$ be the expectations of the process after selection from the population $\nu$. Then:
$$\tilde{f}(\mu') - \tilde{f}(\nu) \geq (1-n\epsilon) \sum_{i=1}^n \hat{f}(i;\nu)^2.$$
\end{lem}
Next, we show that the variance of $\tilde{f}(\nu)$ is at most a small fraction of the expected linear $\nu$-biased Fourier mass of $f$: $\sum_{i=1}^n \hat{f}(i;\nu)^2$---here the expectation is taken over the choice of $\nu.$
\begin{lem}\label{lem:variance-noise}
$${\bf E}_{\nu\sim B} \left[ (\tilde{f}(\mu) - \tilde{f}(\nu))^2 \right] \leq
\frac{1}{N-1} {\bf E}_{\nu\sim B} \left[ \sum_{i=1}^n \hat{f}(i;\nu)^2 \right].$$
\end{lem}
Combining Lemmas \ref{lem:density} and \ref{lem:variance-noise} gives us Lemma \ref{lem:noise-fitness}.
\subsection{Preliminaries} \label{sec:prelim}
Throughout our proofs, we will use the notation and basic facts established at the beginning of Section \ref{sec:monotone}. At times, we will use biased Fourier analysis with different product distributions $\mu$ and $\mu'$ at the same time. To prevent ambiguity, we will refer to the standard deviation of the $i$'th bit as $\sigma_i(\mu) = 1-\mu_i^2$ (similarly for $\sigma_i(\mu'))$, but we will use $\sigma_i$ when the context makes the distribution clear.
Recall that the exentsion of $f:\fbitset^n\to \{1,1+\epsilon\}$,
$$\tilde{f}(\mu) = {\bf E}_{x\sim \mu}[ f(x) ]=\sum_{S\subseteq [n]} \hat{f}(S) \prod_{j\in S} \mu_j $$
is multilinear. Note that its derivative in the $i$'th direction is given by
\begin{align}
\drv{\tilde{f}(\mu)}{\mu_i} &= \sum_{S\ni i} \hat{f}(S) \prod_{j\in S\setminus i} \mu_j ={\bf E}_\mu[ D^{(1/2)}_i f] \nonumber \\
&= \frac{1}{\sigma_i}{\bf E}_\mu[ D^{(\mu)}_i f] \nonumber \\
&= \frac{\hat{f}(i;\mu)}{\sigma_i}.
\label{eqn:russo}
\end{align}
Here (\ref{eqn:russo}) is a straightforward generalization of the Russo-Margulis Lemma~\cite{Margulis:74, Russo:78} for product distributions. Thus, we may write the change in allele frequency from the fitness step as:
\begin{equation}\label{eqn:prob-step}
\mu'_i - \nu_i = \sigma_i(\nu) \frac{\hat{f}(i;\nu)}{{\bf E}_{\nu}[f]}= \frac{\sigma_i^2(\nu)}{\tilde{f}(\nu)} \drv{\tilde{f}(\nu)}{\nu_i},
\end{equation}
where the first equality holds by Equation \eqref{eqn:select2} (derived in Section \ref{sec:monotone})
\subsection{Proof of Lemma \ref{lem:density}}
We will compute the fitness change as each coordinate changes.
Consider the hybrid distributions given by the expectations
$w^i = (\mu'_1,\ldots,\mu'_{i}, \nu_{i+1},\ldots,\nu_n)$
so that
$w^0 = \nu$ and $w^n = \mu'$. We have that
$$ \tilde{f}(\mu') - \tilde{f}(\nu) = \sum_{i=1}^n \tilde{f}(w^{i}) - \tilde{f}(w^{i-1}).$$
Observe that the first increment is easily computed as
$$\tilde{f}(w^1)- \tilde{f}(\nu) = (\mu'_1-\nu) \drv{\tilde{f}(\nu)}{\nu_1} = \frac{\hat{f}(1;\nu)^2}{{\bf E}_\nu[f]}.$$
However, this formula will not be valid for subsequent hybrids as the derivatives of $\tilde{f}$ have changed.
We start by showing that the derivatives in each direction do not change much between the hybrid distributions. The lemma below will allow us to approximate the derivatives of the hybrids by the derivative of $\tilde{f}$ at $\nu$.
\begin{lem}\label{lem:coord}
Let $\nu'\geq \nu \in [+1,-1]^n$ differ only on the $j$-th coordinate, i.e., $\nu_\ell=\nu'_\ell$ for all $\ell\neq j$ and $\nu'_{j} > \nu_{j}$. Then for any $i\geq 2$ and $j<i$,
$$\drv{\tilde{f}(\nu')}{\nu'_i} - \drv{\tilde{f}(\nu)}{\nu_{i}} = (\nu'_j-\nu_j) \frac{\hat{f}(\{i,j\};\nu)}{\sigma_i \sigma_j}.$$
In particular, for the hybrid distributions above with $i\geq 2$:
$$
\drv{\tilde{f}(w^{i-1})}{w^{i-1}_i} - \drv{\tilde{f}(\nu)}{\nu_i} =
\frac{1}{{\bf E}_\nu[f]\sigma_i} \sum_{j=1}^{i-1} \hat{f}(j;\nu) \hat{f}(\{i,j\};\nu).$$
\end{lem}
\begin{proof}
Expanding the derivatives in terms of the unbiased Fourier coefficients, we have that
\begin{align*}
\drv{\tilde{f}(\nu')}{\nu'_i} - \drv{\tilde{f}(\nu)}{\nu_i} & =
\sum_{S\ni i} \hat{f}(S) \left( \prod_{\ell \in S\setminus i} \nu'_\ell -\prod_{\ell \in S\setminus i} \nu_\ell \right) \\
& = \sum_{S\ni \{j,i\}} \hat{f}(S) (\nu'_j- \nu_j) \prod_{\ell \in S\setminus\{j,i\}} \nu_\ell \\
& = (\nu'_j -\nu_j) E_\nu [ D^{(1/2)}_i D^{(1/2)}_j f].\\
\end{align*}
The proof of the first equality in the lemma is completed by noting that:
$${\bf E}_\nu[ D^{(1/2)}_i D^{(1/2)}_j f] = \frac{1}{\sigma_i\sigma_j}{\bf E}_\nu [ D_i^{(\nu)}D_j^{(\nu)} f].$$
For the second equality of the lemma,
we may write the difference between the derivative in the $i$'-th direction under the hybrid distribution $w^{i-1}$ and under the original distribution as a telescoping sum:
\begin{align*}
\drv{\tilde{f}(w^{i-1})}{w^{i-1}_i} - \drv{\tilde{f}(\nu)}{\nu_i} & =
\sum_{j=1}^{i-1} (w^{j}_j-w^{j-1}_j) \frac{\hat{f}(\{i,j\});\nu)}{\sigma_i\sigma_j} \\
& = \sum_{j=1}^{i-1} \frac{\sigma_j \hat{f}(j;\nu)}{{\bf E}_\nu[f]} \frac{\hat{f}(\{i,j\};\nu)}{\sigma_i\sigma_j} \\
& = \frac{1}{{\bf E}_\nu[f]\sigma_i} \sum_{j=1}^{i-1} \hat{f}(j;\nu)\hat{f}(\{i,j\};\nu).
\qedhere
\end{align*}
\end{proof}
We are now ready to prove Lemma \ref{lem:density}:
\begin{lem*}[\ref{lem:density}]
Let $\mu'$ be the expectations of the process after selection from the population $\nu$. Then:
$$\tilde{f}(\mu') - \tilde{f}(\nu) \geq (1-n\epsilon) \sum_{i=1}^n \hat{f}(i;\nu)^2.$$
\end{lem*}
\begin{proof}
We will bound each of the differences between the hybrid densities in the summation:
$$ \tilde{f}(\mu') - \tilde{f}(\nu) = \sum_{i=1}^n \tilde{f}(w^{i}) - \tilde{f}(w^{i-1}) .$$
Since $\tilde{f}$ is multilinear, we have that for $i\geq 2:$
\begin{align*}
\tilde{f}(w^{i})-\tilde{f}(w^{i-1}) & = (\mu'_i - \nu_i ) \drv{\tilde{f}(w^{i-1})}{w_i^{i-1}}\\
& = \frac{\sigma_i\hat{f}(i;\nu)}{{\bf E}_\nu[f]}
\left(\drv{\tilde{f}(\nu)}{\nu_i}+\drv{\tilde{f}(w^{i-1})}{w_i^{i-1}} - \drv{\tilde{f}(\nu)}{\nu_i}\right) \\
& = \frac{\sigma_i\hat{f}(i;\nu)}{{\bf E}_\nu[f]}
\left(\frac{\hat{f}(i;\nu)}{\sigma_i}+
\frac{1}{{\bf E}_\nu[f]\sigma_i}\sum_{j=1}^{i-1} \hat{f}(j;\nu)\hat{f}(\{i,j\};\nu) \right).
\end{align*}
Recalling that $\hat{f}(\{i,j\};\nu) = \sigma_i \sigma_j {\bf E}_\nu[ D^{(1/2)}_i D^{(1/2)}_j f]$ and using the fact that $| D^{(1/2)}_i D^{(1/2)}_j f(x)| \leq \epsilon/4$
for any $x$ and ${\bf E}_\nu[f]\geq 1$, we have:
\begin{equation} \label{eqn:err}
\frac{1}{{\bf E}_\nu[f]\sigma_i}\left| \sum_{j=1}^{i-1} \hat{f}(j;\nu) \hat{f}(\{i,j\},\nu) \right| \leq
\epsilon \sum_{j=1}^{i-1} \sigma_j |\hat{f}(j;\nu)|. \end{equation}
Assume WLOG that $\sigma_i |\hat{f}(i;\nu)| \geq \sigma_j |\hat{f}(j;\nu)|$ for all $j<i$. so that \eqref{eqn:err} is at most
$\frac{\epsilon}{4} (i-1) \sigma_i |\hat{f}(i;\nu)|$.
Substituting into the telescoping sum, we have:
\begin{align*}
\sum_{i=1}^n \tilde{f}(w^i) - \tilde{f}(w^{i-1}) & \geq \frac{1}{{\bf E}_\nu[f]} \sum_{i=1}^n \sigma_i\hat{f}(i;\nu) \left(\frac{\hat{f}(i;\nu)}{\sigma_i} - \frac{\epsilon}{4} (i-1) \sigma_i |\hat{f}(i;\nu)|\right) \\
& \geq \frac{1}{1+\epsilon}\sum_{i=1}^n \hat{f}(i;\nu)^2 (1-\frac{\epsilon}{4}(i-1)\sigma_i^2 ) \\
& \geq (1-n\epsilon) \sum_{i=1}^n \hat{f}(i;\nu)^2.
\qedhere
\end{align*}
\end{proof}
\subsection{Proof of Lemma \ref{lem:variance-noise}}\label{subsec:variance-noise}
Our goal is to show that:
$$ {\bf E}_{\nu\sim B} \left[ \sum_{i=1}^n \hat{f}(i;\nu)^2 \right] \geq (N-1) \cdot {\bf E}_{\nu\sim B} \left[ (\tilde{f}(\mu) - \tilde{f}(\nu))^2 \right].$$
The first key observation is that the Fourier basis with respect to the product distribution $\mu$ is still orthogonal with respect to $B$, i.e.,
we have ${\bf E}_{\nu\sim B} [ \phi^\mu_S(\nu) \phi^\mu_T(\nu)]=0$ for $S\neq T$,
because $B$ is a product distribution and ${\bf E}_{\nu \sim B} [ \phi^\mu_S (\nu)]=0$ for $S\neq \emptyset$.
In particular, Parseval's holds for the extension of $f$ with respect to $B$:
\begin{claim}\label{eqn:parsevals}
$${\bf E}_{\nu\sim B}[\tilde{f}(\nu)^2] = {\bf E}_{B} \left[ \left( \sum_{S} \hat{f}(S;\mu) \phi^\mu_{S} (\nu) \right)^2 \right] = \sum_{S} \hat{f}(S;\mu)^2 {\bf E}_{\nu\sim B}[ \phi_{S} ^\mu(\nu)^2 ].$$
\end{claim}
Our approach will be to consider both sides of the inequality using the $\mu$-biased Fourier basis of $f$. This is straightforward for the variance of $\tilde{f}(\nu)$ using Parseval's. For the right hand side,
we observe that the $\nu$-biased linear coefficients may be viewed as functions in $\nu$. In fact, each linear coefficient can be viewed as the extension of the $\mu$-biased difference operator to $[-1,1]^n$, modulo
a normalization factor.
\begin{claim} \label{eqn:lin-mu}
$$\hat{f}(i;\nu) = \frac{\sigma_i(\nu)}{\sigma_i(\mu)} \sum_{S\ni i} \hat{f}(S;\mu)\phi^\mu_{S\setminus i}(\nu).$$
\end{claim}
\begin{proof}
We rewrite the linear $\nu$-biased Fourier coefficient in terms of the $\mu$-biased difference operator:
\begin{align*}
\hat{f}(i;\nu) = {\bf E}_{x\sim \nu} [ D_i^{(\nu)} f ] &= \sigma_i(\nu) {\bf E}_\nu [ D_i^{(1/2)} f] \\
&= \frac{\sigma_i(\nu)}{\sigma_i(\mu)}{\bf E}_\nu \left[ D_i^{(\mu)}f \right] \\
&= \frac{\sigma_i(\nu)}{\sigma_i(\mu)} \sum_{S\ni i} \hat{f}(S;\mu) {\bf E}_\nu[ \phi^\mu_{S\setminus i}]\\
&= \frac{\sigma_i(\nu)}{\sigma_i(\mu)} \sum_{S\ni i} \hat{f}(S;\mu)\phi^\mu_{S\setminus i}(\nu),
\end{align*}
where the penultimate equality holds by the definition of $D^{(\mu)}_i$, and the final holds because $\nu$ is a product distribution.
\end{proof}
Finally, we will use the fact that the variance of $\phi^\mu_i(\nu)$ grows smaller as the sample size increases:
\begin{fact}
$${\bf E}_{\nu\sim B}[ \phi^\mu_{S\setminus i}(\nu)^2 ] = {\bf E}_{\nu\sim B}[ \phi^\mu_{S}(\nu)^2 ]\cdot N.$$
\label{eqn:lin-var}
\end{fact}
\begin{proof}
Because $B$ is a product distribution, we have:
$${\bf E}_{\nu\sim B} [ \phi_{S}^{\mu}(\nu)^2 ] = {\bf E}_{\nu\sim B}[\phi_{S\setminus i}^{\mu}(\nu)^2]\cdot {\bf E}_{\nu\sim B}[\phi_i^\mu(\nu)^2].$$
Then
\begin{align*}
{\bf E}_{\nu\sim B}[ \phi_i^\mu(\nu)^2 ] & = {\bf E}_{\nu\sim B} [ (\nu_i -\mu_i)^2/\sigma^2_i(\mu)]\\
& = \frac{ {\bf E}_{\nu\sim B}[(\nu_i-\mu_i)^2]}{1-\mu_i^2}\\
& = \frac{{\bf E}_B[ \nu_i^2] -\mu_i^2}{1-\mu_i^2}\\
& = \frac{N^{-1} +\mu_i^2- N^{-1}\mu_i^2 -\mu_i^2}{1-\mu_i^2}.
\qedhere
\end{align*}
\end{proof}
\medskip
\noindent With the previous claims
in hand, we are ready to prove Lemma \ref{lem:variance-noise}:
\begin{lem*}[\ref{lem:variance-noise}]
$${\bf E}_{\nu\sim B} \left[ (\tilde{f}(\mu) - \tilde{f}(\nu))^2 \right] \leq
\frac{1}{N-1} {\bf E}_{\nu\sim B} \left[ \sum_{i=1}^n \hat{f}(i;\nu)^2 \right].$$
\end{lem*}
\begin{proof
We first consider the expected $\nu$-biased linear Fourier weight. Applying Claims \ref{eqn:lin-mu} and \ref{eqn:parsevals}, and summing over $i$ we have:
\begin{align*}
\sum_{i=1}^n E_{\nu\sim B}[ \hat{f}(i;\nu)^2 ] &= \sum_{i=1}^n \frac{{\bf E}_B[ \sigma_i(\nu)^2]}{\sigma_i(\mu)^2}\sum_{S\ni i} \hat{f}(S;\mu)^2 {\bf E}_B[\phi_{S\setminus i}^\mu(\nu)^2] \\
&= \left(1-\frac{1}{N}\right)\sum_{i=1}^n \sum_{S\ni i} \hat{f}(S;\mu)^2 {\bf E}_B[ \phi_{S\setminus i}^\mu(\nu)^2 ] \\
&= \left(1-\frac{1}{N}\right)\sum_{i=1}^n \sum_{S\ni i} \hat{f}(S;\mu)^2 {\bf E}_B[ \phi_S^\mu(\nu)^2 ] N \\
&\geq (N-1)\cdot \sum_{S\neq \emptyset} \hat{f}(S;\mu)^2 {\bf E}[ \phi_S^\mu(\nu)^2] .
\end{align*}
Note in the first equality that the $\sigma_i(\nu)^2$ depends only on $\nu_i$, while $\phi_{S\setminus i}^\mu$ does not, so the expectations may be taken separately.
For the next equality, we calculate
\begin{align*}
{\bf E}_B[\sigma_i(\nu)^2] &= 1-E_B[\nu_i^2] \\
& = 1 - \left(\mu_i^2+ \frac{1-\mu_i^2}{N} \right),
\end{align*}
which gives that
$$\frac{{\bf E}_B[ \sigma_i(\nu)^2 ] }{ \sigma_i(\mu)^2} = 1-1/N.$$
The third equality holds by Fact \ref{eqn:lin-var}, and the final inequality holds since each non-empty coefficient appears in the sum at least once.
Using Claim \ref{eqn:parsevals} and rewriting the variance of $\tilde{f}(\nu)$ using the $\mu$-biased Fourier basis for $f$, we have:
\begin{align*}
{\bf E}_{\nu\sim B} [ (\tilde{f}(\nu) - \tilde{f}(\mu))^2 ] &= {\bf E}_B [ (\sum_{S\neq \emptyset} \hat{f}(S;\mu) \phi_S^\mu(\nu))^2 ] \\
&= \sum_{S\neq \emptyset} \hat{f}(S;\mu)^2 {\bf E}_B[ \phi_S^\mu(\nu)^2]. \qedhere
\end{align*}
\end{proof}
\section{Bounding the cumulative effect of sampling} \label{sec:martingale}
We saw in Lemma \ref{lem:density} that the selection step always results in a non-negative change in fitness when $\epsilon$ is sufficiently small. The sampling steps, however, may decrease fitness. In this section we show that the cumulative effect of sampling on fitness will be small. We use $\mu^{t}$ to denote the initial distribution of the generation at time $t$, and $\nu^{t}$ to denote the product distribution after the sampling step. according to $B(\mu^{t})$. Then the selection step determines the population at time $t+1$ which we write as $\mu^{t+1}$ (determined by $\nu^{t}$).
By Lemma \ref{lem:noise-fitness} at each stage the variance of $\tilde{f}(\nu)$ is a small fraction of the expected fitness increase after selection. Summing over all generations, the total variance from the sampling steps is a small fraction of the total fitness increase from the selection steps. Finally, we bound from above the total fitness decrease due to sampling effects; for this last step we need the following generalization of Bernstein's inequality to martingales with unbounded jumps by Dzhaparidze and Zanten:
\begin{lem}
\label{lem:generalized-bernstein}(Theorem 3.
\footnote{This is a special case of their Theorem 3.3, which corresponds, in
their notation, to the limit as $a\rightarrow0$.%
}, \cite{bernstein_generalized}) Let $\left\{ {\cal F}_{t}\right\} _{t=0,1,\dots}$
be a filtration, and let $\zeta_{1},\zeta_{2},\dots$ be a martingale
difference sequence w.r.t. $\left\{ {\cal F}_{t}\right\} $. Consider
the martingale $S_{T}=\sum_{t=1}^{T}\zeta_{t}$. Define:
\begin{align*}
H_{T} = \sum\zeta_t^2 +\sum\mbox{E}\left[\zeta_{t}^{2}\mid F_{t-1}\right]
\end{align*}
Then, for each stopping time $\tau$,\[
\Pr\left[\max_{T\leq\tau}\left|S_{T}\right|>z,H_{\tau}\leq L\right]\leq2\exp\left(\frac{-z^{2}}{2L}\right)\]
\end{lem}
Using this machinery, we can show that the total decrease of fitness due to sampling is small:
\begin{lem}\label{lem:main}
Let $\beta = \left(\frac{2\epsilon}{(N-1)(1-n\epsilon)}\right)^{1/2}$ and $\alpha = \sqrt{2\beta \ln \frac{2}{\beta}}$.
Then
$$\Pr[ | \sum_{t=1}^T \tilde{f}(\nu^t) -\tilde{f}(\mu^{t-1}) | \geq \alpha ] \leq 2\beta.$$
\end{lem}
\begin{proof}
For each sequence $\left(\mu^{t}\right)_{t=0}^{T}$ of
populations up to time $T$, define its congruence class as a subset
of infinite sequences:\[
\left[\left(\mu^{t}\right)_{t=0}^{T}\right]=\left\{ \left(w^{t}\right)_{t=0}^{\infty}\colon w^{t}=\mu^{t}\forall t\leq T\right\} \]
Now, for a time $T$, consider the space of possible sequences of populations:
\[ \mathcal{F}_{T}=\left\{ \left[\left(\mu^{t}\right)_{t=0}^{T}\right]\right\}.\]
Then $\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\dots$ is a filtration. We will consider the following martingale:\[
S_{T}= \sum_{t=0}^T\zeta_t = \sum_{t=0}^{T}\tilde{f}\left(\nu^{t}\right)-\tilde{f}\left(\mu^{t}\right)\]
Notice that this is indeed a martingale because\begin{eqnarray*}
\mbox{E}\left[S_{T}\mid\mathcal{F}_{T-1}\right] & = & \mbox{E}\left[\sum_{t=0}^{T}\tilde{f}\left(\nu^{t}\right)-\tilde{f}\left(\mu^{t}\right)\mid\mathcal{F}_{T-1}\right]\\
& = & S_{T-1}+\mbox{E}\left[\tilde{f}\left(\nu^{T}\right)-\tilde{f}\left(\mu^{T}\right)\mid\mathcal{F}_{T-1}\right]\\
& = & S_{T-1}.\end{eqnarray*}
To apply Lemma \ref{lem:generalized-bernstein},
we also define the following sequences:\begin{eqnarray*}
M_{T} & = & \sum_{t=0}^{T}\left(\tilde{f}\left(\nu^{t}\right)-\tilde{f}\left(\mu^{t}\right)\right)^{2}\\
V_{T} & = & \sum_{t=0}^{T}\mbox{E}\left[\left(\tilde{f}\left(\nu^{t}\right)-\tilde{f}\left(\mu^{t}\right)\right)^{2}\mid\mathcal{F}_{t-1}\right]\\
H_{T} & = & M_{T}+V_{T}\end{eqnarray*}
For each $T$, we show that $\Pr[H_T \geq \beta] \leq \beta$ by bounding ${\bf E}[ H_T]$ and applying Markov's inequality. Applying Lemma \ref{lem:noise-fitness}, we have that \begin{align*}
{\bf E}[ M_T ] &= {\bf E} \left[ \sum_{t=0}^T (\tilde{f}(\nu^t) - \tilde{f}(\mu^{t}))^2 \right] \\
& \leq \frac{1}{(N-1)(1-n\epsilon)}\cdot {\bf E} \left[ \sum_{t=0}^T \tilde{f}(\mu^{t+1}) - \tilde{f}(\nu^t) \right] \\
& \leq \frac{\epsilon}{(N-1)(1-n\epsilon)},
\end{align*}
where the last inequality holds because
$$
{\bf E}\left[ \sum_{t=0}^T \tilde{f}(\mu^{t+1}) - \tilde{f}(\nu^t) + \sum_{t=0}^T \tilde{f}(\nu^t) - \tilde{f}(\mu^{t}) \right] =
{\bf E}[ \tilde{f}(\mu^{T+1}) -\tilde{f}(\mu^0)] \leq \epsilon,
$$
and ${\bf E}[S_T]= {\bf E}[ \sum_{t=0}^T \tilde{f}(\nu^{t})-\tilde{f}(\mu^{t})]=0.$
Similarly we may apply Lemma \ref{lem:noise-fitness} to each term of $V_T$: \begin{align*}
{\bf E} \left[ (\tilde{f}(\nu^t) - \tilde{f}(\mu^{t}))^2 | \mathcal{F}_{t-1} \right]
&= {\bf E}_{\nu^t \sim B(\mu^{t})} \left[ (\tilde{f}(\nu^t) - \tilde{f}(\mu^{t}))^2 \right] \\
& \leq \frac{1}{(N-1)(1-n\epsilon)} \cdot {\bf E}_{\nu^t \sim B(\mu^{t})} \left[ \tilde{f}(\mu^{t+1}) - \tilde{f}(\nu^{t}) \right].\\
\end{align*}
Summing over $t$ and taking the expectation, we again have that ${\bf E}[V_T] \leq \epsilon/(N-1)(1-n\epsilon)$. Thus, we have that ${\bf E}[H_T] = {\bf E}[M_T+V_T] \leq \frac{2\epsilon}{(N-1)(1-n\epsilon)} \leq \beta^2$.
Finally, applying Lemma \ref{lem:generalized-bernstein} to $S_T$, we have
$$\Pr[ \max_{T \leq \tau} |S_T| \geq \alpha, H_T \leq \beta] \leq 2 \exp \left(-\frac{\alpha^2}{2\beta}\right)
\leq \beta.$$
Combining this with the bound $\Pr[H_T \geq \beta]\leq \beta$ gives the lemma.
\end{proof}
\section{Proof of the main theorem} \label{sec:finalproof}
To complete the proof of Theorem \ref{thm:main-full},
we first show (Lemma \ref{lem:fast-vertex} below) that for sufficiently large $T$, the population $\mu^{T}$ is at a vertex of the Boolean cube with high probability. Finally, we combine this with Lemma \ref{lem:main}, which bounds the probability that $\tilde{f}(\mu^T)\neq 1+\epsilon$.
\begin{lem} \label{lem:fast-vertex}
There is a constant $C>0$ such that
for any $T\geq C\cdot \frac{\epsilon n^8 N^4}{1-n\epsilon}$, we have:
$$ \Pr[ \mu^{T} \notin \fbitset^n ] < 2/n.$$
\end{lem}
\begin{proof}
Note that if $|\nu_j^{t'}| = 1$ for some time $t'$, then $\nu_j^{t} = \nu_j^{t'}$ for every $t\geq t'$. Observe also that if $|\mu_j^{t'}| > 1-(n^2N)^{-1}$ (in this case we say $j$ is $\alpha$-determined with $\alpha= n^{-2}N^{-1}$), we have by Markov's inequality:
$$\Pr[ |\nu_j^{t'}| < 1 ] \leq 1/n^2.$$
We will show that after enough time, it is unlikely that there is any coordinate that was never $\alpha$-determined. More precisely,
let $A_{j;t}$ be the event that coordinate $j$ was not $\alpha$-determined for $\mu^{1},\dots,\mu^{t}.$
To prove the lemma, the above reasoning tells us that it suffices to show that for $T$ as set in the condition of the lemma:
$$ \Pr[ \bigvee_{j=1}^n A_{j;T}] \leq 1/n. $$
We will consider each coordinate separately and show that for each $j$,
$\Pr[ A_{j;T} ] \leq n^{-2}.$
Our proof will use the following simple claims relating $\Pr[A_{j;T}]$ to the selection steps of the process.
\begin{claim}\label{claim:determined-variance}
Fix any time $t_0$ and an interval $T_1,$ such that $t_0+T_1\leq T$. Then:
$$ {\bf E}\left[ \left(\sum_{t=t_0}^{t_0+T_1} \nu_j^{t}- \mu_j^{t}\right)^2 \right] \geq \frac{\alpha\Pr\left[A_{j;T}\right]T_{1}}{2N}.
$$
\end{claim}
\begin{proof}
\begin{align*}
{\bf E}\left[ \left(\sum_{t=t_0}^{t_0+T_1} \nu_j^{t}- \mu_j^{t}\right)^2 \right] & =
\sum_{t=t_0}^{t_0+T_1} {\bf E}_{\nu^{t} \sim B(\mu^{t})} \left[ \left(\nu_j^{t} -\mu_j^{t}\right)^2\right] \\
& \geq \sum_{t=t_0}^{t_0+T_{1}}\Pr\left[A_{j;t}\right]{\bf E}_{\nu^{t}}\left[\left(\nu_{j}^{t}-\mu_{j}^{t}\right)^{2}\mid A_{j;t}\right]\nonumber \\
& \geq \frac{\alpha\Pr\left[A_{j;T}\right]T_{1}}{2N}.
\end{align*}
Note that for $t < t'$, the outcome of $(\nu_j^{(t')} - \mu_j^{(t')})$ has expctation $0$, even given any information about time $t$; this gives the first equality.
The last inequality holds because $\Pr[A_{j;t}] \leq \Pr[ A_{j;t'}]$ for $t\geq t'$ and becaus
$$ {\bf E}_{\nu^{t}\sim B\left(\mu^{t}\right)}\left[\left(\nu_{j}^{t}-\mu_{j}^{t}\right)^{2}\mid A_{j;t}\right] = \frac{\sigma_j(\mu^{t})^2 }{N} \geq \frac{\alpha}{2N}. \qedhere
$$
\end{proof}
The next claim tells us that for any interval of time $t_0\dots(t_0+T_1),$ the change in $\mu_j$ due to the sampling steps cannot be much more than the change from the selection steps.
\begin{claim}\label{claim:noise-select}
$$ \frac{1}{2}\left(\sum_{t=t_0}^{t_0+T_1} \nu_j^{t} - \mu_j^{t}\right)^2 \leq
\left(\sum_{t=t_0}^{t_0+T_1} \mu_j^{t+1}-\nu_j^{t}\right)^2+4.$$
\end{claim}
\begin{proof}
Observe that
$$\mu_j^{t_0+T_1+1} - \mu_j^{t_0} = \left(\sum_{t=t_0}^{t_0+T_1} \nu_j^{t} - \mu_j^{t}\right) +\left(\sum_{t=t_0}^{t_0+T_1}\mu_j^{t+1}-\nu_j^{t}\right)$$
has magnitude at most 2, which gives:
$$ \left| \sum_{t=t_0}^{t_0+T_1} \nu_j^{t} - \mu_j^{t} \right| \leq \left|\sum_{t=t_0}^{t_0+T_1}\mu_j^{t+1}-\nu_j^{t} \right|+2.$$
Squaring both sides, the claim follows from the fact that
$2|x|^2+8 \geq (|x|+2)^2 . \qedhere$
\end{proof}
We now complete the proof of the lemma. First, combining Claims \ref{claim:determined-variance} and \ref{claim:noise-select} tells us that for $t_0+T_1\leq T$:
$$ {\bf E}\left[\left(\sum_{t=t_0}^{t_0+T_1} \mu_j^{t+1}-\nu_j^{t}\right)^2\right] \geq
\frac{\alpha\Pr\left[A_{j;T}\right]T_{1}}{4N} - 4.
$$
By applying Cauchy-Schwarz and Lemma \ref{lem:density}, we can relate the quantity inside the expectation to the change in fitness:
\begin{align*}
\left(\sum_{t=t_0}^{t_0+T_1} \mu_j^{t+1}-\nu_j^{t}\right)^2
& \leq T_1 \sum_{t=t_0}^{t_0+T_1} \left(\mu_j^{t+1}-\nu_j^{t}\right)^2 \\
& = T_1 \sum_{t=t_0}^{t_0+T_1} \left(\frac{\sigma_i(\nu^t)\hat{f}(j;\nu^t)}{\tilde{f}(\nu^t)}\right)^2\\
& \leq \frac{T_1}{1-n\epsilon}\sum_{t=t_0}^{t_0+T_1} \tilde{f}(\mu^{t+1})-\tilde{f}(\nu^t)
\end{align*}
Taking expectations on both sides, we conclude that for a sufficiently long interval, the expected change in fitness on that interval is a good upper bound on $\frac{\alpha \Pr[ A_{j;T}]}{4N}$:
\begin{equation}\label{eqn:interval}
\frac{1}{1-n\epsilon}{\bf E}\left[ \sum_{t=t_0}^{t_0+T_1} \tilde{f}(\mu^{t+1})-\tilde{f}(\nu^t) \right] \geq
\frac{\alpha\Pr[A_{j;T}]}{4N}-4/T_1.
\end{equation}
We can now amplify this bound $T_2$ times while using the fact that the total fitness change is at most $\epsilon$ ($T_1$ and $T_2$ will be set after).
\begin{align*}
T_2 \left(\frac{\alpha \Pr[A_{j;T}]}{4N} - 4/T_1\right) & \leq
\sum_{\ell=0}^{T_2-1} {\bf E}\left[ \sum_{t=\ell*T_1}^{(\ell+1)*T_1} \tilde{f}(\mu^{t+1})-\tilde{f}(\nu^t) \right] \frac{1}{1-n\epsilon} \\
& = {\bf E}\left[ \sum_{t=0}^{T_1*T_2} \tilde{f}(\mu^{t+1})-\tilde{f}(\nu^t)\right] \frac{1}{1-n\epsilon}\\
& = {\bf E}\left[ \sum_{t=0}^{T_1*T_2} \tilde{f}(\mu^{t+1})-\tilde{f}(\nu^t) +
\tilde{f}(\nu^{t})-\tilde{f}(\mu^{t})\right]\frac{1}{1-n\epsilon}\\
& = {\bf E}[ \tilde{f}(\mu^{T+1}) - \tilde{f}(\mu^0)]\frac{1}{1-n\epsilon}\\
& \leq \frac{\epsilon}{1-n\epsilon}
\end{align*}
Finally, we have
$$ \Pr[A_{j;T}] \leq \frac{4N\epsilon}{T_2(1-n\epsilon)\alpha} + \frac{8N}{\alpha T_1}.$$
Taking $T_1= 16N^2 n^4$ and $T_2 = \frac{\epsilon 8N^2 n^4}{1-n\epsilon}$ bounds the probability by $1/n^2.$
\end{proof}
Lemma \ref{lem:fast-vertex} tells us that with probability at least $1-2/n$, the population vector will be at a vertex after at most $T=O\left(\frac{\epsilon n^8 N^4}{1-n\epsilon}\right)$ steps, in which case $\tilde{f}(\mu^{T})\in \{1,1+\epsilon\}$. On the other hand,
Lemma \ref{lem:main} tells us that for any $T$, the probability that the total negative effect of the sampling exceeds $\alpha$ is at most $\beta$; since the fitness change for each selection step is non-negative (for our choice of $\epsilon$), we have that
$$ \Pr[\tilde{f}(\mu^{T})\neq 1+\epsilon] = \Pr[ \tilde{f}(\mu^{T}) < \tilde{f}(\mu^{(0)}) - \alpha] \leq \beta$$
when $\tilde{f}(\mu^0) > 1+\alpha$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,528 |
Q: Detailed implementation of IEEE754 in MATLAB? In MATLAB,
>> format hex; 3/10, 3*0.1
ans =
3fd3333333333333
ans =
3fd3333333333334
>> 3/10 - 3*0.1
ans =
bc90000000000000
Is this result predictable? i.e. I can follow some rules of floating point arithmetic, and get 3/10 = 3d3333333333333, 3*0.1 = 3d3333333333334 again by hand.
A: The rules are:
*
*In MATLAB, unless specified otherwise (via constructors), all literals have double precision in the sense of IEE754 standard: http://www.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html
*All arithmetic operations are executed according to the usual precedence rules: http://www.mathworks.com/help/matlab/matlab_prog/operator-precedence.html
*When mixing numeric types with double in an arithmetic operation, MATLAB converts the double to the other numeric type before executing the operation—as opposed to C, for example, which does the other way around.
By using these rules you can pretty much predict the results of any arithmetic expression (always little endian memory layout, bit patterns are two's complement for signed integers and IEEE754 for floats). The alternative is to let MATLAB apply the rules for you; the results will be consistent and repeatable.
A: The reason is that when creating the binary representation for 0.1 a roundup has occurred, introducing a small error:
>> 0.1
ans =
3fb999999999999a
There should be infinitely many of those 9s in the end but we cut it and round up the last digit. The error is small but becomes significant when you multiply by 3
>> 3*0.1
ans =
3fd3333333333334
When correctly calculated by division this last digit shouldn't be 4:
>> 3/10
ans =
3fd3333333333333
It is interesting to see that this error is not big enough to cause a problem when we multiply by some other number smaller than 3 (the threshold is not exactly 3 though):
>> 2.9/10
ans =
3fd28f5c28f5c28f
>> 2.9*0.1
ans =
3fd28f5c28f5c28f
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,047 |
Una baralla de cartes és un joc de naips o cartes de joc, en general 48. Aquests solen ser unes estampes rectangulars —la major part del temps— fetes de cartó, que tenen un dibuix per una cara i certs objectes en l'altra o un nombre de figures variable.
Les recerques assenyalen que els naips, i per tant la baralla, van ser creats al a la Xina, i que van arribar a Europa d'Orient introduïts pels àrabs cristianitzats a Catalunya i després van passar a Itàlia durant el regnat de Pere el Gran.
Existeixen diferents tipus de baralles, cada regió o país posseeix les seves pròpies, algunes de més conegudes que les altres, com per exemple la baralla anglesa, francesa o de Tarot.
El primer esment que es fa d'un joc de naips és a Barcelona, el 1310 pel Consell de Cent, per tant ja existien anteriorment. Generalment, els jocs requereixen una baralla completa. Per a certs jocs cal apartar les cartes que no s'utilitzen —Durak, Pinacle— i en alguns altres cal que s'utilitzin dos o tres jocs de baralles —Canastra—.
Etimologia
L'Alcover Moll diu que el mot baralla no sembla pas venir de baralla 'renyina' sinó que sembla adaptació del castellà baraja.
En castellà la RAE diu que 'naipe' ve del català naip, i aquest potser de l'àrab ma'íb 'censurable'.
Naips
Els naips, o «cartes» tradicionals, són elements del joc de naips. Són unes estampes fetes sobre cartró o material plàstic que formen una baralla i que s'han de barallar, barrejar, mesclar o fer les cartes abans de jugar. Gairebé sempre tenen forma rectangular, però en alguns països com l'Índia poden tenir formes rodones. Els jocs de naips més coneguts són el Tute, el Mus i la Brisca (a Espanya), el Truc, (a Argentina, Xile, Paraguai, Veneçuela i Uruguai), el Bridge, el Pòquer i la Canastra.
Exemples de naips utilitzats en la baralla espanyola, la francesa, l'alemanya i l'anglesa. Les baralles difereixen pel nombre i la nomenclatura dels naips.
Història
Hi ha diferències d'opinions sobre si els naips es van crear a l'Índia, o si es van usar primer a la Xina o a Egipte, encara que l'opinió majoritària és que haurien estat creades al , a la Xina.
A la Xina es jugava amb un tipus de naip (baralla de cartes xinesa) que derivava del paper moneda i de les fitxes del dòmino. A Pèrsia es va originar el Ganjifa o Gânjaphâ, un tipus de joc de cartes que es va fer popular en l'Índia durant l'imperi mogol al segle XVI. A Índia es jugava el Dasavatara Ganjifa, joc que està format per una baralla amb deu colls basats en els deu avatars o reencarnacions del déu Visnú: peix, tortuga, senglar, lleó, nan, destral, arc i fletxa, raig, caragol de mar i cavall. La major part dels naips indis són rodons, de diferents grandàries i estan fets amb cartó lacat, cartó pedra i a cops ivori. Igualment, al Japó es juga els jocs de cartes Karuta i les seves dues baralles més populars són el Hanafuda i el Uta-garuta.
El més probable és que els naips arribessin a Europa d'Orient, introduïts pels àrabs per la mitjanceria dels regnes cristians d'Espanya, encara que també es diu que van ser portats pels croats. La primera versió es pot sostenir en el fet que la baralla més antiga sigui la que és dita espanyola i que els colls de la baralla àrab eren monedes, copes, simitarres i bastons, que evolucionarien després a ors, copes, espases i bastos.
Pel que sembla, el Consell de Cent, va prohibir els jocs de cartes el 1310 a Barcelona, essent aquesta la menció més antiga del joc de naips a Europa, perquè la prohibició prova que ja feia anys que eren en ús. També en altres parts de la península ibèrica, eren coneguts els naips, puix que per exemple a Castella els estatuts de l'ordre de cavalleria de la Banda fundada per don Alfons XI de Castella el 1331, prohibien als cavallers de jugar a naips. Una mateixa interdicció va dictar don Joan I de Castella el 1387. Del reialme d'Aragó i Catalunya van poder passar a Itàlia, a partir de la conquesta de Sicília per Pere III d'Aragó (1282). Emperò els italians pretenen haver-los conegut ja el 1299 i si, com creuen alguns, els naips van donar ocasió al gravat sobre fusta també s'hauria d'atribuir aquesta invenció als italians. Però el més segur és que, si vingueren d'Orient els naips, també en va venir el mode d'impressió.
A França, el 1337, en les constitucions de l'Abadia de Sant Víctor, a Marsella, s'esmenta —prohibint-ho als frares— un joc que diuen «pàgines» que es podria referir als naips: L'agost de 1381, el fill d'un mercader de Marsella, a punt d'embarcar per a Alexandria, promet d'abstenir-se dels jocs d'atzar, entre els quals anomena la baralla: nahipi.
El pare jesuïta Menéstrier (1631-1705), en un article publicat el 1702, en el Journal del Trévoux, exposa que les primeres cartes franceses es van fabricar a Espanya el 1392 per a entreteniment del rei Carles VI de França, afligit d'atacs de «malenconia», i que el joc simbolitzava l'estructura feudal.
Un decret del Senat de Venècia de 1441 que prohibia la introducció de naips en la Senyoria indica que la seva fabricació després d'haver estat florent es trobava en decadència per les importacions de baralles estrangeres. Els naips del que es conserven a Venècia i altres ciutats són més grans que els d'avui i d'un cartó gruixut semblant al paper de cotó dels antics manuscrits. Les figures que ressalten sobre camp d'or són: tres reis, dues dones i dues sotes (en la baralla espanyola, cada una de les quatre cartes que duen el nombre 10 i una imatge d'un patge o infant, ço és un soldat de peu), una d'elles a cavall. Cada figura porta un bastó, una espasa o una moneda. Les colors semblen aplicades per mitjà d'uns contramotlles.
En alguns naips antics es pintaven dones en lloc d'homes sobre els palafrens. En algunes zones d'Andalusia, els asos estaven representats en forma de nois nus.
Una nova teoria és que els asos de certes baralles estan directament inspirats en les monedes romanes anomenades Aes.
Evolució
A partir de la primitiva baralla espanyola de 56 cartes —quatre colls de catorze cartes— es va crear la del Tarot, afegint-hi vint-i-dos naips, sense coll específic, els arcans majors, i mantenint la resta de la baralla. Semblant tenir origen en el inchiate.
Si bé és molt discutible, sembla que la primera baralla diferent de l'espanyola va ser la baralla alemanya, amb quatre colls —campanes, fulles, cors i glans) amb deu nombres i tres figures. Amb el temps va perdre els nombres 2 al 5, tots dos inclusivament, esdevenint una baralla de trenta-sis cartes.
Com l'alemanya, la baralla francesa, substitueix els nombres de les figures per lletres, R (roi), D (dame) i V (valet), esbiaixant així el malefici del número tretze, i, també com l'alemanya i l'espanyola més moderna, deixa reduïdes les figures a tres. Els seus colls són piques, cors, trèvols i diamants. L'anglesa deriva de la francesa, canviant els noms de les figures, J (jack), Q (queen) i K (king) i també de l'as (A).
Significat dels colls en la baralla francesa
Alguns expliquen que els colls de la baralla tenen el significat dels quatre pilars de la societat en l'edat mitjana:
♦ Diamant o Ors o Cocos o Rombes: economia de l'edat mitjana o també la Monarquia;
♠ Piques o Espases: exèrcit o també la Noblesa de gran importància en l'època en què es van lliurar grans batalles.
♥ Cor o Copes: Església o el Clergat que va prendre gran importància en l'edat mitjana.
♣ Trèvol o Bastos o Flors: agricultura o també els Plebeus de la societat feudal.
Expressions relacionades
Carta falsa o carta blanca: en alguns jocs de naips, la que no és triomf o és de poc o cap valor;
carta falla: la que exigeix que es falli en certs jocs de naips;
carta ferma: la més gran del joc o la que guanya les que encara no s'han jugat.
Vocabulari
acollar, apariar, destriar: classificar les cartes en mà per color.
atot: carta del coll triat que mana per damunt dels altres colls.
banca: en certs jocs de contrapartida, jugador que dirigeix el joc i n'assumeix les juguesques de tots els altres jugadors.
basa o plega: cartes que el jugador recull en guanyar una jugada; en certs jocs de cartes, cadascuna de les partides que són part d'un període de joc més llarg. Fer basa: recollir les cartes guanyades en una jugada.
bola (fer '') : guanyar totes les bases, en certs jocs de cartes.
cantar: proclamar que s'ha assolit una determinada opció de joc o combinació de cartes.
capot: jugada de cartes en què un jugador fa totes les vases i deixa l'altre sense fer ne cap.
capot (fer o donar '') (a algú): guanyar un jugador totes les bases (o un cert nombre de bases al la fi del joc).
carta falla: a la botifarra, nom que es dona al naip que és el sol d'un cert coll entre els que posseeix un jugador.
catxa, catxar (ang: bluff): en els jocs d'envit, juguesca forta que es fa malgrat tenir un mal joc amb l'esperança de guanyar la basa perquè els altres jugadors s'acovardiran perquè pensaran que el jugador que la fa té bones cartes.
catxar: prendre, apoderar-se de cartes d'altri. «Catxa-li un trumfo».
coll: cadascuna de les quatre menes de cartes que formen la baralla i que es coneixen per les quatre diferents figures (oros, copes, espases i bastos). De quin coll anem?
contrafallar: V. Refallar
crupier: persona que dirigeix el joc en un casino o casa de joc.
donada: acte de donar; especialment, acte de repartir les cartes als jugadors (cast. mano).
donar: repartir les cartes als jugadors.
envidar: fer un envit.
envit: juguesca estratègica consistent en augmentar la posada inicial durant el transcurs d'una partida.
escapçar: dividir les cartes en dos munts, després d'escartejades, posant a sota les que eren damunt, per a evitar possibles fraus projectats pel jugador que les ha escartejades.
escartejar, mesclar, barrejar o fer les cartes: remenar i barrejar les cartes d'un joc abans de jugar.i repartir-les als jugadors (cast. barajar).
espitjar: en alguns jocs de cartes, descobrir lentament i amb un cert teatre el número d'una carta mitjançant el fet de fer-hi lliscar girant per damunt una altra carta. El centre del gir és l'angle inferior esquerre de les dues cartes.
falla: mancança d'un dels quatre colls.
fallanca: engany simulat.
fallar: en certs jocs de cartes, estar mancat de cartes del coll que s'ha tirat, i haver de tirar trumfo per a guanyar al contrari; cast. fallar.
ferma: dit d'una carta (o jugada) qualsevol quan se sap segur que és la carta de més valor restant en el joc, normalment perquè les de més valor ja s'han jugat.
girada: acte de descobrir la que ha d'ésser trumfo; cast. voltereta.
mà: jugada; volta en què es juguen les cartes que s'han donat d'una vegada. Tenir la mà o Ésser mà: ser el primer en l'orde dels jugadors; ser qui rep la primera carta o fitxa quan les reparteixen i qui la tira en començar el joc; Perdre la mà: deixar, el primer en l'orde, la preferència a algun dels companys; cedir voluntàriament la preferència a un altre.
palla: cartes que valen molt pocs punts a qui les guanya.
parar, pagar o dur-lo: perdre.
passada: jugada; volta en què es juguen les cartes que s'han donat d'una vegada.
passar: renunciar a prendre part en la jugada, sia voluntàriament, sia per necessitat.
patota (fer la '') : fer un engany als companys de joc (cas. hacer trampa).
pedres: cartes que valen molts de punts a qui les guanya.
pinta: senyal que tenen en els dos caps certes cartes de jugar, perquè es pugui conèixer de quin coll són abans de destapar-les; cast. pinta.
refallar: jugar un atot superior a aquell amb què un altre jugador ha fallat.
retrucar: en el joc de truc, trucar el jugador que ha acceptat el truc d'un adversari.
tallar: en certs jocs, distribuir les cartes; portar el joc i jugar diners o quelcom altre contra tots els altres jugadors.
timba: període que hom ha estat jugant a cartes.
totar d'oros: jugar amb els trumfos a la mà.
trabucar les cartes: posar les cartes de joc amb la figura cap avall.
trumfo: coll que gaudeix d'una determinada preferència, segons cada mena joc.
En alguns països hispanòfons es coneix l'espasa com a «pica», «cor negre» i el diamant com a «rombe», «coco» o «brillo».
Tipus de baralles
Baralles per a jocs de cartes
Baralles europees clàssiques
La baralla espanyola té 48 naips, i de vegades 2 comodins (baralla de 50), repartits el quatre colls: ors, copes, espases i basts. La baralla de 48 cartes està numerada de l'1 (as) al 9, essent les figures el 10 (sota), l'11 (cavall) i el 12 (rei), i molt sovint hi ha en el comprat malls de 40 naips, sense els vuits i els nous, puix que molts jocs així ho requereixen. Altres jocs, com la Manilla, empren la baralla espanyola de 48 naips;
la baralla italiana és igual que l'espanyola de 40 cartes, amb la diferència que la sota està numerada com 8, el cavall com 9 i el rei com 10. Probablement la baralla espanyola va ser adoptada al Regne de Nàpols durant el domini d'Aragó i ha evolucionat posteriorment més a Espanya que a Itàlia, per la qual cosa la napolitana és semblant a l'antiga baralla espanyola;
la baralla alemanya té un total de 40 cartes numerades del 6 al 10, seguides per Unter o Bauer (inferior o pagès), Ober o Dame (superior o dama), König (rei) i Daus o Ass (dos o as). Estan repartides en quatre colls: Schellen (campanes), Gras o Grün o Laub (pastura, verd o fullatge), Herz o Rot (cor o vermell) i Eichel (gla) —existeixen diverses derivacions d'aquesta baralla com la centreeuropea, la franca o la suïssa—;
la baralla francesa té 52 cartes distribuïdes en 4 colls (cors, diamants, piques i trèvols) essent numerades de l'1 (o as) al 10 i seguides per la V (de la veu francesa valet o patge), la D (de dame o senyora) i la R (de Roi o rei). Tradicionalment, tant els cors com els diamants es representen en vermell mentre que les piques i els trèvols en color negre;
la baralla anglesa és semblant a la baralla francesa però l'As porta A (de ace) i les figures porten la J (de la veu anglesa jack o patge), la Q (de queen o reina) i la K (de king o rei).
Baralles que prenen com a model el model clàssic afegint-hi cartes
baralla de Tarot, barreja que està composta de 78 cartes dividides en 22 naips coneguts com a arcans majors, 40 arcans menors i 16 honors o figures. Aquests dos darrers grups estan classificats en quatre colls: espases, copes, bastos i ors, de la mateixa manera que la baralla espanyola;
baralla Minchiate. Semblant a la del Tarot, però amb 40 arcans.
Baralles asiàtiques
baralles Karuta: baralles japoneses clàssiques, essent les seves les baralles més populars la Hanafuda i la Uta-garuta;
baralles Ganjifa o Gânjaphâ: baralles originades a Pèrsia, la variació de les quals d'aquest tipus de joc de cartes, dita Dasavatara ganjifa, es va fer popular en l'Índia.
Altres tipus de baralles
baralles de cartomagia: baralles usades per a l'il·lusionisme;
baralles col·leccionables;
baralles numerades —per a jocs tipus Uno.
Fabricants de baralles
Bicycle, Estats Units
The United States Playing Card Company (USPC), Cincinnati, Ohio, Estats Units
Liceo Gráfico SL, Espanya
Naipes Heraclio Fournier S.A., Vitòria, Àlaba, País Basc
Maestros Naiperos , S.L., València
Naipes Canarios S.L., Espanya
Naipes Comas, l'Hospitalet
France Cartes, França
Heritage, Gran Bretanya
Piatnik, Àustria
ASS Altenburger, Alemanya
Cartamundi, Bèlgica
Copag, Brasil
Protea Playing Cards CO, Sud-àfrica
Saint Petersbought, Rússia
Kuo Kau, Taiwan
Naipes y Pasatiempos, NAPASA, Mèxic
Pasatiempos Gallo, Mèxic
Productos Leo, S.A., Argentina
Naipes Negros, Argentina
Cantaclaro, Colòmbia
Editorial Colina , Medellín, Colòmbia
Referències
Vegeu també
Joc de naips (Clàssics)
collita (joc de cartes)
Mestre de les Banderoles
Joc de cartes col·leccionables (Moderns)
Tarot
Baralla xinesa
Jocs de cartes | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,505 |
When you've been single for as long as I have, it's easy to give up, to stop believing that God has the right person waiting for you. Someone who will complete you in a way that can't even be put into words. (And no, the implication is not that you were not a full person before him, but rather that having that special person in your life will will greatly enhance your life).
So I start thinking that maybe my standards are too high, maybe what I want isn't important in the grand scheme of things. I wonder about the few guys I could have dated and I wonder, did I let a good one go?
When I'm feeling hopeless and full of despair, and wondering if it'll ever happen for me, I watch this slideshow of the engagement of the Atlanta photographer, Ross Oscar Knight. He got engaged last summer to a woman that he adores, whose every accomplishment he seems to take pride in. You can watch the slide show here, and read some of the backstory here.
I deserve a guy who I can be silly with, one who will want to treat me like a queen, knowing full well that I would not hesitate to treat him like a king. I want a man who honours me enough that he would wash my feet, that he'd want to do something over the top to show how much he cares for and cherishes me. I recognize that this was Knight's thing, and I'm not searching for someone exactly like him. Instead, I want the man who will show me in his own way that I am his beloved, that I am cherished and valued above all others in his life.
I continue to wait with hope.
i wish u gud luck.and,i knw u'll get the right guy oneday.
Thanks, Isaiah! I hope you find the right person for you too!
I once thot my ideals were too high…..
I tried (still hoping) to salvage a relationship (oops, my first)…..
I believe so much that the picture one has of the other in their 'minds i' does exist…..
However, I believe patience has great virtue (Surulere); only…..while we wait, we should learn THE lesson we ought to in every relationships/ friendships we have (& still have). There is always a lesson to learn. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,209 |
The GRDC has developed a drift reduction guide for spray operators explaining how new restrictions to the use of 2,4-D will impact on-farm applications. PHOTO GRDC.
Grain growers and spray operators can now access a practical guide explaining how to maintain efficacy when using coarser spray qualities in line with new restrictions to the use of 2,4-D.
The guide has been developed by the Grains Research and Development Corporation (GRDC) to assist industry understand the on-farm implications of the Australian Pesticides and Veterinary Medicines Authority (APVMA) decision to suspend the labels of all products containing the active ingredient 2,4-D from October 4, replacing them with a permit.
The action by the APMVA was taken in response to widespread damage over several years to sensitive crops, such as grapes, horticultural crops, summer pulses and cotton. The APMVA permit will stay in place until the finalisation of the 2,4-D review. Public consultation on the review is expected to start later this year.
Under the permit there are changes to the 'directions for use' for 2,4-D including: changes to application technique, spray quality, timing and the observance of mandatory no-spray buffer zones, as well as increased requirements for detailed record keeping.
Industry spray specialist Bill Gordon, who has done extensive work on best practice application, has helped develop the latest GRDC guide to 2,4-D use, for those working in the paddock.
Mr Gordon said it was important to understand the new changes were primarily targeted at drift mitigation and did not restrict any other aspects of the current approved use patterns as detailed in the new permit (replacing the original product labels).
Downwind buffers now apply (typically less than 50m, subject to rate and product being applied) between application sites, downwind sensitive crops and environmentally sensitive aquatic areas.
Mr Gordon said the new permit also included an advisory statement for 2,4-D use in cereals, fallow and pasture from October 1 to April 15. These statements advise operators to use an Extremely Coarse (XC) or Ultra Coarse (UC) spray quality and to take steps to mitigate the risk of spray drift such as adopting increased water rates and slower application speeds.
"Additional record keeping is also required under these changes, so operators now need to update spray records, with greater detail, within 24 hours of application and to keep these records for a minimum of two years," he said.
"The permit also includes clearer instructions to help identify temperature inversions to reduce off-target spray risk.
Mr Gordon said the changes would mean many spray operators would have to buy additional sets of nozzles to meet the new requirements for VC, XC or UC spray quality.
"In practical terms, many low-pressure air induction nozzles, such as the Teejet™ AIXR or Hardi Minidrift™, are not able to produce VC, XC or UC droplets at useful pressures in the nozzle sizes most commonly used, which range from 02 (yellow), 025 (lilac) and 03 (blue)," he said.
"Therefore, many spray operators will need to change to high pressure air induction nozzles, such as the Hardi Injet™, Teejet™ TTI or TTI-60, or the Agrotop™ TD-XL-D.
"Operators are encouraged to contact their suppliers well before starting spray activities to secure the supply of their nozzle requirements.
If spray operators are using Pulse Width Modulation Systems, Mr Gordon said there were several options to ensure they were meeting the new permit requirements.
"Very coarse spray qualities can be achieved on Pulse Width Modulation systems using Wilger™ MR-04 or SR-06 nozzles at pressures below 2.4 Bar. Other nozzle sizes may be appropriate if using the Wilger™ DR nozzle types," he said.
Mr Gordon said operators would also need to consider adjusting application volumes when using coarser spray qualities.
"When increasing the droplet size, it is important to consider increasing the total application volume to maintain coverage and efficacy," he said.
Mr Gordon said there were additional state and territory restrictions which spray operators and growers must adhere to which may include restricted areas and times of use. Operators are advised to check with their relevant state authority for details.
More information is available from the new GRDC Fact Sheet 'Maintaining efficacy with larger drops' at https://bit.ly/2IT3lND. For more information about best practice spray application go to https://grdc.com.au/spray-drift.
The GRDC also invites growers, advisors and industry stakeholders to join a spray drift webinar on October 16 from 2pm to 3pm AEST.
The webinar will include a presentation by APVMA about the changes and feature a panel of experts to provide practical advice and personal experience to answer your questions. To register for the spray drift webinar go to https://bit.ly/2ybu9nN. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,237 |
Xcel Energy, Colorado regulators weigh…
Xcel Energy, Colorado regulators weigh utility's role in electric-vehicle market
Proposed legislation would clarify utilities' role in serving electric vehicles
ElectricCharging.jpg Mike Angelo installs and constructs an electric charging station for electric vehicles while David Altman takes a photo of the installment at the South Boulder Recreation Center Wednesday, August 10, 2011. Chancey Bush/ The Camera
By Judith Kohler | jkohler@denverpost.com | The Denver Post
Advocates of getting more electric vehicles on Colorado roads say the effort could gain even more traction if Xcel Energy and other utilities take on a larger role, and it appears that could happen.
Xcel Energy officials indicated during a public meeting last week with the Colorado Public Utilities Commission that they will likely file a case this year on rates for public, fast-charging stations and for stations handling vehicle fleets. Another potential case would address providing part of the infrastructure for charging stations.
On another front, three legislators are sponsoring a bill to remove a block on investor-owned utilities like Xcel Energy from owning and operating charging stations.
"Our piece of legislation is going to allow a company like Xcel and, of course, the (Public Utilities Commission), to work together in a public-private way so we can build the infrastructure to try to boost electric-vehicle usage in our state," said Sen. Angela Williams, D-Denver.
The legislation, Senate Bill 19-077, co-sponsored by Sen. Kevin Priola, an Adams County Republican, and Rep. Chris Hansen, a Denver Democrat, would allow investor-owned utilities to own and operate charging stations as part of their regulated services, which would permit a return on their investment.
Similar bills failed in the previous two legislative sessions, but the sponsors said they think chances of passage are good this year.
Although there are different interpretations of the current law, Xcel Energy believes it is prohibited from owning and operating the charging equipment for electric vehicles, Jack Ihle, the utility's director of regulatory and strategic analysis, said in an email.
Xcel Energy is moving forward with projects in Minnesota, which doesn't have a law limiting investor-owned utilities' involvement in the electric-vehicle industry.
Critics say the Colorado law, enacted in 2012, was meant to define electric providers' roles but had unintended consequences and is now a drag on efforts to boost the number of charging stations. The law doesn't apply to rural electric utilities, some of which are working with homeowners and businesses to install chargers.
States across the country are exploring policies to develop the electric-vehicle market and encourage the expansion of private and public infrastructure, according to a report written by a group of industry and environmental organizations as well public agencies. The group presented the report to the state utilities commission last week.
Members of the commission had sought input on the challenges and opportunities of expanding the use of electric vehicles in Colorado. Jeffrey Ackermann, commission chairman, said the panel is trying to determine what its role might be.
During his first few days as Colorado's new governor, Jared Polis signed an executive order reaffirming the previous administration's goal of seeing nearly 1 million electric vehicles on the roads by 2030. His order builds on work by former Gov. John Hickenlooper to build charging stations across the state and to help convert fleets and public transit to electric vehicles with the state's share of a settlement with Volkswagen.
Colorado received nearly $70 million from the deal between Volkswagen and the federal government over allegations that the auto company modified computer software to cheat on federal emissions tests. About $10 million is being used to build 33 high-speed charging stations along the state's major transportation corridors, said Will Toor, executive director of the Colorado Energy Office.
Colorado ranks seventh in the nation in the number of electric-vehicle charging stations — 710 — according to the National Renewable Energy Laboratory. Of those, 650 are open to the public.
The Colorado Air Quality Control Commission is expected to consider a zero-emission standard, which would mandate that a certain percentage of manufacturers' vehicles sold in the state be electric. The requirement would likely be between 6 percent and 10 percent and apply initially to the 2023 model year.
"There's clearly been support for the electrification of the transportation sector (in Colorado) for a number of years," Toor said.
The governor's executive order doesn't attempt to issue directives to the utilities commission, Toor said, but does "encourage the commission and utilities to work toward implementing programs and policy" supporting use of electric vehicles.
"I think it's driven both by a sense that utilities have a really important role to play in being able to achieve a deep penetration of electric vehicles, but also that there are specific benefits to the electric system," Toor said.
During last week's discussion of the electric-vehicle report, the commission encouraged Xcel Energy to submit filings to start charting what the utility could do.
"That's a really great next step. When utilities file an application, the commission and other interested stakeholders can really dig in and review what does this mean, what can utilities accomplish and do to help remove barriers to electrification," said Gwen Farnsworth, senior energy policy adviser for Western Resource Advocates.
Farnsworth said utility customers would benefit as the number of electric vehicles increase, especially if the vehicles are charged during low-demand periods, because utilities could spread their fixed costs across more ratepayers.
Gov. Jared Polis pushes Colorado toward zero-emission vehicles with first executive order
Colorado's plans for $68.7 million Volkswagen pollution settlement finalized
Colorado's Electric Car Dream: More charging stations, less smog and the warm embrace of renewable energy
A 2017 report by consulting firm M.J. Bradley & Associates said widespread use of electric vehicles could generate $43 billion in benefits for Coloradans by 2050 in the form of lower utility bills and reduced greenhouse-gas emissions. Other benefits include lower vehicle maintenance and fuel costs.
While acknowledging the benefits of electric vehicles, commission members voiced concerns about ensuring that low-income and rural areas are not overlooked when building charging stations.
"We still have vast areas of rural Colorado that are unserved," said commissioner John Gavan.() "I look at this as almost analogous to the problem that we had with broadband (internet). How are we really going to drive ubiquitous charging deployment more aggressively in the rural areas? I think that's going to be very key."
Colorado Public Utilities Commission
Judith Kohler
Judith Kohler joined The Denver Post in August 2018 and is part of the business team, writing about energy, aerospace, agriculture and other topics. She spent 21 years with The Associated Press, covering politics, government, energy and the environment in Colorado and Wyoming.
jkohler@denverpost.com
Follow Judith Kohler judith.kohler.12 Follow Judith Kohler @JudithKohler
More in Transportation
Colorado still has a transportation funding crisis. Can Republicans and Democrats agree on a solution?
DIA has fallen behind on maintenance even as it's poured billions of dollars into expansions
Pedestrian traffic deaths in Colorado are up 89% since 2009, AAA analysis shows
Fire forces closure of 6th Avenue at I-25 junction | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 921 |
↑ Carreras E, Bertz H, Arcese W, Vernant JP, Tomás JF, Hagglund H, Bandini G, Esperou H, Russell J, de la Rubia J, Di Girolamo G, Demuynck H, Hartmann O, Clausen J, Ruutu T, Leblond V, Iriondo A, Bosi A, Ben-Bassat I, Koza V, Gratwohl A, Apperley JF (1998). "Incidence and outcome of hepatic veno-occlusive disease after blood or marrow transplantation: a prospective cohort study of the European Group for Blood and Marrow Transplantation. European Group for Blood and Marrow Transplantation Chronic Leukemia Working Party". Blood. 92 (10): 3599–604. PMID 9808553.
This page was last edited 15:44, 28 February 2018 by Omodamola Aje. Based on work by Husnain Shaukat and Irfan Dotani. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,750 |
Okidata OKIPAGE 4w Driver. WindowsXP Driver and Status Monitor for the Okipage 4w. To correctly install the printer follow the steps below.
IMPORTANT: "DO NOT" use Add Printer Wizard to install the driver.
1. Plug printer into local machine.
2. Turn printer ON. If the hardware is not automatically detected, reboot the machine.
3. The "Found New Hardware Wizard" will start.
4. Select the radio button "Install from a list or specific location (Advanced)".
5. Deselect "Search removable media (floppy, CD-ROM...).
6. Select "Include this location in the search".
8. In the pop-up windows that appears, browse to the location of the driver and select "Ok".
10. Windows will begin the installation.
11. In the "Completing the Found New Hardware Wizard" screen click "Finish". | {
"redpajama_set_name": "RedPajamaC4"
} | 3,794 |
{"url":"https:\/\/indico.cern.ch\/event\/686555\/contributions\/2960505\/","text":"# ICHEP2018 SEOUL\n\nJul 4 \u2013 11, 2018\nCOEX, SEOUL\nAsia\/Seoul timezone\n\n## Measurement of D \u2217 Production in Diffractive Deep Inelastic Scattering at HERA\n\nJul 5, 2018, 5:10 PM\n15m\n101 (COEX, Seoul)\n\n### 101\n\n#### COEX, Seoul\n\nParallel Strong Interactions and Hadron Physics\n\n### Speaker\n\nAndrew Mehta (University of Liverpool (GB))\n\n### Description\n\nMeasurements of $D^{\u2217}(2010)$ meson production in diffractive deep inelastic scattering ($5 < Q^2 < 100$ GeV$^2$) are presented which are based on HERA data recorded at a centre-of-mass energy $\\sqrt{s} = 319$ GeV with an integrated luminosity of $287$ pb$^\u22121$. The reaction $ep\\to eXY$ is studied, where the system $X$, containing at least one $D^{\u2217}(2010)$ meson, is separated from a leading low-mass proton dissociative system $Y$ by a large rapidity gap. The kinematics of $D^{\u2217}$ candidates are reconstructed in the $D^{\u2217}\\to K\\pi\\pi$ decay channel. The measured cross sections compare favourably with next-to-leading order QCD predictions, where charm quarks are produced via boson-gluon fusion. The charm quarks are then independently fragmented to the $D^{\u2217}$ mesons. The calculations rely on the collinear factorisation theorem and are based on diffractive parton densities previously obtained by H1 from fits to inclusive diffractive cross sections. The data are further used to determine the diffractive to inclusive $D^{\u2217}$ production ratio in deep inelastic scattering.\n\nEur.Phys.J.C77 (2017), 340 [arxiv:1703.09476]\n\n### Primary authors\n\nStefan Schmitt (Deutsches Elektronen-Synchrotron (DE)) H1 Collaboration (DESY) Andrew Mehta (University of Liverpool (GB))\n\n### Presentation materials\n\n Mehta_H1DiffCharm_ICHEP.pdf Mehta_H1DiffCharm_ICHEP.pptx","date":"2022-08-12 06:49:39","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\": 0.6884087324142456, \"perplexity\": 7912.270061686875}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571584.72\/warc\/CC-MAIN-20220812045352-20220812075352-00633.warc.gz\"}"} | null | null |
\section{Introduction}
Most analytical models of FRII extragalactic radio sources are based
on the existence of two jets emerging from an AGN in the core of the
source; a model first proposed by Rees (1971)\nocite{mr71}. After
passing through a strong shock at the location of the radio hot spot,
the jet material inflates a cavity of hot, rarefied gas. The expansion
of this cocoon is supersonic with respect to the surrounding IGM and
therefore drives a bow shock into this material. This shock will
compress and heat the gas of the IGM, thereby increasing X-ray
emission from thermal bremsstrahlung. Inferred on theoretical grounds
by Scheuer (1974)\nocite{ps74}, this hot layer of gas between bow
shock and cocoon was first observed in Perseus A by B\"ohringer {\em
et al.} (1993)\nocite{bvfen93} and in Cygnus A by Carilli {\em et
al.} (1994)\nocite{cph94}. The extended X-ray emission detected in
the vicinity of 3C 356 (Crawford \& Fabian 1996\nocite{cf96}) may
represent another example of thermal bremsstrahlung of the IGM boosted
by the presence of a powerful radio source.
The first analytical investigation of a cavity expanding
supersonically into a surrounding atmosphere was presented by Sedov
(1959)\nocite{ls59} who considered the case of a strong explosion in a
uniform density profile. Dyson et al. (1980)\nocite{dfp80} extended
this analysis to spherically symmetric winds with a permanent energy
input and Heinz {\em et al.} (1998)\nocite{hrb98} included density
gradients by modeling the external density distribution with a King
(1972)\nocite{ik72} profile. The solutions of Sedov and Dyson {\em et
al.} are both self-similar. The expansion of both the cocoons and bow
shocks of FRII sources is shown to be self-similar regardless of its
geometrical shape by Kaiser \& Alexander (1997)\nocite{ka96b}
(hereafter KA), and this result suggests that a solution based on the
spherically symmetric case may be possible for a radio source provided
that the elongation of the cocoon along the axis of the jet can be
accommodated. The solution for the flow of the shocked IGM between
bow shock and cocoon requires solution of at least a two-dimensional
problem instead of the one-dimensional spherical case of an explosion
or wind.
In this paper we assume the bow shock surrounding the cocoon of FRII
radio sources to be elliptical. We then determine the properties of
the shocked gas between bow shock and cocoon by integrating
numerically the relevant hydrodynamical equations. Analogous to KA, we
will assume that the density distribution of the IGM surrounding the
radio source is well described by a power law of the form $\rho _o
(r/a_o)^{-\beta}$, where $r$ is the radial distance from the centre of
the distribution.
Although we may choose the shape of the bow shock freely without
violating the conditions for self-similar expansion of the cocoon or
bow shock, we will show that another condition for self-similarity may
not be fulfilled by some of the solutions presented here. This
condition is the pressure equilibrium of the jets within the uniform
pressure of the cocoon away from the hot spots. This implies that for
these cases the model may not be self-consistent. One possibility, as
we discuss, is that the assumption of an elliptical shape for the bow
shock must be relaxed and the source expands self-similarly with a
somewhat modified shape. However, we believe that the results obtained
in this paper are an important first step in solving the problems
posed by the flow between bow shock and cocoon of FRII sources.
The general dynamics of the large-scale structure of FRII sources
depends on the details of the confinement of the cocoons of these
objects. It is usually assumed that the pressure of the material in
the cocoon is balanced by the ram pressure of the IGM (e.g. Scheuer
1974\nocite{ps74}, KA). In this paper we show that the assumption of
ram pressure confinement of the cocoons overestimates the ratio of the
pressure in front of the hot spot and that of the cocoon material,
leading to lower hot spot advance speeds in the analysis of KA.
The results of the integration of the hydrodynamical equations allows
us to determine the X-ray surface brightness of the shocked
material. The result of this calculation can be directly compared with
X-ray observations. The only currently available X-ray map of a
powerful FRII source with sufficient spatial resolution to allow such
a comparison, is the {\em ROSAT}\/ map of Cygnus A presented by
Carilli {\em et al.} (1994)\nocite{cph94}. However, the advent of the
new X-ray telescopes such as {\em AXAF}\/ will allow us to extend this
analysis to a greater number of FRII sources.
In Section 2 we transform the hydrodynamical equations to an
elliptical coordinate system comoving with the self-similar expansion
of the bow shock and the cocoon. Section 3 describes the numerical
method used to integrate the equations. The properties of the flow are
discussed in Section 4. In Section 5 we calculate the X-ray emission
as predicted by the model and compare our results with the X-ray maps
of Cygnus A and 3C 356.
\section{The hydrodynamical equations in an elliptical coordinate system\label{sec:hydro}}
The hydrodynamical equations governing adiabatic fluid flow in
three dimensions if gravitational fields and viscosity can be
neglected, are the equation of motion:
\begin{equation} \frac{\partial}{\partial t} \,
{\bf v} + \left( {\bf v} \cdot {\bf \nabla} \right) {\bf v} = -
\frac{1}{\rho} {\bf \nabla} \, p,\label{eqmo}\end{equation}
\noindent the equation of continuity:
\begin{equation} \frac{\partial}{\partial t} \, \rho + {\bf \nabla} \cdot
\left( \rho {\bf v} \right) = 0\label{eqco}\end{equation}
\noindent and the adiabatic equation:
\begin{equation} \frac{\partial}{\partial t} \, \frac{p}{\rho ^{\Gamma}}
+ \left( {\bf v} \cdot {\bf \nabla} \right) \frac{p}{\rho ^{\Gamma}} =
0,\label{eqad}\end{equation}
\noindent where {\bf v} is the velocity vector of the gas, $p$ the
pressure, $\rho$ its density and $\Gamma$ its adiabatic index. We
assume in the following analysis that the bow shock surrounding the
cocoon is a prolate ellipsoid with rotational symmetry about the
$z$-axis defined by the jet, and therefore we rewrite these equations
in prolate spheroidal coordinates ($\eta$, $\xi$, $\phi = \pi /2$, see
Appendix \ref{sec:nonmov}). The rotational symmetry reduces the
problem to two spatial dimensions.
\begin{eqnarray} c \, c_r \
\frac{\partial}{\partial t} \, {\bf v}' + \left( {\bf v}' \cdot {\bf
\nabla}' \right) {\bf v}' & = & - \frac{1}{\rho '} {\bf \nabla}' p',\nonumber\\
c \, c_r \
\frac{\partial}{\partial t} \, \rho ' + {\bf \nabla}' \cdot \left( \rho '
{\bf v}' \right) & = & 0,\nonumber\\
c \, c_r \
\frac{\partial}{\partial t} \, \frac{p'}{\rho '^{\Gamma}} + \left(
{\bf v}' \cdot {\bf \nabla}' \right) \frac{p'}{\rho '^{\Gamma}} & = &
0,\label{prosph}
\end{eqnarray}
\noindent where all dashed quantities are functions of the elliptical
coordinates $\eta$ and $\xi$, $c$ is constant and $c_r = \left( \sinh
^2 \eta +\sin ^2 \xi \right) ^{1/2}$.
As the bow shock expands it is possible for all times to find a
stationary prolate spheroidal coordinate system characterized by a
constant $c$ in which the bow shock is described as a coordinate
surface with constant $\eta =\eta _b$. The aspect ratio of the cocoon,
i.e. the ratio of its length to its width, $R_{b}$, and the `length'
of the bow shock, i.e. the point at which the cut of the bow shock
with the $xz$-plane crosses the $z$-axis, determine the particular set
of values for $c$ and $\eta _b$. KA show that the expansion of the bow
shock and of the cocoon is self-similar; in this case the bow shock
can be described as a coordinate surface with $\eta =\eta _b$ for all
times. This is achieved by setting $c=c_o L_b$, where $c_o$ is a
constant and $L_b=f(t)$ is the `length' of the bow shock as described
above (see Appendix \ref{sec:mov}). It follows that
\begin{eqnarray}
c_o \, L_b \, c_r \left( \frac{\partial}{\partial t} {\bf v}' +
\dot{\eta} \, \frac{\partial}{\partial \eta} {\bf v}' + \dot{\xi} \,
\frac{\partial}{\partial \xi} {\bf v}' \right) & & \nonumber\\
+ \left( {\bf v}' \cdot
{\bf \nabla}' \right) {\bf v}' & = & -\frac{1}{\rho '} {\bf \nabla}'
p',\nonumber\\
c_o \, L_b \, c_r \left( \frac{\partial}{\partial t}
\rho ' +
\dot{\eta} \, \frac{\partial}{\partial \eta} \rho ' + \dot{\xi} \,
\frac{\partial}{\partial \xi} \rho ' \right) & & \nonumber\\
+ {\bf \nabla}' \cdot \left( \rho '
{\bf v}' \right) & = & 0,\nonumber\\
c_o \, L_b \, c_r \left( \frac{\partial}{\partial t} \frac{p'}{\rho
'^{\Gamma}} +
\dot{\eta} \, \frac{\partial}{\partial \eta} \frac{p'}{\rho
'^{\Gamma}} + \dot{\xi} \,
\frac{\partial}{\partial \xi} \frac{p'}{\rho
'^{\Gamma}} \right) & & \nonumber\\
+ \left(
{\bf v}' \cdot {\bf \nabla}' \right) \frac{p'}{\rho '^{\Gamma}} & = &
0.\label{comov}
\end{eqnarray}
Because of the self-similar evolution of both the bow shock and the
cocoon the flow between them must also be self-similar. This allows a
further simplification of equations (\ref{comov}) by introducing
dimensionless variables for the velocity, density and pressure
of the shocked IGM in which the dependencies on time (i.e. on $L_b$
and $\dot{L} _b$) and the spatial coordinates are separated,
\begin{eqnarray}
{\bf v}' & = & \dot{L} _b \, \left[ U_{\eta} (\eta,\xi) \, {\bf \hat{e}} _{\eta} +
U_{\xi} (\eta,\xi) \, {\bf \hat{e}} _{\xi} \right], \nonumber\\
\rho ' & = & \rho _o \, \left( \frac{L_b}{a_o} \right) ^{-\beta} \
R(\eta,\xi), \nonumber\\
p' & = & \rho _o \, \left( \frac{L_b}{a_o} \right)
^{-\beta} \, \dot{L} _b ^2 \ P(\eta,\xi),\label{dimless}
\end{eqnarray}
\noindent where we model the external density distribution with a
power law, $\rho _o (r/a_o)^{-\beta}$ with $\rho _o$ the central
density, $a_o$ the core radius and $r$ the radial distance from the
centre of the distribution.
Using the comoving coordinate system described above and replacing the
flow variables with their dimensionless counterparts we find
\begin{eqnarray}
c_o \, c_r \, \frac{\ddot{L}_b L_b}{\dot{L}_b^2} \, U_{\eta} +
b_{\eta} \, \frac{\partial}{\partial \eta} \, U_{\eta} + b_{\xi} \,
\frac{\partial}{\partial \xi} U_{\eta} & & \nonumber\\
+ \left( c_g \, U_{\eta} - c_e
\, U_{\xi} - 2 \, c_o \, c_r \, c_e \, c_g \right) U_{\xi} + \frac{1}{R} \,
\frac{\partial}{\partial \eta} P & = & 0,\nonumber\\[0.5cm]
c_o \, c_r \, \frac{\ddot{L}_b L_b}{\dot{L}_b^2} \, U_{\xi} + b_{\eta}
\, \frac{\partial}{\partial \eta} \, U_{\xi} + b_{\xi} \,
\frac{\partial}{\partial \xi} U_{\xi} & & \nonumber\\
- \left( c_g \, U_{\eta} - c_e
\, U_{\xi} - 2 \, c_o \, c_r \, c_e \, c_g \right) U_{\eta} + \frac{1}{R} \, \frac{\partial}{\partial \xi} P & = & 0,\nonumber\\[0.5cm]
R \, \frac{\partial}{\partial \eta} U_{\eta} + b_{\eta} \,
\frac{\partial}{\partial \eta} R + R \, \frac{\partial}{\partial \xi}
U_{\xi} + b_{\xi} \, \frac{\partial}{\partial \xi} R & & \nonumber\\ +
\left[ \left( c_e + \coth \eta \right) U_{\eta} + \left( c_g + \cot
\xi \right) U_{\xi} - c_o \, c_r \, \beta \right] R & = & 0,\nonumber\\[0.5cm]
-\frac{\Gamma \, P \, b_{\eta}}{R} \, \frac{\partial}{\partial \eta} R
+ b_{\eta} \, \frac{\partial}{\partial \eta} P -\frac{\Gamma \, P \,
b_{\xi}}{R} \, \frac{\partial}{\partial \xi} R & & \nonumber\\
+ b_{\xi} \,
\frac{\partial}{\partial \xi} P + c_o \, c_r \left[ \left(\Gamma
-1\right) \beta +2 \,\frac{\ddot{L}_b L_b}{\dot{L}_b^2} \right] P & =
& 0, \label{fourth}\end{eqnarray}
\noindent where we have split the equation of motion into two
equations corresponding to components parallel to ${\bf \hat{e}}
_{\eta}$ and ${\bf \hat{e}} _{\xi}$ respectively, and
\begin{equation} b_{\eta} = U_{\eta} - c_o \, c_r \, c_e \ \ , \ \
b_{\xi} = U_{\xi} + c_o \, c_r \, c_g.\label{bdef}
\end{equation}
Furthermore, the ratio of the length of the bow shock, $L_b$, and the
length of the cocoon, $L_c$, must also be a constant. Using the result
\begin{equation} L_c = c_1 \, \left( \rho _o \, a_o^{\beta} \right)^{-\frac{1}{5-\beta}} \,
Q_o^{\frac{1}{5-\beta}} \, t^{\frac{3}{5-\beta}}
\label{linear}
\end{equation}
\noindent from KA, where $Q_o$ is the jet power and $c_1$ is a
dimensionless constant, $\ddot{L}_b L_b/\dot{L}_b^2$ reduces to
$(\beta -2)/3$, and the time dependence in equations (\ref{fourth})
can be eliminated.
\section{Numerical method\label{sec:nummet}}
The set of equations (\ref{fourth}) represent a system of first order
partial differential equations. Although there are no time derivatives
in this set the general form is that of an initial value problem
because the values of the variables are known at the surface of the
bow shock and the solution has to be propagated from this surface at
$\eta _b$ inwards to smaller $\eta$ in the direction of the contact
discontinuity delineating the cocoon.
The projection of the bow shock onto the $xz$-plane is an ellipse with
$\eta _b=$ constant and is described by
\begin{equation} r_b = \pm \sqrt{c_o \, \sinh ^2 \eta _b + \tanh ^2 \eta
_b \, l_b^2}, \label{delin}\end{equation}
\noindent where $l_b$ is the dimensionless coordinate defined by
$l_b=z/L_b$. At the tip of the bow shock where $l_b=1$, $r_b =0$, and
for $l_b=1/2$ one finds $r_b=1/(2 \, R_b)$. These conditions yield
\begin{equation} c_o = \sqrt{1-\frac{1}{3 \,
R_b^2}} \ \ , \ \ \cosh \eta _b = \frac{1}{c_o}.
\end{equation}
To derive the initial conditions at the bow shock we need to know the
velocity of the bow shock perpendicular to its surface,
$v_{\perp}$, and from the appendix of KA we find
\begin{equation}
v_{\perp} = \frac{v_x - \frac{\partial r_b}{\partial z} \,
v_z}{\sqrt{\left( \frac{\partial r_b}{\partial z} \right) ^2 +1}}.
\end{equation}
\noindent Since the bow shock expands self-similarly, $v_x= x/L_b$ and
$v_z = l_b$. By assuming that the bow shock is strong and that the
adiabatic index of the external atmosphere is 5/3 this implies for the
assumed elliptical shape of the shock surface
\begin{eqnarray}
U_{\eta} (\eta _b , \xi) & = & \frac{3}{4} \, \frac{\sinh \eta
_b}{c_r}, \nonumber\\
U_{\xi} (\eta _b , \xi) & = & 0\nonumber, \\
R (\eta _b , \xi) & = & 4 \, c_o^{-\beta} \, \left( \cos ^2 \xi +
\sinh ^2 \eta _b \right) ^{-\frac{\beta}{2}},\nonumber\\
P (\eta _b , \xi) & = & \frac{3}{4} \, c_o^{-\beta} \, \left( \cos ^2 \xi +
\sinh ^2 \eta _b \right) ^{-\frac{\beta}{2}} \, \frac{\sinh ^2 \eta
_b}{c_r^2}.
\end{eqnarray}
Given the symmetry of the problem it is sufficient to integrate the
equations only between $\xi =0$ and $\xi = \pi/2$ which corresponds to
the $z$ and $x$-axes respectively, and the following boundary
conditions must apply along these coordinate lines:
\begin{eqnarray}
U_{\xi} & = & 0, \nonumber\\
\frac{\partial}{\partial \xi} U_{\eta} & = & 0, \nonumber\\
\frac{\partial}{\partial \xi} R & = & 0, \nonumber\\
\frac{\partial}{\partial \xi} P & = & 0.
\end{eqnarray}
The numerical integration was performed using a finite difference
discretisation scheme employing NAG\footnote{NAG is a registered
trademark of Numerical Algorithms Group Ltd.} routines on a grid of
1001 points between $\xi=0$ and $\xi=\pi/2$ separated by equal
intervals of length $\pi/2000$. Equations (\ref{fourth}) are
non-linear and it is therefore not straightforward to assess the
stability of the numerical method used. Trial calculations show that
in particular the solution for $U_{\xi}$ is subject to
oscillations. To suppress these oscillations, it is necessary to
introduce an `artificial viscosity' term of the form $a \, \partial ^2
U_{\xi}/ \partial \xi ^2$ to all equations in the system containing
expressions proportional to $\partial U_{\xi}/ \partial \xi$
(e.g. Richtmyer \& Morton 1967\nocite{rm67}). The constant $a$ is
chosen to be small compared to any other term in the system of
equations in order to keep the influence of the `artificial viscosity'
within the error of the integration method.
The integration has to be stopped at the contact discontinuity. There
can be no gas flow across this surface and therefore the criterion for
stopping the integration is met when the component of the gas velocity
perpendicular to the contact discontinuity is equal to the
corresponding component of the self-similar expansion velocity of this
surface. The shape of the contact surface, $r_c(z)$, and therefore the
self-similar expansion velocity, $z \, {\bf \hat{e}}_z + r_c (z) \,
{\bf \hat{e}}_x$, is not known {\em a priori}. Since the bow shock
will be closest to the cocoon at the hot spot, one expects the flow to
first reach the expansion velocity of the cocoon at $\xi=0$ and some
value $\eta = \eta_c$, i.e. on the $z$-axis. Assuming that $\partial
r_c / \partial z \rightarrow \infty$ at this point, the condition for
stopping the integration on the $z$-axis becomes $U_{\eta} (\eta _c ,
\xi =0) =c_o \, \cosh \eta _c$, where we have again made use of the
fact that the cocoon is expanding self-similarly. A comparison with
equation (\ref{bdef}) shows that at this point $b_{\eta} = 0$ which
means that the system of equations (\ref{fourth}) breaks down, as one
would expect at a contact discontinuity.
The integration was advanced from the bow shock at $\eta = \eta _b$ to
smaller values of $\eta$ in steps of $\eta _b / 5\times 10^4$ and
after each step it was checked whether the flow had reached the cocoon
at the hot spot, i.e. whether the velocity of the flow along the
$z$-axis was within 1\% of the self-similar velocity.
\setcounter{footnote}{1}
The cocoon is not elliptical and therefore does not coincide with the
coordinate surface at $\eta _c$. The integration was continued by
considering the grid point of the finite difference mesh of smallest
$\xi$ for which the solution had not yet reached the contact
surface. All derivatives with respect to $\xi$ in equations
(\ref{fourth}) were approximated by linear extrapolation using two
values of these expressions separated by $\eta _b /5\times 10^2$ along
a line of constant $\xi$. This extrapolation effectively converts
equations (\ref{fourth}) into a system of ordinary differential
equations which were solved using an implementation of the backward
differentiation formulae from the NAG{\footnotemark} library. The
integration was continued until the solution for the last point of the
grid had reached the cocoon.
\section{The gas flow between bow shock and cocoon\label{sec:flow}}
\begin{figure*}
\label{fig:sphecom}
\end{figure*}
An important test of the analysis and integration presented here is to
consider the case of a spherical symmetric wind as discussed by Dyson
{\em et al.} (1980)\nocite{dfp80}. These authors considered only the
case of a wind expanding into an uniform atmosphere, but using the
results of the previous section it is straightforward to extend their
analysis to more complex environments in which $\beta \ne 0$ (for a
solution using King density profiles see Heinz {\em et al.}
1998\nocite{hrb98}).
In the case of a spherical bow shock there is an additional degree of
rotational symmetry about the $x$-axis. This implies that one can set
$\xi =0$; $U_{\xi}$ and all derivatives with respect to $\xi$
vanish. Furthermore it is possible to change from the remaining
prolate spheroidal coordinate $\eta$ to the more familiar spherical
polar coordinate $r$ by using $r=c_o \cosh \eta$. Because of these
simplifications the second equation in (\ref{fourth}) vanishes
completely and we obtain
\begin{eqnarray}
\frac{\beta -2}{3} \, U_r + (U_r - r) \, \frac{\partial}{\partial r} U_r +
\frac{1}{R} \, \frac{\partial}{\partial r} P & = & 0,\nonumber\\
R \, \frac{\partial}{\partial r} U_r + (U_r - r) \, \frac{\partial}{\partial
r} R +2 \, \frac{U_r R}{r} - \beta R & = & 0,\nonumber\\
-\frac{\Gamma P}{R} \, (U_r - r) \, \frac{\partial}{\partial
r} R + (U_r - r) \, \frac{\partial}{\partial r} P & & \nonumber\\
+ \frac{3 \Gamma \beta
- \beta -4}{3} \, P & = & 0,
\end{eqnarray}
\noindent where we have used $R_b = 1/\sqrt{3}$ for a spherical bow
shock and therefore $\eta _b \rightarrow \infty$ which implies that
$\coth \eta \rightarrow 1$ for all $\eta$ close to $\eta _b$. Setting
$\beta =0$ recovers the equations used by Dyson {\em et al.}
(1980)\nocite{dfp80}. The initial conditions at the bow shock at $r=1$
for the spherical case are $U_r = 3/4$, $R=4$ and $P=3/4$.
Figure \ref{fig:sphecom} shows plots of the solution for the case of a
spherical bow shock. The curves for $\beta =0$ are identical to those
presented by Dyson et al. (1980)\nocite{dfp80}. In this case the
contact discontinuity is reached at $r_c=0.86$. For the two other
cases presented here we find $r_c=0.85$ and $r_c=0.84$ for $\beta =1$
and $\beta =2$ respectively. In all three cases the gas is accelerated
away from the cocoon but the acceleration is somewhat less effective
for the cases with a density gradient in the external atmosphere,
hence the greater detachment of the cocoon from the bow shock. While
this difference in the flow pattern is not very significant, the
distribution of the density and the pressure of the gas behind the bow
shock is distinctively different for each case. The gas is expanding
adiabatically after being compressed by the bow shock. This behaviour
is evident from the curves in Figure {\ref{fig:sphecom} for $\beta =0$
but is masked for $\beta = 2$ by the presence of dense gas at high
pressure close to the cocoon which has passed through the bow shock
earlier than the material in the regions closer to the bow shock. The
density and pressure of the external gas in front of the bow shock
decreases faster than that of the material which is expanding after
passing through the bow shock, giving the impression that the gas is
further compressed after passing through the bow shock. In this case
the cocoon is surrounded by a relatively thin, heavy shell of highly
compressed gas, while for $\beta =0$ the gas closest to the contact
discontinuity is rarefied.
Results from the numerical integrations of equations (\ref{fourth})
for the more complicated case of ellipsoidal bow shocks applicable to
FRII sources are summarized in Table \ref{tab:results}. Note, that the
stand-off distances between bow shock and cocoon are small both at
the hot spot and close to the $x$-axis. The shape of the cocoon is
therefore also close to an ellipsoid and this agrees well with
observations (e.g. Leahy \& Williams, 1984\nocite{lw84}). The
following sections describe the results in greater detail.
\begin{table*}
\label{tab:results}
\end{table*}
\subsection{The velocity field}
\begin{figure*}
\label{fig:velfield}
\end{figure*}
\begin{figure*}
\label{fig:velblow}
\end{figure*}
Figure \ref{fig:velfield} shows a cut defined by the $xz$-plane of the
velocity field of the gas flow between bow shock and contact
discontinuity for one half of a wide ($R_{b}=1$) ellipsoidal bow shock
in an uniform atmosphere. Because of the rotational symmetry about the
$z$-axis the vectors are lying fully in the $xz$-plane and there is a
reflection symmetry about the $z$-axis. The vectors show the velocity
of the flow with respect to the comoving coordinate system, with the
bow shock and cocoon at rest.
A closer examination of the flow close to the hot spot shows the flow
bending around the tip of the cocoon and then moving along the contact
discontinuity (Figure \ref{fig:velblow}, left). On the $z$-axis, the
axis of the jet, the flow ends in a stagnation point at the hot spot,
similar to the case of a spherical bow shock. Close to the $x$-axis
the flow is decelerated and there is a stagnation point on the
$x$-axis ahead of the cocoon (Figure \ref{fig:velblow}, right). This
behaviour is imposed on the solution by the assumed symmetry. The flow
pattern is very similar for other values of $\beta$ and $R_b$.
\subsection{Density and pressure}
\begin{figure*}
\label{fig:dengray}
\end{figure*}
The density distribution associated with the flow is shown in Figure
\ref{fig:dengray}. For a uniform atmosphere the density of the shocked
gas decreases towards the cocoon to a value which is below that of the
external density, similar to the behaviour of the gas behind a
spherical bow shock. For $\beta =1$ the density decreases slowly from
the bow shock to the cocoon, again similar to the spherical case
(Figure \ref{fig:sphecom}), however, the density of the gas just
behind the shock now decreases monotonically with increasing $x$
because of the external density gradient. For $\beta =2$ a similar
trend with increasing $x$ is seen, together with an increase in
density from the bow shock to the cocoon.
\begin{figure*}
\label{fig:pregray}
\end{figure*}
The pressure distribution of the gas between bow shock and contact
discontinuity is shown in Figure \ref{fig:pregray}. For a uniform
external density the pressure decreases monotonically from the hot
spot towards the $x$-axis, whereas for a decreasing density profile
the pressure firstly decreases away from the hot spot then increases
again towards the $x$-axis. For $\beta =2$ the pressure close to the
$x$-axis almost reaches the pressure in front of the hot spot.
In the analysis presented in KA which led to the prediction of
self-similar expansion of bow shock and cocoon we showed that it is a
good approximation to assume a constant pressure distribution within
the cocoon due to a high sound speed in this region. Across the
contact surface pressure is of course continuous. The results shown in
Figure \ref{fig:pregray} indicate a pressure variation around the
contact surface which is a minimum at some point between the hot spot
and the $x$-axis. In fact, within the cocoon there must be a backflow
(e.g. Norman {\em et al.} 1982\nocite{nsws82}) which is initially
accelerated from the hot spot and must, given the assumed symmetry, be
decelerated towards the symmetry plane of the $x$-axis. For most
values of $\beta$ this pressure variation is small and therefore we
believe the expansion of the source will remain approximately
self-similar and the results of KA and the analysis presented here
will apply. For $\beta$ close to 2 the pressure variation becomes
significant. The solution presented here will then be valid while the
source has an approximately elliptical shape, however in this case the
source is unlikely to expand self-similarly. This is because one of
the assumptions for self-similarity is the pressure equilibrium of the
jets within the radio cocoon the pressure within which is assume to be
uniform away from the hot spots. At least two possibilities for the
evolution of such a source exist. Firstly, if the source retains
approximate axial symmetry then as the high sound speed in the cocoon
leads to pressure equalization, the excess pressure in the shocked gas
will cause the cocoon to collapse resulting in a `pinching off' of the
cocoon; the bow shock will then not follow an elliptical coordinate
surface, although subsequent expansion may still be approximately
self-similar. Alternatively, the axial symmetry of the problem may not
be retained leading to significant off-axis flow in the cocoon as is
commonly observed.
Note also that with the steeper rise of the pressure in direction of
the core for large $\beta$ the density is increasing as well in these
regions. This implies that radiative cooling due to thermal
bremsstrahlung becomes important and our assumption of adiabatic
conditions may not be valid at this point. Any cooling will decrease
the pressure and including radiative processes in the analysis should
alleviate the problem of the steep pressure gradients towards the
core. However, including these effects is beyond the scope of this
paper and in the present form the existence of the pressure gradients
described above means that the model is not fully self-consistent for
large values of $\beta$.
\begin{figure*}
\label{fig:bigpan}
\end{figure*}
A comparison of the density and pressure distributions for different
aspect ratios of the bow shock but for the same external atmosphere is
shown in Figure \ref{fig:bigpan}. The gradients in the density and the
pressure distribution are greater for higher $R_b$. The variation in
pressure is again seen to be smaller than in density. The stand-off
distance between bow shock and cocoon at the hot spot, $\Delta _h$,
decreases for increasing $R_b$, while it stays almost constant for
constant $R_b$ but increasing $\beta$ (see Table
\ref{tab:results}). For large values of $R_b$ the stand-off distance
at the $x$-axis, $\Delta _x$, decreases with increasing $\beta$.
We conclude that for $\beta$ somewhat less than 2 self-similar
expansion can occur with a flow distribution and source shape well
approximated by the analysis presented here. For $\beta$ close to 2
the source may pass through a phase in which the analysis of the
present paper applies, but this does not represent a self-similar
expansion and deviations from the assumptions of axial symmetry or
elliptical bow shocks must occur.
\subsection{Confinement of the cocoon}
In our analysis presented in KA the ratio of the pressure at the hot
spot, $p_h$, and that of the cocoon, $p_c$, determines the value of
the dimensionless constant $c_1$ which describes the expansion of the
cocoon. In the case of a cylindrical cocoon which is confined by the
ram pressure of the receding IGM a simple estimate gives
\begin{equation}
\frac{p_h}{p_c} = 4 \, R^2.
\label{ram}
\end{equation}
This estimate ignores the variation in external density along the
outside of the cocoon and the small amount of backflow. Taking $p_c$
to be the pressure at the contact discontinuity half way between the
core and the hot spot, we find that our results are well fitted by
empirical relations of the form
\begin{equation}
\frac{p_h}{p_c} = \left\{
\begin{array}{ll}
2.51 \, R^{1.95} & \mbox{; $\beta =0$}\\
1.95 \, R^{1.63} & \mbox{; $\beta =1$}\\
1.03 \, R^{1.57} & \mbox{; $\beta =2$}
\end{array} \right..
\label{ratio}
\end{equation}
\noindent The assumption of ram pressure confinement of the cocoon,
equation (\ref{ram}), therefore leads to an overestimate of the ratio
$p_h/p_c$.
Because of the self-similar expansion of the cocoons, the cocoon
volume is given as $c_3 L_c^3$, where $L_c$ is the length of the
cocoon and $c_3$ is a constant depending on the cocoon geometry. The
values of $c_3$ derived from the analysis presented here are given in
Table \ref{tab:results}. For the assumption of a cylindrical cocoon
$c_3$ equals $\pi / (4 \, R^2)$; we find a maximal deviation of only
28\% of this approximation from our results.
The profile of the density distribution in the environments of
extragalactic radio sources is well fitted by a King
(1972)\nocite{ik72} profile. For distances from the centre of the
density distribution greater than the core radius, $a_o$, the King
profile is well fitted by the power law distribution, and for
distances smaller than $a_o$ a constant density $\rho _o$ is a good
approximation. Consider a radio source which starts out in such a
uniform density environment and expands beyond the core radius. KA
argue that the ratio $p_h / p_c$ is determined by intrinsic properties
of the jet since the cocoon and the jet are assumed to be in pressure
balance. In this case equations (\ref{ratio}) imply that the cocoon of
such a source will become narrower when it emerges from the uniform
density regime into the region with a decreasing density
gradient. This result is in agreement with the observation of Black
(1992)\nocite{ab92} who notes that in his sample shorter sources tend
to be `fatter'.
\section{X-ray emission from the heated IGM\label{sec:xray}}
\begin{figure*}
\label{fig:temgray}
\end{figure*}
Assuming the shocked IGM between bow shock and contact discontinuity
is an ideal gas consisting entirely of ionised hydrogen the absolute
temperature at any given point in the flow is
\begin{equation}
T = \frac{m_p}{k} \, \dot{L}_b^2 \, \frac{P(\eta , \xi)}{R(\eta ,
\xi)},
\label{temp}
\end{equation}
\noindent where $m_p$ is the mass of a proton and $k$ is the Boltzmann
constant. Note, that the temperature of the gas in the flow does not
depend on the temperature of the environment the source is expanding
into.
Figure \ref{fig:temgray} shows a plot of the ratio $P/R$ for various
atmospheres. The differences between the temperature distribution for
different $\beta$ are relatively small. The thin shell of rarefied gas
around the cocoon in an uniform atmosphere is slightly hotter than its
surroundings and the overall temperature of the gas in the flow
decreases somewhat for increasing $\beta$. A comparison of models for
various aspect ratios (Figure \ref{fig:bigpan2}, left column) shows an
increasing prominence of the hot spot region with increasing $R_b$
which can be explained by the faster expansion speed of the bow shock
at the hot spot with respect to the expansion perpendicular to the jet
axis. Again the temperature scale is very similar in all four panels.
\begin{figure*}
\label{fig:bigpan2}
\end{figure*}
The specific emissivity of ionised hydrogen at temperature $T$ due to
thermal Bremsstrahlung is (e.g. Shu, 1991)\nocite{fs91}
\begin{equation} \epsilon _{\nu} = 7 \times 10^{-51} \frac{n_e \,
n_p}{\sqrt{T}} \, e^{-\frac{h \, \nu}{k \, T}} \ \mathrm{\frac{W}{m^3
\, Hz}},
\label{them}
\end{equation}
\noindent where $n_e$ and $n_p$ are the number densities of the
electrons and of the protons respectively. When observed with a finite
bandwidth, $\nu_1 \rightarrow \nu_2$, the integrated X-ray emissivity
is given by
\begin{eqnarray}
\lefteqn{\epsilon = 5.6\times 10^{11} \, \frac{\left( \rho _o \,
a_o^{\beta} \right) ^2}{a_1} \, L_b ^{\frac{-5\beta -2}{3}} \,
\sqrt{R^3 \, P} \, } \nonumber\\
& \times &\left[ e^{-4.0\times 10^{-7}
\, a_1^2 \, L_b^{\frac{4-2\beta}{3}} \, \nu \, \frac{R}{P}} \right]
_{\nu_2}^{\nu_1}\ \mathrm{\frac{W}{m^3}},
\label{emission}
\end{eqnarray}
\noindent with
\begin{equation}
a_1 \equiv \frac{5-\beta}{3} \, c_1^{\frac{\beta -5}{3}} \, \left( \rho
_o \, a_o^{\beta} \right) ^{\frac{1}{3}} \, Q_o^{-\frac{1}{3}},
\end{equation}
\noindent where we have used equation (\ref{linear}). To obtain the
X-ray surface brightness this expression has to be integrated along
the line of sight through the flow. The dependence of $\epsilon$ on
the advance speed of the bow shock, $\dot{L}_b$, is non-linear. To
proceed we therefore have to choose specific values for the external
density profile, the jet power, the linear size of the cocoon, $L_j$
and the observing frequency. Assuming $\rho _o = 5\times 10^{-22}$
kg/m$^3$, $a_o = 1$ kpc, typical values for the gas surrounding
isolated galaxies (Canizares {\em et al.} 1987\nocite{cft87}), $Q_o =
10^{39}$ W (Rawlings \& Saunders 1991\nocite{rs91}), $L_j=100$ kpc and
an observing band from $\nu _1 =0.1$ keV to $\nu _2 =2$ keV
(appropriate for the High Resolution Imager, {\em HRI}\/, of the {\em
ROSAT}\/ satellite) we obtain the results shown in Figure
\ref{fig:brigray}. Here we have assumed that the source is
rotationally symmetric about the $z$-axis, that the source lies
exactly in the plane of the sky and we have neglected the cosmological
redshift of the observing frequency.
\begin{figure*}
\label{fig:brigray}
\end{figure*}
The appearance of the flow in X-ray emission varies significantly with
$\beta$. For a uniform density environment the hot spot region
dominates the emission, while for higher values of $\beta$ the region
close to the $x$-axis is brightest and the emission is concentrated
towards the edge of the cocoon. Together with the observation made
earlier that the temperature variation is not very large within the
flow it is now evident that the X-ray surface brightness for the
frequency band chosen is a good tracer of the density
distribution. X-ray observations of FRII sources with high resolution
can therefore constrain the shape of the density distribution of the
atmosphere the source is expanding into. A similar result was found by
Clarke {\em et al.} (1997) from numerical simulations of a cocoon-bow
shock structure around a jet. For smaller aspect ratios the X-ray
surface brightness is more diffuse (see Figure \ref{fig:bigpan2},
right column), but the general features of the appearance of the
source remain the same. Note however the difference in the ages of the
sources of the same linear size but of different geometry and/or
environment.
Heinz {\em et al.} (1998)\nocite{hrb98} introduce observational
diagnostics for X-ray observations of extragalactic radio
sources. These are based on their assumption of spherical geometry for
the bow shock and cocoon which implies a uniform surface brightness at
all points within the flow at equal distance from the centre of the
source. From the analysis presented here it is clear that the X-ray
surface brightness of the flow between bow shock and cocoon can vary
considerably within the flow region if the source geometry is
elongated and/or the density gradient of the external medium is
steep. In these cases it is not straightforward to infer the absolute
values of source or environment parameters from X-ray observations
alone.
The total radiation due to thermal bremsstrahlung, $dE/dt$, can be
found by setting $\nu _1 =0$ and $\nu _2 \rightarrow \infty$ in
equation (\ref{emission}) and integrating the resulting emissivity
over the volume of the flow. For almost all the cases investigated
here we find the total luminosity to be significantly less than 10\%
of the jet power. We therefore conclude that our adiabatic treatment
of the flow between bow shock and cocoon is justified.
A rough estimate for the local cooling time of the shocked IGM can be
obtained by dividing the local energy density, $P(\eta,\xi)$, by
$dE/dt$. For all cases presented in this paper we find cooling times
of the order of 10\% to a few thousand times the Hubble time which is
much greater than the inferred life times of radio sources. The
presence of a radio source therefore influences the evolution of its
environment far beyond the limited life time of the source itself.
\subsection{Application of the model to Cygnus A and 3C 356}
The high resolution {\em ROSAT}\/ map of Carilli {\em et al.}
(1994)\nocite{cph94} has sufficient spatial resolution to allow a
comparison of source features observed in the radio at low frequency
with those observed in X-rays. For the density distribution of the
material surrounding Cygnus A we find $\rho _o =1.4 \times 10^{-22}$
kg/m$^3$, $a_o = 25.6$ kpc and $\beta = 1.4$ from the fitting of the
X-ray surface brightness distribution expected from a King profile
($\Omega =1$, $H_o =50$ km s$^{-1}$ Mpc$^{-1}$) to the data presented
by Carilli {\em et al.} (1994)\nocite{cph94}. Note, that this is
different from the fit obtained by these authors. Arnaud {\em et al.}
(1987)\nocite{ajfcnsm87} determine the temperature of the gas to be
$4\times 10^7$ K. Carilli {\em et al.} (1994)\nocite{cph94} also
present a 347 MHz VLA map of Cygnus A from which it is possible to
determine the linear size and the aspect ratio of the cocoon. Assuming
that Cygnus A is lying exactly in the plane of the sky we find $L_j
\approx 105$ kpc and $R \approx 2$; the difference in the lengths of
the two cocoons is roughly 15\%. The results of an integration of the
hydrodynamical equations describing the flow between bow shock and
cocoon for these parameters is given in Table \ref{tab:results}. The
aspect ratio of the bow shock of Cygnus A implied by this calculation
is 1.6.
Using the model of Kaiser, Dennett-Thorpe \& Alexander
(1997)\nocite{kda97a} for the radio emission of FRII sources we have
calculated the radio spectrum of Cygnus A for the given
parameters. The model is in good agreement with the observed radio
luminosities in the Gigahertz range presented by Baars {\em et al.}
(1977)\nocite{bgpw77} for a jet power of $Q_o=2\times 10^{39}$
W. Using the value of $p_h/p_c =4.1$ and the model of KA\nocite{ka96b}
we find an age of $2.9 \times 10^7$ years for Cygnus A. The present
hot spot advance speed implied by this is $9.7 \times 10^{-3}$ c. Both
results agree well with those of Carilli {\em et al.}
(1991)\nocite{cpdl91}.
After subtracting the expected X-ray emission due to thermal
bremsstrahlung of a King profile with the parameters quoted from the
map presented by Carilli {\em et al.} (1994)\nocite{cph94}, the
excess or deficit of the emission due to the presence of the radio
source becomes visible. The most interesting feature for this analysis
are the emission enhancements to the sides and just outside the
cocoons because they must be caused by the shocked IGM between bow
shock and cocoon. For the eastern cocoon the peak in the residual
emission lies towards the core of the radio structure and the excess
measured after the King profile has been subtracted is 0.21 counts per
second per 0.5 arcsecond pixel with the {\em HRI}\/ in the band
between 0.1 keV and 2 keV. This implies an X-ray emission of the gas
in this region corresponding to 0.27 counts per second per pixel with
the {\em HRI}\/ because the measured count rate along a line of sight
through the cocoon region, which we assume to be free of X-ray emitting
gas, is 0.06 counts per second per pixel less than that expected from
the King profile. For the western cocoon the situation is less clear
and the emission enhancement may not be significant.
The surface brightness map for the case of Cygnus A is very similar to
the one shown on the bottom right of Figure
\ref{fig:brigray}. Qualitatively the predictions of the numerical
calculation presented here agree well with the observations. The
emission enhancements are at the correct positions, close to the core
of the source just outside the cocoon. However, a closer examination
of the temperature distribution reveals some severe problems with the
application of the results found in the previous section to the case
of Cygnus A. The value of $P/R$ at the point of highest surface
brightness is roughly 0.012. Together with the derived advance speed
of the hot spot we find a temperature of $1.9\times 10^6$ K at this
point. This is well below the temperature of the King profile assumed
for the gas which surrounds the bow shock. Indeed, even the model
temperature just behind the bow shock close to the core of the source
is below the temperature of the unshocked gas. This is caused by the
assumption of strong shock conditions along the entire bow shock which
may not be applicable to the case of Cygnus A.
Using the temperature of the ambient gas the peak in the X-ray surface
brightness predicted by our model is $4\times 10^{-16}$ W m$^{-2}$ per
0.5 arcsecond pixel which corresponds to 0.012 counts per second per
pixel with the {\em HRI}\/. For the conversion we have used the
Internet version of PIMMS\footnote{PIMMS was programmed by K. Mukai at
the High Energy Astrophysics Science Archive Research Center of NASA,
available at {\em http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html}\/}
with a column density for the galactic medium of $3 \times 10^{25}$
m$^{-2}$ towards Cygnus A (Clarke {\em et al.}
1997\nocite{chc97}). This discrepancy may be explained by the
existence of cold, dense clumps of gas embedded in the otherwise
smooth distribution of gas in galaxy clusters (Fabian {\em et al.}
1994\nocite{fcem94}). The X-ray emission of a volume of gas after the
bow shock has passed through it is proportional to $\rho ^2 M
\sqrt{T}$, where $\rho$ is the pre-shock density of the gas, $M$ the
Mach number of the shock within the gas and $T$ the temperature. For
ideal gas conditions the Mach number is proportional to $v/\sqrt{T}$,
where $v$ is the velocity of the shock with respect to the gas. The
ratio of the X-ray luminosity of a cold, dense cloud and that of hot
IGM with the same volume is then given by $\rho_c^2 v_c / (\rho _x^2
v_x)$ , where subscripts $c$ and $x$ are used for the cloud material
and the hot IGM respectively. The cold clouds are small and will
therefore quickly re-establish pressure equilibrium with the shocked
hot IGM surrounding it after the bow shock has passed through it
(e.g. McKee \& Cowie 1975\nocite{mc75}). We can therefore set the
velocity of the bow shock within the cloud material to $\sim
\sqrt{\rho _x / \rho _c} v_x$. With this we find that the ratio of the
X-ray luminosity of a cold cloud and that of the hot IGM occupying the
same volume is given by $(\rho_c / \rho _x)^{3/2}$. The peak in the
X-ray emission observed in the eastern lobe of Cygnus A can thus be
explained if the gas in this region is about 5 times denser than the
otherwise smoothly distributed IGM.
The striking symmetry of the two emission peaks on either side of the
base of the eastern cocoon is difficult to explain in this
scenario. Note however, that there is another peak in the X-ray
emission situated to the south-west of the core well inside the
western cocoon. This could be another clump of cold material with
similar properties to the two discussed above, which is observed in
projection and therefore appears to lie within the cocoon.
Crawford \& Fabian (1996)\nocite{cf96} report the detection of
extended X-ray emission in the vicinity of the FRII radio galaxy 3C
356. They argue that the emission is the signature of an extended IGM
in a cluster around 3C 356 which makes this radio galaxy the most
distant object (z=1.079) in which cluster gas has been detected
directly. The resolution of the X-ray map of this object is
insufficient for any quantitative study. However, the observed
extension is aligned with the northern cocoon of 3C356 which suggests
that the X-ray emission of the IGM surrounding the northern cocoon may
be enhanced by the compression and heating of this material by the bow
shock of the radio source, similar to the case of Cygnus A.
\subsection{An empirical model for the X-ray emission}
\begin{figure*}
\label{fig:empcom}
\end{figure*}
It is straightforward to calculate the total X-ray luminosity of the
shocked IGM between the bow shock and the cocoon for a given frequency
and bandwidth, $L_X$, predicted by the model for a given set of source
parameters by integrating over the surface area of the bow
shock. However, this requires the numerical calculation to be
performed for this particular set of parameters. In the following we
develop an analytical approximation for the X-ray luminosity using
empirical relations of source parameters derived from the numerical
calculations.
The total mass of the external gas swept up by the bow shock during
the life time of the radio source, $M$, is given by
\begin{equation}
M = \int \rho _o \left( \frac{r}{a_o} \right) ^{-\beta} \, dV_b = 4 \,
\pi \, \rho _o \, a_o^{\beta} L_b ^{3-\beta} \, I,
\end{equation}
\noindent where
\begin{equation}
I \equiv \int _0 ^{\pi /2} \int _0 ^{\eta _b} \sinh \eta \sin \xi \,
\frac{\sinh ^2 \eta + \sin ^2 \xi}{\left( \cos ^2 \xi + \sinh ^2 \eta
\right)} \, d\xi \, d\eta.
\end{equation}
If we assume this mass to be uniformly distributed over the volume
occupied by the flow, the density in this region is found to be
$\overline{\rho} = M/(V_b -V_c) \equiv M / (a_2 \, L_b^3)$, where
$V_b$ and $V_c$ are the volume enclosed by the bow shock and that of
the cocoon respectively. By definition $V_b = 2 \pi L_b ^3 \left(
c_o^2 \sinh ^2 \eta _b -1/3 \tanh ^2 \eta _b \right)$ and $V_c = c_3
L_b ^3$.
We define $\overline{T}$ as the average of the temperature of the gas
in the flow and since the temperature variations within the flow
material are small, local deviations from $\overline{T}$ will be small
as well. For an ideal gas $\overline{T} = m_p/k \, \dot{L}_b^2 \,
\overline{P/R}$.
With the two assumptions of uniform density and uniform temperature in
the flow we find from equation (\ref{them})
\begin{eqnarray}
\overline{L}_X & = & 5.2 \times 10^{13} \, \overline{\rho} \,
\sqrt{\overline{T}} \, \left( V_b -V_c \right) \, \left[ e^{-4.8 \times
10^{-11} \, \nu / \overline{T}} \right] _{\nu _2} ^{\nu _1}
\nonumber\\
& = & 9.1 \times 10^{13} \, \frac{I^2}{a_2} \, \frac{\left(
\rho _o \, a_o^{\beta} \right) ^2}{a_1} \, L_b^{\frac{7-5\beta}{3}} \,
\sqrt{\overline{P/R}} \nonumber\\
& \times & \left[ e^{-4.0 \times 10^{-7} \, a_1^2 \,
L_b^{\frac{4-2\beta}{3}} \, 1/\overline{P/R}} \right] _{\nu _2} ^{\nu
_2}.
\label{emp}
\end{eqnarray}
\noindent This expression depends on $c_1$, $\overline{P/R}$ and
$I/a_2$ which are all functions of the exponent of the density profile
of the unshocked IGM, $\beta$, and the geometry of the FRII source
defined by the aspect ratios $R$ or $R_b$. Empirical expressions for
these functions can be derived from the results of the numerical
calculation. We find
\begin{eqnarray}
\overline{P/R} & = & \left\{
\begin{array}{llll}
0.17 \, R^{-1.60} & \mbox{or} & 0.12 \, R_b^{-1.74} & \mbox{; $\beta
=0$}\\
0.13 \, R^{-1.68} & \mbox{or} & 0.088 \, R_b^{-1.71} & \mbox{; $\beta
=1$}\\
0.095 \, R^{-1.67} & \mbox{or} & 0.068 \, R_b^{-1.64} & \mbox{; $\beta
=2$}
\end{array}
\right. , \nonumber\\
I/a_2 & = & \left\{
\begin{array}{llll}
0.029 \, R^{-1.89} & \mbox{or} & 0.019 \, R_b^{-2.05} & \mbox{; $\beta
=0$}\\
0.12 \, R^{-1.00} & \mbox{or} & 0.093 \, R_b^{-1.01} & \mbox{; $\beta
=1$}\\
0.75 \, R^{0.50} & \mbox{or} & 0.83 \, R_b^{0.49} & \mbox{; $\beta
=2$}
\end{array}
\right. , \nonumber\\
c_1 & = & \left\{
\begin{array}{llll}
1.22 \, R^{0.49} & \mbox{or} & 1.35 \, R_b^{0.53} & \mbox{; $\beta
=0$}\\
1.02 \, R^{0.63} & \mbox{or} & 1.18 \, R_b^{0.64} & \mbox{; $\beta
=1$}\\
0.71 \, R^{0.88} & \mbox{or} & 0.85 \, R_b^{0.86} & \mbox{; $\beta
=2$}
\end{array}
\right.
\end{eqnarray}
Figure \ref{fig:empcom} shows a comparison of equation (\ref{emp})
using the empirical relations above, with the results of the numerical
calculation for $\rho _o = 5\times 10^{-22}$ kg m$^{-3}$, $a_o = 1$
kpc, $Q_o = 10^{39}$ W, $\nu _1 =0.1$ kev and $\nu _2 = 2$ keV. For
$\beta > 0$ the empirical model agrees well with the numerical
results. The deviation for large linear sizes and $\beta =0$ is
acceptable in many cases since it is unlikely that a uniform density
region extends to very large distance from the centre of any object
hosting a radio source. Note however, that the fit of the empirical
model worsens for higher observing frequencies since the exponential
part of equation (\ref{emission}), which is not well fitted by the
empirical formula, then dominates this expression.
\section{Conclusions}
We presented a numerical integration of the hydrodynamical equations
governing the self-similar gas flow behind a strong shock in two
dimensions assuming the shock to have an ellipsoidal geometry. The
density distribution in the unshocked environment of the shock is
modeled with a power law. Applying this model to the bow shocks
surrounding the cocoons of extragalactic radio sources we found that
there should be a pressure gradient within the cocoons of sources
located in non-uniform environments which is consistent with the
existence of backflow within the cocoon. In extreme density profiles
non-axially symmetric flow or pinching-off of the cocoon is very
likely to occur. Furthermore, the expansion of the cocoon and bow
shock may not be self-similar. In these cases ($\beta \sim 2$) the
model is not self-consistent. We also find that the assumption of ram
pressure confinement of the cocoon perpendicular to the jet axis leads
to an overestimate of the ratio of the pressure in front of the hot
spot and the pressure in the cocoon.
From the properties of the gas in the flow we calculated the X-ray
surface brightness predicted by the model. Because of the small
variations of the temperature in the flow, the surface brightness was
found to be a good tracer of the gas density. The appearance of radio
sources in X-rays is found to vary significantly with the properties
of the external medium. Simple diagnostic tools have to be refined to
account for the elongation of the bow shock of most sources. The
cooling times of the IGM shocked by the bow shock are much longer than
the expected life time of the radio sources causing the bow shock. The
evolution of any concentration of matter in the universe will
therefore be significantly influenced by the presence of a radio
source in its centre. This is the case even if the activity time scale
of the source is much shorter than the evolution time scale of its
surroundings.
We compared the predictions of our model with the X-ray map of Cygnus
A of Carilli {\em et al.} (1994)\nocite{cph94}. Although our model
predicts the strongest X-ray emission at the position where it is
observed, on the outside of the cocoon closest to the core of the
source, the model can not reproduce the observed luminosity. The
emission may be caused by the presence of overdense clumps of gas
embedded in the otherwise smooth IGM.
The enhanced X-ray emission of the shocked IGM behind the bow shock of
a strong radio source may also explain the observed extended X-ray
emission around 3C 356.
Based on the numerical calculation we also presented an empirical
model for the total X-ray emission expected from the shocked IGM
surrounding the cocoons of powerful FRII sources.
Despite the inconsistencies within the model that arise for large
values of $\beta$ we believe that this analysis is an important first
step in determining the influence of powerful radio sources on the
properties and evolution of their gaseous environments.
\section*{Acknowledgments}
The authors thank S. Rawlings for drawing their attention to the X-ray
map of 3C 356 and the anonymous referee for her/his very helpful
comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,418 |
university of north carolina at greensboro (92)
university of north carolina at greensboro. school of music, theatre and dance (47)
college student newspapers and periodicals -- north carolina -- greensboro (16)
student activities -- north carolina -- history (11)
student publications -- north carolina -- greensboro (11)
university of north carolina at greensboro -- newspapers (11)
uncg (9)
arts -- north carolina -- greensboro -- periodicals (5)
creative writing (higher education) -- north carolina -- greensboro -- periodicals (5)
ua9.2 school of music performances -- programs and recordings, 1917-2007 (47)
oh003 uncg centennial oral history project (26)
the carolinian (11)
oh002 uncg institutional memory collection (8)
charles m. adams american trade bindings collection (4)
oh001 greensboro voices collection (4)
ua100 artifact collection (4)
ua107 textile collection (4)
alumnae news / alumni news / uncg magazine (3)
government documents collection (uncg university libraries) (3)
coy, kacie (11)
atkins, gertrude walton (2)
lancaster, jim m. (2)
uncg department of interior architecture (2)
adams, charles m. (1)
alfred m. burton (1)
allen, james h. (1)
american meat institute. (1)
american meat institute. dept. of public information (1)
: Greek
The Carolinian [April 17, 2013]
The April 17, 2013, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
Oral history interview with Charles M. and Ellen Adams, 1990 [text/print transcript]
OH003 UNCG Centennial Oral History Project
Charles M. Adams (1907-1990) served as director of the library from 1945 to 1969 at the Woman's College of the University of North Carolina now The University of North Carolina at Greensboro. Adams talks about growing up in North Dakota and...
Oral History interview with Karen McKay, 2015
WV0592 Karen McKay Oral History Interview
Primarily documents the life of Karen McKay and her service with the United States Women's Army Corps (WAC) and Army Reserve.
McKay recalls the dismay she felt about anti-Vietnam war demonstrators and draft dodgers, and how this encouraged...
The Carolinian [August 29, 1985]
The August 29, 1985, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
The Carolinian [September 25, 2012]
The September 25, 2012, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
Oral history interview with Louise Dannebaum Falk, 1990 [text/print transcript]
Louise Dannenbaum Falk (1908-1997) graduated from the North Carolina College for Women in 1929. The name of the institution was changed to the Woman's College of the University of North Carolina, and now is called The University of North Carolina...
[DAR scrapbook, 1944]
DAR Scrapbooks Collection
This scrapbook is a month by month collection of newspaper articles from 1944 collected by the Greensboro Chapter of the Daughters of the American Revolution. This volume thoroughly recognizes the achievements and tragedies of men and women in the...
The Carolinian [February 15, 1983]
The February 15, 1983, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
The Carolinian [January 30, 1986]
The January 30, 1986, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
The Carolinian [December 6, 1984]
The December 6, 1984, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro.
The Carolinian [November 6, 2007]
The November 6, 2007, issue of The Carolinian, the student newspaper of The University of North Carolina at Greensboro. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,763 |
# ALSO BY BARBARA GOLDSMITH
_Johnson v. Johnson_
_Little Gloria... Happy at Last_
_The Straw Man_
_Victoria Woodhull, 1868. At thirty, she was sent by her spirit guide on a great mission_.
THIS IS A BORZOI BOOK
PUBLISHED BY ALFRED A. KNOPF, INC.
Copyright © 1998 by Barbara Goldsmith
All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Alfred A. Knopf, Inc., New York, and simultaneously in Canada by Random House of Canada Limited, Toronto. Distributed by Random House, Inc., New York.
www.randomhouse.com
Library of Congress Cataloging-in-Publication Data Goldsmith, Barbara.
Other powers: the age of suffrage, spiritualism, and the scandalous Victoria Woodhull/by Barbara Goldsmith.—1st ed.
p. cm.
Includes bibliographical references.
eISBN: 978-0-307-80035-0
1. Woodhull, Victoria C. (Victoria Claflin), 1838–1927.
2. Feminists—United States—Biography.
3. Women—Suffrage—United States—History.
4. Suffragists—United States—History. I. Title.
HQ1413.W66G65 1998
305.42′092—dc21
[B] 97-49464
v3.1
_To my dearest friend_
To preach the doctrine you must live the life.
—VICTORIA WOODHULL
Do as I do. Consult the Spirits.
—COMMODORE CORNELIUS VANDERBILT
Orthodoxy is _my_ doxy and heterodoxy is _your_ doxy....
—THE REVEREND HENRY WARD BEECHER
# _Contents_
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
_Introduction_
PROLOGUE _A Page of History_
ONE _Born Again in the Lamb's Blood_
TWO _A Child Without a Childhood_
THREE _The Spiritual Telegraph_
FOUR _My Long-Accumulating Discontent_
FIVE _You Ugly Creature_
SIX _True Wife_
SEVEN _Willfully Did Kill_
EIGHT _God Bless This Trinity_
NINE _An Accident of Fate_
TEN _Draw Its Fangs_
ELEVEN _A Dangerous Man_
TWELVE _Written in Fire_
THIRTEEN _To Equal Account_
FOURTEEN _Consult the Spirits_
FIFTEEN _We Are Ready. We Are Prepared_.
SIXTEEN _Soup for Three_
SEVENTEEN _A Hard Place_
EIGHTEEN _The Evangel_
NINETEEN _Your Child Is Not My Child_
TWENTY _The Yawning Edge of Hell_
TWENTY-ONE _The Woodhull Memorial_
TWENTY-TWO _Silence, Time, and Patience_
TWENTY-THREE _The Worst Gang_
TWENTY-FOUR _This Girl Is a Tramp_
TWENTY-FIVE _Yes! I Am a Free Lover!_
TWENTY-SIX _A Heavy Load_
TWENTY-SEVEN _Mrs. Satan_
TWENTY-EIGHT _Burst Like a Bombshell_
TWENTY-NINE _I Can Endure No Longer_
THIRTY _What Have We Done Now?_
THIRTY-ONE _A Monstrous Conspiracy_
THIRTY-TWO _Human Hyenas_
THIRTY-THREE _Daniel in the Lion's Den_
THIRTY-FOUR _A Meteor's Dash_
THIRTY-FIVE _The Last Enemy_
_Abbreviations_
_Notes_
_Bibliography_
_Acknowledgments_
_Illustration Credits_
# _Introduction_
DO AS I DO. Consult the Spirits!" Commodore Vanderbilt told a _Tribune_ reporter in 1870, when asked how he'd made his millions. Then he added, concerning the stock of the Central Pacific Railroad, "It's bound to go up.... Mrs. Woodhull said so in a trance."
Two decades ago I came across this exchange in connection with a book I was writing on the Vanderbilt family. I had no idea what it meant, though it piqued my curiosity. Soon afterward I discovered that the Commodore was a dedicated Spiritualist, one of an estimated to million or so in the United States in the post–Civil War decade. Also, I learned that "Mrs. Woodhull," another dedicated Spiritualist, was the notorious Victoria Woodhull, whom a tabloid referred to in 1872 as "The Prostitute Who Ran for President."
These fragmentary details led me to look further into the connection between Vanderbilt and Woodhull and especially into Spiritualism. From the study of Spiritualism, I was inevitably led to the woman's rights movement, both before and after the Civil War, for one of the many ways women managed to relieve the burdens imposed upon their gender was by seeking empowerment through the spirits. Not all members of the woman's rights movement were Spiritualists, but woman's rights were inseparable from Spiritualism.
What interested me most was how the social and sexual mores, the pressures and events of that time, affected these people, particularly women. I have relied mainly on such primary sources as letters, diaries, conversations recorded in shorthand, the public and private writings of the principals concerned, trial transcripts, and, of course, the newspapers of the day. In 1870, there were thirty-five daily newspapers in New York and another eighteen in Brooklyn. The papers of those icons of the woman's movement, Elizabeth Cady Stanton and Susan B. Anthony, are contained in 110 reels of microfilm that I have been fortunate enough to live with for the past few years. Much of the research for this book has been culled from the thousands upon thousands of pages of brittle, yellowing trial transcripts: the Great Vanderbilt Will Contest; the case brought by Roxanna Hummel Claflin, Victoria Woodhull's vengeful mother, against Victoria's second husband, Colonel James Blood; the murder trials of Daniel McFarland, Hester Vaughn, Edward (Ned) Stiles Stokes; and, especially, the trial involving the distinguished journalist Theodore Tilton, who sued the leading churchman of his day, Henry Ward Beecher, for having seduced his wife.
_Victoria presents_ The Woodhull Memorial _to Congress, January 11, 1871_.
The Beecher-Tilton Scandal Trial, as it came to be known, captured the imagination of the American public much as the O. J. Simpson trial has in our time. For two years, it dominated headlines and was exhaustively analyzed in private homes, public auditoriums, and pulpits throughout the nation, as well as in several books. Although the trial seemingly revolved around an alleged seduction, it also raised in the most vivid way the issue of sexual relations at that time and the role of church and state in defining and regulating these relations.
As I was led deeper and deeper into the world of Victorian America, I was surprised to find that many of the books I read, especially the biographies, had little to say about the larger historical context surrounding their subjects. To see how the people and events associated with Victoria Woodhull interacted, I created, with the help of a computer, a chronology that eventually extended to four hundred pages. From this I was able to see what the various characters, whose lives intersected with Victoria's, were doing and often thinking at specific times. People who seemed in the books I'd read to dwell in separate worlds could now be seen as parts of a larger drama. This inevitably led to a better understanding of the texture of the age and the lives of women than I had previously encountered.
Major events in this book have, of course, been previously recorded. The Civil War draft riots, the collapse of the gold market on Black Friday, September 24, 1869, the impeachment proceedings against President Andrew Johnson, have been treated in numerous books. My contribution, I believe, was to look at these and similar events through the lives of some of the people involved, a process that was both enlightening and arduous because the relevant materials were not easily found.
Some of the documents that proved most valuable to me have only recently become of interest to historians. Lists of household furnishings, descriptions of food, the wardrobe of a "woman of society," helped me grasp the quality of these lives. Such details as Catharine Beecher's numbingly precise instructions on how to press a shirt, involving the use of some twenty items, brought the ritualistic drudgery of domestic existence into sharp relief.
In many of the books I read, particularly those of the period, important material that revealed the actual character of these people had been expurgated. For example, the letters of Susan B. Anthony were altered by her official biographer, Ida Husted Harper, to make her seem decorous and pure, rather than the direct, insightful person she undoubtedly was. Pages from Anthony's diary relating to the sexual excesses and blackmail schemes of Victoria Woodhull and her sister, Tennessee Claflin, were ripped out and destroyed by Harper, as were the pages relating to the crucial controversy of this book. Harper boasted that she kept a bonfire going for a week with just such unacceptable material. Fortunately, from the letters of Anthony's colleagues and friends that refer to these missing pages, I was able to reconstruct much of what Harper suppressed. Again and again, in the letters I quote, one finds the directive "Burn this!" Fortunately, we are the beneficiaries of the repeated failure to obey this command.
Perhaps the most startling discovery was the extent to which Spiritualism and the inception of woman's rights were intertwined. At a time when women had no power to achieve equal rights, they relied on the "other powers" provided by Spiritualism to sustain their efforts. Through the mouths of trance speakers came words of wisdom from long-dead seers, and from the spirits came the courage to go forward. The rise of Spiritualism coincided with Samuel B. Morse's invention of the telegraph, an invisible means of communication, and expanded at a time when a devastating war had imposed the unbearable loss of husbands, sons, and lovers.
The Spiritualist influence on Victoria Woodhull has in most accounts of her life been largely dismissed or ignored. But as Victoria herself said, to tell her life without her spirits would be "as if you were writing _Hamlet_ and decided to leave out his fathers ghost." I can unequivocally say that Victoria Woodhull's belief in the spirits and their great powers was genuine. This belief shaped her life and informed her existence. The title bestowed on her in her heyday, the "Joan of Arc of the Woman's Movement," was apt, for there is no doubt that she too heard the voices that defined her mission.
My research on Victoria Woodhull was complicated by the fact that after she was forced to flee to England in 1877, she forged many documents in order to obscure her past. In some of these bogus documents she changed her own name as well as the names of her relatives and assigned them reputable occupations and royal ancestry. More reliable are the materials available in this country. In newspaper, family, and court archives, I have been able to obtain such rare documents as the original indictment brought against Tennessee Claflin for manslaughter and the papers of Victoria's second husband, Colonel James Blood, relating his service in the Sixth Missouri Regiment during the Civil War.
It has been deeply rewarding to come to know Victoria Woodhull, and I am grateful for the opportunity to share this knowledge with my readers. I have endeavored to supply some of the historical context in which Victoria and the women of her era struggled, for without this setting, her career, indeed her entire life, would seem merely eccentric or even aberrant. The religious ferment that accompanied the so-called second Great Awakening and the movements it spawned—temperance, abolition, and the struggle for woman's equality with man—is at the heart of this story.
Victoria Woodhull achieved her brief renown in a world where the most famous preacher in America, Henry Ward Beecher, was said to preach to "at least twenty of his mistresses every Sunday" but a woman wore a "scarlet letter" for the act of adultery. This was a world where prostitution was a widely accepted social custom, wives were considered property, unwanted infants were wrapped in rags and deposited on doorsteps or in the river, and working women made barely enough to keep themselves alive. In this world Hester Vaughn lay in a pool of blood next to the frozen corpse of her newborn child, and Abby Sage McFarland saw her former husband murder her fiancé and be acquitted by an all-male jury.
In this society, Victoria Woodhull pictured herself as "the evangel," but she was, in fact, a woman before her time. Her views on marriage and divorce and the sexual equality she called "free love" are relevant today. Her spirit is with us still.
# PROLOGUE
A PAGE OF HISTORY
MAY 5, 1892: As the delegates to the National American Woman Suffrage Association convention filed into the Auditorium Theatre in Chicago, they found on the seats a leather-bound pamphlet titled "A Page of History." On the first page was an announcement that Victoria Woodhull would run for president of the United States against Grover Cleveland and Benjamin Harrison. The rest was a compendium of extravagant praise from such leaders of the woman's rights movement as Susan B. Anthony, who called Victoria a "bright, glorious, young and strong spirit"; Elizabeth Cady Stanton, who predicted, "In the annals of emancipation the name [Victoria Woodhull] will have its own high place as a deliverer"; and Paulina Wright Davis, who said, "I believe you were raised up of God to do wonderful work and I believe you will unmask the hypocrisy of a class that none others dare touch." Isabella Beecher Hooker lent the religiosity for which her family was famous by stating that Victoria was "Heaven sent for the rescue of woman from the pit of subjection." What the pamphlet did not say was that this praise had been written a quarter of a century earlier.
The following day, about a dozen reporters waited impatiently in Parlor K of the Wellington Hotel for Victoria Woodhull's press conference to begin. It had been scheduled for ten, but by eleven she still had not appeared. In order to placate the newsmen who were threatening to leave, her husband, John Biddulph Martin, the wealthy head of a family-owned bank in London, ordered the waiters to serve whiskey and ham sandwiches. Thus they were standing about eating and drinking when Victoria swept into the room holding the arm of her sister, Tennessee, now Lady Cook, Viscountess of Montserrat. They were dressed identically, in blue velvet gowns trimmed with Venetian lace. Victoria wore her trademark white rose at her throat, just as she had done twenty years earlier, when she had first run for the United States presidency. The sisters made a stunning pair: At fifty-four Victoria retained the fine, chiseled features and ramrod posture that reminded some of her admirers of Queen Victoria. At forty-six, Tennessee was still an ivory-skinned beauty with red hair and a delicate cleft chin.
Victoria Woodhull Martin greeted each of the reporters and then announced that her nomination was sponsored by the NAWSA. She said she had composed a letter of acceptance that would be distributed at the end of the conference and added, "If my political campaign for the Presidency is not successful, in fact, it will be educational for women."
Actually, both Martin and his wife knew that this was a costly but hopeless campaign, but they didn't care. They were interested not in American politics but in British society. John Martin was convinced, even if his British friends were skeptical, that his wife had once been the respected leader of the woman's rights movement in America. If she could recapture that position now and prove it, with scrapbooks full of praise from the American press, then Martin might be able to convince his social peers, who had long shunned and reviled his American wife, that the scandalous rumors surrounding her were merely the spiteful gossip of misinformed bigots.
Though few Americans any longer remembered who Victoria Woodhull was, her old antagonist Lucy Stone, the Boston woman's rights leader who was in Chicago for the convention, could never forget. She felt that two decades earlier, Victoria had almost wrecked the movement. The following day, when Lucy read of Victoria's press conference, she called one of her own and told the reporters, "The statement that Mrs. Biddulph Martin is our candidate for president is wholly without foundation. We have no presidential candidate, and we do not even know the persons who are said to have nominated her."
After Stone's statement, the younger members of the NAWSA ridiculed Victoria's claim. Many of them denounced her as a self-aggrandizing charlatan who had long since been abandoned by the movement. These women assumed the endorsements were bogus, and one representative, Mary Frost Ormsby, sent her copy to Susan B. Anthony and wrote, "Knowing your love of _truth_ and _justice_... I take the liberty of sending this out to you.... I was deceived by Mrs. Martin into the belief she was a philanthropist and an honest woman. My eyes are now opened, I know to the contrary." There is no record that the seventy-two-year-old Anthony replied to Mrs. Ormsby, but she carefully pasted Ormsby's letter in her scrapbook. It would have been difficult for Anthony to explain to someone of Mrs. Ormsby's generation that the praise of Victoria Woodhull, attributed to her and the others, was genuine.
In his effort to influence the NAWSA delegates, John Martin suggested contacting Isabella Beecher Hooker, the eminent Spiritualist, woman's rights advocate, and member of the prestigious Beecher family, who had written Victoria that she would be in Chicago for the convention. To her husband's surprise, Victoria said she wanted nothing to do with Mrs. Hooker. Martin protested. Hadn't she told him that Isabella was her closest American friend, the one who had supported her in what she referred to as her "Gethsemane"? Still Victoria was adamant.
Two days later, hoping to find a strong ally, John Martin, without his wife's knowledge, arranged to meet Isabella Beecher Hooker in a parlor off the main lobby of the Sherman House. He was accompanied by a male secretary who recorded the ensuing conversation in shorthand, and later typed it. At seventy, Isabella was frail, suffered from arthritis and had trouble hearing, but she was still prominent in the movement. After a brief exchange of pleasantries, Martin said, "Mrs. Hooker, I am glad Mrs. Martin is not here. Some things are easier to say outside of her presence. My wife does not seem to recollect why it is that her work of twenty years ago on the suffrage movement has been forgotten while her attack on your brother Henry Ward Beecher is not forgotten."
Of course Isabella knew the answer to John Martin's question, but she would not tell him about his wife's catastrophic involvement with her brother, the great Brooklyn preacher, which had led to Victoria's imprisonment and exile. Instead she changed the subject and recalled a happier time a quarter of a century earlier when Isabella first saw Victoria Woodhull standing against a corridor wall of the Capitol building in Washington, nervously waiting to address a joint session of Congress, an honor that had been awarded to no other woman.
"Her dress was peculiar," commented Isabella.
"How so?" asked Martin.
"She wore a felt hat such as men wear. When she rose to speak I thought she would have fainted. Her face flushed in patches. I was fascinated by Mrs. Woodhull."
How could John Martin understand that fascination and all that had happened as a result? And who would have thought that what started with Victoria's assertion of equal rights for women would eventually cause them both to become reviled outcasts? "You know Mrs. Henry Ward Beecher is writing recollections of my brother in a Philadelphia paper in which I am treated unkindly," she said.
"I know this is a painful subject," replied Martin, who seemed concerned only for Victoria. "But you know that my wife was attacked in Henry Ward Beecher's biography. I tell you plainly we will have it all out, come what may to any of his family."
Turning his threat aside, Isabella continued to recall the tangled past. "My brother Henry called on me to denounce Victoria. I refused. I said that I knew nothing against her and all that I knew of my own knowledge was in her favor. Then my friends began to fall away. I was estranged from my family. My daughter, now in Heaven, told me I had no right to imperil my husband's life for the sake of Mrs. Woodhull. Because I would not denounce her they tried to make me out insane. I was left alone. Quite alone. No one can tell what I suffered." The secretary's notes of the conversation are so precise that one almost hears her despairing voice.
John Martin replied, "You forget, Mrs. Hooker, what my wife suffered."
Isabella chose not to answer this accusation, but said wearily, "All this is an old story. Mrs. Martin forgets how long she has been away from America. If she wants supporters for her campaign she should spend ten thousand dollars and put out a special edition of _The Arrow_ , the great organ of the Spiritualists. They will support her."
Abandoning his efforts to win Isabella over, Martin replied with evident annoyance, "I will tell her what you say, but after this interview it is hardly necessary that you should meet with her." With this he dismissed Isabella, the loyal friend who had once stood with Victoria on what appeared to be the brink of a new world.
Despite his failure to recruit Mrs. Hooker, John Martin persisted. He arranged to meet with Joseph R. Dunlop, the publisher of the _Chicago Mail_ , and asked him to run an article in his newspaper about Victoria Woodhull and what she had done for women. Dunlop obliged. On May 8, 1892, an article appeared, "Tennie and Her Vickie," which began mildly enough, "The Woodhull and Claflin campaign for the presidency is being launched and delegates have arrived in Chicago to participate in the convention in which the gentle Victoria is to be nominated." But the _Mail_ added that when Victoria and her sister, Tennessee, lived in Chicago they operated "a _house_ in a grand and peculiar style" and practiced spurious fortune-telling and healing techniques. Their mother, Roxanna (Roxy) Claflin, was described as a bogus fortune-teller. Their father, Reuben (Buck) Claflin, was a charlatan and a thief. The article concluded, "And all America knows that Victoria Woodhull was solely responsible for the greatest scandal of the century."
Victoria told her husband that these "lies" were ruining her health and threatening her life. Would she never be free of her malicious enemies? The day after the _Mail_ printed its story, John Martin appeared, over his wife's objections, at the Cook County Circuit Court and lodged a suit against Dunlop and the _Mail_ for $100,000 in damages.
Undaunted, on May 10, Dunlop ran another article stating that Tennessee Celeste Claflin, though now a titled English lady, was still under indictment in Ottawa, Illinois, on a charge of manslaughter dating back to 1863. "It was so long ago. It was another world," Tennessee told John Martin by way of explanation. Evidently, all of this proved too much for the conservative banker. He dropped the case against Dunlop and the _Mail_ and took Victoria and Tennessee back to New York, where they boarded the _Persian Monarch_ to Southampton.
But who was Victoria Woodhull? John Martin knew her in only one of her many roles, as the adored wife who shared his life. But she was also the Spiritualist, the "high priestess" of free love, the crusading editor, the San Francisco actress and part-time prostitute, the founder of the first stock brokerage firm for women, the disciple of Karl Marx, the blackmailer, the presidential candidate, the sinner, the saint.
She had been all of these and more, for in her many aspects she combined in abundance many of the influences that shaped the women of her world. Her compassion for their suffering was the most persistent and most genuine of her feelings, and Victoria never stopped believing that the spirits had brought her into the world to lead a "social revolution." She said that from her birth, and even before, she had been marked for this fate. As a Spiritualist and a clairvoyant, she claimed to remember every event in her life, even to the moment of her conception, even to her birth when her mother clasped her to her breast and "the look of pain and anguish... was burnt into my plastic brain as she suckled me."
# CHAPTER ONE
BORN AGAIN IN THE LAMB'S BLOOD
ROXY CLAFLIN stood on the frozen, rutted road, shivering in her threadbare calico dress as the late December winds lashed the frosted fields of Homer, Ohio, and resolutely awaited the messenger of God. When at last the black-cloaked rider, whom she had seen in one of her visions, thundered past on his mud-spattered black horse, she felt the beginning of exaltation. She would be reborn as fresh to creation as Eve.
In 1837, in the farm towns of Ohio, most religious revivals occurred in the long, dormant period from winter to spring. As if summoned by an unseen force, the isolated farmers, storekeepers, and laborers assembled in remote barns and churches to yield themselves up in pain and ecstasy, to obliterate themselves, to emerge, born again, from a crucible of fire. This young nation was ruled externally by its government but internally by rigorous Calvinist doctrine. Those who denied God's power were no more free of His iron hand than those who affirmed it. In these years heaven and hell were awesomely present, and revivals swept the nation. But the old Puritan faith that prescribed harsh laws of predestination and infant damnation had begun to yield to the less stringent "new Calvinism" of the kind preached by the great Lyman Beecher, who proclaimed that, through conversion, one could forge one's own destiny. If men and women chose a life of virtue, both they and their offspring might be spared damnation. Beecher's new, gentler Calvinism was intended to save souls for his church, for he knew that the world was changing, and no longer were people content to be consigned to an immutable place in society or in the hereafter over which they had no control. In nearby Cincinnati, Beecher preached this doctrine at revivals over the objections of the Presbyterian Church Synod.
In October 1835, the synod charged Lyman Beecher with heresy. At his trial, Beecher's twenty-two-year-old son, Henry Ward, stood by his side, handing him the books and other documents to which he referred in his defense. According to contemporary accounts, Lyman Beecher's defense was so equivocal and protracted that the weary church synod disbanded before it got around to asking his opinion on predestination. Later, when he too began to preach, Henry Ward Beecher would learn to follow his father's example of shading the truth to save his popularity, if not his soul.
Henry Ward Beecher was the fourth and dullest of Lyman's six sons. His mother, the former Roxanna Foote, was a brilliant, lighthearted woman who died when Henry was three. In a sermon, he would recall that as a child he had not even one toy. In school, Henry's slow wits and his stammer brought punishment. Often he was confined to the girls' corner, where he sat on a stool for hours at a time wearing a peaked paper dunce cap. Yet he laughed easily and knew how to flatter and charm.
At fourteen, Henry was sent to the Mount Pleasant Classical Institution in Amherst, Massachusetts, where he met Constantine Fondolaik, whose parents had been killed by the Turks in the massacre of 1822. Henry saw Constantine as "a young Greek God" and wrote, "We are connected by a love which _cannot_ be broken." At nineteen, both young men were sent to Amherst College, and Henry declared, "We will love and watch over one another, seeking by all means in our power to aid and make each other happy." Throughout his life Henry Ward was able to feel such love for both the men and women who attracted him. As for Constantine, he died fifteen years later of cholera. Henry named his third son in memory of his dear companion.
In May of his freshman year at Amherst, Henry met Eunice Bullard, the daughter of a stern doctor who lived near Worcester, Massachusetts. Dr. Bullard discouraged his daughter's high spirits and gave her a strict Puritan upbringing. Once, when she appeared at the dining room table in a low-cut silk dress, her enraged father remarked that she must be cold, then picked up a tureen of soup and flung it at her. After a two-year sporadic courtship, Henry bought Eunice an engagement ring for which he paid 85 cents. That done, for the next three years he saw Eunice only twice. His constant companion, however, was Constantine, and on school vacations they traveled around the country demonstrating phrenology, the new so-called science that showed how the size and shape of the skull defined character and intelligence. At last, after a six-year, on-and-off-again courtship, Henry and Eunice were married.
Twenty-six-year-old Henry Ward Beecher secured his first pastorate as minister of the Second Presbyterian Church in Indianapolis in May 1839. It was a poor congregation, and the parish house was only ten feet wide. Eunice said the bedroom was so small that she was "obliged to make the bed on one side first, then go out on the veranda, raise a window, reach in and make the bed on the other side." They had arrived in the midst of a malaria epidemic, which Henry knew about but had not disclosed to his pregnant wife for fear she would not agree to the move. Soon Eunice came down with the disease and gave birth to a stillborn child. Her next child died at fifteen months.
_Eunice Bullard, 1837. Henry bought her an engagement ring for 85 cents_.
_Henry Ward Beecher, the fiery young preacher at twenty-six_
In the five years that followed this death, Eunice aged prematurely from the effects of her repeated pregnancies but produced three living children. By then her husband had become a fiery revivalist preacher and had discovered that he could reach souls through raw emotion in a way that he never could do through intellect alone. He conducted revivals throughout Ohio and Indiana. In the woods near Terre Haute he whipped his audience into such a frenzy that he was able to bring a thousand men, women, and children into the fold in a single day. This burgeoning new country provided fertile ground for the second wave of what was known as the Great Awakening. So fiery was the spirit of revival that upstate New York and New England, where it was strongest, came to be called the Burned-Over District.
Beecher preached to a country on the brink of a dramatic change, both economic and ideological. John Deere had just invented the steel plow. The Pitts brothers had patented the first steam-powered threshing machine. The first transatlantic crossing by steamship was about to take place. Yet the "peculiar institution" of slavery still fueled the economy of the South, and even in the North abolitionism was not yet popular or safe. In 1837, Elijah Lovejoy, an Illinois editor who attacked slavery, was met by an enraged mob who smashed his presses. As he tried to prevent them from setting fire to his warehouse, Lovejoy was shot dead. Wendell Phillips, the twenty-six-year-old scion of a wealthy and prestigious Boston family, had been drifting through life as a dilettante lawyer. He was so affected by Lovejoy's death that he decided to dedicate himself to continuing this man's mission to free the slaves.
In Abolition Hall on Sixth Street in Philadelphia, Angelina Grimké spoke of the senseless death of Lovejoy and urged the immediate emancipation of the slaves. She condemned "a respectable woman of Charleston" who, during recent revivals, held thrice-daily devotions in the same room that only hours before had been filled with the screams of her slaves as she lashed them with cowhide. Grimké stood her ground while the audience smashed every window in the hall. The next day the mob returned to the hall and burned it down. They then proceeded to Thirteenth Street and set fire to African Hall, a Negro orphanage.
_Henry and Eunice with two sons. Eunice aged prematurely, as she had nine children and five miscarriages_. (Illustration Credit 1.1)
This was an age in which men were free to treat women with the same detached cruelty as they did their slaves. For women, regardless of class, marriage meant the surrender of every right to property and person. A woman's wages were given directly to her husband. If she sought a divorce her husband was legally entitled to sole custody of the children. Women had no legal rights to the disposition of their personal possessions upon death. Women could not testify in court or serve on a jury and were barred from universities, law schools, and medical schools. Women could not vote.
No laws protected a woman from physical abuse at the hands of her husband or father unless such abuse resulted in death, though a few states stipulated the size of the instruments that might be used to inflict punishment. One judge upheld the right of a husband to beat his wife, writing that to do otherwise would "disrupt domestic harmony." A wife's body was to be used at her husband's will; she could not deny him sexual access. If she ran away, the law of the nation supported her recapture and return. The United States followed Sir William Blackstone, the authority on English common law, who held that "the husband and wife are one, and that one is the husband."
Women's lives were an endless round of pregnancy, childbearing, and domestic duties. They worked hard and died young. In the East in the year 1837, women worked in more than one hundred low-paying industrial occupations. The recently invented spinning jenny and the power loom provided employment in factories where women worked for fifteen or sixteen hours a day, for which the wage was $3 a week. In the West, in addition to their domestic duties, women often worked beside their men at hard manual labor. The other occupations that yielded money to women were severely limited: teaching, midwifery, prostitution, and, for the select few, writing.
The church supported this repression. The doctrine of John Calvin preached that Eve had caused the expulsion from Eden, that women brought sin into the world and thus condemned all "mankind" thereafter to toil and suffering. The marriage ceremony, a sacrament of the church, specified the word _obey_. From the pulpit, male ministers laid down the laws of personal conduct for women and declared themselves the sole intermediary between God and human beings.
But by the late 1830s the old Calvinist belief that one's fate was predestined and the doctrine of infant damnation were becoming increasingly intolerable, particularly to women of all classes. At a time when one in every two babies died, women were sorely in need of solace, the promise of salvation, and a better world beyond. The loosening of the old Calvinist grip on the American religious psyche was typified by Henry Ward Beecher, who took the more permissive doctrine of his father, Lyman, to an extreme. Henry Ward's revivalist preaching was a cauldron of emotionalism compared to the steely Puritanism it was displacing. To Victoria Woodhull's mother—the ignorant, deprived Roxy Claflin—revival offered up the hope that if only she could yield her soul up to God, she would, by her own will, be saved.
The life of Roxy Claflin, born Roxanna Hummel, was crude and harsh. Illegitimate, poor, uneducated, brutalized, she enjoyed the unintended benefit of being free of the constraints of respectable society, free to find her own lonely, superstitious way. Years later, the proper ladies of the East and the titled ladies of Europe would inquire of Roxy Claflin's daughters, Victoria and Tennessee, about their antecedents, and over the years the family history grew more illustrious: Roxanna was the daughter of a wealthy Dutchman. Roxanna was a descendant of Pocahontas. In Roxanna's veins flowed "the royal blood of Germany." Eventually, Victoria Woodhull proclaimed that the family was descended from Alexander Hamilton and George Washington.
What was the truth? A _Hartford Courant_ reporter once asked Roxy where she came from, and she replied, "Nowhere." She was squat and fat, with stubby legs, and spoke with a thick German accent. She could neither read nor write. A common rumor was that Roxy was the illegitimate daughter of Captain Jake Hummel, a German immigrant who owned a tavern, The Rising Sun, in Hummel's Wharf, Pennsylvania. But it was also said that Captain Jake had simply taken the name Hummel from the town itself, which had been named for much older settlers. Another tale was that Roxy was the illegitimate daughter of "a real Hummel" mother and that her father was Simon Snyder, a three-term governor of Pennsylvania, who had farmed her out to the local tavern keeper.
It is known that at thirteen Roxy Hummel was working as a maid in Selinsgrove, Pennsylvania, in the home of Governor Simon Snyder's son John, a rich rake with a love of horseflesh and women. Once, John had served as best man at a friend's wedding, and when the ceremony was over he plunked the new bride in his carriage and sped off with her to Utica, New York. Several months later he brought her back, but when her husband refused to accept his belated and pregnant bride, John installed her in his own household, where she lived with her illegitimate daughter, whom John named Utica. John Snyder remained a bachelor all his life. Their status brought no stigma in this free-wheeling town with twenty taverns to every church, where stolid middle-class Methodists rubbed elbows with roisterers and gamblers.
On a June morning in 1824, Reuben Buckman "Buck" Claflin galloped into Selinsgrove with a string of jet-black horses that, after several scrubbings, turned out not to be black at all. No one doubted that Buck was a horse thief who had hastily disguised his booty, but this only enhanced his reputation. Buck was callous, cold, cruel, and smart, all traits that John Snyder found handy. He promptly hired this horse thief to run his private racing stable. Together, they gambled and drank and visited brothels. The pounding of hooves at dawn often marked the reckless riders' passage home.
Buck came from East Troy, Pennsylvania, where the fractious Claflin brood, who were famous for their strength and cunning, had built a timber cabin and carved out a crude farm in the dense pine wilderness. This was cruel country where hard work yielded a meager existence. Buck's brother Carrington could shoulder a 283-pound barrel of salt from the Eagle Hotel to the Sugar Creek Bridge, a quarter mile away, and back again. But the brilliant, erratic Buck had little use for manual labor. He taught himself to read and write and had a natural aptitude for mathematics. As an adult, he read law with a local attorney but was too restless to stick to a desk job. He traveled from one settlement to another, playing cards, betting on horse races, and trading horses. When Buck arrived in Selinsgrove with his string of stolen horses, he was twenty-eight, tall and lanky, with an unkempt beard and a prominent nose already covered with a red web of broken capillaries. He wore a black patch over his right eye. Some said he lost his eye in a childhood accident; others that it had been poked out in retribution for an unpaid gambling debt.
The towns along the Susquehanna River swarmed with rough raftsmen. Buck thrived in this atmosphere, taking the suckers for their cash with a crafty cut of the cards (it was said that he could remember the order of every card in the deck) or some other confidence scam. When he lost at the races, sometimes he paid his debts with counterfeit money. One sheriff caught him holding a counterfeit $20 bill, but Buck stuffed the evidence in his mouth and swallowed it.
At John Snyder's, Buck was given a room next to Roxy Hummel's. In December 1825 they were married. Four months later Margaret Ann was born. John Snyder said Margaret Ann might be his daughter, since Roxy had served both his own and Buck's sexual needs. If indeed both Roxy and John were the issue of Governor Simon Snyder, the incestual implications of impregnating her seemed not to trouble him. Such was the casualness with which these matters were treated.
During the day in these river towns, men congregated in the carnival atmosphere of horse races and horse sales. At night they crowded the taverns to drink, gamble, and brawl till dawn. As the men amused themselves, their women, whom they liked to call their "chore horses," would spend what little money they had to buy a bit of hope. Buck put Roxy to work telling fortunes in temporary tents where the ground was covered with sawdust, damp from tobacco juice. When the women crossed Roxy's palm with their pennies, she would comfort them with her visions. She assured bereft mothers that their dead infants were happy in the bosom of the Lord.
Roxy's nervous energy and superstitious nature assisted her. She would fall into a trance, and with the help of the spirit of the Virgin Mary, whom in her broken English she called "the holy mudder in Israel," would speak of what had been and what was to come. On days when the force seized her, Roxy would run to a hilltop and shout to God to forgive her sins and those of her neighbors, which she would then reveal at the top of her lungs. Buck was said to make use of these revelations in his blackmail schemes.
As Buck traveled from one Pennsylvania river town to another, his wife and family trailed along. They moved to Troy, to Beech Creek, to Glen Union, to Sinnemahoning (which local folk said should simply be called Sin, since there wasn't much else there). Buck ran a tavern and worked on riverboats. Usually he stayed in one place just long enough to clean up and clear out.
In eleven years of wandering, Roxanna Claflin gave birth to six children. Two girls, Delia and Odessa, died in infancy. The surviving girls were Margaret Ann and Polly; the boys, Maldon and Hebern. In 1837, Buck and Roxanna arrived in Homer, Ohio, their pockets bulging with cash from Buck's various scams. They put all their money down on a ramshackle one-bedroom frame house with wild bittersweet vines climbing over the broken-down porch. Behind the house was an untended apple orchard and, across the field, a gristmill with a flowing millrace. Roxanna, with no friends and no company except her children and her spirit visions, waited as she did each year for the revivalist preacher to come and set up his tent for what would be a winter-long series of ecstatic meetings.
To women repressed in every way, the release provided by a revival was intense. The line between religious and sexual ecstasy was often crossed in the excitement. The Reverend John Humphrey Noyes, a New England preacher, was among the first to recognize this phenomenon, especially among women. "Religious love is a very near neighbor to sexual love and they always get mixed in the intimacies and social excitements of revivals," he wrote. "Hence these wild experiments and terrible disasters."
Noyes first encountered this unacknowledged sexuality in the revival experience as a young minister. In February 1837, he preached at a revival in Brimfield, Massachusetts, and watched with chagrin as the power of his words loosed a flood of emotion. Prim young maidens in the congregation fell "into a state of frenzy over which I could exercise no control," he wrote. That night Noyes, sensing that he had triggered a dangerous force, fled Brimfield and trudged through the snow until he reached his father's house, sixty miles away in Putney, Vermont.
After his departure, two of the young women, Mary Lincoln and Maria Brown, found their way to the bed of the Reverend Simon Lovett "and suffered themselves to be taken in the act." When Mary's father, a respected physician, heard of her sin he cast her out of his house and she fled to the home of her friend Flavilla Howard. Within hours, many other young women flocked to Flavilla's home "to praise the Lord, to sing and dance and kiss each other in a frantic way." Dr. Lincoln arrived to reclaim his daughter, but Mary would not go with him. She cried out that her father was possessed by a devil and "smote him on the face in order to cast it out."
The following day, after prophesying that Brimfield was to be burned to the ground in a great conflagration as described in Genesis, Mary and Flavilla fled the anticipated "fiery hail." It was past midnight when these two stumbled into an isolated farmhouse begging shelter. They were barefoot and naked. Mary's toes were so frozen that several had to be amputated. For over a week it was thought that the girls would die of exposure but they survived and, with the revival spirit gone out of them, meekly returned to Brimfield.
Seven hundred miles from Brimfield, on Christmas Eve of the same year, 1837, Roxy Claflin was caught up in the third night of a turbulent Methodist revival. Outside the night was frigid, but inside the hastily constructed tent it was warm with the heat of bodies as they swayed and prayed and called out to the preacher, who conjured up visions of souls writhing in hell. The Calvinist conversion in Brimfield and this Methodist revival in Ohio provided an identical ritualistic transformation. First came the proclamation of sin—the filth and evil clinging to the human soul. Then the mounting fear and terror—the look into the abyss. Then the casting of oneself upon God's mercy—the throwing off of all personality in abject surrender of self. Finally, in heightened consciousness and new understanding, the cleansed soul emerged.
Driven by the preacher's fiery words, Roxy rose from the bench, her face damp with sweat. She swayed in place, then moved into the aisle and, clasping her hands together in front of her face, began to spin round and round, twisting and turning in torment until the bonnet tumbled from her head and, held by its strings, bounced along behind her. "Glory hallelujah," she shouted, and, as her frenzy grew, "Clasp me to you," "Save me," and "I am coming to you." Her eyes glazed over, and she began to babble in tongues. A staccato cacophony of unintelligible syllables spilled from her lips amid a white foam of spittle until she fell to the ground and cried out, "I am born again in the Lamb's blood!"
Buck pulled his wife to her feet and dragged her to the back of the tent, where he threw her down behind a sturdy bench, pulled her skirt over her head, and forced himself into her. All around them people were falling to the ground, groaning, crawling, barking like dogs. Religious ecstasy and sexual fervor became as a single flame, sweeping the revival. And thus, according to her own account, was Victoria Woodhull conceived.
# CHAPTER TWO
A CHILD WITHOUT A CHILDHOOD
AS THE nineteen-year-old Queen Victoria read _Oliver Twist_ , thereby learning for the first time of the "squalid vice" and "starvation in the workhouses and schools" of her own realm, far away in Homer, Ohio, on September 23, 1838, Roxanna Claflin gave birth to her seventh child and named her for the queen, whose coronation had taken place that year. Three more children followed: Utica in 1841 (named for the illegitimate daughter of Roxy's former employer and perhaps half-brother, John Snyder); a nameless son who died; and, in 1846, Tennessee Celeste. After ten births and three deaths, the family was complete.
The Claflin children lived in squalor under their parents' chaotic rule. Buck's only money came from odd jobs and various deceptive schemes. Homer, Ohio, was a town of about four hundred inhabitants. Middle-class Methodists ran the local stores and grew wheat and corn. Day after day, eight distilleries produced boatloads of corn whiskey. Excessive drinking—a disease masked as a social pleasure—was a part of life: an anodyne, a means of controlling slaves or blighting Indians, an excuse to beat one's wife and children. The dirt roads in Homer were pockmarked by tree stumps, and wagons often sank into the deep mud. Ditches, dug along the sides to contain the spring rains that flooded roads and fields, were clogged with refuse. Malaria, cholera, and typhus regularly swept the town. People still believed that maggots spawned inside the bellies of dead animals. The sources of infection were a mystery.
Bands of pigs roamed the streets and nosed into outhouses. The Claflins, who had no outhouse, dug holes in the backyard. In the spring they retrieved the human excrement to nourish their garden. Like others in the town, they drew their water from a backyard well. Traditionally, wash day was on Monday, the day after dressing up for church. The rest of the week people worked and slept in their everyday clothes. Wood was chopped and carried into the house to a cookstove. The ashes were emptied into barrels and used to absorb the pools of water that gathered around the steps. Smoke from lard-oil lamps darkened the rough wood walls of their house.
For a time Buck's twin brother, Robert, his wife, and their nine children lived with them. Regardless of sex, they slept three or four to a bed or sprawled on the floor in the bedroom, kitchen, and cellar. The children had no regular routine or chores. They distressed the neighbors by using the same unwashed green flask to fetch both beer and milk. At mealtimes they often begged at back doors. The Scribners next door could hear Roxy screaming at the squabbling children. Fierce quarrels were followed by equally fierce reconciliations. Roxy had a crude, mystic sense of family unity and insisted that no one go to bed angry.
Victoria would later say that her mother was a Spiritualist before such a movement existed. Roxy told her children of her visions, of how she had seen Jesus extend his bloody hands toward her and how she had recognized the devil because he had a small red tail and a cloven hoof. Victorias biographer, Theodore Tilton, later wrote that Mama Roxy "tormented and harried her children until they would be thrown into spasms, whereat she would hysterically laugh, clap her hands, and look as fiercely delighted as a cat in playing with a mouse.... This lady, compounded in equal parts of Heaven and Hell will pray till her eyes are full of tears and in the same hour curse till her lips are white with foam."
As for Buck Claflin, he ruled absolutely over his family. His children were his property, and near-worthless property at that, to be broken like wild horses. He beat them for no reason at all. After a particularly brutal beating, Victoria's favorite brother, Maldon, ran away and was never heard from again. Roxy said that he was working on a railroad somewhere—maybe in heaven. Theodore Tilton wrote that Victoria "was worked like a slave—whipped like a convict. Her father was impartial in his cruelty to all his children; her mother, with a fickleness of spirit that renders her one of the most erratic of mortals, sometimes abetted him in his scourgings and at other times shielded the little ones from his blows."
After Roxy lost a child to typhus, she began to doctor her children with something new on the American scene—mesmerism. Roxy saw her first demonstration of mesmerism at a local fair where Buck had set her up as a fortune-teller alongside the two-headed cow and the snake charmer. Mesmerism emerged in the late 1700s, after the Viennese physician Franz Anton Mesmer treated a female patient suffering some fifteen ailments, including blindness and paralysis of her left hand. Mesmer decided that these symptoms were hysterical in origin, somehow relating to this woman's highly restricted life. Mesmer instructed his patient to swallow a solution containing traces of iron and then he attached magnets to her legs and stomach. Instantly, she felt tumultuous waves of energy and fell into a crisis state similar to a seizure. After several such treatments, the woman was totally cured.
_Roxy (Roxanna) Hummel Claflin and Buck (Reuben Buckman) Claflin. She was an illegitimate, illiterate fortune-teller. He was a con man and a thief_. (Illustration Credit 2.1)
Therapeutic magnets dated back to the ancient Greeks, who used them to halt bleeding, soothe inflammation, purge infection, and promote general healing. Magnets themselves, according to the theory, did not heal but induced the body to heal itself. Mesmer said that the curative power of magnets derived from their stimulation of an invisible electrical energy or a fluid in the body. He named this process "animal magnetism."
To apply this animal magnetism for curative purposes, Mesmer ordered the construction of a _baquet_ , an oak tub twenty feet in diameter, outfitted with a series of iron rods connected to "magnetized" jars of water. As many as two dozen patients could occupy the tub at one time, and as they did so Mesmer wrapped his small frame in a voluminous, lilac-colored cape and pranced about waving a metal wand. He also played a glass harmonica to stimulate more vibrations. During the treatment some patients would be seized by an uncontrollable force as in the revival experience. Mesmer's patients often emerged "cured" of insomnia, Saint Vitus' dance, rheumatism, blindness, stammering, and other ailments.
It was Mesmer's disciple the Marquis de Puységur, however, who developed what came to be known as mesmerism. Puységur professed to cure ailments by controlling the flow of electrical energy throughout the body but as a side effect, he sometimes awakened in his subjects supernormal powers. Puységur's magnetized patients would fall into deep sleeplike trances in which they frequently could perform extraordinary feats of clairvoyance, telepathy, and precognition. Once awakened, his patients had no memory of these accomplishments, but often their former ills had vanished. Mesmerism had reached the previously undiscovered unconscious.
Both mesmerism and revivalism seemed to demonstrate that by abandoning oneself to a powerful force, one could effect immediate renewal of body and spirit. From earliest childhood both Victoria and her sister Tennessee remembered their mother going into a trancelike state and transmitting electrical energy to their bodies through her hands to cure them when they were ill. Roxy said she used her "other eyes" to read their minds. Mesmer's animal magnetism frequently was identified by its proponents as "the power of the soul," and when Roxy doctored her children with mesmerism she pronounced them "sanctified."
No one has yet fully explained the extraordinary feats mesmerized subjects could perform, feats achieved by Mama Roxy and some of her children. From childhood Victoria, Tennessee, and their brother Hebern said they experienced clairvoyant visions and healing powers. Of these powers, Tennessee seemed to have the largest share, but Victoria also apparently possessed skills demonstrated by mesmerized subjects, such as mind reading, finding lost objects, and describing events taking place where she had never been or before they occurred.
When mesmerism crossed from Europe to America, it found no medical auspices under which to establish its claims, and so the emphasis on healing gave way to sensational demonstrations at fairs and carnivals. Most often the chosen subject was a young woman. At the Homer, Ohio, fair of 1845, Rose Scott, an attractive girl of fifteen, was put into a trancelike state by a male mesmerist. According to a newspaper report, a bottle containing ammonia was then passed under her nose with no visible effect. Her finger was pricked with a hat pin, but there was no reaction and, mysteriously, no blood. The mesmerist then produced a red handkerchief and announced it was a snake. Rose recoiled in terror. When he suggested that she was covered with vermin, she frantically clawed at her bodice. Here was a passive, compliant female with no will of her own; the sexual implications were obvious.
Experiments in mesmerism took place throughout the country. In 1843, Henry Ward Beecher, having polished his skills as a revivalist, began practicing mesmerism as well. He mesmerized his sister Harriet Beecher Stowe with repeated success. During the first session she noted, "He succeeded in almost throwing me into convulsions—spasms and shocks of heat and prickly sensation ran all over me.... This strange tempestuous effect was occasioned simply by our sitting opposite to each other with our eyes fixed and our thumbs in contact for about thirty minutes and it was dissipated by making reverse passes which relieved first my head, then my lungs, then my lower limbs and lastly my arms."
A new age was dawning, one of electrical impulses, invisible energy, and other unseen, unexplained phenomena. In 1842, Samuel F. B. Morse petitioned Congress for $30,000 to construct a telegraph line between Washington and Baltimore. In the debate that followed, the representative from Tennessee suggested an amendment mandating that half the appropriation go to experiments in mesmerism. The chairman of the congressional committee noted that the suggestion was not out of order, because "it would require a scientific analysis to determine how far the magnetism of mesmerism was analogous to the magnetism employed in telegraphs." Thus faith in invisible powers, mediated not by preachers in their Calvinist pulpits, but accessible to all believers, was becoming a feature of everyday life. And for women, revivalism and mesmerism presented heretofore unimagined opportunities to exert personal power.
BEYOND A THICK hedge that separated them from the Claflins lived the Scribners, a respectable Methodist family. In their bright kitchen, red-and-white gingham curtains framed the windows, and gleaming copper pots hung above the woodstove. Victoria would later describe this kitchen as typical of those she saw "in Heaven." What a different world was this cozy household from the chaos and instability next door. Vickie, the interloper, longed to be a part of the Scribners' world, but her disreputable background shut her out. In adulthood she would come to despise the word _antecedents_.
Mrs. Scribner forbade her daughter Laurie to play with the shiftless Claflins, and one day when the child returned from the Claflins' orchard carrying a basket of green apples, her mother beat her with a willow switch. Stable and secure, the world of the Scribners was bound by conformity to a set of rigid principles. Catharine Beecher, the eldest child in the famous family of preachers, wrote a _Treatise on Domestic Economy_ in 1841 and dedicated it "To American Mothers." Her book dictated the rules for households like the Scribners', where a woman's proper role was to make a man's home his castle. In her widely accepted view, a woman must sublimate all personal ambition and desire, which were evil, and devote herself to perfecting the hearth and home. This was "woman's sphere."
_Catharine Beecher. She was unmarried and a virgin but was considered America's prime expert on child-rearing and "domestic economy._ " (Illustration Credit 2.2)
Catharine Beecher preached a gospel of perfect domesticity elevated to godliness. Throughout the nation, middle-class white women deferred to her rules. System and order were Miss Beecher's holy creed, accompanied by specific instructions on laundering, housekeeping, cooking, personal hygiene, and female education. With absolute clarity and conviction, this unmarried virgin of forty-one, who had no home of her own, prescribed how to raise a generation of women.
The physical and domestic education of daughters should occupy the principal attention of mothers... and the stimulation of the intellect should be very much reduced.... Much less time should be given to school and much more to domestic employments, especially in the wealthier classes....
Needle-work, drawing, and music, should alternate with domestic pursuits. One hour a day devoted to some study, in addition to the above employments, would be all that would be needful to prepare for a thorough female education.... Those who have the most important duties of society to perform—men—should have well-disciplined and well-informed minds.
When Victoria's brothers and sisters begged at the Scribners' back door, these righteous Christians sometimes handed them a plate of scraps or, on rare occasions, milk and johnnycake. But Victoria wouldn't beg. It was Mr. Scribner's sister, twenty-one-year-old Rachel, who answered the door on the day when five-year-old Vickie inquired if there were any chores she could do. Rachel saw that under the grime the child was beautiful: brown ringlets, arresting blue-gray eyes, and a quiet pride that made her different from the rest of the raucous clan next door.
Rachel Scribner took Vickie into her warm kitchen and into her heart. For a year Vickie visited almost daily, and Rachel taught her to read and write and saw that the child never went hungry. She washed her hair and praised Vickie's quick intelligence. One morning Vickie arrived at the Scribners' home to find the draperies drawn and a pine coffin in the parlor. Rachel Scribner had been stricken with cholera and died, all within twelve hours.
Years later, Victoria recounted her first spiritual vision—her first escape from an intolerable world. She was standing in the apple orchard, unable to cry, when the spirit of Rachel Scribner, a few hours dead, came to her. The spirit took Victoria by the hand, and she "felt herself gliding through the air." They traveled on a spiral path, engulfed in an intense white light, until they arrived at the spirit world. Heaven appeared to this small, perplexed child to be like the ordered world in which the Scribners lived, a world of well-kept homes, well-dressed men and women, of laughing and singing children, of peace and love. She later wrote,
I saw the spirits descending to earth and mortals ascending to the Spirit World and mingling in common unity. The people seemed to be much engaged as people are. They were coming and going as if they were very busy, and the scenery of that world was a counterpart of this.
I met some spirits who I since know to be Demosthenes, Bonaparte and Josephine. They told me that I was to be in their charge, and that they were to constantly guard, guide, instruct and care for me so that I should be, when grown, fitted to do their work on earth.
Towards the close of the visit, for such it really was, I was shown what I now know to have been a panoramic view of the future. The mountains and valleys changed places with the seas, the entire face of the nation underwent a transformation. Cities sank and people fled before appalling disasters in dismay. Then a wondrous calm settled over everything. Confusion, anarchy and destruction were replaced with a scene of beauty and glory which is beyond the power of language to describe. The earth had been changed into a paradise and had become the common abode of people of both spheres. The spirits said that all this would be realized during my life and that in making it possible I would bear a prominent part.
When I returned to physical consciousness, I told my mother, as well as I could in my childish way, all that had happened to me. My body had been unconscious for two hours. I had been among the angels.
By her own account Victoria was "a child without a childhood." From the time she could remember she had been farmed out as "a household drudge" to begin at dawn on endless rounds of washing and ironing, running errands, spading gardens, chopping wood, laying fires, tending infants. Buck collected her meager wages. Retreating perhaps from the cruelty and preternatural excitement of her surroundings, Vickie turned to the spirits. She later told Theodore Tilton that since childhood she had been aided by them. He wrote, "She has been lifted over the ground by angelic helpers lest she should dash her feet against a stone. When she had too heavy a basket to carry, an unseen hand would sometimes carry it for her. Digging in the garden as if her back would break, occasionally a strange restfulness would refresh her and she knew the spirits were toiling in her stead."
Victoria believed that the spirits guarded and sustained her. "I often performed the most laborious tasks without fatigue and mastered studies only by magic. Another more singular experience was when walking it seemed to me my feet did not touch the ground. While I saw my body going along, I was two or three feet above in the air. This phenomenon has followed me all my life."
It was Roxy herself who set her daughters on the road to the spirit world. She predicted that there would be better days for them. She had seen it in a vision: One day they would be rich and would "ride around in their own carriages." When they were shunned by their more respectable neighbors, she told her girls that they were "different" and what she did, or they did, was right simply because they did it. She told them that God was within them, and so any action they took could not be wrong. This feeling of entitlement was to persist throughout Vickie's long life.
When Roxy took her to the log schoolhouse, Vickie would ignore the children who laughed at her tattered clothes and the teacher who clipped a clothespin on her nose to discipline her. She would console herself with what her mother had told her. Vickie was a quiet child, unlike the other Claflins, and she absorbed knowledge effortlessly. She could look at a page once or twice and know it by heart, an ability that would serve her well in later life.
Buck's main occupation at this time was the sale, for $1 a bottle, of a "Life Elixir," which contained alcohol and laudanum, an opiate, mixed with herbs and molasses. Roxy brewed it and began using it herself. Once while Roxy was mixing a batch of elixir in a kettle strung up in the fireplace, Vickie saw the door open to admit a tall, lean man. "A red silk handkerchief concealed his face," but she knew he was the devil because "I saw not a booted foot but a cloven hoof." The devil stared into her eyes, Vickie later recounted, but it was he who flinched and abruptly left the house.
When she was ten, she also recalled, her two-year-old sister, Tennessee, lay in her cradle unable to move, near death from pneumonia. As Vickie sat rocking her sister, two angels appeared and pushed her away from the cradle. They fanned the baby's face with "pale white hands and blew sweet breath upon her." When Roxy entered the room, Vickie lay on the floor in a trance while Tennessee lay rosy and healthy in the crib.
Buck Claflin was never a man to ignore an opportunity to make money. As the revival fires raged throughout Ohio, child preachers became a popular attraction. A girl of seven preached hellfire and damnation while perched on her father's shoulder. A towheaded boy of five stood on a crate, and at the end of his sermon cast a black handkerchief to the ground and shouted, "So shall sinners fall in Hell." From the time Vickie was eight, Buck Claflin took her around Ohio to preach, and the child captivated audiences with her magnetism and ecstatic fervor. Vickie would stand on two crates, one on top of the other, and stare down at the faces in the crowd. "I am the Word.... Sinners, repent," she commanded. "Listen to me, for I know things you do not know. I can cure or I can smite!" No doubt she had heard this admonition from revivalist preachers and she repeated it with such passion that her listeners cringed. Even in her youth she was a consummate showman, addicted at an early age to the attention she received.
After nine years in Homer, Buck began long treks; sometimes he'd walk twenty miles and return home with "bleeding feet and haggard face." In the fall of 1847, he insured his gristmill against fire for $4,000, which seemed odd because he never operated it. One Saturday night he set out on one of his walks and spent the night in a town ten miles away. The following morning he headed home, but in the late afternoon stopped at the Brandon Tavern, only two miles from his house. He was still there at eleven that night when he walked to the window and said to the landlord, "There is a red glow in the sky to the south. I have a presentiment that's my gristmill on fire." Buck borrowed a horse and galloped into Homer to find a heap of smoldering ashes where his mill had been.
When Buck went to collect the insurance, the people of Homer began to wonder about the unused gristmill. Soon there was talk of arson and even more talk of tar and feathers and cleansing the town of this primitive, cheating clan. Roxy was suspected of torching the mill, and so was ten-year-old Vickie. Buck disappeared, leaving Roxy and her brood without food or money.
The previous year, Buck had worked for a few months as a postmaster. After he disappeared, a trunk filled with empty envelopes addressed to Homer residents was found in the post office, Buck having pocketed the money and whatever else of value he found in them. In January 1848, the upstanding Christian women of Homer organized a bazaar, and the proceeds were given to Roxy Claflin on the condition that she leave town and rejoin her husband, wherever he might be. Meanwhile, far away in the heart of the Burned-Over District near Rochester, New York, the powers that had been nurtured in Victoria were about to be given a name, and that name would soon identify a new religious movement dominated by women: Spiritualism.
# CHAPTER THREE
THE SPIRITUAL TELEGRAPH
AT SIX o'clock on the night of March 31, 1848, John and Margaret Fox shooed their daughters, eleven-year-old Catherine, called Kate, and thirteen-year-old Margaretta, called Maggie, upstairs to the bedroom in their tiny saltbox house with two windows in front and three on each side, in Hydesville, New York. All week strange nocturnal noises had disturbed them—mysterious raps and bumps that could not be explained. Tonight they were determined to put the girls to bed early to make up for lost sleep. No sooner had the Fox sisters gone upstairs than the banging began again, this time with great force and rapidity, resounding throughout the house. John and Margaret rushed upstairs, entered the room, and stood transfixed. The children were in bed, but the raps were now louder than ever.
Kate Fox was a year older than Victoria Claflin and gave off the same aura of intense energy. As her parents stared, she jumped out of bed and began talking to an unseen presence. "Follow me," Kate commanded, snapping her fingers as she paraded around the room. A steady progression of raps seemed to follow her. "Now, do as I do," she said, and clapped her hands three times. Three distinct raps answered her. "Three raps mean yes," Kate announced to her astonished parents. Her mother then asked, "Are you a disembodied spirit?" Three rapid raps followed.
Within the week, twelve people had squeezed into the Foxes' bedroom. William Duesler, a neighbor, sat on the bed next to Mrs. Fox, whose questions were answered with such distinct raps that Duesler felt the bedstead shiver. Then Duesler began pointing to various people in the room; he asked how many children they had, how many sons, how many daughters, how many children had died. Each time the number of raps was correct. Duesler asked the ages of different people in the room and finally asked his own age. Thirty-one raps was the answer. He was confounded: "I did not think anyone about here knew my age except myself and my family."
By May 1848, so many people had made pilgrimages to the Fox home—peeking in the windows, camping in the yard, lighting bonfires at night—that Maggie and Kate fled. Maggie went to the nearby house of her married brother, David, and Kate to her thirty-three-year-old married sister, Ann Fish (known by her middle name, Leah), who lived in Rochester.
Rochester, in the heart of the territory that had been burned over by the fiery streak of the second Great Awakening, was known for its freethinking atmosphere, and many of its inhabitants supported abolition, including a group of Quakers who believed that the enslavement of body or soul was degrading and unjust. Of the six hundred Quakers in the Rochester area, Amy and Isaac Post were among the most prominent. Isaac was a prosperous druggist, and he and his wife met and entertained other abolitionists in the parlor of their comfortable home at 36 Sophia Street. Their cellar was a station on the Underground Railroad that transported runaway slaves to Canada.
William Lloyd Garrison, publisher of _The Liberator_ , who had been dragged half-naked through the streets of Boston for his antislavery views, and Wendell Phillips, now a dedicated abolitionist, frequently stayed with the Posts. So did Frederick Douglass, an escaped slave who had become an eloquent abolitionist lecturer and with help from the Posts and others had started his own newspaper, _The North Star_. Douglass so cherished the Posts' friendship that while away on a lecture tour he wrote, "I should love much to look into your home... about once in [every] twenty-four hours."
The Posts' radicalism came gradually. At their Quaker meetings, Isaac and Amy said they were "moved by the spirit" to speak out against slavery, but they were forbidden to do so by the strictures of these gatherings. Therefore, they and several other antislavery Quakers withdrew from this traditional group and became followers of Amy Post's cousin, Elias Hicks. But prejudice followed them, and within the year the "Hicksites" had censured Isaac and Amy Post for participating in antislavery societies with non-Quakers. Once again the Posts withdrew, and with a small group of liberal followers founded the Waterloo Congregational Friends.
One of their group was Daniel Anthony. Once the well-to-do owner of a cotton mill, Anthony had lost his savings in the financial panic of 1837, and retreated to a "six-penny farm" in Rochester, where he scratched out a laborious living. To supplement his meager income he'd secured a job as Rochester district salesman for the newly formed New York Life Insurance Company and traveled extensively throughout the area soliciting customers. The second of his seven children, his twenty-eight-year-old daughter, Susan B. Anthony (in her teens she abbreviated her middle name, Brownell), had secured a position as headmistress of the Female Department at the Canajoharie Academy and was self-supporting. Daniel Anthony greatly influenced his daughter's thinking. In a typical letter he wrote Susan, "Of what use is preaching and all this pretended and blind devotion so long as this horrible business of trafficking in the bodies of men, women and children is sanctioned and actually carried on by those making the highest pretensions to goodness... sanctioned by both the Civil and Religious Institutions of this Republican and Christianized land."
_Maggie (Margaretta) Fox, Kate (Catharine) Fox, and Leah (Ann Leah) Fox Fish. They penetrated the barrier to communicate with the dead_. (Illustration Credit 3.1)
_Amy Post. She concluded that "an unbroken chain of communication exists] between the Infinite and all beings._ " ([Illustration Credit 3.2)
Six miles from the Posts' Rochester home was that of Thomas and Mary Ann McClintock, and it was there that in June 1848 the Posts and several other members of the Waterloo Congregational Friends gathered around a mahogany tea table in the parlor to draw up an agenda for their group. The tilt-top table with three legs was a common fixture in many parlors but now, according to those present, a remarkable occurrence took place. As members of the group presented their ideas, the table began to vibrate with raps of approval from the spirits. As word of this phenomenon spread, the McClintocks' table became famous as the first "spirit table." Soon it was believed that certain tables served as catalysts to transmit the thoughts of the spirits.
In the Rochester area rappings began to be heard in many homes. Rapidly the methods of spirit communication became increasingly sophisticated. At first, spirit messages were spelled out by a series of raps, from one to twenty-six, corresponding to the letters of the alphabet. Then people began speaking the letters of the alphabet, and the spirits rapped at the appropriate letter. Spirit messages were further speeded up when it was discovered that both Kate Fox and her sister Leah's daughter, Lizzie Fish, could fall into trances and directly communicate with those who had "passed over."
Kate Fox, then living at Leah's house, sent a note to Amy Post that she had contacted the spirit of the Posts' five-year-old daughter, Matilda, who had died the previous year. The respected Posts promptly appeared at the Fish residence and became enthusiastic participants in these new discoveries. Acts of mediumship and clairvoyance quickly became the order of the day.
At Leah Fish's house, family and visitors felt the cool touch of invisible spirit hands. As Leah played the piano, she later wrote, those present said they felt "the deep throbbing of the dull accompaniment of the invisibles, keeping time to the music as I played." At dinner, with guests present, loud raps would often be heard, and one end of the table would rise up and thump down, sending plates and glasses flying. Isaac Post discovered that the spirits could take over his hand and produce "automatic writing." In this manner, he insisted, he wrote _Voices from the Spirit World_ , a book with an introduction by Benjamin Franklin and containing messages from Washington, Jefferson, and Voltaire, all long dead.
The following November, Amy Post accompanied Maggie Fox and Leah Fox Fish to Corinthian Hall, the largest auditorium in Rochester, where they were to undergo the first of three arduous public tests of their strange powers. People began lining up at dawn, and by nine o'clock, when the box office opened, more than seven hundred had gathered. There were four hundred tickets, and at 75 cents each they went as fast as the cashier could hand them over. By six in the evening some tickets were being scalped for as much as $10. When Maggie and Leah arrived, they were taken to a back room to face a committee of sour-faced ladies who instructed them to strip off all their clothing. Amy Post, who was shocked at this demand, stood guard over them and draped their naked bodies in borrowed shawls while their garments were minutely examined. Once this procedure was completed, they were taken to the stage to face a committee of men chosen for their skill at exposing frauds.
As the inquisition began, Maggie and Leah's skirts were twisted around their ankles and tied securely in place with handkerchiefs. Then a member of the committee moved forward and placed his right hand on Maggie's feet, anchoring them in place. He put his left hand on the floor. Though Maggie could not move, raps came from the floor. The procedure was repeated with Leah, with similar results. Several other tests were conducted, but no one could locate the source of the raps.
On the second day of tests no further progress was made. Some members of the committee postulated that ventriloquism or some such device was causing the raps. Handkerchiefs were tied around the sisters' mouths, but the rapping continued. On the third and final night, the committee decided to test for electrical energy. Almost a century before, Benjamin Franklin had said that electricity represented the force of "disembodied spirits." An early Spiritualist described electricity as "the God principles at work." Hoping to demonstrate that these so-called spiritual phenomena were caused by the sisters directing electrical energy, it was advertised that they would be examined while standing on materials that did not conduct electricity.
Anticipation was keen. The audience packed the auditorium, anxiously awaiting the tests that would finally expose Maggie and Leah. So sure was a reporter from the _Rochester Democrat_ that a fraud would be revealed that he had written an article to that effect, which already was set in type. One member of the examining committee vowed he would throw himself over Genesee Falls if he could not discover some trick. Another said he would forfeit his new beaver hat.
The sisters were once again examined by the ladies' committee before appearing onstage, where the spirits rapped answers to a number of questions, each time correctly. Next, they were asked to stand on a mound of pillows to halt the transmission of electrical impulses. But when they called upon the spirits, time and again the raps were heard. No trick was exposed, but no man committed suicide or lost a hat. The reporter rewrote his article. And a new religious movement was under way.
Members of the Posts' immediate group of Waterloo Congregational Friends were among the first converts to Spiritualism. These liberal, antislavery freethinkers had carried to an extreme the Quaker principle that God's laws were written in every human soul. Their declaration of principles maintained that "an unbroken chain of communication exists between the Infinite and all beings." This conviction seemed to be confirmed in the year 1848, when telegraph lines finally reached the Rochester area, making it possible to send messages by means of a little understood electrical force. The concept formulated by Isaac Post that one could utilize a "Spiritual Telegraph" to establish a line of communication with those who had "passed over" was no more difficult for these people to accept than the telegraph itself. Thus the barrier between the mortal and the spirit worlds had been penetrated.
The rise of Spiritualism came at a time of rapid territorial and economic expansion. Not only Morse's telegraph but other technological advances—canals, railroads, and steamships—led to a new America whose boundaries seemed both limitless and God-given. In 1845, when Texas joined the Union as the twenty-eighth state, Congress affirmed the "right of our Manifest Destiny to overspread and to possess the whole of the continent." But as the country grew, Ralph Waldo Emerson spoke for a lost security when he wrote, "Instead of the social existence which all shared, now there was separation." The Calvinist ethic that once had held everyone in place was fading, and the search for alternative meaning in a transformed world had begun.
Many Spiritualists turned to the teachings of Emanuel Swedenborg, a highly influential Swedish philosopher. In 1750, while in a trance, Swedenborg claimed to have seen the spirit world and recorded what he saw there. In the heavenly society, all goods and services were shared, and men were permitted both wives and concubines. Swedenborgs vision not only fulfilled male fantasies but also exploited the nascent interest in science. He wrote that "the Spirit World is derived from atoms... the disintegrating chemical action of electricity and magnetism which throws out ethereal particles into the great ocean of unindividualized spirit." In a like manner, in the 1840s a French disciple of Swedenborg, Charles Fourier, propounded "a common system of movement for the material and the spiritual world." Fourier's theory of "passional attraction" advocated free sexual relationships based on "elective affinities."
Religion, sex, and the role of women were being reshaped by forces and theories that seemed mysterious and difficult to comprehend. Perhaps the strangest synthesis of these elements was to be found in a so-called machine built in 1853 by a Spiritualist and trance medium, John Murray Spear. With "spirit guidance," Spear constructed a contraption of zinc batteries, metal balls, and thousands of copper wires, encased in a wooden frame. He named it the New Motor. The purpose of this machine was as vague as the man himself, but he claimed that it not only harnessed spiritual electricity but also housed an as-yet-unborn soul.
With the Virgin Mary in mind, Spear declared that this new force was to be given life through a woman. Soon after, a Boston Spiritualist said that she had experienced an immaculate conception and bore in her womb the "living principle" of Spear's machine. The woman, large of stomach, prostrated herself before the machine, and, having undergone labor pains, she reached "a crisis," and her stomach flattened. Then Spear announced that "at precisely the time designated and at the point expected, motion appeared in the New Motor corresponding to embryonic life."
This announcement was met with skepticism and fear. The New Motor, or the Wonderful Infant, as Spear now referred to it, was called the work of the devil. When Spear moved his creation to Randolph, New York, because of its "lofty electrical position," panic spread throughout New England. Within the week, a mob broke into the shed that housed the Wonderful Infant and smashed it to bits. Spear related how the frenzied group "tore out the heart of the mechanism, trampled it beneath their feet, and scattered it to the four winds."
Spear's bizarre experiment acknowledged the life-giving force of women. And it was to women that the appeal of Spiritualism was especially potent. Louise Chandler Moulton, an early convert to Spiritualism, was typical—born to strict Calvinist parents who banned the reading of romances, dancing, and playing games of chance, including backgammon. Throughout her childhood she was plagued by "an awful foreboding of doom and despair." She would wake "in the depth of the night, cold with horror" and think, "Why, if I'm not among the elect I _can't_ be saved, no matter _how_ hard I try." Through Spiritualism her fate was now in her own hands.
Death too inspired Spiritualism. "Woman's sphere" was confined to the home, where death was an all too frequent visitor. In New York State, as the first spirit rappings were being heard, half of the deaths were of children under five. Beloved relatives died at home, most often attended by women. Beyond loneliness and grief, women, particularly middle-class Christian women who had been plagued by doubt as to the fate of the departed, could now see these dear souls safely nestled in the hand of God. Spiritualists responded to death as a transforming event: Rather than mourn, they wore white at funerals, where mediums delivered messages from the newly arrived spirits in Summerland (the Spiritualist name for heaven).
The attribution of purity and passivity to the female nature made women ideal vehicles to channel messages. By the 1850s, a group of female trance speakers were among the first women permitted to speak in "promiscuous assemblies," which meant gatherings of both sexes. Speaking with the authority of the spirits but without personal responsibility for what they said, these women could not be censored for their statements. Since the spirits were guiding them, they had courage, for they spoke the truths of a greater power. Women, no matter how ill-educated, could now transmit the wisdom of spirits as diverse as Socrates and Benjamin Franklin: Not surprisingly, the rights of women were very much on the minds of these great thinkers.
Spiritualism was dominated by women, and its manifestations most frequently occurred in places within their control. "Not in Church, not in Capitol, but in the family came the first demonstrable recognition of immortal life and immortal love—the holiest truth to the holiest place," wrote Andrew Davis, an early convert to Spiritualism. And the preacher Charles Beecher observed, "It is not in its published literature, its periodicals, its lectures, and its public mediums that the strength of the movement lies. It is in its family or home circles." Séances were conducted around parlor tables where men and women held hands to form Spirit Circles. Ouija boards with planchettes were common accessories. (The planchette was a heart-shaped piece of wood mounted on three gliders. It was said to respond to magnetic forces passing through the fingers, which guided the point of the instrument to a letter of the alphabet. In this manner spirit messages were spelled out.)
_A spiritscope. In 1855 Dr. Robert Hare invented this machine to prove that spirits exist_.
Revivalists and even traditional preachers described the beauties of heaven and the torments of hell so vividly that they seemed very real places. A Spiritualist named William Hayden noted that the very ministers who comforted parishioners by assuring them that the deceased "are not gone but are often with you, watching over you, loving you" were shocked when he told them that he had verified this statement by communicating with the dead. These thoughts were echoed by Warren Felt Evans, the Spiritualist minister and author of _The New Age and Its Messenger:_ "If a person becomes sufficiently spiritual before his departure to a higher realm, to see, hear and even converse with those who walk the velvety soils of the land of perpetual spring... his sanity is seriously called into question."
After the death of his wife, Roxanna, Lyman Beecher impressed upon his children that she was still with them. Although she had died when Henry Ward Beecher was a child of three, he would later say his mother was the guiding influence and inspiration of his life. Because of her, he professed to see God not as a strict father but as a loving, nurturing, and forgiving mother. In Roxannas spirit, Henry Ward Beecher would conceive his _Gospel of Love_ , which two decades later would make him the most popular preacher in the nation.
This new Calvinism, with its revival ritual, the powers released by mesmerism, and puzzling unseen electrical forces, had pointed the way to Spiritualism. Twenty years later, Theodore Tilton's wife, Elizabeth, called Lib, who was to play a pivotal role in the life of Victoria Woodhull, said that as she knelt by the corpse of her dead son, Paul, she saw two angels descend through the stained-glass dome of her Brooklyn Heights home and in a blaze of white light carry his spirit up into the realm of heaven. And the "Spiritual Telegraph" hummed with the words of comfort, reassurance, and power that women, even the strongest, so desperately needed.
# CHAPTER FOUR
MY LONG-ACCUMULATING DISCONTENT
In 1848, as spirit raps swept the Rochester area, they served not only to communicate with the dead but to give courage to the living. Just one month after the three-legged spirit table in the McClintocks' parlor reverberated with raps as the Waterloo Congregational Friends were formulating their declaration of principles, Elizabeth Cady Stanton and four other women assembled in the same parlor to discuss the plight of women. Three of these women—Mary Ann McClintock, Lucretia Mott, and Jane Hunt—were Waterloo Congregational Friends. The fourth, Mott's youngest sister, Martha Coffin Wright, had been expelled by the Quakers for marrying "out of meeting." To these women Elizabeth Cady Stanton, a frustrated thirty-three-year-old mother of three sons, "poured out... the torrent of my long-accumulating discontent with such vehemence and indignation that I stirred myself, as well as the rest of the party, to do or dare anything." Stanton wrote that she and the other women were determined to call a "convention to discuss the social, civil, and religious condition and rights of women," to be held July 19 and 20 in the Wesleyan Chapel in Seneca Falls, New York.
It was essential that they prepare a statement of their beliefs, but Stanton said she and her companions "felt as helpless and hopeless as if they had been suddenly asked to construct a steam engine." Lucretia Mott, a well-known Quaker preacher and abolitionist, was the only one who had any experience as an organizer. The rest of the women were occupied solely as wives and mothers. Reports from antislavery and temperance conventions were suggested as models for their declaration, but these documents seemed dull and imprecise. Time grew short until only five days remained before the convention, and still no paper had been written. Stanton, desperately seeking inspiration, sat down at the McClintocks' spirit table, picked up the Declaration of Independence, and began reading it aloud. At once the women decided to use it as the basis for their own Declaration of Rights and Sentiments. "We hold these truths to be self-evident, that all men _and women_ are created equal...." Miraculously, the ideas began to flow as Elizabeth Cady Stanton placed her paper on the spirit table and wrote the declaration for "the first organized protest against the injustice which has brooded for ages over the character and destiny of one-half the race."
Stanton wrote from experience, for she knew full well the oppression of women. After nine years of marriage she had found herself isolated on the outskirts of Seneca Falls, suffering from "mental hunger." Her husband, Henry Stanton, had wanted to move there to relieve his lung congestion, but he was gone most of the time, pursuing his legal and political career and organizing New York's Free Soil Party. In fact, he had been absent for the birth of each of their three sons. Elizabeth was now occupied with household chores and the time-consuming homeopathic cures she administered to her three boys—five, three, and almost two—all of whom had contracted malaria. She had learned this method from her brother-in-law, Edward Bayard, who had left a promising law practice to become a homeopathic physician. Elizabeth approved of his career, although her sister Tryphena, Edward's wife, who was active in homeopathic medicine herself, felt he was giving up all chance of wealth. Edward longed for understanding and affection, but Tryphena thought his ideas and aspirations impractical.
Homeopathy was practiced as a more benign alternative to morphine and alcohol, the most common medications, and to the purging, blood-letting, blistering, and cutting administered with dubious results by practitioners of the "heroic medicine" of the day. According to the first homeopathic physician, Dr. Samuel Hahnemann, a minuscule amount of a disease-causing agent could be diluted with liquid, creating a "spiritlike" essence. If this substance was then introduced into the body, the patient could be cured of that disease.
In accepting homeopathy, as in other aspects of her life, Elizabeth Cady Stanton chose to think for herself. She gave birth to each of her children attended by a midwife rather than a doctor. The morning after her fourth son was born, she wrote, "I got up, bathed myself in cold water and have sat by the table writing several letters." She noted with some humor that others were waiting for her to die to prove her theories wrong. In a time when the infant mortality rate in America was 50 percent, Stanton bore seven healthy children, whom she raised to adulthood. In a world of seven-course meals she was to declare, "We know what not one woman in ten thousand does know—how to take care of a child, make good bread, and keep a home clean.... Our children... are healthy, rosy, happy and well-fed. Pork, salt, meat, mackerel, rancid butter, heavy bread, lard, cream of tartar and soda, or any other culinary abominations are never found on our table." Stanton's thoughts on sewing were equally unconventional: "As an amusement it is contemptible; as an education of head or heart, worthless... as a support the most miserable of trades."
LEFT: _Lucretia Coffin Mott—the Quaker pioneer of woman's rights_
RIGHT: _Martha Coffin Wright. She joined her older sister, Lucretia, in rebelling against the subjection of women_. (Illustration Credit 4.1)
_Elizabeth Cady Stanton. Her rage against the male establishment fueled her fight for woman's rights_. (Illustration Credit 4.2)
Although she asserted her independence by adamantly refusing to be addressed as Mrs. Henry Stanton, Elizabeth felt keenly her subordinate role as a woman. She recalled the moment when this feeling of inferiority had begun: In July 1826, the eleven-year-old Elizabeth Cady tiptoed into her parents' parlor to find it a "silent chamber of death." The drapes were drawn, the darkened room lit only by two sputtering candles. The mirrors and pictures were draped in white linen. Beside the open casket of her brother Eleazer sat her father. The eminent judge Daniel Cady was pale and immobile. "As he took no notice of me, after standing a long while, I climbed upon his knee.... He mechanically put his arm around me and, with my head resting against his beating heart, we both sat in silence, he thinking of the wreck of all his hopes in the loss of a dear son, and I wondering what could be said or done to fill the void in his breast. At length he heaved a deep sigh and said, 'Oh, my daughter, I wish you were a boy!' Throwing my arms about his neck, I replied, 'I will try to be all my brother was.' "
And she did. Elizabeth learned to ride a horse with great skill and daring. At Johnstown Academy she excelled in Latin, Greek, and mathematics. "I resolved... to be at the head of all my classes and thus delight my father's heart.... I surprised even my teacher, who thought me capable of doing anything. I learned to drive and to leap the fence and the ditch on horseback. I taxed every power, hoping some day to hear my father say, 'Well, a girl is as good as a boy, after all.' But he never said it." Elizabeth became keenly aware that no matter how she tried, "father... prefers boys." She suffered from what William Blackstone had referred to in his _Commentaries_ as "the defect of sex."
There were to be no Cady men. Of Margaret and Daniel Cady's eleven children, all five boys died, while five of the six daughters lived to adulthood. Elizabeth was the seventh child born into this socially prominent and affluent family in Johnstown, New York, forty miles northwest of Albany. Her father became a congressman in 1815, the year of her birth, and subsequently a New York Supreme Court judge. Her mother was the daughter of Colonel James Livingston, who had fought under Washington in the Revolution. Margaret Cady was almost six feet tall, an imposing, cool woman whose energies went to producing children and to overseeing a supremely well-organized household. Elizabeth usually saw her mother at teatime and dinner; she thought of her as a distant queen.
As a child Elizabeth would sometimes sit unnoticed in a corner of her father's office in her red flannel dress, the stiff linen collar scratching her neck while she listened to women pleading with Judge Cady to right the wrongs against them. She heard about the husband who took his wife's hard-earned wages to buy liquor while their children starved; about unremitting beatings; about children taken away by husbands, their mothers forbidden ever to see them again. Old Mrs. Brown came to Judge Cady to help her keep the farm that her grandfather and father had owned before her, one that had been willed by her late husband to a stepson. And Flora Campbell came crying to Judge Cady because her husband had mortgaged her farm, over her objections, and now creditors were taking it away.
Under the law there was nothing Judge Cady could do. He would take down his heavy law books and read the statutes aloud. Later, unobserved, Elizabeth would mark the pages he'd read and replace the books in the bookcase upside down so she could find them again. She planned one night to tiptoe into her father's office with a scissors and cut out all the bad laws that made women cry. She told this to Flora Campbell, and soon her father was explaining that cutting the laws from his books would do nothing to change them.
When Elizabeth was ten, Edward and Henry Bayard, the sons of Delaware senator James A. Bayard, came to read law in Judge Cady's office. A few months after her brother Eleazer's death, Elizabeth's eldest sister, Tryphena, eleven years her senior, married Edward Bayard. Edward loved to tease Elizabeth by reading her the most vexing laws relating to women (glass, eggs, and women could not be insured), passages from the Bible ("But I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence"), and _The Taming of the Shrew_ (Elizabeth hated Petruchio and sympathized with Kate). Elizabeth's anger smoldered, but "I soon noticed that, after losing a few games of chess, my opponent talked less of masculine superiority," she later wrote. A particularly arrogant student in Judge Cady's office was taken by Elizabeth on a ten-mile gallop. "I left him... half dead at his hotel, where he will be laid out, with all his marvelous masculine virtues, for a week at least," she exulted.
Elizabeth's mother, Margaret, was forty-one when Eleazer died, but her husband still demanded a son. In the next five years she suffered seven miscarriages. At last she gave birth to another son, also christened Eleazer. The child died at four months. Margaret, physically and emotionally exhausted after eleven births, retreated to her bed in her darkened bedroom. Escaping into illness, she determined to remain in a condition where her husband could not impregnate her again. (When she ceased menstruating, Margaret recovered her health.) As she languished, Tryphena and Edward took over as surrogate parents to Elizabeth and her two younger sisters, Margaret and Catherine.
At fifteen Elizabeth graduated with highest honors from Johnstown Academy and hoped to go to Union College in Schenectady with the boys. Her ambivalent father, at once proud of her accomplishments but feeling that they were not befitting a girl, who should be trained for the role of mother and housekeeper, forbade his daughter any more formal education. As she was to complain sometime later, "To think that all in me of which my father would have felt a proper pride had I been a man, is deeply mortifying to him because I am a woman." When her brother-in-law Edward Bayard interceded for Elizabeth, the judge agreed to send her to Emma Willard's Female Seminary in Troy, New York, where she spent two years studying the essential female graces: music, dancing, and French. The academic courses were at a level far below what she had already mastered. At Emma Willard she felt purposeless, bored, and full of vague longings.
The revival season began that spring in Troy, and the great reverend Charles Grandison Finney arrived to bring souls to salvation. Finney was considered a leader of the new Calvinists, a man so persuasive and feared by the more restrained clergy that Lyman Beecher had tried to keep him from bringing his "streak of fire" into Beecher's own pastorate in Litchfield, Connecticut. The rebellious Elizabeth easily came under Finney's sway:
I can see him now, his great eyes rolling around the congregation and his arms flying about in the air like those of a windmill.... He described Hell and the Devil and the long procession of sinners being swept down the rapids about to make the awful plunge into the burning depths of liquid fire below and the rejoicing hosts in the Inferno coming up to meet them with the shouts of the Devil echoing through the vaulted arches. He suddenly halted and pointing his index finger at the supposed procession, he exclaimed:
"There, do you not see them?"
I was brought up to such a pitch that I actually jumped up and gazed in the direction to which he pointed, while the picture glowed before my eyes and remained with me for months afterwards.
Yet Elizabeth could not achieve conversion, an essential step in finding salvation, because she was unable to follow the revivalist ritual that demanded she expunge her entire personality and feel herself "a monster of iniquity." She found the process "puzzling and harrowing to the young mind." She sought out Finney and confided, "I cannot understand what I am to do. If you should tell me to go to the top of the church steeple and jump off, I would readily do it, if thereby I could save my soul, but I do not know how to go to Jesus."
Finney's answer didn't help. "Repent and believe. That is all you have to do to be happy here and hereafter." Later Elizabeth was to say, "Such preaching worked incalculable harm to the very souls he sought to save. Fear of the Judgment seized my soul. Visions of the lost, haunted my dreams. Mental anguish prostrated my health.... Returning home, I often at night roused my father from his slumbers to pray for me, lest I should be cast into the bottomless pit before morning."
Alarmed by his daughter's condition and shocked by what he considered Finney's vulgar camp-meeting conversions, Judge Cady took his daughter to Niagara Falls to regain her health, accompanied by Tryphena and Edward Bayard. Edward explained to Elizabeth "the nature of the delusion" she had experienced in the revival process, "the physical conditions, the mental processes, the church machinery by which such excitements are worked up and the impositions to which credulous minds are necessarily subjected." The new Calvinism, he told her, was as hypocritical as the old, and he encouraged her to pursue truth in her own way. Edward took her for long walks. At night, his face illuminated by the campfire, he read to her from Scott and Dickens. He delighted in Elizabeth's fine intellect and found her beautiful. Soon she forgot about the fires of hell. Two months later, Elizabeth visited the Edward Bayards. The walks began again, the teasing, the frolicking. One day, caught off guard, Edward told Elizabeth the secret burning in his breast—he loved her. Elizabeth in return admitted what she had previously not admitted even to herself—she was in love with her brother-in-law.
He begged her to run away with him. She resisted. On several occasions Edward, to promote his secret suit, cast himself as chaperone when Elizabeth and her sister Margaret went to New York to visit another sister, Harriet. One moonlit night Elizabeth and Edward strolled along the Battery as he spoke openly of his love. New York was far from Johnstown, and walking with this handsome, attentive man—her friend, her soul mate, her love—she realized that with him she might have the adventurous life she wanted. Seeing the ships glittering in the harbor, he begged her to sail away with him. She was tempted, but she realized that to be with Edward she would have to betray her sister Tryphena, and they would both become outcasts. There was no way for them to have a life together.
To escape this clandestine romance Elizabeth fled to Peterboro, New York, to visit her cousin Gerrit Smith. There she met a lawyer and fellow houseguest, Henry Stanton. Like Edward Bayard, he was ten years her senior; like Gerrit Smith, he was a famous abolitionist. On bright fall mornings two carriages would start off from the Smiths' house for antislavery meetings. They would return at midnight, their passengers tired but exhilarated by what they had heard. Henry Stanton was a clear thinker and a forceful speaker. On a crystalline October day, Henry and Elizabeth went riding through a blaze of autumn colors. As the horses walked "slowly through a beautiful grove, he laid his hand on the horn of the saddle and, to my... pleasure and astonishment," proposed marriage. Elizabeth, by now twenty-four, accepted.
Edward Bayard was infuriated by this turn of events and used his influence with Judge Cady, who was not an abolitionist, to discredit Stanton. The judge threatened to disinherit Elizabeth if she married him. The pressure was relentless, and after a visit to the Bayards in February 1840, Elizabeth broke off her engagement. Henry Stanton told Gerrit Smith he "dreaded the influence of Mr. Bayard," but he would not abandon his suit and continued to write to Elizabeth. In May 1840, Henry told Elizabeth that he would be abroad for eight months, working for the American Anti-Slavery Society. As his departure approached, she saw herself being drawn back into the untenable situation with her brother-in-law, Edward.
Elizabeth made up her mind. There was no time for a trousseau; on Friday, May 10, 1840, she and Henry Stanton were married. She wore a simple white gown and insisted the word _obey_ be stricken from the ceremony. The following day they sailed for England and the World Anti-Slavery Convention. It was said that after her marriage Edward Bayard never allowed himself to be in a room alone with Elizabeth and that he never stopped loving her. As for Elizabeth, thirty-one years later, on a moonlit walk with Isabella Beecher Hooker, she "in an unguarded moment unveiled... as never to mortal eyes before" that even after all the years of marriage and seven children, she was still in love with Edward Bayard and to him went "the steadfastness of my affection." Elizabeth was to say that instead of "seeking solace elsewhere" she had channeled her energies into the fight for woman's rights.
ON THE MORNING of July 19, 1848, the first woman's rights convention was called to order in the Wesleyan Chapel in Seneca Falls. Although the first day was reserved for women, men came and were allowed to stay. As Elizabeth watched the chapel fill, she became fearful. In addition to writing the Declaration of Rights and Sentiments, she had drafted resolutions for the convention. On her own initiative she wrote, "RESOLVED—That it is the duty of the women of this country to secure to themselves their sacred right to the elective franchise."
For Henry Stanton this was too much. He opposed his wife's revolutionary claim to the ballot. He told her that she would turn the proceedings into a farce and that he wanted no part of it. He would leave town rather than attend her meeting. She refused to budge, and he left. Elizabeth Cady Stanton later maintained that her husband had supported her resolution, but her letters and diary show that friction between them grew in proportion to her independence. Judge Daniel Cady came to Seneca Falls and questioned his daughter's sanity, once again threatening to disinherit her if she did not withdraw her outrageous demand for suffrage. Even the ardent woman's rights advocate Lucretia Mott opposed Stanton's resolution, telling her, "Oh, Lizzie! If thou demands that, thou will make us ridiculous! We must go slowly."
Elizabeth, resolution in hand, turned to Frederick Douglass, who was attending the meeting. She had met him four years earlier at a meeting of the American Anti-Slavery Society and had written of him at that time, "He stood there like an African Prince, conscious of his dignity and power, grand in his physical proportions, majestic in his wrath, as with keen wit, satire, and indignation he portrayed the bitterness of slavery, the humiliation of subjection to those who in all human virtues and capacities were inferior to himself."
Stanton asked Douglass what single act would most benefit his own people. "The ballot," he replied. "Yes," said Stanton, "I see that the ballot is exactly what we women need!" Emboldened by Douglass's response, Elizabeth presented her resolution to give women the vote. Amy Post, who had arrived at the Seneca Falls convention directly after a séance with Leah Fox Fish, forcefully endorsed the resolution, as did Mary Ann McClintock. Frederick Douglass too supported the resolution, stating, "Our doctrine is that right is of no sex." After a heated debate at the third and final session of the convention, Elizabeth's resolution passed—barely.
When the convention adjourned, the participants agreed to reconvene at the Unitarian church in Rochester two weeks later. This time the convention was organized by Amy Post. Empowered by Spiritualism, Post was even bolder than the Seneca Falls women. Lucretia Mott's husband, James, had chaired the first convention, but Amy Post persuaded the women to elect a female president, a move so unprecedented that even Elizabeth opposed it. Nevertheless, Abigail Bush (who had become a Spiritualist after witnessing the Fox sisters' powers at the Posts' home) became president. And Elizabeth apologized for her "foolish conduct."
_Susan B. Anthony at twenty-eight. She rejected the "draining trap" of marriage_. (Illustration Credit 4.3)
With a Quaker sense of obligation, Daniel Anthony attended the Rochester convention and signed the Declaration of Rights and Sentiments, as did his wife, Lucy, and youngest daughter, Mary. His daughter Susan B. Anthony would return from the Canajoharie Female Academy the following year and shortly thereafter join Elizabeth Cady Stanton in the battle for woman's rights, one that would last until their deaths and beyond. When their relationship began in May 1851, Susan was thirty-one, almost five years younger than Elizabeth. Susan B. Anthony liked to dress in fashionable plaid and had a number of suitors. However, she suffered from a wandering right eye (and therefore preferred to be photographed in profile). This self-consciousness about her condition contributed to her later image as an "old maid," but it was a conscious choice on her part to avoid the draining trap of marriage. Susan had intense feelings for her female friends and loved them with a passion that others reserved for the men in their lives.
In the next few years, what had begun at Seneca Falls became a powerful movement. First came word from Ohio that a woman's rights convention was being planned there as well. Then Lucy Stone, one of the first women to attend Oberlin College and one who had refused to take the name of her husband, allied herself with Paulina Wright Davis, who lectured on anatomy to women, whose own bodies were often as foreign to them as the moon. These two planned yet another woman's rights convention, to be held the next fall in Worcester, Massachusetts.
In those early years, Stanton often called upon Anthony to come and "make the puddings and carry the baby while I ply the pen." Stanton most often stayed at home to raise her family while Anthony presented her friend's position papers at conventions. Anthony was not a natural public speaker, and even she sought help from spiritual sources. While on the road she wrote to Stanton, "Oh dear, dear! If the _spirits_ would only just make me a _trance medium_ and put the _rights_ into my mouth. You can't think how earnestly I have prayed to be made a speaking medium for a whole week. If they would only come to me thus, I'd give them a hearty welcome."
Spiritualism and the inception of woman's rights were inextricably intertwined. Elizabeth Cady Stanton heard spirit raps in her home. And when Gerrit Smith's daughter Elizabeth (Libby) Smith came for a visit, she too heard them. (Three years later Libby Smith would introduce "bloomers;"—the Turkish pantaloons and short skirt named for the newspaper editor Amelia Bloomer—as a protest against clothes that constricted a woman's movement and impaired her health.) In a letter to his cousin Isaac Post, George Willets chronicled how belief in the spirits had become widespread among the first woman's rights advocates. He wrote,
As to the rapping heard at Rochester and which I heard when then away from the [Fox] girls... I hear it very frequently, and much more now than then.... Thomas McClintock's folks are very sure that they have heard the same. Also Elisabeth Stanton. Gerrit Smith's daughter was on a visit to E. Stanton and heard about it. She went home and told her mother who had full faith in it and the daughter wrote to E. Stanton, a day or two since, that her mother had heard it several times so if it is Humbug it seems to spread fast.
Spiritualism and woman's rights drew from the same well: Both were responses to the control, subjugation, and repression of women by church and state. Both believed in universal suffrage—the equality of all human beings. For women—sheltered, repressed, powerless—the line between divine inspiration, the courage of one's convictions, and spirit guidance became blurred. Not all woman's rights advocates became Spiritualists, but Spiritualism embraced woman's rights.
# CHAPTER FIVE
YOU UGLY CREATURE
BUCK CLAFLIN saw money in the tremendous success of the Fox sisters and the swift spread of Spiritualism. In 1848, after Buck's flight from Homer, he rejoined his family in nearby Mount Gilead, Ohio. The Claflins moved in with Vickie's older sister Margaret Ann, who had married Enos Miles, the town druggist. The Claflin clan descended on Mr. Miles, fighting and squabbling as always. As for Margaret Ann, she had never been exposed to proper middle-class living and was not about to change her ways. Her marriage ended soon after her family arrived, when Enos caught his wife in a local hotel room with a man. The usually mild-mannered druggist pursued Margaret Ann down Mount Gilead's main street brandishing a butcher knife.
A penchant for unrestricted sex seems to have been a family trait. Polly Claflin too had married respectably. Her husband, Ross Burns, was a lawyer who later became lieutenant governor of Kansas. But Polly, a true Claflin, sent notes to admirers via boys in the town. When Ross Burns found out, he divorced her. Polly claimed that Burns had walked out on her while her baby was dying, and after her son's death, for two days she followed Burns everywhere, carrying the corpse in her arms.
Five-year-old Tennessee, who had been farmed out to relatives near Williamsport, Pennsylvania, was soon reading the thoughts of her neighbors. She told a farmer who had lost his calf exactly where to find it. When a Mrs. McDonell offered the child some fruit, Tennessee inquired, "But where is the best fruit?"
"What fruit?" asked Mrs. McDonell.
"The fruit you put away," answered Tennessee.
It was true. Mrs. McDonell had hidden the best fruit before the child's arrival.
All of this skeptics might explain away, but there was more: Newspaper accounts at the time relate how Tennessee awoke one night screaming in terror. She had seen a vision of a building in flames. She described the fire to her relatives and then to reporters in minute detail, including what the firemen were wearing and how they had been unable to reach the blaze. A month later, just such a fire took place at the Dickinson Seminary in nearby Williamsport. So precise was the child's description that she was accused of starting the fire. A subsequent investigation proved her innocence.
When Buck heard of the ruckus his child had caused and how her fame was spreading and people were flocking to her for advice, he bought himself a wagon with a bright red canopy trimmed with bouncing blue cotton balls, picked up Tennessee, and took her on the road, billing her as the "Wonderful Child." Tennessee told fortunes, located objects, predicted the future, and delivered spirit messages. After a time, he took her back to Mount Gilead and teamed her up with Victoria in imitation of the Fox sisters. Buck moved his daughters into Mrs. Webb's boardinghouse, where they conducted séances at tables that resounded with raps, channeled spirit messages, and produced spirit music—for a price. Buck collected the money.
Utica was of little use to her father. Although she was very pretty, she was slow, self-occupied, and could not convince an audience that she was in touch with the spirit world. She was consigned to help her mother brew Buck's "Life Elixir." Utica was soon addicted to corn whiskey and opium.
While Tennessee accepted her powers with a light heart, often resorting to generalities about love and money when no spirit inspired her, Victoria was intense and erratic. Buck, expecting only a performance, was dismayed when Vickie lost herself in deep trances. She would emerge from this state debilitated and unaware of what had happened. Many years later she wrote that in one such trance she had told a man, on his way to New York to meet his family, that his wife's spirit said that three hours previously their ship had foundered in a storm and both she and the children had drowned. The man was alarmed but continued on his way only to find that the ship had sunk at the precise time Vickie had specified. On another occasion she announced that she'd encountered the spirit of a young girl who said she had come for her brother. Only one man at the séance table had a deceased sister, but he was discounted because he was only twenty-eight. The following morning he was found dead in his bed.
In order to intensify his daughters' performances, Buck starved them for days at a time. Victoria would later tell her biographer, Theodore Tilton, that "in my own home I was treated with a cruelty that still beclouds the memory of my early days." Buck brutally beat her with braided willow withes or a stick of firewood, and Vickie often intimated that he sexually abused her as well. Years later, Vickie would say that Buck made her "a woman before my time." And she would tell Lucretia Mott and her sister Martha Coffin Wright, "All that I am I have become through sorrow."
By her own account, after a particularly "savage" episode, Victoria went down to the stream behind their house and began to wash herself clean when she saw a phosphorescent sparkling in the water that materialized into a powerful spirit, "a majestic guardian" next to whom Buck Claflin was but a puny toy. She saw the spirit clearly—a stalwart young man wearing a white toga and a laurel wreath in his tightly curled brown hair. He spoke to her with words of comfort, telling her that she would "rise to a great distinction, that she would emerge from her poverty and live in a stately house, that she would win great wealth in a city crowded with ships, that she would become the ruler of her people."
At fourteen, finding no way out of an intolerable situation, Victoria manifested the behavior that would typify the rest of her life. Unable to protect herself from the parents who enslaved her emotionally, abused and exhausted, this child-woman retreated to a safe world. Wracked with rheumatism, shaking with fever, and unable to leave her bed, she conversed daily with her two dead sisters, Delia and Odessa. "I would talk to them as a girl tattles to her dolls." Angels also came and went, and these "gracious guests" became her "constant companions."
When Buck began losing money because of Victoria's illness he consulted a local physician, Dr. Canning Woodhull, who, at twenty-eight, had a reputation for being a rake and too fond of alcohol. Woodhull visited Victoria and was taken with the beautiful, fragile young girl. He prescribed a nourishing diet, regular school lessons, and walks in the fresh air to revive her spirit.
Victoria later told Tilton that when she recovered, Canning Woodhull asked her to the Fourth of July picnic, and upon returning her to her door said, "My little puss, tell your father and mother that I want you for a wife." Roxy, like many wives of child abusers, had closed her eyes to her husband's worst offenses but instinctively saw in Woodhull's proposal a way to end an unexpressed problem. Victoria, with unusual delicacy, later wrote that her mother was "not unwilling to be rid of a daughter whose sorrow was ripening her into a woman." Buck, however, was determined to hold on to his daughter and told Dr. Woodhull to stay away. Four months later, in November 1853, fifteen-year-old Victoria Claflin eloped with Canning Woodhull. "My marriage was an escape."
WHILE VICTORIA WOODHULL embarked on a traditional marriage, a radical fringe seeking political, religious, and personal equality was pursuing alternative ways of life. Inspired by European thinkers, so-called utopian communities began to proliferate in the United States. After his New Motor was trampled by the mob in Randolph, John Murray Spear founded a commune at Kinatone, New York, where children were raised in a common house and marriage was abolished in favor of complete sexual freedom. Three decades after the Reverend John Humphrey Noyes had incited the sexual frenzy at Brimfield, he founded the community of Oneida in upstate New York. In 1869, Noyes wrote, "Since all the saints are on a par together in equal dedication to faith, communism among them in a sex relation as in all others is the only logical arrangement." Therefore, "sexual intercourse should be no more restrained by law than eating and drinking." Other such communities included Brook Farm in Massachusetts, New Harmony in Indiana, Memonia and Berlin Heights in Ohio, and Modern Times, founded by Stephen Pearl Andrews in the pine barrens of Long Island, forty miles east of New York City.
The brilliant Stephen Pearl Andrews had mastered thirty-two languages. At thirty-one, on a trip to England in 1843, he became interested in the phonetic shorthand system of Isaac Pitman and subsequently introduced it to America. In 1849, Andrews became a correspondent for the _New York Tribune_ , owned and edited by Horace Greeley, for which he recorded congressional debates in his loops and dashes. Stephen Pearl Andrews wanted to reform all aspects of society and in 1851, founded the community of Modern Times as a model to demonstrate what he termed his "New Age" philosophy. Andrews proposed a new domestic arrangement in which sexual relations were based solely on "spiritual affinities." The cofounder of Modern Times, Josiah Warren, thought Andrews's free love views were "more troublesome than a crown of thorns," but Andrews insisted that they were nothing more than the affirmation that free love was "the antithesis of enslaved love."
For Andrews, a cool intellectual, the idea that no government, church, or law should encroach upon an individual's freedom to choose a mate seemed logical, but in practice his experiment led his followers into a thicket of sexual aberration. Modern Times's colonists tied a red thread to their index fingers to announce that they were married and untied it when they decided not to be. No one questioned a child's paternity. In time this community became the refuge of fanatics and faddists. Men and women paraded about naked. One woman lived solely on beans until she died.
Andrews, undaunted, then founded the New York Free Love League and held twice-weekly meetings to propound his philosophy. Horace Greeley, appalled by these liberal sexual views, attacked the league in his newspaper. Shortly thereafter Andrews's clubhouse was raided, and four people were arrested for disorderly conduct. Greeley then carried on a written debate with Andrews in the _Tribune_ on the subject of the relationship between the sexes. Greeley wrote that no matter what the feelings or circumstances, the marriage bond was indissoluble. Andrews argued the utopian view that the only relationships that should endure were those based on love. Andrews's further statements about sex were so inflammatory that Greeley refused to print them, so Andrews published his side of the debate in a book, _Love, Marriage, and Divorce, and the Sovereignty of the Individual_ , a virtual textbook on free love.
_Stephen Pearl Andrews. He believed in the "science of Spiritualism" and in free love_. (Illustration Credit 5.1)
The overwhelming majority of Americans favored Greeley's view that traditional marriage, sanctioned by church and state, was the only proper arrangement. For women, a middle-class marriage was depicted in books and sermons as an entry into a safe world of hearth and home. Married women were taught to be the power behind the throne, responsible for men's moral conscience, raising sons to run the world. Horace Greeley, in the debate with Andrews, wrote that sexual relations should be for the purpose of procreation only and declared, "All sexual relations that do not contemplate this end are sinful." He believed that divorce, or even remarriage while a former spouse was alive, was "adultery" and that "every form of sexual relationship except the union for life of one man with one woman" was a sin. Greeley practiced what he preached, but his own marriage was far from a typical, traditional union, for his unfeeling adherence to the conventions of the day tipped his wife's fragile constitution into madness. And the life they lived represented, in a monstrous fashion, the dark side of the celebrated American dream of "woman's sphere."
IN AUGUST 1849, Horace Greeley arrived in Rochester, and Elizabeth Cady Stanton took advantage of his visit to invite him to meet with her to discuss coeducation. The conversation was inconclusive, but Greeley's real purpose lay elsewhere: His wife, Mary Cheney Greeley, had instructed him to fetch Kate Fox. Mary was convinced that this remarkable eleven-year-old was the person she needed to contact her recently deceased child, Arthur, whom the Greeleys had called Pickie.
Horace Greeley met Kate, persuaded her to accompany him to New York, and escorted her to his ugly, barren Turtle Bay farm on the East River. Kate was to provide the means by which the inconsolable Mary could keep her child with her in death as in life. In return, Greeley promised Kate that the _Tribune_ would support her. Greeley was "convinced beyond the shadow of a doubt of the [Foxes'] perfect integrity and good faith" and assured his readers that the sisters themselves did not cause the rappings to occur.
The first séance Kate Fox conducted at the Greeleys' farm was attended by Jenny Lind, "The Swedish Nightingale," the operatic soprano whom P. T. Barnum had imported to New York. As raps began to shake the Greeleys' table, Lind demanded of Horace, "Take your hands from under the table!" Greeley obliged by clasping his hands on top of his head, but the raps did not stop. Within minutes, Kate Fox fell into a trance and summoned the spirit of Pickie.
With Kate Fox as the trance medium, Mrs. Greeley and her deceased five-year-old "dream boy," Pickie, were soon communicating daily. However, in transmitting messages between Pickie and his mother, Kate sensed something ominous. Like others who had come to live under Mary Cheney Greeley's roof, Kate felt contaminated by this woman's bizarre behavior, a mania that demanded obedience to her intractable whim. The Greeleys' dark and unfurnished home was oppressive. The food, through slavish devotion to the regimen of Dr. Sylvester Graham, was confined to bran bread, overcooked rice and beans, porridge (often rancid), and puddings. Kate Fox, undoubtedly aware that the mighty _Tribune_ could make or break her career, knew she must remain until winter, but she wrote to Amy Post in Rochester, "I am very lonely. Oh, how I miss you." To her brother David she lamented, "Why did I leave my mother?... My head aches so I can hardly write." Perhaps Kate's powers revealed to her the truth of Pickie's nightmare life and violent death and that of the four Greeley infants who had died before him. For this was no average grieving mother and no loving, protective father. In fact, Mary Cheney Greeley and her husband, despite their public image of marital harmony and rectitude, played out a vicious drama of domestic derangement.
Mary Cheney had met Horace Greeley in 1834, when she was twenty-two—a thin, nervous girl with prominent, hyperthyroid dark-brown eyes and rapid speech in which the words tumbled out. Bright and well educated, she had left the boredom of her Cornwall, Connecticut, home to teach at a girls' school in New York City. Mary boarded at a "Graham House" on Barclay Street. Sylvester Graham, the founder of these houses, had been a sickly orphan child. As an adult he'd studied physiology and diet, cured himself, became a popular lecturer, and established boardinghouses that adhered to his principles: no alcohol, caffeine, meat, or butter or other shortening. Dr. Graham was particularly known for his tasty, ginger-colored rectangular crackers. Emerson called him "the poet of bran bread and muffins." He believed in exercise, loose clothing, drinking lots of water, and confining sexual intercourse to twelve times a year. The diet at Graham House featured bread, rice, molasses, milk, vegetables, and pies made with fruit juice as a sweetener. Mary Cheney was Dr. Graham's dedicated disciple. At Graham House the food was excellent, but it was often gray and nearly inedible when Mary cooked it.
Horace Greeley, then twenty-three, was a fellow boarder. The son of impoverished, alcoholic parents who owned a small farm near Amherst, New Hampshire, Greeley had escaped to New York and the world of the intellect, starting his own small weekly publication, _The New Yorker_. They seemed a good match: Horace Greeley was lost in a world of ideas, and Mary was "crazy for learning." They were married in July 1836. Mary came into the marriage with savings of $5,000, but in the first year of their marriage _The New Yorker_ nearly foundered in the great financial crisis that gripped the country. In 1837, one out of seven people in New York City was receiving alms. The city was filled with beggars and homeless people who slept in the streets. By the following Christmas, Mary's money was gone, and she resumed teaching to help support them both.
_Horace Greeley. He was admired by the public but despised by his wife_.
_Mary Cheney Greeley, 1834. At twenty-two she was "crazy for learning" but was cast into a domestic hell_. (Illustration Credit 5.3)
In the winter of 1838, Mary's firstborn son died at three months, and five months later she suffered a miscarriage. At the time, Horace Greeley was in Albany and wrote a friend, "My wife is in bad health and is left utterly alone in New York through the worst part of the year—a circumstance which pleases neither of us." For Mary the horror had begun: She felt herself trapped in a world from which she was desperate to escape. When Horace returned from Albany that winter, he found a note from Mary awaiting him at his _New Yorker_ office, informing him that she had abandoned wedded life "because her fire would not burn." But she did not run away, or perhaps she left and then came back. Her husband found her at home.
Six months later Mary suffered another miscarriage. This woman who had been "crazy for learning" now entered a nightmare world of medical quackery. For the sake of propriety, female bodies were not examined by male doctors, who prescribed treatments nonetheless. Mary was bled and purged and given calomel and so much morphine that the world became a blur. At this time Horace wrote to Margaret Fuller, a famous writer and woman's rights activist whom Mary Greeley greatly admired, "Mary is terribly ill and downhearted—a miscarriage of the worst kind and a great danger of the loss of her eyesight. She is now unable to bear the light and must be kept so all winter, without reading or doing anything, but I think she will ultimately recover."
The following year another son was born to this distracted, apathetic woman whose fragile psyche and body were being torn apart by a sexual routine she loathed and over which she had no control. Day after day, Mary sat listlessly on a bare mattress in a darkened room, unable to attend to herself, much less her infant. The baby lay in a crib in that bleak room and soon died. A pattern had been established: "Mrs. Greeley is in her usual bad health," wrote her husband in the winter of 1841. In the same year Greeley launched his penny daily, the _New York Tribune_ , which was housed in a second-floor attic at 30 Ann Street in lower Manhattan, some three-and-a-half miles south of the Turtle Bay farm. Often Greeley slept there at night. On Christmas, he stayed all day in his office. He became increasingly obsessed with work.
When he was at home Horace referred to his wife as "Mother" and slavishly waited on her, ignoring her deep depression, her tantrums. An accommodation was made: Mary Greeley, though confused and demented, was left to be supreme ruler of a nightmare realm in which her husband cast himself as an ineffectual observer. Being impotent in this domestic world, Greeley asserted the one aspect of virility remaining to him by regularly impregnating his wife.
For Mary Greeley sexual intercourse with her husband led to disastrous results. One morning after she had submitted to an unwanted conjugal visit, she informed Horace that she was leaving. That same afternoon she sailed for Europe and did not return for five months. "Mrs. Greeley's life is all absorbed in the pettiest, meanest cares—and going off for months," wrote Horace. When she returned, her husband again impregnated her in what was now a pattern of creating children to be neglected and ignored by his distracted wife. A public that was to find Horace Greeley's eccentricities amusing—his absentmindedness, his one blue sock and one brown, the way he sometimes mistakenly sent _The New Yorker_ to _Tribune_ subscribers—could not know that these traits extended to an increasing diminution of awareness of the environment in which he lived, a blunting of his own responsibility for the horrendous acts that took place in his home.
The Greeleys' Turtle Bay farm was nicknamed "Castle Doleful" by Horace. Here a desperate Mary, seeking a way out of her domestic hell, abandoned the medical doctors who had damaged her mind and body and turned to phrenology, symbols, omens, and finally to the spirit world. Mary began conversing with the spirits that she said inhabited the house and banned almost all visitors, saying that they created "bad magnetism." She sent people away if they failed to acknowledge the spirit raps that she heard. Servants invariably quit after a few days, no matter how much they needed the money. She would let no light into certain rooms, no fire could be lit on certain days, no tree was to be cut down because it might harbor spirits, no animal was to be slaughtered—old cows fell down and died in her overgrown pasture.
On the rare occasions that Horace Greeley was at home, it was he who built the morning fire, milked the cows, and cleaned the house. Greeley wrote that Mary kept house in a "Castle Rackrent" fashion. The house remained bare save for beds and chairs—there were no curtains, no rugs, no pictures. Meals were sparse, erratic, dismal.
Mary would observe laundry day by washing a single garment over and over perhaps a dozen times, after which, hands red and raw, she would fly into a screaming fit. Catharine Beecher's _Treatise on Domestic Economy_ , which had become the bible of domesticity, included twelve pages "On Washing," eleven pages "On Whitening, Cleansing, And Dyeing," and seven pages on preparing to iron a garment. Her list of some twenty items required for doing laundry was so demanding and detailed that it took massive concentration just to read it. In addition to the ironing board itself, she specified
_A séance in 1871. Thousands of Americans gathered around parlor tables to experience spirit manifestations_. (Illustration Credit 5.2)
a woolen ironing-blanket, and a linen or cotton one, to put over it; a large fire, of solid wood and charcoal, (Unless stoves or furnaces are used); a hearth, free from cinders and ashes, and a piece of sheet-iron, in front of the fire, on which to set the irons;... three or four old holders, made of woolen, and covered with old silk, as these do not take fire like cotton or linen; a ring, or iron-stand, on which to set the irons, and a piece of board to put under it, to prevent very hot irons from scorching the sheet; a linen or cotton wiper, one for each ironer, and a piece of beeswax, to rub on the irons when smoked. There should be, at least, three irons for each person ironing. There should also be one or two large clothes-frames, on which to hang the clothes, when ironed; and a small one for the smaller articles....
Mary Greeley had neither time nor patience for Catharine Beecher's idealized, endless list. She would thrust her iron into a fireplace full of ashes and vigorously press soot into a garment. Then, observing her work, often she would hurl the iron across the room. Her husband found that laundry day was a good time to steer clear of Mary and remarked on her "inability to accomplish anything." Often Horace would return home on the eleven o'clock Harlem stagecoach, arriving at midnight to find his wife crouched in a corner, obsessively scrubbing the floor. Sometimes she would scrub the same spot till dawn, before falling into an exhausted sleep on the floor.
It was in 1844 that their blond, blue-eyed son, "Pickie," had been born. Mary determined to keep this child far from the world. When Pickie began to walk, she locked the gates of the farm so no visitor could arrive unannounced. No children were permitted to play with Pickie. He was kept on a strict Graham diet and allowed no "dirty food"—meat or sweets or condiments. Mary insisted that he see her visions, hear the spirit raps she heard, and talk to angels as she did. Eventually, the child complied.
To keep Pickie to herself and presumably to avoid sexual intercourse, Mary insisted that her husband stay away much of the time. He began lecturing throughout the Midwest, often sleeping in his red-and-blue blanket roll on the floor of railroad stations. When Pickie was two, Greeley addressed a letter to his son, "Your sore, sick father will return home this week to his own crazy house and his boy's bright smiles." Then, addressing his wife, he wrote, "My piles are ugly, especially a great new one, and my general health is not good.... I hear nothing to interest you." Under such circumstances, their reunions could hardly have been joyful or sexually pleasurable.
But return he did, and nine months later Mary gave birth to another child—a girl. But she wanted only Pickie and the hermetic world they shared. In a series of letters to Margaret Fuller, Horace Greeley chronicled the horrors in his own home as if he were a detached observer. He made no effort to correct the situation or to find help for his wife's mental condition. In these letters he observed that his life had become so bleak that he had taken to boarding in the city. The little girl lay in her basket, "while Mother often said she wished her dead on account of the labor and anxiety she caused." The baby became so weak from hunger that she could barely cry but lay staring with eyes exaggerated by her gaunt countenance. "Pickie regarded her as a rival and an obstacle to his enjoyments, urging that she should be given away." No one lifted her from her basket for days at a time as she lay in her own excrement. She died on May 6, 1847. Wrote her father, "I never saw a creature so patient under suffering and so grateful for kindness, and her love for me was a precious wonder.... It is awful to think how friendless and often, I fear, consciously so, our blessing was.... With proper treatment and care, I am sure my darling would have survived.... A week after the funeral Mother said to [Pickie], 'Are you not sorry for the death of little sister?' 'Yes,' he replied, and after an instant's thought added, 'but you couldn't take care of us both.' "
Pickie dictated a letter to Margaret Fuller: "I had a pretty little sister. We called her Dotty. She was so little. She had very bright eyes. She looked at me so still. I did not like her eyes. She has gone to another world and I am very sorry."
Pickie was then three and a half, but Mary, in her effort to keep him as her own, still dressed him in a long white linen shift and diapers. His blond curls remained uncut. But Pickie had begun to assert himself. When he showed the least tendency to disobey, his mother whipped him, sometimes several times a day. Wrote Greeley, "When I came home at 10½ p.m. he had been hours in bed but was still broad awake (his mother had whipped him twice). His mother whips him often but never rules him, and I have no voice in his management and never can have." Greeley added the observation that Pickie was "sick of hope deferred."
The child eked out his lonely existence playing with blocks or pebbles. Increasingly he came to resent his mother. He fantasized that he had been adopted and that his real mother was a kindly Dutch woman who had been too poor to keep him. As Pickie's temper grew more erratic, the beatings grew more savage. Mary took to threatening him with spiritual terrors: phantoms that would plunge him in boiling oil, invisible devils that lived in their upper chamber. Horace wrote, "He is governed and restricted and cramped till he hates all law and all authority.... When beaten, as he was when we rode out last Saturday, he looks her in the eye with an aspect of indignation and grief, yelling, 'O you ugly creature!' He does not get whipped so often as he did, for his Mother has adopted the plan of shutting him up in the upper chamber, which speedily brings him to subjection."
When Pickie was five, Mary Greeley encouraged her husband to spend the winter in Washington. But before departing, once again he impregnated her, and once again a daughter was born. Within the first eight months of her life baby Ida suffered "malaria, diarrhea, sprue and cholera." Miraculously she survived, but Pickie too contracted cholera, and he died within twelve hours.
Pickie had been swept away, but even then Mary refused to loosen her iron grip. It was at this time that she conceived the idea of summoning the eleven-year-old Spiritualist Kate Fox to establish communication with her son. In the four months that Kate lived with the Greeleys, every day Mary spoke to her "angel" Pickie through Kate. Spirits occupied the house. Kate wrote a letter to Amy Post describing one séance: "The piano was sweetly played upon by spirit fingers and the guitar was played, then taken up and carried above our heads. Each person in the circle was touched." She related how the spirits "ring bells and move tables all when our feet are held. We have convinced many skeptical people." But of Mary Cheney Greeley, Kate Fox wrote, "How I hate her!"
# CHAPTER SIX
TRUE WIFE
BY THE time Victoria Woodhull became a famous woman's rights advocate, she would use the example of her own first marriage to condemn all oppression in marriage. Her detractors would object that, just as her many spiritual "predictions" were revealed in retrospect and could not be proved, so she had invented incidents in her own marriage simply to illustrate the subjection of women. Her critics claimed that Dr. Canning Woodhull was a gentleman who had taught this wild child manners and the ways of polite society. That was true. But it was also true that he was an alcoholic and a morphine addict.
According to Victoria Woodhull's highly colored account, Canning had not shared her bed since the third night of their marriage. He slept at the local brothel. She knew that because when the money ran out and she went to look for him, following his trail from tavern to tavern, she finally found him eating pigeon pie and drinking champagne, surrounded by whores.
At night she would stand by the window, waiting. Sometimes in the early morning hours she would hear the faltering steps of her husband as he stumbled along the cobblestones. One night she saw Canning Woodhull in a vision, "walking unsteadily across a road that I instinctively knew was on the outskirts of the city. With my soul I watched over him as he went into the Freemont Hotel and lay down on a sofa. As he lay down, his billfold fell on the floor." A clerk moved quickly, picked up the billfold, and furtively stuffed it behind the sofa cushion. When Dr. Woodhull returned home the following morning, he could not remember where he had been. Vickie went to the Freemont Hotel, identified the clerk, and retrieved her husband's money.
After only six weeks of marriage she discovered a letter in her husband's jacket pocket. "Did you marry that child because she too was _en famille?_ " the writer asked. Reading further, Vickie realized that this letter was from her husband's former mistress, whom he had sent to Terre Haute, where she gave birth to a boy. Woodhull was also sending her money. Although Vickie was married to a man "mostly with his cups and his mistresses," she still sought his love. When he was sober Canning Woodhull had a certain sweet passivity and he was attentive to her. She called him "Doc" and marveled at how much he knew. He was less harsh than her father.
Winter of 1854 found the Woodhulls living in a tenement in Chicago. Just after Christmas, Vickie, attended by her "half-drunken" husband, gave birth to a son. For years thereafter, she dwelt on her "mortal agony" and insisted that she remembered "icicles clinging to her bed post." After the delivery, Dr. Woodhull promptly left. A next-door neighbor miraculously appeared to wash and feed the new mother. When Vickie produced no milk, the neighbor wrapped the baby in a blanket and carried him to another woman with a newborn son who nursed both babies. On the third day after her son's birth, Vickie became delirious. Childbed fever raged through her body. Three days later Mama Roxy appeared, declaring that she had been summoned by the spirits. Victoria remembered nothing of her mother's arrival. That same evening Woodhull returned from a drunken bender and both husband and mother—one employing medicine, the other the spirits—snatched Vickie back from "the iron door."
The son, Byron (named for the poet, who had died thirty years earlier), was an imbecile who never developed teeth and whose speech was never more than a series of grunts. Sometimes Victoria said Byron was an idiot because Doc had kicked her in the stomach while she was pregnant. At other times she said that, as an infant, Byron had fallen from a second-story window. Sometimes, she blamed herself.
Victoria still believed that marriage was a sacred institution, and she "wrestled with God" to find a way to live with this arrangement. The next winter, when Doc once again deserted her, Vickie left Byron with her mother and set out in search of her husband. Melodramatically, she later described herself as wearing a thin calico dress, no underwear, and only India rubbers on her feet as she ran through the frozen streets of Chicago. In that condition, she burst into the dining room of the boardinghouse where Doc sat at dinner with his fellow boarders and the mistress whom he called his wife. When she saw her husband so comfortable, so warm in the light of the fire, her rage exploded. She was lonely and hungry, she said, and she was his "true wife." She told of all that had befallen her, until there were tears in the eyes of the boarders and the mistress's face turned scarlet. Then and there the listeners forced "the harlot to pack her trunk." Vickie's shamed husband took her by the hand, and together they returned to their rented hovel.
They needed money and a new life. "Go West, young men, and grow up with the country!" Horace Greeley commanded in the _Tribune_. In 1849, gold was discovered in California, which a year later became the thirty-first state. In the next seven years the population of California grew from fifteen thousand to three hundred thousand as prospectors mined $450 million worth of gold. Commodore Cornelius Vanderbilt, seeing a way to increase his shipping fortune, provided a quick route to the goldfields across the isthmus of Nicaragua. When his crew found the San Juan River too dangerous and threatened mutiny, the Commodore took the wheel, tied down the safety valve, and ran his boat up the river to its destination. In 1858, with gold still attracting people from across the nation, Vickie and Doc set out for San Francisco with four-year-old Byron.
This was a place of wealth, teeming with men as rough and raw as the roustabouts of Buck Claflin's youth. But soon Victoria found herself and her family "beggars in a land of plenty." No one had time to listen to her visions and predictions unless they concerned the location of the precious ore they were seeking. Men outnumbered women twenty to one and found their pleasures in taverns and brothels. Desperate for money, Victoria answered an ad to become a "cigar girl" in a tavern. After one day of crude remarks, undisguised groping, and bold sexual propositions, she was fired by the owner, who told Vickie that her reticence was bad for business and that he needed to have "somebody who can rough it." But he walked her home, and on meeting Doc Woodhull, discovered that he was a fellow Freemason. He gave Vickie a $20 gold piece and wished her good luck.
Day after day Vickie sat sewing, the only work she could find. Her average wage was 38 cents a day. Anna Cogswell, an actress, hired Vickie to alter her costumes, and they became friends. When Vickie confessed, "It is no use. I am running behind, and I must do something," Anna suggested that she appear onstage. Vickie's unusual facility for instantly memorizing a text served her well here, as it would later in her career. She said that she could be given a new part in the morning, rehearse it by day, and be letter-perfect by night. She also boasted that overnight her wages jumped from $3 to $52 a week, but that was quite a different matter.
To be a so-called "actress" was not as difficult or degrading as being a "cigar girl," but it was an entry into the world of the demimonde. In theaters, in America and abroad, from across the gas footlights affluent men chose the women who would later occupy their beds. The ballet and opera choruses also provided such opportunities for sexual selection. The tutus of the ballerina fully exposed the leg, while in the theater, low décolletage and tightly laced bodices emphasized breasts that were semi-exposed, elevated, and served up like ripe melons. Victoria wore such costumes for her part in _The Country Cousin_ and in _New York by Gaslight_. When the evening's performance finished, the revels began.
It was at one of these so-called parties that Anna Cogswell introduced Victoria Woodhull to another destitute "actress," Helen Josephine Mansfield, called Josie. The daughter of a reporter for a Boston newspaper, Josie had come with her parents in 1852 to Stockton, California, where her father was killed in a duel. Her mother then moved to San Francisco and married a professional gambler named Richard Warren. Josie would later say that her stepfather sexually molested her when she was twelve and continued to do so for three years, threatening to maim her if she ever spoke of it. Josie did not tell her mother, who by this time was drinking heavily. By the time Josie was fifteen, her stepfather was selling her to other men. Once he forced her to have intercourse with a wealthy attorney named D. W. Perley, after which he popped out from under the bed, pointed a loaded pistol at Perley's head, and demanded $500.
To escape her stepfather's blackmail schemes, Josie eloped with a strolling actor named Frank Lawlor, who often performed onstage with Anna Cogswell. Victoria Woodhull's and Josie Mansfield's paths would cross again, a decade later and three thousand miles away, in New York. By then, neither would be an actress and neither would be destitute.
Soon Vickie came to hate what she euphemistically referred to as "the stage." At twenty, she was supporting a feebleminded son and a husband who, now lost in a fog of drugs and alcohol, had become equally dependent on her. After two years of acting and casual prostitution Vickie was becoming increasingly desperate. "I am meant for some other fate," she told herself. She later wrote that one evening while performing in the ballroom scene of _The Corsican Brothers_ , suddenly she heard her sister Tennessee's voice calling out, "Victoria, come home," and saw a vision of Tennie beautifully dressed in a striped red-and-white French silk frock, standing beside Mama Roxy. "Come home!" her sister called out again, beckoning her. Vickie bolted from the stage and, still wearing her pink silk ball gown and silver slippers, ran through the night to her lodgings and threw her few belongings into a bag. The next morning Vickie, Doc, and Byron, then six, boarded a steamer heading for New York.
Whether Victoria was a reliable witness to the events of her own life is debatable. She was never to explain why the spirits summoned her or why she fled, but abrupt escapes "guided by the spirits" were to become a pattern. On the voyage she recalled that she was "thrown into such vivid spiritual states" that she "produced a profound excitement among the passengers" with her trance speaking and her visions. Victoria claimed that when at last she landed in New York and took the train to Columbus, Ohio, Tennessee greeted her at the door wearing the frock Vickie had seen in her vision. Mama Roxy said that she had instructed Tennie to use her powers to "send the spirits... to bring her [Vickie] home."
The Claflins were prospering: By 1860, Tennessee had become her father's golden goose. He would ride into town in his bright wagon with his daughter seated at his side, take rooms at a boardinghouse or hotel, and place an ad in the local paper:
**A WONDERFUL CHILD!**
**Miss Tennessee Claflin
Who is only fourteen years of age!!**
This young lady... has been endowed from her birth with a super-natural gift to such an astonishing degree that she convinces the most skeptical of her wonderful powers. She gives information of absent friends, whether living or dead, together with all the past, present and future events of life.
She can see and point out the medicine to cure the most obstinate diseases—even those that for years baffled our best physicians—and can direct salves and liniment to be made and used that will cure old sores, fever sores, cancers, sprains, weakness in the back and limbs and other complaints....
She will point out to ladies and gentlemen their former, present and future partners... and when required will go into an unconscious state and travel to any part of the world, hunt up absent friends, whether dead or alive, and through her they will tell the inquiring friend their situation and whereabouts, with all the events of life since they last met....
She may be consulted at her room [address inserted] from the hours of eight o'clock a.m. to nine o'clock p.m. Price of consultation, $1.00.
Tennessee worked thirteen hours a day while Buck collected the money and sold his alcohol- and opium-laden cure-all, now renamed "Miss Tennessee's Magnetic Life Elixir," at $2 a bottle, double the old price. "It was a hard life," Tennie would later declare. "I was forced to humbug people for the money."
When Victoria rejoined the family, she found that her father had amplified his daughter's real powers with his own chicanery. When Buck arrived in town, he would visit the local cemetery and copy information from the tombstones. Also, there was a lively black market in what came to be known as the "Blue Book," which listed the local people most likely to attend séances and gave their family histories, including the names and dates of those who had died, as well as information on present romantic or domestic attachments and personal habits.
Fortune-tellers and thought-readers were known to use a technique called "muscle-reading": If a member of the audience called out the alphabet, the mind reader could guess the correct letter by watching his expression. In another trick, if a person was asked to write six names on a slip of paper, one of which was that of a dead relative, the writer usually did not hesitate on the correct name and thus revealed which one to choose.
Then there was "one ahead," which made it possible for fraudulent mediums to read messages written inside tightly folded pieces of paper. The medium drew a folded piece of paper from a bowl and, with a look of intense concentration, pressed it against her forehead and read the message "through the power of her mind." The medium then unfolded the paper and supposedly read the message aloud. The trick lay in staying one message ahead of the audience. The first message was made up by the medium, then confirmed by an accomplice planted in the audience. What the medium had in fact unfolded and memorized was the next message she was to recite.
Aside from Buck's tricks, real powers manifested themselves in the twenty-two-year-old Victoria as they did in the fourteen-year-old Tennie. In a state of almost ecstatic excitement, Victoria would see auras around her clients and know immediately when someone was sick. Once she told a judge, the father of a seemingly healthy fifteen-year-old daughter, that the girl would die within the year and her prediction came true.
The first records of Victoria's magnetic healing go back to this time. She derived her techniques from her mother's early doctoring, which Roxy in turn had learned from observing carnival demonstrations of Franz Anton Mesmer's animal magnetism. The right hand was thought to contain positive magnetic energy, the left negative. Between the two hands a current was generated similar to that observed by Mesmer and the Marquis de Puységur. By passing one's hands along the body without actually touching it, healing magnetic fluid was said to be stimulated. If one believed.
By 1860, magnetic healing had become a popular alternative medical treatment. In Elmira, New York, sixteen-year-old Olivia Langdon (called Livy) was returning home on a winter's day when she slipped on the ice. She was carried to her bed. By morning she was paralyzed. For two years Livy Langdon remained in this condition, lying flat on her back in a darkened room. A parade of physicians visited her, but all hope of recovery slipped away until, as a desperate measure, a Spiritualist healer, Dr. Jarvis Rogers Newton, was called in. He marched into Livy's bedroom, pulled back the heavy draperies, and flung the windows open. He said a brisk prayer, then put both of his arms around Livy's shoulders and announced, "Now we will sit up, my child." Livy sat up. Newton then declared, "Now we will walk a few steps, my child." And she did.
Years later Livy's husband, the author Samuel Clemens, known as Mark Twain, asked Dr. Newton the secret of his success. Newton replied that he wasn't sure himself, but he suspected that some form of electricity passed from his body into that of the patient. After this explanation, Clemens began to recommend faith healing, although he was aware of the frauds perpetrated by many so-called healers. In _Huckleberry Finn_ the king, a charlatan, declares, "I've done considerable in the doctoring way.... Layin' on o' hands is my best holt—for cancer and paralysis, and sich things and I k'n tell a fortune pretty good when I've got somebody along to find out the facts for me."
One thing was sure: Faith in the healer was essential to the cure. And Victoria inspired faith. Women came to her to cure their diseases, but just as often, to unburden themselves of their deepest secrets with tales of sexual abuse and perversion, of sickness induced by maltreatment and neglect, of poverty and oppression. Women confided their cares, their hopes, their fears and longings, and Victoria tried to help them with clairvoyant advice and predictions. All her life she would carry with her the memory of the miseries of the "thousands of women" who came to her for hope.
Victoria believed in her own power to transmit magnetic energy that would put the person she touched under her control. When she greeted people, she took both of their hands in hers. It was often reported that in her grasp one felt a current of energy, a remarkable warmth and light flowing throughout one's body. Victoria also believed that the animal magnetism she generated could disable her enemies. She had learned this early on, when people mocked her preaching and she would stare at them, unblinking, until they turned away. One woman was so affected by her gaze that, in front of a crowd, her false teeth popped out of her mouth.
Victoria would tell fellow Spiritualists that on the periphery of her vision she could see objects—eggs, small bottles, lorgnettes with metal frames—that tumbled and turned slowly in space before alighting at her feet. She maintained that suddenly her water glass would fly out of her hand, her pot of lip rouge would whirl away and plop down in the middle of her wineglass. She said that before her concentrated gaze rings cracked open and forks bent out of shape, their prongs twisting and braiding.
Victoria's boasts concerning her powers extended far beyond those of most Spiritualists. She wrote in several documents that on one occasion when she returned home, Roxy told her that Byron had been seized by a fit and had died two hours earlier. "No. I will not permit his death," Victoria exclaimed, and ran to him. Tearing her dress open to the waist, she clasped her son's cold body and concentrated all her energy on generating heat. She "glided insensibly into a trance.... I beheld Jesus standing in the doorway. His arms were outstretched showing me His hands and feet. I saw the scars of His crucifixion.... My whole future was foreshadowed in that time." When Victoria regained consciousness, she reported, the boy, soaked with perspiration, was alive in her arms.
Victoria, who by now had learned from her father's showmanship, decided to strike out on her own. Leaving her family in Ohio she set out for Indianapolis, Indiana, with her husband and son, where she rented rooms at the Bates House and placed an announcement, or "card," in the local newspaper advertising herself as a "Clairvoyant Medium and Magnetic Healer." But these powers alone did not provide sufficient income: After two months in which "servant girls" patronized Mrs. Woodhull, she was accused of using the extra rooms she had rented for assignations and was ejected from the Bates House for "disorderly conduct."
Now seven months pregnant and once again destitute, Vickie begged her husband to borrow money from his father. She later claimed that she and Canning visited Dr. Woodhull's father, a judge in Rochester, but he refused to help them. (However, there is no record of a Judge Woodhull in or around the Rochester area at this time.) In April 1861, in yet another rented room, Vickie gave birth to her second child. After a protracted labor the baby was delivered at four a.m. by the shaking hands of Doc Woodhull, who, totally inebriated, half-severed the umbilical cord without tying it off, placed the baby girl on the pillow next to Vickie's head, and left the house. Vickie lapsed into an exhausted sleep from which she awoke to find her hair and the pillow soaked with blood. Drop by bloody drop, her baby's life was oozing away. With the last of her strength she bit through the cord and tied it herself.
Unable to move from the bed, she reached out, found a broken chair rung that lay nearby, and banged on the wall, calling for help. She repeated this again and again until a woman who lived in the next house heard the rapping through the wall. The front door was locked, but the woman removed a grate in the basement and climbed the stairs to rescue the new mother.
Three days later, while Vickie sat in bed holding little Zulu Maud in her arms, she looked out the window and caught sight of her husband "staggering up the steps of a house across the street, mistaking it for his own." It was then that she finally asked herself, "Why should I any longer live with this man?"
# CHAPTER SEVEN
WILLFULLY DID KILL
AS VICTORIA lay in the tenement room holding her newborn daughter, at four-thirty a.m. on April 12, 1861, Confederate shore batteries under the command of Brigadier General Pierre G. T. Beauregard opened fire on South Carolina's Fort Sumter. After a thirty-four-hour bloodless bombardment, Major Robert Anderson instructed Private Hart to lower the Union flag, which was then carefully pointed, put in a mailbag marked "Major Anderson, Fort Sumter, April 14, 1861," and dispatched to the Bank of Commerce vault in Washington for storage. Anderson then surrendered to Confederate troops.
For many the Great Rebellion came as a surprise: Only the previous month in his inaugural address Lincoln had declared, "I have no purpose, directly or indirectly, to interfere with the institution of slavery in the States where it exists." But the cauldron of controversy concerning several issues had boiled over—states' rights against federal authority, the expanding industrial North against the agrarian South, slaveholders against those who had no need for slave labor. As the country mobilized, Northern experts predicted victory within ninety days. Lincoln issued a call to arms, and seventy-five thousand volunteers responded.
Women quickly stepped out of their traditional roles. For the past decade many women had fought for their rights, but now they tacitly agreed to postpone equality and enfranchisement for the war effort. Lucy Stone gave up the income from her woman's rights lectures and made shirts for Union soldiers. Mary Livermore, a Chicago woman's rights advocate, began what was to become four years of visiting camps and hospitals, organizing sanitary aid societies, and meeting countless times with President Lincoln. She overcame the opposition to women nurses at the front and raised more than $100,000 for medical supplies. She was responsible for wiping out scurvy by bullying farmers into donating fresh fruit, which was then rushed by rail to the troops. Traveling through Illinois, Livermore saw "women in the fields everywhere, driving the reapers, binding and shucking and loading the grain." Susan B. Anthony returned to her father's New England farm and did hard manual labor so her brothers could be released for the fight.
_Mary Livermore was a heroine of the Civil War who fought on for woman suffrage_. (Illustration Credit 7.1)
In Brooklyn, the Plymouth Church Ladies Guild provided clothing and food packages for Union soldiers. Largely because of its flamboyant and charismatic preacher, Henry Ward Beecher, Plymouth Church was the most prominent of the scores of churches in this, the third-largest city in the country. Beecher's patron was Henry Bowen, the owner and publisher of the _Independent_ , a popular religious newspaper. His wife, Lucy, had organized the Ladies Guild but shortly thereafter fell mysteriously ill. On a fall day in 1862, Lucy Bowen—thirty-eight, the mother of ten, still a beauty—lay on her deathbed. She spoke in a whisper as her husband bent over her, "I cannot meet my Maker unless you know," she breathed. She told her husband how her preacher had seduced and abandoned her. For more than a year Lucy Bowen had gone to Beecher's private study at Plymouth Church, and he had called her his love. He had given her a key to the door so that she could enter, lock the door behind her, and wait for his triple rap. One day when Beecher was not expecting her, Lucy approached the door and saw another woman insert a key in the lock and move quickly inside. Lucy heard the bolt snap into place. The realization that she was not the only woman with whom Henry Ward Beecher was consorting broke her heart. But for Henry Bowen the worst of his wife's confession was that she vowed she loved Beecher still.
_Henry Ward Beecher's Plymouth Church (and below) during a Sunday sermon_
_The church is located on Orange Street, between Hicks and Henry Streets in Brooklyn_.
Bowen swore himself to silence, for to expose Beecher was also to expose the fact that he had been cuckolded. Furthermore, Bowen had an enormous stake in the financial stability of Plymouth Church. He was a founding member and owner of the land on which the church stood. Plymouth Church bonds yielded him 7 percent interest, and he received part of the revenues from the pew auctions. Knowing of Beecher's power, he allowed his wife's seducer to officiate at her funeral. In reprisal, however, Bowen used his wife's confession to force Beecher to become his puppet by endorsing all manner of merchandise advertised in the _Independent_ , from Chickering pianos, to Waltham watches, to the indignity of trusses. As Bowen's demands escalated, Beecher sought escape. Abruptly, he left his pulpit, wife, and children, and embarked for England to lecture for the Union cause.
In Beecher's absence, Henry Bowen's fortunes began to rise, and to a great extent he had Theodore Tilton to thank for it. Tilton, a gifted writer of twenty-eight, was Beecher's protégé. Henry Ward Beecher had thought Lincoln an unfit candidate who would be defeated for the presidency in 1860 and had criticized him from the Plymouth Church pulpit in August 1861, flatly declaring the war "a failure." For the first time, however, Tilton disagreed with Beecher, and he unequivocally supported the president in Bowen's _Independent_. As a result, Bowen was appointed collector of the Port of Brooklyn and became a powerful political figure, the king of local patronage, devoting himself to licensing ships and collecting duties on their cargo. It was up to Bowen to determine if such cargo was legal or illegal. If illegal, he confiscated the goods, which were stored in warehouses owned by himself and other prominent political figures.
As the war progressed and the harbor filled with supply ships, Bowen and his cronies had an unlimited opportunity to collect kickbacks and bribes. His greed even extended to the insignificant: He hired a gang of Irish street urchins to peddle, on Chatham Street, the groceries and clothes that had been confiscated. In one year, Bowen's income reportedly increased from $23,000 to $183,000. In the second year of the war, Bowen built the largest and most imposing house in Brooklyn on the corner of Willow and Clark Streets, a huge white colonial mansion with terraced Italianate gardens, wide lawns, and a magnificent view of the river and the bay. He had the faces of each of his ten children carved into the frames of his mahogany armchairs.
_Henry Bowen was said to be the richest and most treacherous man in Brooklyn_.
From England came word that Henry Ward Beecher had taken the country by storm. As he toured Liverpool, Birmingham, and Manchester, even before he spoke a word he was greeted with enthusiasm. This was a happy time for him, for he received the "soul-food"—the unconditional approbation—he craved. And at home Tilton published article after article on Beecher's success, though Bowen tried to shake his confidence in his mentor. One day as the two men stood at the rail of the Fulton Street ferry, Bowen told Tilton of Lucy's deathbed confession. Soon after, from his summer home in Woodstock, New York, he wrote Tilton, "I sometimes feel that I must break silence.... One word from me would make a revolution throughout Christendom... and Beecher would be driven from his pulpit and from Brooklyn in twelve hours."
On July 21, 1861, soldiers both blue and gray carried their dead and dying from a battlefield in Virginia: The Union soldiers called it Bull Run; the Confederates, Manassas. In Philadelphia, a nineteen-year-old Quaker girl, Anna Dickinson, speaking at the Academy of Music, accused General George B. McClellan, commander of the Union forces, of "cowardice and treason" for having delayed so long in engaging the enemy. As a result she was fired from her position at the United States Mint in Philadelphia, where she had worked as an adjustor six days a week, eleven hours a day, for $28 a month. But the fervor of her speech so inspired the famous abolitionist publisher William Lloyd Garrison that he hired Anna to speak for his cause. Thus Dickinson's brilliant career was launched, one that would meaningfully affect both the woman's rights movement and the life of Victoria Woodhull.
By early 1862, President Lincoln was convinced that if he issued an emancipation proclamation freeing the slaves, he could create economic and social chaos in the Southern states, but he hesitated to do so until the Union won a decisive battle. During the darkest days of the war, the president made a trip to Brooklyn to consult with Theodore Tilton, whose opinion he had come to value, though Tilton was neither a politician nor a soldier. The president intended his visit to be secret, but soon all Brooklyn knew that he had come to Lib and Theodore Tilton's Livingston Street home and that, when their daughter Florry presented Lincoln with a bouquet of flowers, he bent down and kissed her on the cheek. It was said the president left Brooklyn more determined than ever that the issuance of an emancipation proclamation was necessary.
At Antietam in Sharpsburg, Maryland, the North won a battle, but at a terrible price: Seven thousand of General Joseph Hooker's men were killed in fourteen minutes. Nevertheless, this Pyrrhic victory gave the president the confidence to draft an emancipation proclamation to take effect on January 1, 1863. Many doubted that Lincoln had the courage to fulfill his promise, but on December 31, Frederick Douglass was among the hopeful who attended a midnight freedom-watch service at Tremont Temple in Boston. When a telegraph messenger arrived with the news that Lincoln had indeed issued the Emancipation Proclamation, there was cheering and then absolute silence. Then Douglass raised his voice in song, and the crowd followed: "Sound the loud timbrel o'er Egypt's dark sea. Jehovah hath triumphed, his people are free."
Lincoln was a reluctant emancipator. Though the proclamation technically freed 3 million blacks, it excluded slaves in Union-occupied territories and in the four border states that had remained loyal to the Union. A London newspaper observed that "a human being cannot justly own another... unless he is loyal to the United States."
A Spiritualist medium, Mrs. Nettie Colburn Maynard, who had conducted séances in the White House at which Lincoln was present, claimed that the president had been moved to emancipate the slaves through a spirit message she had transmitted to him. Mary Todd Lincoln, a devoted Spiritualist, declared that she had been present when the message was transmitted. Their son Robert later denied that his father was in any way affected by such communications. As death on the battlefield became an ever-present reality, Spiritualism grew stronger. Its adherents, according to several contemporary sources, swelled from about 2 million in 1850 to about 7 million by 1863.
In that same year, with the Union army depleted, Congress passed a conscription act. But a rich man need not go to war; he was allowed to buy a substitute to serve in his place for $300. In the heat of July, in New York City, a protest against America's first forced draft, led by impoverished Irish immigrants—who did not want to die for the sake of freeing black men to compete with them for jobs—soon expanded into a riot. Elizabeth Cady Stanton, who was now living in New York, where her husband, Henry, was deputy custom collector for the Port of New York, witnessed the riot. Her eldest son, Daniel (called Neil), who worked as a clerk in his father's office, was standing in front of their house on the night of July 13 when a "brutal mob" of Irish ruffians strode down the street and seized him. "Here's one of those three-hundred-dollar fellows!" one shouted. Elizabeth thought her son was about to "be torn limb from limb." But as she later wrote, "Neil with great presence of mind, as they passed a saloon said, 'Let's go in, fellows, and take a drink.' He treated the whole band.... 'Oh,' they said, 'he seems a good fellow, let him go.' "
That night, Elizabeth, thinking that the rioters might attack her for her antislavery views, sent her "servants and the children to the fourth story... opened the skylight and told them, in case of attack, to run out on the roof into some neighboring house." The sky turned red as the Colored Orphan Asylum at Fifth Avenue and Forty-third Street, less than two blocks from her house, was burned to the ground by the rioters. A bystander saw a black child "three years of age thrown from a fourth-story window and instantly killed. A woman one hour after her confinement was set upon and beaten with her tender babe in her arms.... Children were torn from their mother's embrace and their brains blown out in the very face of these afflicted women. Men were burnt by slow fires." It took four days for soldiers and the police to disperse the mob. More than one hundred black people had been killed and scores more were maimed.
_On the night of July 13, 1863, an enraged Irish mob looted and destroyed the Colored Orphan Asylum_. (Illustration Credit 7.2)
Henry Stanton returned from Albany the following day. Although his job at the Custom House was a powerful one, his superior wrote to the secretary of the Treasury, Salmon P. Chase, that his work was lackadaisical, and that "his mind is more absent than his body." He was referring to Stanton's careless supervision of Custom House bonds. In wartime, every ship that entered New York harbor was required to secure its cargo with a bond filed at the Custom House. Once it was determined that the ship's cargo was not contraband, the bond was returned. Cargoes without bonds were seized by the government. By the spring of 1863, it was discovered that bonds being held for certain shipments had disappeared. The Treasury Department investigated, and by fall Henry Stanton was implicated. He questioned his son Neil, who confessed to taking the bonds and selling them.
Because Henry Stanton refused to speak about his son's crime, he was accused of complicity and even of allowing munitions to reach enemy lines. Henry insisted that he was innocent and that his son had been the "weak victim of wily scoundrels." He placed much of the blame for the public scrutiny he had received on his wife's increasingly visible radicalism. She in turn became more determined than ever to become an individual in her own right.
IN THE midst of the war, the Claflin family arrived in Ottawa, Illinois. Eight years later, Victoria was to write that the war had come as no surprise to her. "For years before the Rebellion, I frequently saw armies marching in the air, heard the roar of cannon and the rattle of musketry, and the negroes fleeing before them." Of her sister Tennie's powers she wrote,
When Lincoln was on his way to Washington, to be inaugurated President, my sister Tennie was with father and mother at a prominent hotel in Pittsburgh. Thousands of the inhabitants had assembled in the street.... As the carriage... came in sister's sight, she... exclaimed, "He will never leave the White House alive. I heard a pistol shot and saw the blood running from his head, and his coffin in the carriage with him."
While others were swept up by the war, Buck Claflin thought only of profit based on human need. What disease most plagued mankind? Cancer! He decided that a cancer cure would make him a fortune. Within six months of his arrival in Ottawa, "Doctor" Reuben B. Claflin and his family were doing well enough to rent the entire Fox River House, the town's oldest hotel. Here, on the second floor, Buck Claflin set up an infirmary. He advertised himself as the "AMERICAN KING OF CANCERS... Dr. Claflin guarantees a cure in all cases where patients live up to directions.... Cancers killed and extracted, root and branch, in from 10 to 48 hours without instruments, pain or use of chloroform, simply by applying a salve of the doctor's own make." The salve that purportedly cured cancer was a home brew. In an iron vat in the backyard, the coarse Roxy, assisted by the comely Utica, could be seen stirring a mixture that contained scent, sheep's fat, and lye.
The Fox River House overflowed with Claflins. After ten years of marriage, Victoria had divorced Canning Woodhull and returned to her family with her two children, the idiot Byron and the thriving Zulu Maud. Margaret Ann, divorced from Enos Miles, was there with her four children. So was Polly Burns and her daughter, Rosa, as well as Utica and brother Hebern, who now called himself Dr. Hebern Claflin.
All the Claflin girls were beautiful, especially the three youngest—Victoria, Utica, and Tennessee. To Roxy it seemed a miracle that so plain a woman should be the mother of such daughters. To Buck, their beauty was yet another opportunity to make money. He advertised in the _Ottawa Free Trader_ that on the first floor of the Fox River House lessons in the "cult of love" were to be taught. In the summer of 1863, there were several complaints that the extra rooms at the Fox River House were being used for assignations and that Buck's daughters were prostitutes. The charges were never proved.
In addition to whatever sexual services were offered, Tennessee's clairvoyant powers were becoming well known. Three weeks after the battle of Gettysburg, a local newspaper carried an account of a soldier with a bullet still embedded in his foot who "hobbled in on crutches" to the Claflin infirmary. As Tennie felt his toes she exclaimed, "Why, Captain, you were not wounded in battle!"
"How then did I receive my wound?" he challenged her.
"When the rebels evacuated and the federal army moved forward in pursuit, you lingered behind and were shot by a rebel concealed in a tree."
The soldier was "thunderstruck," for, by his account, this event was known only to himself.
Since Tennie was the most famous of the family, her name was used in Buck's advertisements. They specified that she alone would apply her father's cancer treatment. Buck's salve was so strong that it could strip paint from the side of a barn. After the treatment, the screams of the patients sometimes could be heard throughout the neighborhood. Victoria wrote that once she looked under the bedclothes and saw the damage: ragged, livid areas, pus, blood, and exposed cartilage. She was reminded of a slaughtered animal. One evening, risking her father's anger, she asked Buck why he used this treatment, and he answered, "There are only three cures for cancer—cut it out, poison it with arsenic, or burn it out. I burn it!"
Buck's testimonials appeared in local newspapers, luring more sufferers. One read, "Mrs. Rebecca Howe, recovering from a dangerous situation after treatment by MISS TENNESSEE CLAFLIN wishes to thank this remarkable child and recommends [she] be consulted for cancer treatment." The canny Buck had used Mrs. Howe's name because he knew she had breast cancer. In these strict times, women rarely spoke of their bodies. At dinner parties hostesses sometimes referred delicately to the "bosom" of the turkey. But Rebecca Howe read the testimony and, knowing she was dying, wanted to tell the truth. She wrote to the _Ottawa Republican_ that in fact, her breast had been "wholly eaten away" by the lye salve applied by Tennie. She was racked with unbearable pain and prayed only for a quick death. She concluded that "Miss Claflin is... an imposter, and one wholly unfit for the confidence of the community." Two local doctors attested to the truth of Rebecca Howe's statement and to the hideous pain she had endured.
Four days later, the cancer infirmary was raided by a local marshal and two doctors. Buck Claflin was not present when they pushed past Utica and Mama Roxy and charged up the staircase. Private rooms on the second floor housed new patients undergoing treatment. At the end of the hall was the infirmary ward. The door was open, and there, on rows of unmade cots, were found dying, unfed wrecks of humanity lying in their own excrement. Dr. Joseph Stout stated that not even in the newly opened infamous Andersonville prison camp in Georgia, where one out of every three Union prisoners died, had there been such neglect. Rebecca Howe died that same day.
Within a fortnight, nine separate complaints of disorderly conduct, blackmail, and medical quackery had been lodged against the Claflins, but all these paled beside the charges against Tennessee Claflin: She was indicted for manslaughter. The indictment read, in part:
On the first day of November in the year of our Lord one thousand, eight hundred and sixty-three, in the county of La Salle, and on divers other days and times... one Tennessee Claflin did feloniously and willfully place upon the right side of the breast of one Rebecca Howe divers quantities of deleterious and caustic drugs by means of which a large amount of flesh was... consumed and destroyed... and that Rebecca Howe then and there became mortally sick, sore and distempered in her body and the aforesaid did languish in mortal sickness and thereby died on the seventh day of June, eighteen hundred and sixty-four.... We say that Tennessee Claflin in the manner aforesaid feloniously and willfully did kill and slay contrary to the statutes... one Rebecca Howe.
It was thought that a local lawyer who fancied Tennessee had warned her when he was commissioned to draw up the indictment. In any case, by the time an arrest warrant was issued, the Claflins had disappeared, and the matter soon vanished in the chaos of war. On March 4, 1865, Lincoln was inaugurated for the second time, and by the first week in April, Robert E. Lee's army had dwindled to fewer than thirty thousand men. Grant requested a surrender. Lee asked for terms. On Palm Sunday, April 9, 1865, on the Appomattox road in Virginia, the firing stopped as out of the Confederate line rode a lone soldier, a white flag fluttering from the end of his staff. As Generals Grant, Schuster, and Ord trotted down the road to meet General Lee at the courthouse, the two armies waited in silence. It was over.
The Fort Sumter flag that had been lowered four years before was taken from the Bank of Commerce vault in Washington, D.C., and shipped in its original mailbag to the government transport _Arago_ , docked at the foot of Beach Street in Brooklyn, New York.
# CHAPTER EIGHT
GOD BLESS THIS TRINITY
THE POST Office van, covered with red, white, and blue bunting, its horses' collars decorated with gold stars, came hurtling at full gallop down Brooklyn's Beach Street pier, lined on both sides by a crowd estimated at two thousand. After an abrupt halt, two blue-coated soldiers jumped down, unloaded a mailbag inscribed "Major Robert Anderson, Fort Sumter, April 14, 1861," and presented it to the now general Robert Anderson, as his wife, son, and three daughters looked on with pride. From the bag, Anderson withdrew the American flag and gave it to Sergeant Hart, the soldier who had lowered it during the evacuation of Fort Sumter.
Henry Ward Beecher stood on the pier arm in arm with Theodore Tilton, once his protégé and now the editor in chief of the _Independent_. Standing there in the bright April sunlight, they seemed the most successful of men—men in whom reposed the American future. Beecher had not thought to include his wife, Eunice, in the trip to Fort Sumter, but Tilton had invited his wife, Lib, who refused. Though Tilton never asked her reason, she volunteered that she was not well enough to attend. Most likely she was intimidated by so auspicious a gathering. "I could never be a lady of fashion," she had once admitted.
Henry Bowen, though publisher of the _Independent_ , had not officially been invited to the ceremonies, but he had chartered a boat and arranged his own party to follow the government transport _Arago_ on which Sergeant Hart, bearing the flag, and sixty-one dignitaries and their families were to set sail from Brooklyn. One by one the governors of Maine, New Hampshire, Vermont, Rhode Island, Connecticut, and Massachusetts came aboard to be joined by members of Lincoln's cabinet when the ship docked at Fortress Monroe, Virginia.
Good Friday, April 14, 1865: At Fort Sumter the day was clear and cool, the sky a brilliant blue. Henry Ward Beecher stood upon a heap of stones, the wind ruffling his long, graying hair as he gave the keynote speech to an assemblage that included some of the most prominent men in the nation. After the flag was raised to the accompaniment of a single bugler, Beecher said, "My friends, fellow citizens, and brother soldiers, I am here to fulfill the cherished wish of my heart through four long, long years of bloody war, to restore to its proper place this dear flag.... I thank God that I have lived to see this day. 'Glory to God in the highest.' "
In the hourlong address that followed, Beecher referred several times to his "brother soldiers," but in fact he had been absent for eighteen months during the "Great Rebellion," in luxurious exile in England, where he had fled after Henry Bowen's late wife, Lucy, confessed to the affair with him. Members of Plymouth Church, reading in the _Independent_ of Beecher's great success abroad and seeing that in his absence their membership had fallen off and the price of pews had dropped precipitously, pressured Henry Bowen to bring him back. The greedy Bowen, though busy accumulating graft from his position as collector of the port, still missed his share of the pew auction funds. And so Beecher returned—triumphant.
Henry Ward Beecher was now the most famous preacher in America. The special ferries that ran from New York to Brooklyn every Sunday were known as "Beecher's Boats." When the original Plymouth Church burned down in 1849, two years after Beecher's arrival, the new church on Orange Street in Brooklyn Heights was designed according to his specifications. Here Beecher stood, not behind an altar, but on a rectangular platform jutting out into the audience, surrounded by his parishioners. Beecher was a consummate showman; he could imitate any dialect, impersonate men or women, make his audience laugh or cry or feel the fire of God. But above all, it was _what_ he said that was particularly suited to these uncertain times. Gone were the Calvinist tenets of his father, Lyman Beecher. In their place was the warm, self-indulgent "Gospel of Love." Beecher declared that to be truly religious, one must sin: "Christ can save you, because you _are a sinner_ , not because you aren't one." He preached, "What is terribly and dangerously heterodox this year may be accepted as the very essence of orthodoxy next year. What is orthodoxy? Orthodoxy is _my_ doxy and heterodoxy is _your_ doxy, if _your_ doxy is not like _my_ doxy."
Although his mother had died when he was a toddler, he credited her with his moral teachings. Beecher called himself "wife and mother" to his parishioners. He referred to God as the "Mother of the Church." This feminized, romantic conception was a part of his custard-cream vision, as he told a gathering, "When questions of justice and humanity are blended, woman's instinct is better than man's judgment. From the moment a woman takes the child into her arms, God makes her the love-magistrate of the family, and her instincts and moral nature fit her to adjudicate questions of weakness and want."
And above all there was love—unquestioning, abounding love. The permissiveness of his pulpit utterances, the lavish decoration of his home, his fine silk handkerchiefs, square-toed gleaming leather shoes, his expanding waistline, the unpolished gems in his pocket, were all of a piece. Pleasure, comfort, wealth, were what God had willed for Henry Ward Beecher and for his parishioners. To a new group of capitalists, who had thrown morality to the winds during the war, this was a self-justifying doctrine. From his pulpit Beecher roared, "Ye are Gods. Your faces are crystalline." An entire flock followed Beecher's teachings and none more faithfully than Theodore Tilton's tiny, impressionable wife, Lib, who said of herself, "I cannot think, only feel."
Victoria Woodhull's ragtag existence and her quest for bare survival were far removed from the concerns of the ordered society in which Lib Tilton lived. Here was a life as circumscribed as that of a perfectly executed needlepoint. At the beginning of the war, Brooklyn had been a city of one hundred thousand people and fifty churches. It had five banks, a public library, a city hall, and a police force fifty strong. This city provided a refuge for substantial, white, middle-class families bent on escaping New York's increasing congestion, rising costs, pollution, and influx of immigrants. "Cool, fragrantly airy and no mobs," proclaimed a real estate advertisement. New York City was only a penny ferry ride away.
In this valentine of a world, appearance was all important. For the women there were fashionable clothes, perfectly appointed homes, seven-course dinners, church every Sunday. These women played Chopin, embroidered, darned socks, washed clothes, became pregnant yearly. Lib (Elizabeth) Richards Tilton was a woman of her time and environment: a woman of malleable clay in a world where men did the molding.
On October 2, 1855, in Brooklyn's Plymouth Church, Theodore Tilton and Lib Richards had been joined in holy matrimony by the Reverend Henry Ward Beecher. They were a striking couple—the bride scarcely five feet tall, with jet-black hair and intense, luminous dark-brown eyes; and towering over her, the groom in the traditional morning coat and top hat, six-foot-three, with a mane of blond curls, clear blue eyes, and an abstracted air that masked a fierce intelligence. Lib was engulfed in a sea of white peau-de-soie with a bertha of Brussels lace. At her throat was a gold and pearl heart, a gift from her late father, who had been a jeweler. The groom later stated that he had chosen the "most ideal woman in the world." The bride brought to the altar the desire "to fulfill his every need and dedicate my life to being worthy of his love. We will be as one," she declared fervently.
_The Beecher home on Columbia Heights in Brooklyn_
Lib was two years older than her groom, who had chosen this, his twentieth birthday, as their wedding day. Although Lib's mother would have liked a more ambitious match, she thought her only daughter was growing rather "long in the tooth," and Tilton was said to be a man with a future. The son of a Monmouth County, New Jersey, shoe manufacturer with a comfortable business, at only sixteen he had secured a job as a reporter on Greeley's _Tribune_ , but in 1853 had transferred to the _New York Observer_ , a religious newspaper. There, his major assignment was to report stenographically (using the new system introduced by Stephen Pearl Andrews) the sermons of Henry Ward Beecher. Tilton crossed the river to the city of Brooklyn, joined Plymouth Church, and moved into Mrs. Richards's stolidly middle-class Harrison Avenue boardinghouse.
It was generally acknowledged that Lib and Theodore were well suited. Both were devout. Theodore wrote that he had been brought up "in the old school Presbyterian Church.... My earliest religious bent was toward extreme Calvinism." Tilton admired Beecher for the unbridled emotionalism of his sermons—passages of which he committed to memory. Since childhood, Lib had found in Beecher's Plymouth Church "a home," a place of friendship and warmth. She attended Sunday school, then became a teacher there, and every Sunday she listened to her preacher. And what a preacher he was. Henry Ward Beecher presented a God and a Christ as he envisioned himself—unconditionally loving, caring, understanding. Hearing his words Lib often felt a "kind of ecstasy," which lifted her out of the strict discipline and stubborn grind of her boardinghouse existence.
To the Tiltons, Plymouth Church was more than a place of worship; it defined one's social standing and business opportunities in the Brooklyn community. While at a Plymouth Church service, Joseph Richards, Lib's only brother, had met Henry Bowen, who hired him as assistant publisher of the _Independent_. Directly after the Tiltons' marriage, Joe Richards persuaded Bowen to hire Tilton, who was pleased to leave the _Observer_ because that newspaper would not endorse the abolition of slavery.
Beecher rarely wrote out a sermon and indeed usually began his preparations only two hours before he spoke. He liked the thrill of the extemporaneous and said that he would rather "serve my sermons hot." Tilton recorded Beecher's words as he spoke them and, after editing, printed them in the _Independent_ , thereby providing Beecher with a new source of revenue and increasing the preacher's audience by some four hundred thousand readers. Many people bought the paper solely for Beecher's sermons, articles, and columns, but often he was lax about deadlines and failed to deliver. These lapses were legendary. When Beecher had written for _Bonner's Weekly_ , Robert Bonner had posted boys in shifts outside his door, twenty-four hours a day, until they returned with his copy.
Robert Bonner, who had spent years coddling Beecher, knew how hard it was to get him to produce. In order to win his star contributor's gratitude, Bonner gave him a set of matched bays worth $3,500. Within hours of their arrival, Beecher, with a child's glee, hitched them to a carriage, and with his wife, Eunice, at his side galloped in record time to his Peekskill home. Two hours later, one of the horses dropped dead. The other, lame for life, was given as a work horse to a local farmer.
LEFT: _Lib (Elizabeth) Richards Tilton, 1863. At thirty, Mrs. Tilton was docile and domestic_. RIGHT: _Theodore Tilton, 1863. At twenty-eight, Tilton was the powerful editor of the_ Independent. (Illustration Credit 8.1)
Beecher's gestures were frequently more theatrical than brave. As a young revivalist in the Midwest, he had not come forward against slavery. But the men who brought him to Plymouth Church, financier John Tasker Howard and Henry Bowen, who was the son-in-law of Lewis Tappan, a leader of the abolitionist movement, were militant antislavery advocates. In May 1854, when the Kansas-Nebraska Act became law, Beecher pledged that Plymouth Church would supply twenty-five Sharps rifles to Northern settlers heading for Kansas. These rifles became known as "Beecher's Bibles." Lucy Tappan Bowen had, in her intimate relationship with Beecher, goaded him to write antislavery articles in the _Independent_ , and Tilton had been happy to ghostwrite them for him.
One Sunday, while the Reverend Richard Salter Storrs's neighboring Church of the Pilgrims was mounting a quiet campaign for funds to buy slaves and give them their freedom, Henry Ward Beecher produced on the platform of Plymouth Church a beautiful mulatto girl of twenty and told his congregation she was "to be sold by her own white father... for what purpose you can imagine." Beecher held up iron shackles and cast them to the floor, trampling them underfoot. He ordered the girl to take down her hair and, as it streamed about her shoulders, called out like an auctioneer, "How much for her?" Women began throwing their jewelry onto the platform. Men emptied their pockets and unfastened their watches, tossing the contents into the ushers' baskets until they overflowed, were emptied, and overflowed once again.
Realizing Beecher's stunning appeal, in 1858 Bowen offered him the editorship of the _Independent_ and assured him that young Theodore Tilton would, in Beecher's words, "relieve me wholly from routine office work." Tilton continued to ghostwrite for Beecher, but he was allowed to have his own byline on his other articles and was promoted to assistant editor. Not only Beecher's words and style but also his ideas deeply influenced Tilton. He was "dazzled" by Beecher and later said, "I came to love him as I had no other man."
In the _Independent_ offices at 5 Beekman Street in New York City, Beecher would stretch out on a couch, rattling off his ideas while Tilton feverishly took the notes he would later shape into articles. Then the two would stroll arm in arm down the cobblestone streets, heady with idealistic conversation. They would drop in at art galleries, antiques shops, rug dealers. (Beecher purchased so many Oriental rugs that he would pile them one on top of another, often four deep, in the various rooms of his house.) Beecher admitted that if he saw something he desired, he had to possess it. Often he went on spending sprees, bringing home paintings, bric-a-brac, leather-bound books with uncut pages, and bolts of silk, all of which he would smuggle past his disapproving wife, Eunice. At dusk, Beecher and Tilton would often find a restaurant and the conversation would continue over quail and claret. Finally, like father and son, they would board the ferry for the trip home to Brooklyn. Beecher basked in Tilton's idolatry, saying, "His mind was opening freshly and with enthusiasm upon all questions. I used to pour out my ideas of civil affairs, public policy, religion and philanthropy. Of this he often spoke with grateful appreciation."
In the first six years of their marriage the world expanded for Theodore Tilton, but for Lib Richards Tilton little changed, save for the shattering of her hopes that she and Theodore would "be as one." Even by the standards of the day their life was not that of a normal married couple. As newlyweds, Lib and Theodore lived at Mrs. Richards's boardinghouse, where Lib's widowed mother had married one of the boarders, Nathaniel B. Morse, a respected Brooklyn judge. In this full house the young couple were never alone. Theodore, enthralled by words and the prospect of moving ahead, worked day and night, churning out articles and columns with amazing facility. Lib relied on her mother and the chatter of the boarders for company.
Then came the children, three in four years, and Lib took to her bed for months at a time, debilitated by the seemingly unending cycle of birth and nursing, depressed and in pain from a prolapsed uterus. She later said, "At the birth of my second little girl I was sick in bed from the middle of April until September, when I sat up for the first time. My husband never gave me any sympathy at all." After the birth of the third child, she recalled, "I had a very severe and prolonged sickness but when he saw me he never felt that I was sick because I always tried to seem well, I felt so desirous of his presence. I had no attention whatsoever from him. I do not think it was from neglect so much as from an inability on his part to understand that I was sick and suffering. My doctor, Dr. Portman, said, 'There is care and trouble in that woman's mind and I cannot help it with medicine.' " The following year, Lib was delivered of a fourth child, Matilda, "beautiful Mattie," who wasted away—no one knew why—and was dead at six months. In all that time her husband expressed no word of sympathy and no regret. The gulf between them grew wider.
There is every indication that the root of the Tiltons' discontent was their disappointing sexual relationship. Whatever her dreams of romantic love, the sexual act itself was for Lib, as she was later to explain, one of dutiful submission. She feared the pregnancies that resulted from their infrequent couplings. Theodore found Lib "cold" and "inadequate." He was irritable and critical of her. She "scolded and chided" him. Lib lamented that "he spent a great deal of his time at home in moods of dissatisfaction with the surroundings, yearning and wanting other ministrations. There was nothing in our home that satisfied him."
With each rebuff and imagined inadequacy, Lib Tilton's religious fervor grew until she was spending hours each day on her knees, praying beside the green damask chaise longue in her bedroom. Mrs. Morse, Lib's mother, was a termagant who lost no opportunity to berate her son-in-law for his neglect of Lib. She would work herself into "manias and frenzies," said Theodore. Her marriage to Judge Morse soon became unhappy, and he too came under fire. One evening she was so enraged that she beat the venerable judge on the chest and began to strangle him until his face turned blue. Theodore and two other boarders were able to free the judge only after one of them clasped a chloroform-soaked handkerchief over Mrs. Morse's face.
Shortly after Beecher fled to England, Bowen gave Tilton a raise and appointed him editor in chief of the _Independent_. Unlike his employer, Tilton was naive about business and was delighted and grateful when Bowen doubled his salary to $6,000. But he preferred not to introduce the new people he met in his more exalted position to the "mildew," as he called it, of the boardinghouse. He thought a man determined to "make a name for himself" needed a proper setting.
Theodore insisted the family move from Harrison Avenue to the more fashionable Brooklyn Heights, where he purchased a house at 174 Livingston Street. To Lib, the three-story white clapboard house with its broad steps was too large. There was a front parlor with an organ and a melodeon for herself and daughters Florry and Alice, and a back parlor with a stone fireplace that warmed the entire downstairs. The most unusual feature of the house was a stained-glass dome over the third-floor sitting room. Around the arch surrounding the dome was the inscription, "And into whatsoever house ye shall enter, first say peace be unto this house."
Beecher had taught his protégé well. Tilton wanted nothing but the best. He bought expensive furnishings and decorated the house, as a local newspaper noted, with "luxurious carelessness; the wallpaper was of a dainty cream color, picked out with gold.... The carpets were of the finest, hanging baskets at the windows were replenished every day with the choicest of flowers." The library was furnished with a red velvet chaise longue and matching tufted club chairs. Scattered about the living room were damask sofas and heavy walnut chairs carved with cherubs' faces. On the walls hung dozens of etchings, including a copy of Theodore Tilton's favorite painting, Jacques Louis David's _Oath of the Horatii_ , a parable of war and love in which three brothers are about to fight to the death three adversaries with no regard for the fact that one of them is betrothed to their sister, seen weeping in the background. A scene of Sixtus returning home from war to find his wife dead also appealed to Tilton's taste for melodrama. Despite the extravagant luxury Theodore permitted himself, he frequently berated his wife for overspending on the children's clothes and household management. Having come from a boardinghouse, Lib had no experience running a home of her own and often vexed her husband by her inability to cope with her newly acquired servants and possessions.
In the houses Theodore Tilton now visited, the linen and silver were always just so, the meals perfectly and silently served. He chastised Lib in front of the butcher for not knowing which cut of meat to order. When she begged him to help her with the niceties of domestic life he saw in other homes, saying, "Alone I can do no better, but with you I think I can," he replied angrily, "I do not call upon you to go to the office to do my work. This is yours."
There was no doubt that Theodore found Lib inferior and woefully inadequate. He corrected her grammar. He told her that she did not dress with sophistication. He confided to his closest friend, Francis De Pau Moulton, known as Frank, that his wife was a "small woman, without presence, not a woman of society, not a woman of culture." He regretted that "he had married her young and that he had grown and developed and that she had not." Frank Moulton and his wife, Emma, spent a great deal of time with the Tiltons. Although Frank belonged to no church and believed in no religion, Emma had been brought up in Plymouth Church and taught in the Sunday school with Lib Tilton. Later Frank was to observe of the Tiltons' marriage, "Here is Theodore marrying at twenty a woman like a Spanish nun. He is a bold, frowning, gifted man, a product of Plymouth Church. She is the female product, an idealist, interesting by fervent sincerity, but she made religion a bore and became irksome to her husband."
Frank Moulton's own marriage was a happy one. Unlike Theodore and most men of his day, Frank treated his wife as an equal and even held that she more than earned half his wages, which he gave her for the long hours she spent keeping house. He often said that he had married Emma for her honesty and that they were good friends. Frank Moulton was employed at the dry goods firm of Woodruff & Robinson, which had prospered during the war selling cloth to the Union Army. Perhaps chosen for his imposing presence—as tall as Tilton and with flaming red hair and beard—Moulton had been put in charge of the firm's warehouses and frequently was seen striding along the waterfront with a gun tucked prominently in his belt. When he was not on tour in dangerous areas, the gun was hidden in a shoulder holster under his jacket. Moulton possessed an air of authority even without a firearm. It was this quality no doubt that made Theodore trust and confide in him as he had in his mentor, Henry Ward Beecher.
When Beecher returned from England in the winter of 1863, he found Tilton in the editorial chair of the _Independent_ and Bowen surrounded by a powerful clique of Plymouth Church members who were in charge of Brooklyn's political patronage. Beecher's flock, especially the ladies, looked upon him with adoration, but he needed to carve out a niche for himself. It had been a year and a half since his sermons had appeared in the _Independent_ , a considerable financial loss. Beecher went to Robert Bonner, who gave him the extravagant sum of $24,000 to write a novel, _Norwood or Village Life in New England_ , to run in fifty installments in _Bonner's Weekly_. But Beecher could not pull himself away from the lively life with his parishioners to get down to work. He tried writing a chapter or two but threw them away, despairing of his own wooden prose.
In Beecher's absence, the Tiltons' home had become a gathering place for the great political and intellectual minds of the day. Mary Cheney Greeley spent so much time in Europe that, at Theodore's invitation, Horace began to stay with them. Lib set aside a room for him, and Theodore designed a desk two feet higher than normal, because Greeley liked to raise his arms up and write at chin level. Horace Greeley had purchased a farm in Chappaqua in 1853, but he was so comfortable that even when Mary and their daughters, Ida and Gabrielle, the only two children who had survived, returned from Europe, he continued to live at the Tiltons' during the week.
Theodore Tilton had flourished in Beecher's absence, and he was eager to repay his dear mentor. He advised Beecher to mend his fences with Lincoln, which the preacher did, praising from his pulpit Lincoln's courage in having issued the Emancipation Proclamation and calling him "a great statesman." Once again, Tilton recorded Beecher's sermons and printed them in the _Independent_. With $500 of his own money, Tilton commissioned a portrait of Beecher by William Page of the National Academy. Beecher posed fifty times and Page's studio became another pulpit for him. Sitting in a yellow velvet chair on a raised platform, his gray Puritan cloak tossed rakishly over his shoulders, he reigned as king-prophet over an assembly that included such visitors as Horace Greeley, Wendell Phillips, General Benjamin Butler, and Frank Moulton.
When Beecher had finished a session posing for William Page, he would try to induce Tilton to stay out as late as possible. Beecher confided that he dreaded "going back to my own house," saying that his life with Eunice was a "hell on earth" and that their conversation at the breakfast table was "the vainest, the most vapid, the most juiceless, the most unsac-charine of all things." Theodore, proud of his new home and the people who visited there, urged his mentor to drop in. At first, Beecher found Lib Tilton all but speechless in his presence. Though she was now thirty-one and the mother of three living and one dead child, Beecher described her as "childlike in appearance... she was a child in nature, delicate in health, with a self-cheerful air." Plymouth's pastor was given the sitting room with the spectacular dome in which to work, and Lib faithfully checked his ink and other supplies and hushed the children when he was writing. After an afternoon at the Tiltons', Beecher exclaimed, "Oh, Theodore, God might strip all other gifts from me if he would only give me a wife and a home like yours."
Beecher had been severely blocked and unable to proceed with his novel. "I was almost in despair," he later confessed. "My team would not pull.... I needed something that would give me the courage to go on with it." He masked his apprehension with humor, telling Elizabeth Cady Stanton that he was considering a train wreck to end the whole thing. And when he spoke to Lib about the two chapters he had written, he said jovially, "People have said that I, not sister Harriet, wrote _Uncle Tom's Cabin_ , but they will never say it again after reading _Norwood._ "
One day Theodore Tilton mentioned that his wife had "a remarkable gift as a critic" and that he took everything he wrote to her. He urged Beecher to do the same. Beecher read Lib a part of a chapter and she began to comment on his work. He noted that she could "tell whether a speech put into [his heroine] Rose Wentworth's mouth was one a woman would be likely to say." Soon Beecher was writing with ease.
Even in recommending his wife as a critic, Tilton had been grudging. Lib was, he said, "so domestic a woman that this talent was concealed." As for Lib herself—whose husband made her feel so inadequate that she could barely function, who had taken to her bed for months at a time with a malady "in her soul"—the gratitude of the exalted Reverend Beecher gave her the confidence she had lacked. "I never felt a bit of embarrassment with Mr. Beecher, but to this day I never could sit down with Theodore without being self-conscious and feeling his sense of my inequality with him," she later said.
In the final months of the war, Theodore Tilton went to Washington to help plan the peace that now seemed certain. Lib wrote him there of Beecher's salutary effect on her health. "I took my first walk to the Court Street cars without much difficulty so that I feel free again and will walk out every day." And Theodore replied, "I am glad Mr. Beecher called on you. I will write to thank him." Lib regarded the relationship between her preacher, her husband, and herself as a religious entity, and she wrote, "God bless this Trinity."
At ten-fifteen p.m. on Good Friday, April 14, 1865, Henry Ward Beecher, Theodore Tilton, and a host of other dignitaries boarded the steamer _Quaker City_ to return from the flag-raising ceremony at Fort Sumter. At about the same time, in Brooklyn, Lib Tilton knelt in prayer beside her bed. Passing through Philadelphia that evening, Tennessee Claflin, by her own account, was seized by a vision and blurted out, "Tomorrow the whole nation will drape itself in mourning."
And in Washington, D.C., John Wilkes Booth—a derringer in his right hand, a dagger in his left—entered the president's box at Ford's Theatre and pumped a bullet into Abraham Lincoln's skull. The following morning Vice President Andrew Johnson took the oath of office as president of the United States.
# CHAPTER NINE
AN ACCIDENT OF FATE
BY AN ACCIDENT of fate, the control of the executive branch of the government, with all its patronage and the extended powers granted during the war, now fell into the hands of a Southerner, a former Democrat and a strong supporter of states' rights. "I had thought that states' rights, which interfere with Congress enforcing the rights of citizens of the United States, were buried forever in the red sea of blood that has flowed south of the Potomac!" bellowed Senator Benjamin Butler. He was wrong. Whether the individual states could choose their own governance, superseding federal law, was to become one of the major issues of this Reconstruction period.
During the recent war, Butler had been the occupying general of New Orleans. "Beast Butler" he was called for hanging a rebel who tore down the Union flag and for issuing an order that women who treated his troops disrespectfully should be regarded as prostitutes. His most ingenious edict declared that every escaped slave under his jurisdiction was to be considered the "contraband of war" and therefore free.
On the morning after Lincoln's assassination, Benjamin Butler organized a meeting of radical Republican senators and congressmen to make sure that Andrew Johnson would not lose all they had fought for by negotiating a "soft peace." Johnson, a self-made man who had risen from poverty to the governorship of Tennessee and the United States Senate, had not been opposed to slavery and had owned five slaves himself, but in 1864 Lincoln chose him as his running mate on the Union Party ticket, hoping to unify the nation at the war's end.
The radical Republicans stood on the party's left and had shared the common goal of freeing the slaves. In reality, this freedom came not through the Emancipation Proclamation but through the Thirteenth Amendment to the Constitution, which had been proposed in February 1865, but had yet to be ratified. The amendment specified,
_General Benjamin Butler, 1862. In New Orleans, they called him "Beast Butler._ " (Illustration Credit 9.1)
1. Neither slavery nor involuntary servitude, except as a punishment for crime, whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction.
2. Congress shall have power to enforce this article by appropriate legislation.
But the question of the black people still stuck like a bone in the throat of a shattered society. The problem was how to integrate into this new nation "property" that had been kept ignorant, poor, and governed by the lash. As early as 1863, Lincoln had made it known that he would accept from the South "temporary arrangements for the freed people." He felt that black men who had fought in the war should be enfranchised but hoped that at the war's end the whole problem could be solved by persuading black people to emigrate to other countries. The previous year, through an agreement with the Republic of Haiti, two American promoters had settled a group of Negroes off the Haitian coast on Ile à Vache. Within a year, half the settlers were dead of forced labor and starvation, and Lincoln brought the survivors back to the United States. Radical Republican George W. Julian of Indiana declared in a burst of frankness to his fellow congressmen, "The real trouble is that _we hate the negro_. It is not his ignorance that offends us, but his color."
Although Andrew Johnson supported the Thirteenth Amendment, and initially the radical Republicans were optimistic about Reconstruction, by late summer Johnson's Southern sympathies became apparent. The president's Reconstruction plan (he preferred the term "Restoration") for the South was to form loyal state governments under provisional governors. His only demands were that government officials take a loyalty oath, repudiate Confederate debt, and ratify the Thirteenth Amendment. Johnson granted mass pardons to former Confederates (twenty thousand within two years). He announced that as soon as the Southern state governments were formed, he would revoke martial law and withdraw federal troops. Reconstruction, he declared, would then be completed.
Not yet begun! responded the appalled radical Republicans in Congress. This was no Reconstruction but a virtual return to prewar conditions. Johnson's policies confined the Negro to a permanent underclass and a cheap source of labor. The charismatic champion of Negro rights Wendell Phillips declared that if Johnson prevailed, then the Union had fought a "murderous and wasteful war... for no purpose at all." Just as the Republicans feared, eight Southern states quickly established "Black Codes" specifying, among other provisions, that black children could be "bound out" as apprentices for no wages. Blacks who were not employed could be arrested and then hired out to pay their prison fines. If such a hired hand left his job, he forfeited his wages and was rearrested. Though they were now emancipated, black people had no civil rights.
Theodore Tilton and Henry Ward Beecher took opposite positions in regard to the legal protection of black people. Tilton allied himself and the _Independent_ with the radical Republicans in Congress. He wrote a series of unsigned articles attacking Johnson for his indulgence of the South and his "wretched treatment of black people." Tilton perceived that only if the federal government acted with all due speed "while the defeated South lies supine at the feet of the North" could equal rights be secured.
Beecher, however, wholeheartedly endorsed Johnson and his policies. Now it was Beecher, not Tilton, who had the ear of a president, and there were rumors that he would assume a Cabinet post. Indeed, one Sunday he ended the morning sermon dramatically with, "And may I not be seduced away from this pulpit!" On Sunday night, October 22, 1865, with three thousand people filling the pews and overflowing into the aisles of Plymouth Church, the Reverend Beecher delivered what the _Times_ called "A Great Political Sermon." Although he did not mention Theodore Tilton by name, Beecher began by saying that articles defamatory to President Johnson had appeared in the _Independent_. He wanted it clearly understood that someone else wrote those unsigned critical articles, and he wished it known that he felt Andrew Johnson should have "unstinting praise" as a "great American president."
Beecher said Johnson's plan for state governments was enlightened and he pleaded for "love and good will" toward the defeated South. He said, "I hold that it is not possible for us in the North, except in a remote way, to affect the condition of the black man in the South," and suggested that America rely upon "the kindness of the white man in the South, which is more important to the blacks than all the policies of the nation put together.... The laws and intents of the government and of ourselves will prove to no avail if they are unpleasant to the white people of the South."
Tilton was angered to the very marrow and retaliated with an essay in the _Independent_ , objecting to Beecher's reference to reliance on the "kindness" of Southern white men. "There is no such kindness," he wrote. "Does any man suppose that the Southern states of their own kindness will ordain equal rights? Why is the South annulling her ordinances of secession? Because she is coerced. Why is she abolishing slavery? Because she is coerced. And when will she ordain equal suffrage? Only when she is coerced." Tilton also took exception to the statement that the North was "remote" and powerless to change the situation. "On the contrary we hold that the power of the North in a direct way, may not only greatly affect but entirely revolutionize the condition of the black man in the South, that is, by the intervention of Congress to secure equal suffrage." Tilton wrote that the consensus of Congress was that if Johnson had, during the first months of his presidency, declared to the South, " 'I want the negroes to vote,' the South would have acquiesced without a murmur because it was expecting such a policy to be enforced."
Lib Tilton was dismayed by the conflict between her husband and her preacher. She wrote to Theodore, who was in Washington working with the radical Republicans, expressing her concern. He replied, "I have not seen Mr. Beecher and I suppose his difference is a difference only of opinion and not good will. But I am right and can't be driven from the rock under my feet." The first crack in Tilton's idolatry of Beecher had appeared.
Tilton vowed to keep his personal and political relationship with Beecher separate. But he was clearly torn. In November, returning home for a brief visit, he wrote to Beecher, "My friend, from my boyhood up you have been to me what no other man has been—what no other man can be. While I was a student the influence of your mind on mine was greater than all the books and all the teachings. By you I was baptized—by you married. You have been my minister, teacher, father, brother, friend, companion." At the same time, however, he warned his wife to be careful of her preacher's visits and told Lib about Beecher's alleged affair with Lucy Tappan Bowen and certain other loose conduct with women. Lib had observed how the women of Plymouth Church flung themselves at their preacher when, she said, what this lonely man needed was "soul-food." She was determined, her husband later noted, "to demonstrate that there was a woman who was superior to the silly flatteries with which many ladies in his congregation had courted his society—to demonstrate the honor and dignity of her sex."
FOR FOUR LONG years, woman's rights advocates had put their own dream of equal rights aside and had worked ceaselessly for the Union cause. With victory, once again they were ready to fight for their rights. Many of their friends from abolitionist days supported them. Among these was Henry Ward Beecher, who wrote an editorial, "Woman's Influence in Politics," recommending that women be given the vote. One rainy afternoon he saw Susan B. Anthony wearily walking along Columbia Heights. He clapped his hand on her shoulder and asked, "Well, old girl, what do you want now?" When she told him she was on her way to a woman suffrage meeting he volunteered to take up a collection for her. The following Sunday he raised $200 for her cause. Theodore Tilton, although deeply dedicated to Negro enfranchisement, also championed the cause of woman's rights. He wrote in the _Independent_ that both groups were entitled to the rights and privileges of white males because a "re-examination of the Declaration of Independence has powerfully suggested the equality of all human beings."
Elizabeth Cady Stanton, however, while she believed that the climate was now excellent for woman's enfranchisement, felt that little could be done for the Negro. In January 1866 she wrote, "When Andrew Johnson began the work of reconstruction, the negro's opportunity was lost. Politicians will wrangle over that question for a generation. Our time is now." Many, however, took the opposite view, contending that there was little practical chance that the complex issue of woman's rights, which cut deep into the social, religious, and economic structure of the nation, could be dealt with at this time. Right now the needs of the Negro were pressing; still more important: If the Republicans were to secure a majority in Congress, they needed black votes.
_Henry Ward Beecher preaching to the adoring women_
Though she believed in the primacy of woman's rights, the practical Stanton, realizing that the force of Congress was behind the Negro, decided that it would be politic to strive for the enfranchisement of Negroes and women simultaneously. Stanton and Anthony sought the advice of Theodore Tilton: Was there no way to grant suffrage to, as Stanton phrased it, "both classes of disenfranchised citizens"?
Tilton studied the question and came up with the idea of combining the Anti-Slavery Society and the Woman's Rights Society into a powerful coalition to espouse both causes, to be called the American Equal Rights Association. He diplomatically suggested that Wendell Phillips, who believed in the primacy of Negro rights and therefore might be a potential adversary, act as president of the new organization and that Frederick Douglass serve as a member of the coalition. Douglass accepted in good humor, writing to Stanton, "I have about made up my mind that if you can forgive me for being a negro, I cannot do less than to forgive you for being a woman." Phillips was less obliging and wrote, "While I could continue arguing for woman's rights, just as I do for temperance every day, still I would not mix the movements. That in my view is where... you and I differ. I think such a mixture would lose for the negro far more than we should gain for the woman."
Stanton was appalled by this response and wrote,
We have fairly boosted the negro over our heads, and now we had better begin to remember that self-preservation is the first law of nature. Some say, "Be still, wait, this is the negro's hour." But I believe this is the hour for everybody to do the best thing for reconstruction. A vote based on intelligence and education for black and white man and woman—that is what we need... and press in through the constitutional door the moment it is open for the admission of Sambo.
"Sambo"! With this bigoted rhetoric, Stanton had unknowingly taken the first step toward alienating the very people she most needed as allies.
Just as Amy and Isaac Post's home had been a center for abolitionists, the Tiltons' home was now a center for woman's rights advocates. At first, Lib Tilton had little to say to the "strong women" who gathered at her home, but later she recalled, "It mattered very little to me who they were—I took an interest—the house was open and I really feel that you should give me credit for that one gift of mine, if it is a gift, of seeing something in almost every one to be interested in—even those women who have troubled me so much."
Theodore Tilton, now the powerful editor of the _Independent_ , found himself in a world of ideas that Lib had difficulty comprehending, and her husband consistently made her inadequacies painfully clear to her. One night he took her from what she termed her "sick-bed" to a woman's rights gathering. When Lib hesitated in the doorway, clutching her husband's arm, he admonished her gruffly, "Don't come near me tonight." Once before at a similar function, she said that he told her, " 'I would give five hundred dollars if you were not by my side,' meaning that I was so insignificant that he was ashamed of me."
Among these women, Lib was comfortable at first only with Susan B. Anthony, in whose quiet strength she found reassurance. She admired Stanton but initially was cowed by her intellect. Often at night Lib would leave her husband alone at the chessboard only to wake in the morning to find him in the same position. Lib could not grasp the game but when Mrs. Stanton came to stay, she played chess with Theodore till three in the morning and bested him more often than not. Stanton enjoyed chess but said of it, "You seldom meet a woman who knows the game. They all say it's too hard work, as if thinking were not one of the pleasures of life." In time, Lib would become close to Mrs. Stanton, Paulina Wright Davis, and especially Laura Curtis Bullard, but that would come later, when she was "a changed woman."
_Tennessee Claflin, 1869. She was forced to prostitute her "other powers" for financial gain_. (Illustration Credit 9.2)
IN JANUARY 1864, when the Claflins fled Ottawa to avoid the manslaughter charge against Tennessee, Victoria decided to take her children, ten-year-old Byron and three-year-old Zulu, to Chicago. Within the week an advertisement appeared in the _Chicago Mail_ announcing the opening of an establishment for magnetic healing at 265 Wabash Avenue, Chicago, proprietress Victoria Woodhull. Along with the treatments, Victoria dispensed advice on health, love, and money—everything she was asked came down to these three subjects. She became unusually keen in anticipating questions and in establishing an invisible bond with her clients. At about the same time in Cincinnati, the rest of the Claflins moved into a boarding-house on Mound Street at the corner of Sixth and placed a sign in that window advertising Tennessee Claflin—Fortune-Teller and Magnetic Healer. Two months later, when they'd made a little money, the family rented a two-story house. As usual there was trouble with the law. A police officer followed the spouse of a prominent lumber merchant to the Claflin house, where he found the merchant's wife in an upstairs bedroom in the company of a gentleman not her husband. She was arrested and charged with adultery.
Four complaints were lodged against the Claflin family for running a house of prostitution but they managed to stay put until they met one Dr. James Kerr, who manufactured a cure-all tonic called "Kerr's System Renovator." Kerr, a frequent visitor at the Claflins' establishment, would later say that Tennessee had tried to blackmail him with evidence of his sexual escapades. He said that she had him "dead to rights" but that he told her, "Not me. I'm just like you!" Kerr lived openly with a Madame English, "the most notorious woman in Cincinnati." Together they turned the tables on the Claflins, obtaining depositions from the servants that revealed much of what went on in their bedrooms, naming the women who occupied the rooms but not the men. Kerr presented this evidence to the chief of police, James L. Ruffin. Overnight, the Claflins disappeared. Three weeks later they turned up in Chicago and moved in with Vickie.
In the summer of 1865, Victoria, Tennessee, and the self-proclaimed doctor Hebern Claflin took to the road with their traveling medicine show, driving Buck's wagon through Missouri, Arkansas, and Tennessee. For Victoria Woodhull it was business as usual and if she took notice of a society that was irrevocably being transformed, there is no record of it. The black population of Memphis, Tennessee, had been 3,882 in 1862. In 1865, the 16,509 blacks outnumbered the 11,194 white residents. The Spiritualist Elizabeth Meriwether, who had fled in 1862, returned to find that
_Victoria Woodhull, 1871. Her mission was to create a social revolution_. (Illustration Credit 9.3)
Negro squatters were everywhere. As we drew near Memphis the farm houses on the road side had been either deserted or burned to the ground. Some of the troops were black. They spoke no word to me, I spoke no word to them; their black faces and blue uniforms frightened me. In those days, just after the war, we did not know what the Yankees meant to do to the crushed and conquered Confederate soldiers. Those negroes were armed; they would get leaves of absences; they could walk from Fort Pickering to Ridgeway in half an hour; no friends or neighbors were near me. Can you wonder, my children, that I was uneasy?
A new group of politicians was taking over in Memphis. They frequented the local taverns, where Vickie and Tennie read their palms and occasionally left to spend the night with one or another of them. One evening Tennessee spotted a young gambler, John Bartels, who had just won a fistful of gold pieces in a faro game. She married him the next week. Scarcely a month later, on the way back to Chicago, Bartels asked his wife about a ten-dollar bill she had mysteriously acquired after disappearing for several hours. "Where'd you get it?" he demanded. Tennie wouldn't tell him. This led to a bitter quarrel, after which she agreed to give him her share of the proceeds from the trip on the condition that he go away and divorce her.
Their return to Chicago was flamboyant; if Victoria could not be accepted by society she was nevertheless determined to be noticed. Even as a child when she had preached on The Mount of Olive, her longing for recognition was evident. The wagon rolled into Chicago accompanied by a four-piece brass band and outriders who, to the consternation of the neighbors, galloped up to the door whooping Indian war calls.
By the fall of 1865, the Chicago house was teeming with Claflins. Victoria told them that the spirits had commanded her to go to St. Louis, and she left her own house and her family behind. She used the alias Madame Holland and once again advertised herself as a magnetic healer and clairvoyant fortune-teller. It was in St. Louis that Colonel James Harvey Blood entered the life of twenty-eight-year-old Victoria Woodhull.
At the start of the war, Blood had been a city auditor and president of the St. Louis Railroad Company; a conservative, prosperous man of impeccable reputation who lived in a whitewashed house with his wife, Isabel, and two daughters. Then Blood saw Missouri ripped apart over the question of slavery. Three-quarters of the white men fought for the North, one-quarter fought for the South—brother against brother, friend against friend.
Blood was one of the first to enlist for the Union in the Sixth Missouri. At Sink Pole Woods in March 1862, Colonel Blood received a bullet in his right shoulder. In May, he fought in Corinth, Mississippi, and received another bullet in the right arm. In December, he was wounded in the left hand at Chickasaw Bluff and took two bullets in the left thigh during the assault on Vicksburg. As he waited his turn in the surgeon's tent, he saw limbs being hacked off and bullets pried out with no anesthetic. He returned to his own tent, doused his thigh with whiskey, and with a hunting knife cut the two bullets out and bound the wounds. Unable to fight further, Blood returned to St. Louis a hero. Although he appeared untroubled, his soul had been as ravaged as his body. In 1864, he became the president of the St. Louis Society of Spiritualists. Blood claimed to be in constant communication with his brave companions who had died in the war.
Victoria Woodhull's version of her first encounter with Colonel Blood is romantic: The tall stranger strode into her darkened chamber and sat before her in a low chair. Immediately, she fell into a trance and announced, "I see our futures linked. Our destinies are bound together." When she emerged from her trance, these two were united "on the spot by the powers of the air." In fact, Blood's wife had asked him to accompany her to the Washington Avenue office of one "Madame Holland," who advertised "wonderful cures of female complaints by means of clairvoyance." Soon it was James Blood, not his wife, who frequented Madame Holland's establishment.
That summer Colonel Blood and Vickie fled. Blood left everything behind—wife, children, job, reputation, and a debt of $3,700. He bought a brightly colored wagon with a ball-fringed top like Buck Claflin's, and he and Vickie set out on the road through Missouri and the Ozarks, calling themselves Dr. and Mrs. James Harvey. They repeated a familiar routine: The colonel became the advance man and manager; Vickie read the past and future, gave advice, and provided treatment for physical and emotional problems. When they had made enough money they returned to St. Louis just long enough for Blood to obtain a divorce, give his wife all his possessions, and pay off his debt. Then they moved on.
Meanwhile, the Claflin family in the Wabash Avenue house in Chicago were up to their old tricks amid an atmosphere of illicit sex and blackmail. Young women, many of whom were servant girls, came to have their fortunes told. Some stayed late and along with the four unattached Claflin girls—Tennessee, Margaret Ann, Polly, and Utica—entertained men.
_Colonel James Harvey Blood, 1868. Victoria told him, "Our destinies are bound together._ " (Illustration Credit 9.4)
Blackmail had always been Buck's most lucrative sideline. Five years later, Victoria described how her father blackmailed more than one man who visited his daughters by pretending to be the distraught father of a ruined virgin.
Father, at times a Mephistopheles, waits till the inspiration of cunning overmasters his parental instinct... and lodges an indictment against his own flesh and blood, takes out his handkerchief to hide a few well-feigned tears, clasps his hands with an unfelt agony and money in hand hobbles off smiling sardonically at the mischief he has done.
Tennessee, at twenty, was still the star of the show. Since the age of eleven she had been the main support of her disreputable relatives. Victoria said that the family was united only in their determination that Tennie "should earn all the money." Utica was jealous of her younger sister, which may be why she provided—or created—this description of her activities at this time: "Tennie has had ten men visit her in one night and after each, I've bathed her, given her a new night-robe, and perfumed her for the next."
Once again, neighbors complained that the Claflins were running a house of prostitution, but as usual the charge was difficult to prove. Instead, the neighbors lodged two suits for fraudulent fortune-telling. The landlord, eager for an excuse, canceled the Claflins' lease and evicted them. It was then that Tennessee rebelled. She packed a small suitcase, found Victoria and Colonel Blood, and, with tears in her eyes, asked her elder sister, "My God, have I got to live this life always?"
Victoria could no longer suppress her anger. Because of their father, Tennessee had been branded as a prostitute and still faced a charge of manslaughter. Victoria, in times of pain and terror, had been able to retreat into the world of the spirits. They permeated her soul and comforted her and she believed in them the way others believed in the mercy of Christ. But for Tennessee there was only the reality of her squalid life. "I was almost lost," she was later to admit.
Victoria provided for her younger sister the protection she had been unable to provide for herself. Under Blood's aegis, for the first time, both women were treated decently and allowed to think for themselves. Tennessee admired the colonel and was grateful to him for his part in rescuing her. But Roxy Claflin hated James Harvey Blood for the influence he had over her daughters and Buck hated him for taking away his source of income.
# CHAPTER TEN
DRAW ITS FANGS
AS THE YEAR 1866 began, the radical Republicans were still a minority in the Thirty-ninth Congress; nonetheless they were determined to diminish the president's power and defeat his so-called Restoration. In February, Congress passed an act to extend the duration and grant additional federal enforcement powers to the Freedmen's Bureau, which had been established to help the emancipated slaves. Johnson vetoed the act. Although the Thirteenth Amendment freed the slaves, it did not give them the right to vote or protect them from abridgements of their civil rights by state governments. In April, Congress passed a sweeping civil rights act that specifically empowered all native-born persons over the age of twenty-one (except for "Indians," who were not considered citizens) to enter into contracts, sue, testify in court, and serve on juries. It granted federal jurisdiction over state courts and invalidated discriminatory state laws. Maine senator Lot M. Morrill, a supporter of the act, said, "This species of legislation is absolutely revolutionary. But are we not in the midst of a revolution?" Johnson vetoed this act too.
Theodore Tilton, who had been in Washington to help draft this legislation, was enraged by Johnson's veto and in the _Independent_ declared that if President Johnson recklessly continued to usurp Congress's function, he should be impeached. Tilton was the first to suggest impeachment in an American newspaper, a suggestion praised by Republican newspapers across the country and endorsed by such politicians as William Lloyd Garrison, Thaddeus Stevens, Wendell Phillips, and Benjamin Butler. The _Argus_ called Tilton "the rising young star of America."
Asserting the power of the federal government, the moderate Republicans, the largest group in Congress, banded with the radical Republicans and passed the Civil Rights Act over Johnson's veto. On April 30, the Fourteenth Amendment, intended to bolster this legislation, was introduced into Congress. The first and second sections of the amendment granted the freedmen citizenship and, while not specifically mandating the vote, severely reduced the representation in Congress of any state that denied any _male_ citizen this privilege. The third section, written under the guidance of Stevens and Butler, was added to curb Johnson's presidential pardoning power. It specified that a two-thirds vote of Congress was required before any former Confederate could hold public office.
Stanton, Anthony, and President Johnson opposed the Fourteenth Amendment for very different reasons. The previous summer, as the amendment was being drafted, Stanton had written Anthony, "I have argued constantly with Phillips and the whole fraternity but I fear one and all will favor enfranchising the negro without us. Woman's cause is in deep water." For the first time the word _male_ had been introduced into an amendment to the American Constitution. President Johnson for his part objected to the amendment because the first section, designed to protect the newly emancipated Negro from any abridgement of his rights by the states, implicitly asserted the power of the federal government over states' rights. But his fury was aimed at section three, which transferred his pardoning power to Congress. Although it did not require his signature, Johnson denounced the Fourteenth Amendment and urged the Southern states not to ratify it. Ten of the eleven sided with him in rejecting the amendment.
The Republicans were now convinced that the Southern states had not accepted defeat, while the Democrats felt that the Republicans were ready to use any means to control the federal government "even if it meant to Africanize the South." The Civil Rights Act had gone into immediate effect and "Black Codes" were being voided throughout the South. The Southern state governments, subject to federal military authority, were powerless. In Pulaski, Tennessee, six former Confederate soldiers banded together to take matters into their own hands and formed a secret society. Corrupting the Greek word _kyklos_ , meaning circle, they added Klan for alliteration: the Ku Klux Klan. Throughout the South, white men governed by pistol and rifle.
Bitter in defeat, militant Southerners afflicted the Negro with the hatred they felt for the mighty Yankee. During the war, Memphis had been the headquarters of the Third United States Colored Heavy Artillery at Fort Pickering and became a haven for the black "contrabands" who lived in a ghetto in South Memphis known as Licksville. By 1866, the emancipated Negroes had set up twenty-two schools in this vicinity and farmed land that had been leased to them with the promise that one day they would be permitted to buy these properties. It was soon rumored that these blacks planned "to take control of the city," a belief strengthened by the crowds of tattered but neat Negroes who, having earned their first wages, spent them in Memphis shops where no black people had previously dared to venture.
On April 30, 1866, by order of the War Department, the remaining black regiments in Tennessee were mustered out and ordered to turn in their arms. With these troops officially disarmed, what happened next, although later said to be "spontaneous," was in fact a planned exercise in terror. By Tuesday, May 1, 1866, only 150 black soldiers remained at Fort Pickering when members of the largely Irish police force began a race riot. Many of these men had been Confederate soldiers in the Third Calvary Regiment and had taken part in the massacre at Fort Pillow, where every black Union soldier taken prisoner was shot.
Special targets of violence were the hospitals and schools operated by the Freedmen's Bureau, where the rioters taunted their victims as they beat them, telling them to ask their friends at the bureau for help. More than $100,000 in black property was destroyed. Four schools were burned and a black student, Rachael Hatcher, was shot and then set afire by John Pendergast and his sons Pat and Mike. After two days, federal troops finally arrived and ended the violence. No white person had been killed or injured by a black person. Forty-six black men, women, and children lay dead, and 285 more were maimed. Although there were hundreds of witnesses, no white man was arrested. And Elizabeth Meriwether observed, "One good result of the Memphis riot and massacres was the improved behavior of the negroes."
The horror of the Memphis riot strengthened the position of the radical Republicans in Congress. It also made the enfranchisement of the Negro a question of life and death. An exasperated Theodore Tilton recommended in the _Independent_ that Negro troops be reactivated and used to occupy the South. "They are the natural terriers to watch such rats. They know every trick of the rattlesnake. Therefore, let them be the chief charmers to tame it and draw its fangs."
The following week, on May 10, 1866, a joint meeting of the Anti-Slavery Society and the Woman's Rights Society convened at the Church of the Puritans in Union Square, New York. Although the atmosphere was one of hope and cordiality, there was a division over the primacy of Negro rights. Equally divisive was the mounting political and personal rivalry between Beecher and Tilton.
The Anti-Slavery Society met first. Susan B. Anthony pointed out that the membership of the two organizations was virtually identical and, following Tilton's advice, put forth a resolution that the groups amalgamate. Wendell Phillips vigorously opposed the resolution and it was voted down. However, at the Woman's Rights Society meeting that followed, with Stanton presiding, Wendell Phillips spoke in favor of woman suffrage with the caveat that it would be pursued _after_ Negro rights were secured. Anthony, still hoping to align the groups, now proposed that the Woman's Rights Society change its name anyway, calling itself the American Equal Rights Association (AERA), to emphasize the goal of universal suffrage for Negroes and women alike. The resolution passed.
Late in the day Tilton mounted the platform. This strikingly handsome man was at the height of his powers: "Mrs. President, this convention is called to consider the most beautiful and humane idea which has ever entered into American politics—the right of woman to that ballot which belongs equally to all citizens." Then with a jab at Johnson, he continued, "At least one President is right—I mean _this_ President," and he pointed to Stanton. "She does not claim the ballot for women as women, but for women as citizens. One asks, shall we have a woman for President? I would thank God if today we had a _man_ for President."
When he concluded, Tilton turned his attention to Beecher, his ridicule masked as humor. "According to the programme, it is now my friend Mr. Beecher's turn to speak, but I observe that this gentleman, like some of the rest of the President's friends, occupies a back seat. While, therefore, he is sitting under the gallery, I will occupy your attention just long enough to give that _modest man_ a chance to muster nerve enough to make his appearance in public." Laughter erupted throughout the hall at this absurd vision of Beecher as a shrinking violet.
Beecher rose from his seat, walked the length of the hall, and mounted the platform. Greatly animated, his face aglow with perspiration, he spoke for an hour to a rapt audience of women that included Lib Tilton. Beecher spoke of woman being in every way man's equal and even his superior. He preached a gushing gospel of loving words but they contained little practical advice.
You may... ask me, "What is the use of preaching to us that we _ought_ to [vote] when we are not permitted to do it?"... But the reason you have not voted is because you have not wanted to. It is because you have not felt that it was your duty to vote.
I stand on far higher ground in arguing this question than the Right of Woman. I go to the Supreme bench and argue it on the ground that the nation needs woman, and that woman needs the nation, and that woman can never become what she should be, and the nation can never become what it should be, until there is no distinction made between the sexes as regards the rights and duties of citizenship—until we come to the twenty-eighth verse of the third chapter of Galatians.
Beecher, carried away by his own rhetoric, faltered. Turning to Tilton, he asked, "Galatians—what is it?"
"I don't know," replied Tilton icily, refusing to help the preacher. The insults began. During Tilton's opening speech he had mentioned Lucy Aikin's _Life of Addison_ but inadvertently had referred to the author as Lucy Western, a well-known actress of scandalous reputation.
"If it was Lucille Western, you would," Beecher said jovially, and, extracting a small white Bible from his pocket, he read from Galatians, "There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female; for ye are all one in Christ Jesus...."
Once again Beecher turned to Tilton and with a flourish handed him the Bible. "Theodore was a most excellent young man when he used to go to my church. But he has escaped from my care lately and now I don't know what he does."
Tilton spoke over the chuckling:
In the midst of the general hilarity produced throughout the house by my friend's speech, I myself have been greatly solemnized by being made, as you have witnessed, the public custodian of his New Testament. At first I shared in your gratification at seeing that he carried so much of the Scripture with him. But I found, on looking at the fly-leaf, that the book after all was not his own, but the property of a lady—I will not mention her name. I have, therefore, no right to accept my friend's gift of what is not his own.
With that Tilton handed the Bible back to Beecher. Warming to the audience, he continued:
I remember that when he came home from England, he told me a story of a company of ten ministers who sat down to dine together. A dispute arose among them as to the meaning of a certain passage of Scripture—for aught I know the very passage in Galatians which he just now tried to quote, but couldn't. Some one said, "Who has a New Testament?" It was found that no one had a copy. Pretty soon, however, when the dinner reached the point of champagne, some one exclaimed, "Who has a corkscrew?" And it was found that the whole ten had, every man, a corkscrew in his pocket!
_Wendell Phillips. This aristocratic abolitionist had a secret lover_. (Illustration Credit 10.2)
Beecher interrupted with equally false joviality, "Now I know enough about champagne to know that it don't need any corkscrew!"
"How is it that you know so much more about corkscrews than about Galatians?" Tilton replied.
The mutual antagonism was apparent despite the laughter.
THE FOLLOWING DAY Anthony and Stanton met with Tilton and Phillips. Anthony resented Wendell Phillips's public objection to amalgamating the Anti-Slavery and Woman's Rights societies. But Phillips, more than anyone else, could be credited with awakening the nation to the evils of slavery and he was not about to jeopardize the black vote by joining forces with these women. The discussion concerned the women's campaign to strike the word _male_ from both the Fourteenth Amendment and the proposed revision of the New York State Constitution. Phillips emphasized that the New York State Constitution also included the word _white_ and argued that that was the sole word which must be eliminated. Phillips—so magnetic that a close friend observed, "Whoever came into personal relationship with him felt the charm and spell of his power"—managed to convince Tilton that he was right. Theodore then recommended that the newly renamed American Equal Rights Association circulate petitions throughout the state "praying for the enfranchisement of the negro and postponing [women's] demands until the next revision of the New York State Constitution." Tilton added that "the question of striking out the word _male_ shall of course be presented as an intellectual theory but not as a practical thing to be accomplished at this convention."
To Anthony's dismay, Stanton did not object. Anthony was staggered and exclaimed, "I would rather cut off my right hand than ask for the ballot for the black man and not for women!" As she turned to leave the meeting, she overheard Tilton whisper to Stanton, "What does ail Susan?" Elizabeth replied, "I cannot imagine. I never before saw her so unreasonable and absolutely rude."
Anthony was convinced that Phillips had triumphed over Stanton's reason. She had succumbed to the charisma of this aristocrat whose family had arrived aboard the _Arbella_ , and in 1630 had been among the Puritan founders of the Massachusetts Bay Colony. That evening a hurt and outraged Anthony, ready to do battle, arrived at Stanton's house only to find her friend pacing up and down. "Oh, Susan," she berated herself, "do tell me what is the matter with me!" Elizabeth said that Phillips had publicly betrayed them, had treated her like a manipulable child, and she had gone along with it. She vowed that this would never happen again.
Anthony too had become wary of Phillips. His actions at the convention had been a political betrayal, but another factor influenced her as well: Anthony's attachment—later called "the most passionate affection of her life"—to Anna Dickinson, the fiery Quaker orator. Twenty-four years younger than Anthony, Anna was the child of an abolitionist father who, when she was four, suffered a fatal heart attack while giving an antislavery speech. At fourteen, she began to teach school to aid her impoverished, widowed mother, and four siblings. She was nineteen in 1862 when she made her official debut at the Boston Music Hall. Both Anthony and Phillips had attended and were entranced by Dickinson. Anna Dickinson had long worshiped Wendell Phillips. When she was fifteen, she spent one afternoon on her knees scrubbing the sidewalk in front of the Philadelphia Academy of Music in exchange for free admission to hear Phillips speak. After her Music Hall speech, the usually restrained Phillips came forward, grasped both of Anna's hands in his, and said, "My dear Anna, you brought tears to my eyes—they had almost forgotten the sensation."
_Anna Dickinson—the Quaker girl who became America's most persuasive speaker for woman suffrage_ (Illustration Credit 10.3)
In Boston, Phillips saw Dickinson "almost every day." Anthony, also smitten, soon became convinced that of all "our girls" Anna Dickinson would be her successor in the leadership of the movement. Anna was thought to be a great beauty, plump and rosy, with an hourglass figure, dark eyes, dark hair, a youthful glow. Whether she was truly beautiful is debatable: Some pointed to her hatchet chin, her flat nose, and deep-set eyes. But when she spoke, Wendell Phillips said, it was with a "demoniac-like eloquence." Mark Twain wrote that "her vim, her energy, her tremendous earnestness, could compel the respect and attention of an audience, even if she spoke in Chinese." Phillips tried to enlist Anna solely in the cause of Negro suffrage, but she also supported woman suffrage. And so a powerful emotional and political triangle began to form, as Anthony and Phillips contended for "the possession of her soul."
In the congressional election year of 1866, Elizabeth Cady Stanton declared herself an independent candidate from the Eighth District of New York. Although women could not vote, if elected by male voters, they could serve. Stanton found the campaign a "merciless duty," for she also had the responsibility for a full domestic household. "I must buy butter and meat, hear youngsters spell and multiply, coax parted threads in stocking heels and toes to meet again... and smooth down the ruffled feathers of imperious men or cross chambermaids and cook," wrote Stanton. "Then comes Susan, with the nation on her soul, asking for speeches, resolutions, calls, attendance at conventions."
This, of course, was a reference to the unmarried Anthony's ceaseless campaigning while Stanton was more or less housebound. Susan looked on with dismay as Elizabeth's family increased, and disapproved of her failure to use abstention as a means of birth control. Unlike other women of her day, Anthony did not think that only the man enjoyed sexual intercourse. When Stanton became pregnant with her seventh child, Anthony wrote a friend, "I only _scold now_ that for a _moment's pleasure_ to herself or her husband, she should thus increase the _load of care_ under which she already groans." In November, after months of campaigning in an election where women could not cast their ballots, Stanton received only twenty-four votes.
Meanwhile, President Johnson toured the country carrying a message of his own to "the people." In New Orleans a "much excited" Johnson told a reporter he was certain "the people of the South... were to be trodden underfoot to protect niggers." He hinted that Northern radicals meant to assassinate him and called them traitors. When asked to name these men he cried out, "I say Thaddeus Stevens of Pennsylvania is one. I say Mr. Sumner of the Senate is another, and Wendell Phillips is another!" The _Times_ called Johnson's speeches "vulgar, vindictive and loaded with self-pity," and Tilton wrote in the _Independent_ , "The people have been witness to the mortifying spectacle of the President going about from town to town, accompanied by the prominent members of the Cabinet on an electioneering raid, denouncing his opponents, bandying epithets with men in the crowd, and praising himself and his policies. Such a humiliating exhibition has never before been seen, nor anything even approaching to it."
In July a riot broke out in New Orleans that took place during a political gathering for universal suffrage attended by black and white men and women. Forty-eight black men, women, and children were killed, and 250 more were injured. Against the background of this struggle for the future of the nation, Beecher and Tilton found themselves in a major political fight of their own. Johnson was scheduled to appear in Cleveland at the Soldiers and Sailors Convention, which enthusiastically supported Johnson's allies in this congressional election year. Beecher, who was invited to attend as chaplain, received the invitation at his farm in Peekskill, New York, where he had taken his family to avoid the cholera epidemic sweeping New York and other cities. Fearful of making the trip to Cleveland, he begged off, saying that it was mowing time and his hay fever had been acting up. But he wrote a letter, which was reprinted in newspapers throughout the country, endorsing the convention's recommendation that the South be allowed to rely on state governments with regard to their treatment of the Negro. "If they [the Negroes] have the stamina to undergo the hardships which every uncivilized people has undergone in its upward progress, they will in due time take their place among us," wrote Beecher. "That place cannot be bought, nor bequeathed... it will come to sobriety, virtue, industry and frugality."
"The Cleveland Letter," as it came to be known, met with a storm of criticism both for urging postponement of Negro enfranchisement to some hazy future and for what the _Times_ called Beecher's "unqualified endorsement" of the faltering President Johnson. Beecher's letter infuriated many radical Republicans and clergymen but most of all Theodore Tilton. The intensity of Tilton's anger was, no doubt, influenced by his personal feelings about his wife's growing attachment to Beecher. Tilton excoriated his former idol in the _Independent_ , attacking him for having misused the great power of the church for base political ambition. "He has done more harm to the American Republic than has... any other citizen except Andrew Johnson," Tilton wrote.
The power of Tilton's fiery rhetoric threatened Beecher in a manner he had not thought possible. Horace Greeley, and even Henry's brother, the preacher Edward Beecher, attacked him for his intimations of white supremacy and his support of Johnson. Henry Ward Beecher called a meeting at the vast Brooklyn Academy of Music. Abandoning his usual extemporaneous style, he read a carefully prepared statement disassociating himself from the Democratic Party. "During the first three weary, dark and disastrous years there was an utter want of outspoken sympathy with our Government on the part of the Democratic Party. I cannot belong to that party."
But Tilton would not let the public forget that Beecher's words had a hollow ring, for during that time he had constantly attacked the government and called Lincoln "the worst candidate possible." Tilton wrote, "The spectacle at the Academy of Music had in it a touch of humiliation which even the noblest passages of the address did not redeem.... Something of true moral grandeur is wanting in the position of a veteran who, after twenty-five years of service as a pioneer of political opinion, has nothing nobler to say than simply, 'I am not a Democrat.' "
Tilton's blistering criticism bewildered Beecher, but nothing disturbed him so much as the committee formed within Plymouth Church itself to consider his dismissal. Beecher had been sure of the personal loyalty of his parishioners and considered Plymouth Church his own property. The committee was organized by Henry Bowen, and his son Henry Jr. was appointed chairman. The group included several church members who had benefited from Bowen's patronage during the war. Bowen at last saw an opportunity to avenge himself for Beecher's affair with his late wife. More important—for this was a game of wealth and power—if Beecher were forced to resign, Bowen could put a pawn in his place, and the political patronage emanating from Plymouth Church would be his.
Henry Ward Beecher reeled under the shock of this blow and went for advice to John Tasker Howard, one of the investors in the J. B. Ford publishing syndicate and a founder of Plymouth Church. Howard owed Beecher a considerable debt. During the war his son Joseph had been a _Times_ correspondent known for such antics as filing his story then tying up the wire for hours so his rivals could not reach their newspapers with their own accounts. For Joe Howard, an unprincipled rapscallion, the war had held unlimited possibilities for gain. On the night of May 17, 1864, he and a friend had filched Associated Press memo paper, envelopes, and a stylus and delivered to several New York newspapers the news that General Grant's Virginia campaign was faltering and that Lincoln had issued an order to draft four hundred thousand men.
Howard knew that when this grim news was circulated, the price of gold was bound to rise. Indeed, the following morning, after several newspapers printed the bogus dispatch, the gold market jumped eight points, yielding Joe Howard a $40,000 profit. By afternoon, however, the hoax had been discovered and soldiers with drawn bayonets arrived in the editorial rooms of the participating newspapers and demanded that they issue retractions. It was not long before Howard was found out and dispatched to Fort Lafayette prison.
John Tasker Howard pleaded with Beecher to secure his son's release. Without changing his clothes, Beecher left for Washington and returned the following afternoon with a full pardon. By evening Joe Howard was once again in what Beecher was later to call "the weeping bosom of his family."
Now, on John Howard's advice, Beecher wrote a letter disclaiming any attachment to Johnson. Howard then carried Beecher's letter to the uncommitted committee members and lobbied on the preacher's behalf. In a powerful act of persuasion, Howard caused the committee to disband. Bowen's coup had failed, but had left opposing political factions within Plymouth Church.
Beecher vowed to let Bowen know the price of disloyalty. He withdrew his sermons from the _Independent_ , and local advertising dropped precipitously. But this was a small gesture compared to what happened next: Within three weeks President Johnson had removed Henry Bowen from his newest political appointment as tax collector for the Third Ward, which encompassed the greater part of the city of Brooklyn. He dismissed Bowen's staff and canceled contracts with Bowen's cronies. As this vast political patronage was snatched from his grasp, Bowen, who had so far unstintingly supported Tilton and the popular forces behind him, began to reassess the situation. What should he do to recapture position and power?
# CHAPTER ELEVEN
A DANGEROUS MAN
IN SEPTEMBER 1866, Theodore Tilton was once again on the road as a delegate to the Loyalist convention in Philadelphia, organized to combat Johnson's congressional electioneering. (This convention was composed of delegates from both Northern and Southern states who had remained loyal to the Union during the recent war.) The choice of five women delegates seemed radical enough, but when Frederick Douglass was appointed from Rochester, several radical Republicans found themselves in a precarious political position. Facing a congressional election in only two months, they needed support in the Southern and border states that opposed black enfranchisement. Many Republicans feared that the presence of a black delegate would cost them this support and possibly the election.
Douglass had made headlines when he'd visited the White House to remind President Johnson that empowering his people to vote would give them the means "with which to save ourselves.... The fact that we are... subject to taxation, to volunteer in the service of the country, subject to being drafted, subject to bear the burdens of the State, makes it not improper that we should ask to share in the privileges of this condition." According to his private secretary, Johnson answered "in a manner which indicated repressed anger," saying that giving Negroes the vote might lead to "a conflict of races" and he suggested emigration as a solution to the problem. "The President no more expected that darkey delegation yesterday than he did the cholera," the secretary told a _World_ reporter, and he added that Johnson had remarked when they left, "Those damned sons of bitches.... I know that damned Douglass. He's just like any nigger and he would sooner cut a white man's throat than not."
Douglass would find himself and his cause equally unwelcome at the convention in Philadelphia. Ironically, even some of the delegates who had been steadfast abolitionists feared that the very success of their crusade to end slavery might grant too large a measure of power to the newly freed slaves, a view typified by Henry Blackwell, who, with his wife, Lucy Stone, led the powerful New England woman suffrage contingent. Returning from a recent visit to the South, Blackwell said that he was willing for the Negro to have suffrage, "but not under such conditions that he should rule the South."
On the train to Philadelphia, a group of delegates to the Loyalist convention banded together to ask Douglass to resign his appointment in the interest of granting Congress control over Johnson. Douglass answered, "Gentlemen, with all respect, you might as well ask me to put a loaded pistol to my head and blow my brains out."
Theodore Tilton and Anna Dickinson, a Philadelphia delegate, were warned by Thaddeus Stevens not to ally themselves with Douglass. The capitulation to political expediency was clear: As a young man, Stevens had given his life's savings of $300 to buy the freedom of a slave who was being sold away from his family. He had been among the first to endorse Negro suffrage. Now ill and close to death, he had instructed that he be buried in a nonsegregated graveyard. Yet the pragmatic Stevens was willing to do whatever was necessary to elect a Republican majority in Congress.
The following morning, Philadelphia newspaper headlines declared of the convention, "Philadelphia Full of Miscegenists." "Negro Insurrection To Be Incited." "First Grand National Convention of Spiritualists, Woman's Rights, and Negro Worshipers." Douglass strode through the crowd of delegates in front of Philadelphia's Independence Hall amid tense silence. He observed that the other delegates seemed "ashamed and afraid" of him, turning away as if repelled by a negative magnet. Among the few who greeted him, he recognized only Benjamin Butler.
Douglass realized that the delegates were lining up two by two for a march through the city but no one approached him. "I was the ugly and deformed child of the family." He stood in increasing isolation as the line of pairs grew longer. "Who of my brother members would consent to walk with me?" he asked himself. None seemed willing. He would have to walk alone—a pariah. But as the line began to move out, a tall, lanky man with blond curls cascading to his shoulders appeared at Douglass's side. "He came to me in my isolation, seized me by the hand in a most brotherly way, and proposed to walk with me in the procession. I have been in many awkward and disagreeable positions in my life, when the presence of a friend would have been highly valued, but I think I never appreciated an act of courage and generous sentiment more highly than I did that of this brave young man." This man was Theodore Tilton.
Tilton, who had not been able to amend his humanity for political expediency, came in for sharp criticism. An angry Thaddeus Stevens wrote, "A good many people here are disturbed by the practical exhibition of social equality in the arm-in-arm performance of Douglass and Tilton. It does not become radicals like us particularly to object but it was certainly unfortunate at this time. The old prejudice now revived will lose us some votes. Why it was done I cannot see except as foolish bravado."
_Frederick Douglass. As an escaped slave he lectured for abolition but was reprimanded for sounding like a refined white man_. (Illustration Credit 11.1)
Because of its incendiary nature, discussion of the Negro vote had been excluded from the convention. But as Frederick Douglass noted, "Miss Dickinson, Mr. Tilton and myself felt that any reconstruction of the South leaving the freedmen without the ballot would leave them in the absolute power of the old master class." On the final day, through Anna Dickinson's urging, Tilton introduced the subject, which, according to Douglass, caused "tumult and confusion in the midst of which the president... and other officers left their places on the platform, declaring the convention adjourned." When Tilton called out at the top of his lungs, "This convention is _not_ adjourned, as the majority remains in this hall," the delegates from the border states also stood up and stormed out. Even so, a majority remained.
Once calm had been restored, Tilton and the remaining delegates nominated a new president, and Anna Dickinson made a forceful plea for a constitutional amendment that would, once and for all, guarantee Negroes the vote against any restrictions that might be imposed by the individual states. Then Tilton formally put forth resolutions that would become the basis of the Fifteenth Amendment:
The right of citizens of the United States to vote shall not be denied or abridged by the United States, or by any State, on account of race, color, or previous condition of servitude.
The Congress shall have power to enforce this article by appropriate legislation.
In the absence of the hostile delegates, the resolutions passed with little opposition, marking Theodore Tilton as a hero to some and a dangerous man to others.
In November 1866, a Republican Congress swept into power, its cause aided by the atrocities in the South and the seemingly out-of-control bigotry of Andrew Johnson. With more than enough votes to override every presidential veto, the plans for Reconstruction could at last be implemented. Theodore Tilton's dream of a new society of equality and justice appeared within reach, but the problems in the way of this goal were massive and required more selflessness than those in power possessed. By law, December 18, 1866, marked the deadline for freeing the slaves. On that evening, thousands of bewildered black people were turned out by their former owners onto the roads of the South. The plight of these abandoned souls weighed heavily on Theodore Tilton's mind.
Tilton was dismayed at politicians who had little regard for the equality of black people or of women. As his disillusionment grew he cut himself adrift to explore new possibilities and new ways of looking at the world. From his writings at this time we know that he felt "alone and despairing," but also that he was "striving to tear off, little by little—since I cannot do it at a stroke—some of the bandages that have hitherto bound me blind."
As if trying to find a touchstone, after a lecture in Springfield, Illinois, he visited Abraham Lincoln's grave. Chilled to the bone, he bent down and scooped up an oak leaf that lay near the headstone. Later that night he put the leaf in an envelope and wrote to his wife,
I never thought that Abraham Lincoln was one of the great men of the world but I had always a tender feeling towards him, akin to personal affection. The great fault of our American statesmen and public leaders is a lack of that moral courage which, of all endowments, makes men most truly great. More and more I believe in absolute fidelity to Liberty, Justice, and Equality. Every public man who compromises these great principles retards the progress of his country.
Mr. Lincoln's fame rests on his Proclamation—a measure to which he was driven somewhat against his will. He was willing to save slavery if he could thereby save the Union. This willingness to compromise with wrong showed a mind not great in the highest sense. But he was a noble, grand, and illustrious man after all... death canonizes a great character! The world magnifies its favorite names into colossal fame. Abraham Lincoln's name and fame will probably overtop the records of all other men of the nineteenth century.
Among the beliefs that Theodore Tilton had begun to question was the traditional religion that previously had sustained him. He wrote to Lib,
The more I think of the whole subject of religion, of theology, of the church, of doctrines, of creeds, I am inclined to undervalue or rather see little value in everything but the Christian character.... I see so much in my travels that goes to show how men content themselves with low lives instead of the high... with selfish greed instead of generous self-sacrifice.... Hereafter I mean to take pattern, not after men, but after the Great Teacher.
The sexual conventions of the day also came under Tilton's scrutiny. In Beecher's church he had been taught that the highest love transcended passion. (This, of course, implied that wives were chosen for their purity and high-mindedness, while sexual pleasure was to be found in the company of "vile women," the common term for prostitutes and other women of easy virtue.) As Tilton had explained to Lib, "Men and women who have the mere natural _instinct_ for loving, love with the heart, but they who have a true _genius_ for loving, love with the soul." In his novel _Norwood_ , Beecher wrote that sexual passion needs "rein and curb." But in practice both Tilton and Beecher struggled with the sacred and the profane. Beecher, by way of justifying his own licentious behavior, later explained to Frank Moulton, "If love was proper and not wrong therefore it followed that any expression of that love, whether by a shake of the hand or a kiss of the lips or even bodily intercourse was not wrong." Yet despite this endorsement of "free love," the self-indulgent Beecher deplored his own sexual excesses, and his sexual gratification was followed with the bathos of guilt.
This pattern was not unlike Tilton's own and typified the dilemma of many religious men of the day. Tilton too found himself in a constant spiral of sin and repentance, and he confided to his wife,
I believe that I have less self respect than in former days.... I find myself a constant sinner. Lately I have many times bowed my head like a bulrush. I have once or twice done right under strong provocation to do wrong, and I have several times done wrong under a mere gossamer of temptation.
Like her husband, Lib Tilton was caught in the vise of sexual hypocrisy: Women of her class were trained to transcend carnal desire, thereby fulfilling the religious ideal of spiritual superiority. Catharine Beecher's guide to behavior specified,
Do not suffer your hand to be held or squeezed, without showing that it displeases you by instantly withdrawing it. If a finger is put out to touch a chain that is round your neck, or a breast-pin that you are wearing, draw back, and take it off for inspection. Accept no unnecessary assistance in putting on cloaks, shawls, overshoes, or anything of the sort.
When Theodore was leaving for a lecture in Altoona, Pennsylvania, Lib stood on the train platform and asked plaintively if her love had "totally and thoroughly" satisfied him. Tilton replied in writing, "I believe that my love... is as much as it is possible for a man to bear toward a woman.... Do you still chide yourself for a fancied failure in filling your husband's ideal of a wife?"
Indeed she did. For Tilton, in trying to justify his promiscuity, blamed Lib for his "falling into sin" and finally told her that because of her coldness, he was not above "taking a woman while on the road." For her part, Lib meekly accepted his judgment. She later wrote, "I understood very well that I was not to have the attention many wives have. I gave him to understand that what might be regarded as neglect under other circumstances would not be so regarded by me. I realized that his talent and genius must not be narrowed down to myself. To a very large extent I attribute to that the later sorrows of my life."
But slowly Lib too began to change, for while her husband was away, she found herself increasingly attracted to her preacher. Early on, undoubtedly sensitive to the extent of Mrs. Tilton's repressed sexuality, Beecher emphasized the purity of their relationship. He told her of Fourier's view that an affinity of souls marked the genuine love that was available only to those of the highest nature. True marriage was one in which a man selected his spiritual wife. Lib wrote her husband, "I live in profound wonder and hushed solemnity at this great mystery of soul loving to which I have awakened." For the present, Mrs. Tilton controlled her powerful sexual attraction to Beecher by praying for hours at a time. Yet a part of her recognized the impending danger of her relationship with Beecher, for she also wrote her husband, "No temptation or fascination could cause me to yield my womanhood." And the words, "I am afraid!"
Lib gradually came to see that she became more attractive to her husband because of her relationship with Beecher. And, although he denied it, Theodore was becoming increasingly jealous. In November 1866, from Washington, Tilton wrote his wife, "I think any man is a fool to be jealous. If he is jealous _without cause_ he is foolish, if with cause more foolish." But he showed his true colors when he ended the letter with, "Up till midnight after my lecture last evening talking with Edward Beecher about his backsliding Brooklyn brother."
When Tilton returned home between lecture tours, he spoke incessantly to Lib of Beecher's misguided politics and profligate ways, and he accused her of enticing Beecher with her sensuality. Lib was horrified at the thought and denied it, but she spent hours locked in her bedroom examining her face and body for telltale evidence that she was a wanton. She later reflected,
When Theodore said I had a sensual influence, I used to become impregnated with this idea of his myself. I wondered if it was so and would think it over and over.
He would often talk to me that way by the hour and try to persuade me it was true, but then when I used to get from under his influence I was perfectly sure that no man ever felt that way toward me. Hearing that, day after day, week after week, it made me sick and caused me to distress myself. It kept me in embarrassment. It was a hard thing to live under.
_Theodore and Lib Tilton—the picture of domestic bliss_
Now Tilton questioned his wife after every pastoral visit. Later she said, "I tried to give him a full accounting.... I would throw out a remark which Mr. Beecher had made and Theodore would say, 'You didn't tell me that.'
" 'Oh yes, I did mean to tell you, but I forgot,' I'd say.
"And he'd say, 'You lie!' "
At Christmas, feeling the need of his wife's affection in a world that seemed increasingly hostile, Theodore asked Lib to meet him in Chicago and chided her about her growing attachment to Beecher.
Now that the other man has gone off lecturing you can come to me, leave home, children, kith and kin, and cleave to whom you originally promised to cleave. Oh, frailty, thy name is woman.... If you can get anybody to pour tea for you, and to take sauce from the servants, and to receive pastoral visits, I shall expect to meet you.... I don't expect to be lonesome much longer.
But even this domestic anchor had pulled free. Lib refused to come, giving their daughter Alice's illness as the reason. In fact, under Beecher's influence, Lib had begun to feel herself a "full woman" and therefore to question the passionless role thrust upon her by Theodore. When they were apart, the concept of her pure spiritual nature was bearable, but when they were together the role became troubling. Theodore addressed her as "My espoused saint" and wrote her, "If you should ever appear to me anything less than the ideal woman, the Christian saint I know you to be, I shall not care to live a day longer." She answered,
During the early part of your absence it was well enough to suffer you to believe in my perfection, but as you near home, it is wise to dispel the infatuation little by little, and convince you of the humanity and frailty of your loving
Wife
On Christmas Eve, when Tilton returned home, his wife lavished on him the newly found sexuality that her preacher had awakened in her. This sudden passion in an eleven-year marriage perplexed Tilton. Three days later, when Theodore left to resume his lecture tour, Lib wrote him in the revealing terms one might use to describe one's first orgasm and made a tortured effort to justify her relationship with her husband and her preacher.
My Own True Mate:
... I have been thinking of my love for Mr. B. considerably of late and those thoughts you shall have. Remember, "My heart in this new sympathy for one abounds towards all." Now, I think I have lived a richer, happier life since I have _known_ him. And have you not loved me more ardently since you saw that another high nature appreciated me? Certain it is, I never in all my life had such rapture of enthusiasm in my love for you—something akin to the birth of another babe—a new fountain opened enriching all.... For many years, I did not realize the blessing. What remorse it brings to me! Memories bitter, awful!
But to return to Mr. B. He has been the guide of our youth, and, until the three last dreadful years when our confidence was shaken in him, we trusted him as no other human being.... It is not strange then, darling, that on a more intimate acquaintance my delight and pleasure should increase. Of course, I realize what attracts you both to me is a supposed purity of soul you find in me. Therefore, it is that never before have I had such wrestlings with God that He would reveal Himself to me.... I live in an agony of soul daily.
And Tilton replied,
I like Mr. Beecher in many aspects as well as I ever did but he has ceased to be my soul's prop—ceased to inspire me to my best life. I believe he is not as morally great as he once was.... There was an older virtue which has gone out of him—an influence which used to brighten my life when I came under its ray; an influence, however, which became gradually quenched like a vanishing sunbeam.
# CHAPTER TWELVE
WRITTEN IN FIRE
IN MAY 1867, as the Tiltons were wrestling with the question of marriage in a modern world, Mary Cheney Greeley returned from a seven-month sojourn in Europe. In fact, between 1858 and 1866, she had spent the equivalent of four years abroad. Only with an ocean separating Mary from her husband did she gradually regain her equilibrium and overcome the paranoia that had been induced by the opiates and other harsh treatments administered to her by eminent doctors. At fifty-five, Mary had passed the age of childbearing and, no longer a recluse, she now directed her febrile energy to the cause of woman's rights. Mary had allied herself with Stanton and Anthony in combating the male domination that she felt had ruined her life. Though she despised her husband as much as ever, she enjoyed the company of their daughters, Ida and Gabrielle. Rather than give Horace Greeley any kind of comfort, the women in his family seemed to have an unspoken pact that his home should be as unwelcoming as possible. After fourteen years, the Chappaqua farmhouse remained almost as bare as the day Horace had acquired it. In Paris, Mary had purchased crates full of embroidered Belgian linen, Limoges china, and weighty silver, but stored them unopened in the basement.
Mary Greeley soon learned that her husband had been appointed chairman of the suffrage subcommittee of the New York State Constitutional Convention to take place in Albany in June. Stanton and Anthony, she also discovered, had been conducting a massive campaign to collect thousands of signatures to amend the Constitution, which presently granted the vote solely to _male_ and _white_ citizens. Mary arranged a formal meeting with her husband in the offices of the _Tribune_ and arrived with Elizabeth and Susan. When Greeley heard of their campaign, he squinted through his wire glasses and cautioned, "This is a critical period for the Republican party and the life of the nation. The word _white_ in our Constitution at this hour has a significance which _male_ has not. Your turn will come next." When they protested, Greeley threatened, "If you persevere in your present plan, you need depend on no further help from me or the _Tribune._ "
That June, Stanton and Anthony were nevertheless granted a hearing before Greeley's suffrage subcommittee in Albany. When Stanton gave her testimony asserting the right of women to vote, Greeley interrupted her: "The ballot and the bullet go together. If you vote, are you ready to fight?" She replied, "Yes, we are ready to fight, sir, just as you did in the late war, by sending our substitutes."
Petitions signed by twenty-eight thousand men and women advocating woman suffrage were then presented. The galleries were crowded awaiting the chairman's report. Stanton and Anthony, knowing of Greeley's animosity, held one petition to be presented last, but they informed the press of its contents in advance. As Greeley stood to address the convention, a delegate called out, "Mr. Chairman, I hold in my hand a petition signed by Mrs. Horace Greeley and three hundred other women of Westchester asking that the word _male_ be stricken from the Constitution." Mary Cheney Greeley had openly defied her husband.
Over shouts and female cheers, an angry and obviously embarrassed Greeley read his report, his hand shaking as he held the papers: "Your Committee does not recommend an extension of the elective franchise to women. However defensible in theory, we are satisfied that public sentiment does not demand and would not sustain an innovation so revolutionary and sweeping, so openly at war with a distribution of duties and functions between the sexes as venerable and pervading as government itself, and involving transformations so radical in social and domestic life."
The next time Horace Greeley saw Stanton and Anthony was at a Sunday night reception at the Tiltons' home in Brooklyn. Stanton watched Greeley making his way across the room and whispered to Anthony, "Prepare for a storm."
"Good evening, Mr. Greeley," she said, and extended her hand.
Greeley did not take her hand. He told her, "I saw the reporters prick up their ears and knew that my report and Mrs. Greeley's petition would come out together with large headings in the city papers and probably be called out by the newsboys in the street. You are always so desirous in public to appear under your own name, why did you in this case substitute 'Mrs. Horace Greeley' for 'Mary Cheney Greeley,' which was really on the petition?"
"Because I wanted all the world to know that it was the wife of Horace Greeley who protested against her husband's report," Stanton replied.
"Well, I understand the animus of that whole proceeding.... You two ladies are the most maneuvering politicians in the state of New York.... Now let me tell you what I intend to do. I have given positive instructions that no word of praise shall ever again be awarded you in the _Tribune_ and that if your name is ever necessarily mentioned, it shall be as Mrs. Henry B. Stanton." Abruptly, he turned away. _Greeley's Tribune_ , a newspaper that supported woman suffrage, was now closed to Stanton and Anthony. They had made a formidable enemy.
Other powerful allies and friends were also alienated by the aggressive way these two women furthered their cause. Anticipating that the final wording of the Fifteenth Amendment would include the word _male_ , they drew up a petition to the House of Representatives stating that they would oppose any amendment "to extend or regulate suffrage" unless it made "no distinctions... between men and women." But when they solicited signatures from their former supporters, they were rebuffed. Stanton's cousin Gerrit Smith, for whom she had named her third son, wrote, "I can not sign the petition you send me. Cheerfully, gladly, can I sign a petition for the enfranchisement of women. But I can not sign a paper against the enfranchisement of the negro man unless at the same time woman shall be enfranchised. The removal of the political disabilities of race is my first desire—of sex my second."
At meetings and conventions Stanton and Anthony were asked by former allies to change their position that unless the Fifteenth Amendment included women they would oppose it, but they refused. In her frustration, Stanton once again descended to racism.
Would Horace Greeley, Wendell Phillips, Gerrit Smith, or Theodore Tilton be willing to stand aside and trust their individual interests, and the whole welfare of the nation, to the lowest strata of manhood? If not, why ask educated women, who love their country... who feel that their enfranchisement is of vital importance... to stand aside while 2,000,000 ignorant men are ushered into the halls of legislation?
Stanton's superior attitude toward those she termed "Sambo," coupled with her criticism of the church, the laws governing marriage and divorce, and the male establishment in general, severely damaged her political effectiveness. She was conflicted. She knew that she needed men to further her cause and to provide the money to go forward, but she resented their domination. Three years previously she had written Anthony, "I am not alone, as my niece is with me. She is a beautiful woman. I wish you and I had been beautiful; then we could have carried all men with us to heights divine and entrenched them on principle." Still, Stanton realized as well that a society that forced women to manipulate men degraded both sexes. "Man has been molding woman to his ideas by direct and positive influences, while she, if not a negation, has used indirect means to control him and thus in most cases developed the very characteristics both in him and herself that most needed repression."
_A woman's rights convention, 1850s. Strong women felt they could change their "social, civil, and religious condition._ " (Illustration Credit 12.1)
Stanton smarted over the wounds inflicted by the men of her class who had kept her an outsider and betrayed her. "I wish you were a boy!" had been her father's dictum. Elizabeth Cady Stanton appeared a capable housewife and mother; benign, becurled, fashionably dressed. Her appearance and manner posed no threat, and yet her internal rage at being sealed out of the "halls of legislation" by men fueled her continuing fight for woman's rights.
The next battlefield was Kansas, where separate amendments to the state Constitution had been proposed to enfranchise women and Negroes. Stanton and Anthony arrived in late August 1867. An endless expanse of prairie rolled out in front of them, accented by sparse clusters of sod shanties. This was the land where John Brown and his sons had spilled their blood. Anthony set up headquarters near her brother's home in Leavenworth. Stanton, campaigning separately, was warmly accepted by the pioneer women, who gave her what they had—a place in an unkempt bed shared with several children, cold water, bacon floating in grease, coffee sweetened with sorghum, biscuits green with mold, dried fruit. Much of the state was inaccessible by rail, so day after day she rode to remote prairie towns in a carriage drawn by two mules.
Yet with all the physical discomforts, the open spaces of Kansas and the welcoming, honest people invigorated Stanton, and she came to envision a life for herself and Henry where they might be together in a way that had not previously been possible. Stanton's marriage had deteriorated. Henry Stanton was now an editor of the _New York Sun_ , but he resented the fact that his wife's increasing fame eclipsed his own. Here, she thought, was a place where they might reconcile and not compete. She wrote to Henry, "This is the country for us to move to.... We could build [a house] for $3,000.... You would feel like a new being here. You could be a leader here, as there is not a man in the state that can make a really good speech."
In the Kansas campaign, both Stanton and Anthony were becoming emotionally and financially drained. Greeley's _Tribune_ and Tilton's _Independent_ , Stanton noted, were " _the_ papers out here." Stanton knew that Greeley would oppose her, but she wrote to Tilton for help. He wrote a single editorial advocating the passage of both amendments. In October the flamboyant George Francis Train arrived in Kansas to campaign with Anthony. Train was a showman in the P. T. Barnum mode. He dressed in a purple brocade jacket with a lime-green satin vest and red boots, and people stared at him wherever he went. Train had already achieved fame by traveling around the world in eighty days—a record—thereby inspiring the novel by Jules Verne. Train was the owner of New York's most successful horse-car line but he had made most of his fortune as the prime organizer of the Credit Mobilier, a holding company for the stock of the Union Pacific Railroad. It was into this entity that profits flowed at the rate of almost $20 million a year, as the government ceded more than 450 million acres for settlement and underwrote the westward expansion of the railroads with tax dollars: $16,000 a mile through the plains, $32,000 a mile through the desert, and $48,000 a mile through the Rocky Mountains. The previous inhabitants, the Indians, were driven off their land or killed as the West became the land of the future.
_George Francis Train. Was he a visionary or a madman?_
Train's speeches advocating the vote for women but not for "low-down nigger men" won the votes of the Irish and other Democrats. But his blatant racism soon alienated many woman's rights advocates. Lucy Stone termed Train "a charlatan" and "a lunatic." William Lloyd Garrison, editor of _The Liberator_ , wrote that he was "mortified and astonished beyond measure in seeing Elizabeth Cady Stanton and Susan B. Anthony traveling about the country with that crack-brained harlequin." Anthony was also criticized for using $1,000 of the American Equal Rights Association's funds to cover Train's speaking expenses when he could easily have afforded to pay for himself. Finally, both amendments were defeated by a two-thirds majority in Kansas, leaving blacks and women equally disfranchised.
After the Kansas campaign, Anthony wrote to Anna Dickinson, "Not one leading politician stood by us in the deadly breach. They all mean to delude us into silence and use us to serve party ends." And she confided to her diary, "All the old friends, with scarce an exception, are sure we are wrong. Only time can tell but I believe we are right."
One day Train asked Anthony why she did not start her own publication. "Not lack of brains, but money," Anthony told him.
"I will give you the money," volunteered Train.
Desperate for support, Stanton defended the alliance with Train. "Mr. Train... has some extravagances and idiosyncrasies but he is willing to devote energy and money to our cause when no other man is," she wrote. "It seems to me it would be right and wise to accept aid even from the devil himself, provided he did not tempt us to lower our standard." Stanton's words were to prove prophetic, for in the not too distant future her association would be with the woman soon to be known as "Mrs. Satan"—Victoria Woodhull.
THEODORE TILTON TOO was coming to distrust the politics of Reconstruction. His utopian dream of a new society was fast vanishing as he became increasingly disillusioned with politics and public life. Determined to find his own way, he fancied himself a brave explorer taking nothing for granted, no cant, no fixed dogma. "The old religious teachings, the orthodox view, the dread of punishment, the atonement, have less and less power over my mind," he wrote. Ironically, at the very time Tilton turned from conventional religion, his immense power emanated from his editorship of the _Independent_ , a newspaper that influenced a nation of churchgoers. Wary of Henry Bowen's growing use of the power of his paper for personal and political gain, and perhaps also concerned by his own descent from orthodoxy, Tilton confided to Lib that he no longer knew if he wished to make the _Independent_ his life's work. There even had been a groundswell to nominate him for lieutenant governor of New York State, but Tilton refused to run. He wrote Lib, "I have no ambition to be rich—I never have, none to be in political office, none for social or fashionable preeminence, none that I can detect for oratorical distinction and not a great deal for literary reputation."
To fill the gap left by the increasing cynicism he saw about him and his alienation from Beecher's church, he began to turn to Spiritualism. After reading the biography of Mrs. Pomeroy, a dedicated Spiritualist and clairvoyant, he wrote to Lib,
I was struck with a little passage as follows: "It was," said she [of a near-death experience], "a prophetic state. I have looked into the future. I know of things that are to be hereafter." Such experiences are not uncommon with good men and women near death. I never can read of such an experience except with a thrill. The future is not so far from us. The Kingdom of God is around us. The veil is thin which divides our eyes from the vision of Jordan. Sometimes in sleep, sometimes in wakefulness, there comes to us a glimpse "of the glory that is to be revealed." Our customary notions of the other life are distorted, misshapen, and deformed. Our prevailing theologies have covered our eyes with stained windows.
Theodore Tilton was among the millions who found comfort in Spiritualism in the postwar society. The movement that had begun with the spirit-rapping of the Fox sisters three decades before was at its zenith at the end of the Civil War. The unbearable death of a generation of young men—the hope of America—encouraged the belief that these souls had simply "passed over." They were reachable in a land just beyond the living where they could be called upon to comfort and support the bereft they had left behind. Spiritualist publications flourished and one of them, _The Banner of Light_ , optimistically estimated that Spiritualism had gained a million adherents a year for the previous three years and they now numbered ten million.
Victoria Woodhull and Colonel Blood combined their belief in Spiritualism with a doctrine equally at odds with conventional religion: free love. Utopian communities and other attempts to reform the sexual customs of society had met with little success but still many Spiritualists held free love as a panacea. Of all the radical ideas then current, free love was the most controversial. It represented the ultimate expression of female liberation and profoundly threatened a male-dominated society.
For Blood and Woodhull, Spiritualism and free love provided a rationale for the way they lived. Blood, having seen the horrors of war, knew the fragility of life. He believed that those who had died so wastefully were hovering near. What he did not believe was that sexual exclusivity was a matter of concern and he accepted with equanimity the exchange of sexual favors for politics, pleasure, or profit. Both Tennessee and Victoria were later to say that they had received money for sex. And though it was said that Victoria once threw her shears at Blood when she found him with another woman, on most occasions she was not given to jealousy.
Victoria Woodhull and James Harvey Blood were married in 1866, but they obtained a divorce two years later to protest the confinement of the marriage laws. Five years after the divorce, when Blood was asked, "How long have you and Mrs. Woodhull been separated?" he replied, "We were never separated...." Colonel Blood arrived at his theories intellectually, while Victoria Woodhull led life as she must. But both philosophies, the theoretical and the empirical, were identical.
When, in order to survive, Victoria read fortunes and performed magnetic healing and occasionally sold her body to keep her soul intact, there were two Victorias: one protected and empowered by the spirits and one who experienced the world's cruelty. Therefore, it is not surprising that there are two accounts of her life at this time: the one that Theodore Tilton would later include in his biography of her, and the actual life recorded in newspapers, court records, and other documents. Tilton's flattering portrait represented Victoria's dream of empowerment and her escape from the oppressive life of an exploited and exploiting woman. According to Tilton, when people came to Victoria,
By divine inspiration, she would instantly know the names, residences, and maladies of these strangers.... Her purse grew fat. She reaped a golden harvest.
First at Indianapolis and afterward at Terre Haute, she wrought apparently miraculous cures. She straightened the feet of the lame, she opened the ears of the deaf, she detected the robbers of a bank, she brought to light hidden crimes, she solved psychological problems, she unveiled business secrets, she prophesied future events.... Benedictions... were lavished upon her; money flowed in a stream toward her.... The sum total of the receipts of her practice... to the time of its discontinuance by direction of the spirits, was $700,000. The age of wonders has not ceased!
In the real world Victoria achieved few such wonders. With Tennessee and Colonel Blood, she moved constantly from town to town, scrambling for a living. The sum of $700,000 would not come her way for another three years, and then it would be the largesse of one of the most powerful men in America. As she grew older, Victoria had, by her own account, become discouraged, although she clung to her belief that everything she suffered was preparation for a mission for which the spirits had selected her above all others.
Victoria later said that on a "temporary sojourn" to Pittsburgh, she was sitting in the parlor of yet another boardinghouse when the spirit in the white toga once again came to her. "Your work is about to begin," he said. "All these years we have been preparing you for a great mission.... Go to New York City, to 17 Great Jones Street. There you will find a house ready and waiting for you and yours." Then a vision of a house filled with comfortable furniture floated before her eyes. She saw the bedrooms, the parlor, the stairs, and a front entry hall, where a large black book rested on a mahogany table. Because the book lay flat upon the table, she could not read the title. Gradually, the picture blurred and vanished.
Then the spirit reappeared before her. "Who are you?" Victoria asked. "All these years you have never said. Tell me now."
The spirit pointed to a white marble-topped table that stood nearby and with his index finger traced letters across its surface. At first the letters were invisible, then faint; then they became brighter and brighter until they burst into flame. Victoria read the name written in fire—Demosthenes.
# CHAPTER THIRTEEN
TO EQUAL ACCOUNT
THE DARK BLUE railroad car with New York & Harlem—Albany RR emblazoned in gold on its side rolled through the green fields of the village of Harlem, burrowed through the deep Mount Prospect tunnel from Ninety-sixth to Ninety-second Street, and emerged on Fourth Avenue. From the train Victoria and Tennessee could see shanties, herds of goats grazing near the dusty tracks, occasionally a factory or brewery, its chimneys belching black smoke against the blue sky.
Pollution and noise threatened to ruin the city. To protect the fashionable downtown areas, citizen pressure for a clean environment had forced through a city ordinance in 1850 banning street-level steam locomotives below Forty-second Street. Thus Forty-second Street became the transfer point for railroad passengers. Here the train halted abruptly and the steam locomotive was detached. The cars were uncoupled and hitched one by one to teams of four horses for the journey to the Twenty-seventh Street terminus. Though this process took some forty minutes, few passengers left their seats to descend into the sea of mud in which pigs, goats, and chickens roamed free and climbed onto the steps of the nearby newly constructed hospital of the Society for the Relief of the Ruptured and Crippled.
In the previous six years the population of the city had doubled. Now, in 1868, it was nearing 1 million inhabitants. The train yard at Fourth Avenue and Forty-second Street bisected the city, providing an invisible wall behind which poverty, filth, and pollution were confined to the east while the affluent new city expanded to the west. On the east were factories and at least a dozen breweries, including Gillig's, Ahle's, Neidlenger, and Schmidt. Immigrants—German, English, Irish—crowded the four- and five-story tenements lining Third and Second Avenues. Along the East River were the slaughterhouses to which cattle were driven across the length of Forty-second Street from the Hudson River docks. And on the rocks above the river was a gang-ridden slum called Dutch Hill.
A different city lay to the west: On Fifth Avenue from Thirty-fourth to Forty-second Street stood the high-stooped brownstone houses of New York's wealthy. On Sundays, fashionable couples would stroll on the elevated walkway of the Reservoir at Fifth and Forty-second Street. Real estate experts who had predicted that the rich would never live farther uptown than Forty-second Street were being proved wrong. The brutal but convenient destruction of the Colored Orphan Asylum at Forty-third and Fifth by Irish protesters during the 1863 draft riots had left the avenue wide open to the postwar expansion.
Seeing it before others, Commodore Cornelius Vanderbilt, on whose line Victoria and Tennessee arrived in New York, had purchased large tracts of land around his Forty-second Street train yard as far uptown as Forty-eighth Street, as well as various parcels on Madison and Fifth to Fifty-second Street. Vanderbilt was said to be the richest man in America. Now seventy-four, the Commodore had bought his first share of railroad stock in 1862, only six years before, to show his superiority to his eldest son, William Henry, who had also purchased railroad stock. But almost immediately the Commodore, who had risen from being the impoverished captain of a Staten Island ferryboat to owning the largest fleet of ships in America, saw that steamboats and railroads were about to change places in the future of transportation in America. Eighteen sixty-eight was the year that railroads finally webbed the nation from coast to coast, and by that time Vanderbilt owned the New York & Harlem, the Hudson River, the Albany, the New Haven, the New York Central, and seven other railroad lines.
Of the many entrepreneurs who saw the future, Vanderbilt was perhaps the most visionary. And literally so. He said the glittering new city of New York had come to him in a vision and that he had purchased land because he had seen that eventually the city would extend uptown along Fifth, Madison, and Fourth Avenues to Ninety-sixth Street. With remarkable foresight, Vanderbilt eventually buried the tracks to the north of his depot under Fourth Avenue (now called Park Avenue), all the way uptown to the Mount Prospect tunnel, so that his trains would not disturb the affluent residents who he said would one day live there.
The Commodore devised a plan to bring together his railroad lines and build a central terminal at Forty-second Street. He planned a magnificent facade for this terminal facing in a westerly direction, and although he hired civil engineer Isaac C. Buckhout and architect John B. Snook to carry out his wishes, the Commodore said his vision had revealed exactly what Grand Central Station would look like.
_Commodore Cornelius Vanderbilt. Known for his business acumen, he also experienced visions and believed in Spiritualism_.
_The Reservoir at Forty-second Street and Fifth Avenue. Fashionable couples paraded around the elevated walkway_. (Illustration Credit 13.1)
_Ann Lohman, alias Madame Restell, "The wickedest woman in New York_ " (Illustration Credit 13.1)
Another investor who believed that upper Fifth Avenue inevitably would be the place to live was Charles Lohman. Although he did not believe in visions, he saw signs that this neighborhood would become fashionable: A magnificent new Central Park was emerging from 59th to 106th Street with a new northern extension to 110th Street. Work on the park had proceeded even during the Rebellion, with miles of lanes set aside for trotters and carriages.
Lohman heard that John Hughes, the first Catholic archbishop of New York, planned to buy ten lots on the northeast corner of Fifth Avenue and Fifty-second Street for his official residence. A decade earlier the block from Fiftieth to Fifty-first Streets had been purchased for the construction of Saint Patrick's Cathedral. Work had been suspended because of the war and lack of funds, but now construction was about to resume. Lohman, who hated the archbishop because he had denounced his wife for her popular surgical practice, quickly purchased the lots himself for the grand sum of $36,500. Virtually next door to Saint Patrick's, the Lohman mansion arose, an imposing four-story brownstone with stables and a walled garden, and every window decorated identically with window shades in a gaudy floral print. In May 1867, a discreet silver plaque engraved Office Female Physician appeared on the low iron railing in front of the basement door to One East Fifty-second Street. Ann Lohman, better known as Madame Restell, "the wickedest woman in New York... the infamous abortionist," was open for business.
The horse-drawn railroad car bearing Victoria and Tennessee proceeded briskly down Park Avenue through the prosperous Murray Hill district. As they passed Thirty-eighth Street, Victoria, had she glanced to the right, would have seen the forty-foot-wide brownstone mansion that, two years later, would become her residence. Four blocks to the south, at Thirty-fourth Street, also just off Fifth Avenue, was Annie Woods discreet brownstone, the fanciest house of prostitution in the city.
It was late in the afternoon when the sisters piled into a carriage for the final leg of their journey. The gaslights were already lit in the Women's Bureau at 49 East Twenty-third Street, where Susan B. Anthony and Elizabeth Cady Stanton were closing an issue of _The Revolution_. Both women were known to work until dawn, delighted to be able at last to fight for woman's rights in their own publication. At Fourteenth Street and Union Square the carriage turned onto the great thoroughfare of Broadway. Lining both sides of the street were magnificent marble, brownstone, and cast-iron buildings. Here the rich in top hats and ermine-trimmed capes mingled with ragged beggars. The division between the haves and the have-nots was becoming increasingly evident. The carriage almost slowed to a halt. The sixty-foot-wide street was crowded with vehicles of every description—carriages and pushcarts, buses drawn by teams of eight horses, lone riders—all desperately trying to maneuver though the jam.
At the Bowery the carriage rolled by the Cooper Institute, where the previous week Anna Dickinson had made an impassioned speech on the subject of equal wages for women. The great star of abolition and woman suffrage, wearing a rich, pearl-gray silk dress trimmed with cherry satin and a necklace with a magnificent ruby "fit for an Empress's crown," had been escorted to the stage by Peter Cooper himself and introduced by Henry Ward Beecher. At last the sisters' carriage turned off Broadway and into the quiet charm of Lafayette Place, graced by a colonnade of ancient elms, then made another turn into Great Jones Street.
Victoria wrote her remarkable accounts of her fulfilled visions well after they occurred. Thus it was many years later that she described that when, as weary travelers, they arrived at 17 Great Jones Street, it was nearly dusk. Victoria's hand trembled, she later wrote, as she banged the knocker on that darkened door. She saw a light in an upstairs window and heard someone on the stairs. Then the door opened to reveal a woman who smiled and exclaimed, "Oh, you've come about the rooms." Then the woman instructed them to wait in the entrance hall while she readied their beds. The house was identical to the one in Victoria's vision. On the table rested the same book but this time she could read the title: Impressed in gold letters on the black leather cover was "The Orations of Demosthenes."
_Broadway, 1868—looking north from just below Fulton Street_ (Illustration Credit 13.2)
VICTORIA WOODHULL ARRIVED in Manhattan in 1868 at a watershed moment in the history of America: Many people had lost belief in the capacity to control their own lives, while others, who only a decade earlier had been hod-carriers, stevedores, and red-necked farmers, were today's millionaires. This was the New York of Boss William Marcy Tweed, the grand sachem of Tammany Hall, who came to power by naturalizing 41,112 Irish immigrants in one year on the proviso that they vote for his candidates. On a salary of $10,000 annually, the boss had purchased a Fifth Avenue mansion worth $2 million. Though barely begun, his courthouse—promised to the people early in 1867 for $250,000—was rumored to have cost millions already.
From the crucible of war a ruthless new society was emerging, one in which the delicate question of civic virtue was overwhelmed in the grab for money and power. A society in flux had slipped over the line until excess and corruption were accepted and their practitioners revered and exempt from judgment. The institutions of so-called morality—the law, the church, the government—were easily bent to serve these powerful newcomers. "What do I care for law, hain't I got the power," bellowed Commodore Vanderbilt.
For upper- and middle-class women, the war had expanded life's horizons. During this period many of them had functioned as individuals, often brilliantly. But while men returned from the war to careers and travel and the exhilaration of power, these women were once again relegated to "woman's sphere," idled by their affluent husbands who thought of them as ornaments or servants. Many women reacted strongly against these constraints. The birthrate in New York City fell precipitously. Infanticide was common, abortion rampant. Newspapers carried advertisements for "female physicians" like Madame Restell, who guaranteed to alleviate "disturbing problems," a euphemism for unwanted pregnancies.
For both women and men, the hypocritical illusion of domestic bliss was challenged by women's new awareness. The male establishment resisted accordingly: The homeopathy practiced by Edward Bayard and his sister-in-law Elizabeth Cady Stanton came under attack by members of the New York Academy of Medicine, who called it "quackery and charlatanry." Their own "heroic medicine," they said, was the only legitimate practice. Male doctors attacked midwives and drove many of them from the profession.
With little to occupy their time except mindless pursuits, women of means often became languid, pale, and subject to a condition that had not previously existed: neurasthenia, defined as a hysterical weakening of the constitution. Medical doctors often dealt with nervous or rebellious women by administering a "rest cure," isolating women patients for months at a time and forcing them to remain in bed with no distractions on a diet consisting solely of liquids. Supervised by a male doctor, the patient once more became docile and compliant, her body, as one physician explained, "recycled and taught to make the will of the male her own."
Female neurasthenia was found to be "allied to the sexual organs." A woman's sexual drive was perceived as a danger both to men and also to herself, since the capacity to "over-excite the sexual appetites" could bring about male violence. It was believed by many members of the male medical establishment that women were unclean and infectious. Dr. Augustus Kinsley Gardner, a leading gynecologist of the day, warned that women communicated venereal disease to men who had intercourse with them "in close proximity to the period of their monthly courses.... I do not believe that a man often communicates disease to a woman," he wrote.
In 1868, American gynecological surgeons began performing clitoridectomies to quell sexual desire in women, which was considered a form of derangement. Upper- and middle-class white women who had been taught that any sexual urges were sinful, willingly surrendered their bodies to these male doctors, who tested them for this abnormal arousal by stimulating the breast and clitoris; if there was a response, they surgically removed the clitoris. Bearing the guilt of the victim, some women who had felt desire became complicitous in their own bodily mutilation. One observed that her clitoridectomy scar was "as pretty as the dimple in the cheek of sweet sixteen."
The idea that men had strong sexual urges, whose gratification was socially acceptable, while women were pure and passionless seemed absurd to Victoria Woodhull. She advocated the same sexual standards for both men and women. It would take two more years before she publicly proclaimed a hidden truth that few women of the day would admit or even imagine.
Some women seem to glory over the fact that they never had any sexual desire and to think that desire is vulgar. What! Vulgar! The instinct that creates immortal souls vulgar? Who dares stand up amid Nature, all prolific and beautiful, where pulses are ever bounding with the creative desire, and utter such sacrilege. Vulgar rather must be the mind that can conceive such blasphemy. No sexual passion, say you. Say, rather, a sexual idiot, and confess that your life is a failure... and no longer bind your shame upon your brow or herald it as purity. Bah! Be honest, rather, and say it is depravity. It is not the possession of strong sexual powers that is to be deprecated. They are that necessary part of human character... they are the foundation upon which civilization rests.
Shortly after their arrival at Great Jones Street, Victoria and Tennessee were joined by Colonel Blood, who began working for his brother George on a small New Jersey newspaper. Soon Buck Claflin appeared, for their father was not about to lose a good source of income. Victoria and Tennie already had begun visiting the houses of prostitution and assignation on Fifth Avenue and in Murray Hill, and Buck was soon accompanying them, selling Miss Tennessee's laudanum-laden "Magnetic Life Elixir." Laudanum, an opium derivative, was popular with prostitutes, some of whom consumed as much as fifteen grams a day. Those who preferred not to deal with Buck and were able to venture outside their establishments could go to the drugstores on Mercer, Greene, Wooster, Laurens, and Thompson Streets where large quantities of opium were sold without a prescription and buyers were not asked to register by name.
Victoria and Tennie were medical practitioners of a different sort from their father. Sharing the knowledge they had learned under Buck's aegis, they sold prostitutes such medications as sponges soaked in vinegar that, inserted in the vagina, absorbed a man's discharge and helped to destroy it; sweet-scented water that cleaned the area; and a clove anesthetic that prevented pain and made the women seem warm and eager. These products made life easier for the women and provided the sisters with much-needed cash.
In 1868, New York City alone was estimated to have more than twenty thousand prostitutes. Two blocks from Victoria's house was the Midnight Mission for the Rescue of Fallen Women. Nightly, Christian ladies served soup, read Scripture to these women, and advised them where to obtain respectable jobs that provided a bare subsistence. The mission drew its women mostly from the infamous houses of prostitution on Greene Street, where half-clad women displayed themselves in the open windows.
No effort was made to conceal the city's many houses of prostitution. In fact, _A Gentleman's Guide_ , a popular book, listed the various houses and the services they made available. Some houses sent business cards to men registering at hotels. The police commissioner reported that there were more than six hundred houses of prostitution, another hundred assignation houses for illicit trysts, and thousands of available women who worked as cigar girls and waitresses in saloons and dance halls. Prostitution was an integral part of society. An evening's entertainment for men often included a fine dinner at Delmonico's, a vintage brandy, and a visit to a house of prostitution.
During the war the women who had followed General Joe Hooker's troops were nicknamed Hooker's girls or simply _hookers_. As families in the South disintegrated, many women migrated to Northern cities and supported themselves through prostitution. By the last year of the war, half the prostitutes in New York City were said to be from the South. The social observer George Templeton Strong noted in his diary, "According to the talk of the New York Club the harlotry of the city is largely reinforced by Southern refugee women who were of good social standing at home but find themselves here without means of support and forced to choose between starving and whoring." He added a commonly held view that allowed these victimized women no sympathy: "It seems a just retribution of the Southern slaveholding chivalry who have been forcing their female slaves—black, mulatto, and quadroon—to minister to their pleasures that their rebellion should drive their wives and daughters to flee northward and prostitute themselves to Northern 'mudsills' and plebeian 'Yankees.' "
Men acted out their sexual fantasies and desires with prostitutes on the assumption that their wives were too pure for such actions, and men justified their conduct by condemning these women as "vile," blaming them for the base passions they aroused. To most men and many women as well, these prostitutes deserved their destruction. By enticing men they had committed a sin that would lead down the steep path to ruin, disease, and death. George Ellington, a social chronicler of the day, wrote,
The fashionable _demi-monde_ soon lose conscience and are as dead to its value as to the value of virtue. Let them lead gay lives while they may, they will go down. Their beauty will fade... before many months you may see them in a den in Greene or Mercer Street... a year will not pass before they may want for a crust of bread. Today, they ride in a gilded coach to Central Park, the time may not be far distant when they will ride to Blackwell's Island in the "Black Maria."
Victoria Woodhull would later write in a speech (that she gave on 150 consecutive nights) that in her career she had met "hundreds upon hundreds" of prostitutes and that the common wisdom of the day was wrong.
There are many popular fallacies about prostitution... that the average life of prostitutes is about four years... but it is caused by dissolute living, and drinking, and by the diseases which usually accompany promiscuous intercourse.
The real truth about this is that those prostitutes who never drink, who never permit themselves to become diseased are among the healthiest of women and hold their beauty and vigor to an advanced age.
I know hundreds of wives who confess privately that they would not live another day with their husbands if they had any other method of support and yet pass the poor prostitute as though her touch were leprous.... There may be prostitution in marriage and proper commerce in the bawdy house. It depends upon the specific conditions attending the act itself and not where or how it is obtained.
_A visit to a first-class house. In 1868 there were more than twenty thousand prostitutes in New York City_.
Victoria was welcome in the netherworld of prostitutes and it is through her that we are able to gain a realistic picture of their lives. To Victoria, prostitution was a business. She would later write an analysis of the day-to-day finances of these women. A first-class prostitute paid her madam $40 a week for the use of her room and gave her 20 percent of her profits. Towels were extra. The madam paid politicians from the district about $100 a week for protection. Once a month police officers arrested a few prostitutes who were released the same day after paying a fine of $100. Each prostitute paid the patrolman "$3 to $10 a week and granted him the privilege of visiting her gratis." Police captains and sergeants were also "accorded the privilege of frequenting without charge such inmates as they may select." Victoria commented, "The amount of degradation and bodily injury to which [these women] must daily submit to meet these combined charges and demands may be imagined." Some of these women found permanent or semi-permanent arrangements with their wealthy and influential clients and graduated from prostitute to mistress.
Once Victoria Woodhull became a popular figure, she decried the corruption of a society that sanctioned this double standard:
We are told that prostitution is a "necessary evil."... Necessary indeed! Isn't it rather your shame and my shame and the dishonor of womanhood and the disgrace of manhood that should make the stones weep to contemplate—innocent, virgin girls from twelve to sixteen years of age—your daughters, mine perhaps—sacrificed to this "necessary evil."
Statistics inform us that there are two hundred and fifty thousand professional prostitutes in the country, one-tenth of whom are in New York City, and that these are visited and supported by not less than two and a half millions of men—one-third of the voting population of the country. Think of it! A quarter of a million professional female prostitutes and two and a half millions of professional male prostitutes, ten men to one woman.... When you condemn the poor women, whom you have helped to drive to such a life, remember to visit your wrath upon the best men of the country as well....
Until women come to hold men to equal account as they do the women with whom they consort, or until they regard these women as just as respectable as the men who support them, society will remain in its present state of moral excellence. A man who is well known to have been the constant visitor to these women is accepted into society, and if he be _rich_ is eagerly _sought_ both by mothers having marriageable daughters and by the daughters themselves. But the women with whom they have consorted are _too vile_ to be even acknowledged as worthy of Christian burial, to say nothing of common Christian treatment.
Two months after her arrival in New York, Victoria received a visit from a young actress she had known in California. The actress had arrived in the city penniless and asked Victoria to introduce her to a house of prostitution. Together they went to Annie Wood's "house," where an assignation was arranged with a prominent judge, a deacon of his church. When the judge gave the girl half the usual fee, Victoria intervened, demanding the full fee, and the judge paid the rest. Annie Wood was impressed with Victoria's management of the situation.
From the outside, Annie Wood's house was similar to other Murray Hill brownstones, except for its drawn curtains. Inside, however, according to _A Gentleman's Guide_ , the house was furnished like a Southern mansion, and served French and New Orleans cuisine. The service was said to be perfection. Copies of Rembrandt, Van Dyke, Hogarth, and Reynolds hung on the walls. The bedrooms featured four-posters draped in Brussels lace, and the ceilings were mirrored.
Since Annie Wood admitted that she used an alias, the account of her past that she provided in a sworn affidavit cannot be verified. However, if true, it would explain the taste and luxury displayed in her establishment. Annie claimed to have been born into the family of a wealthy plantation owner in Savannah, Georgia. She said she had lost her entire family and fortune in the war. Like so many other Southern women, she migrated to New York and, after a brief career on the stage, purchased her present establishment with the help of an unnamed investor. She described her house as a kind of club frequented by "only the best gentlemen." As an example, she said that during the evening's entertainment, matters of political and financial importance could be discussed in a discreet atmosphere.
Although the men who visited her house undoubtedly were unaware of it, Annie recorded, in two big black books, their names, professions, and pertinent information about their sexual preferences. Annie was to prove herself a good friend to Victoria and Tennessee, for as a later letter reveals, she would make these books available to them to do with what they might.
According to _A Gentleman's Guide_ , Annie specialized in mulatto women. (In the census of 1860, five hundred and eighty-eight thousand mulatto women were counted in what were then the slave states, a clear demonstration of the white man's mastery over his property.) In addition to these women, Annie provided a parody of what she herself might once have been. She dressed her most special whores as Southern belles who seemed aristocratic and virginal. They wore white silk gowns decorated only with thick yellow, blue, or pink satin sashes and received visitors in a conservatory with white latticework covering the windows and forming a cathedral dome. Annie Wood would later tell Victoria Woodhull that the way she felt about her own life was expressed in the copy of Bernini's statue of Daphne and Apollo, which she placed in the fountain in the center of this room. The white marble statue was of a nude Daphne in flight, writhing in fear and stretching her slim arms upward. Apollo's muscular arms grasp the nymph in the moment of capture. One can see the roots springing from Daphne's toes as, desperately seeking escape, she transforms herself into a laurel tree.
# CHAPTER FOURTEEN
CONSULT THE SPIRITS
IN THE world of the demimondaine the most open of secrets was Commodore Vanderbilt's insatiable sexual appetite. In the world of the Spiritualists the most open of secrets was the Commodore's support of any medium, fortune-teller, or healer who could aid him in his insatiable search for riches and immortality. He paid Mrs. Tufts, a medium from Staten Island, enough money to retire to Vermont when she rid him of two spirits: a boy of seven who had been crushed under the hooves of Vanderbilt's four white-footed trotters as they sped around the reservoir in Central Park, and a railroad worker who had been mangled under the wheels of the Commodore's _Flying Devil_ and appeared before him in a bloody, shredded, and oil-stained condition.
Vanderbilt instructed his barber to collect his hair and burn it, for fear that someone who secured a lock would have power over him. He believed that through portraits one could communicate with those who had "passed over," and kept a miniature of his late mother in his breast pocket above his heart. He despised doctors and followed the advice of Spiritualist healers, one of whom had instructed that saltcellars be placed under each leg of his bed to ward off evil spirits.
It was common knowledge that Commodore Vanderbilt saw all callers at his town house at 10 Washington Place, no matter who they were. He usually dispatched them promptly and rudely, but said he never knew where a good idea might come from. Within a month of his arrival, the ever alert Buck Claflin arranged a meeting between his Spiritualist daughters and the Commodore. When the crusty Vanderbilt peered over the balcony of his Greek revival mansion and saw the two women standing below, he could not have failed to be impressed: There stood Victoria with her delicate cameo features and Tennessee with her overblown figure. Vanderbilt had an eye for women, so much so that the housemaids frequently quit to escape his prurient advances.
The Commodore, for all his bluster, was old and lonely. Despite his fortune he was not welcome in society. He swore like a stevedore and spat tobacco juice onto his hostesses' Persian carpets. He was nearly illiterate. His home life was bleak; he and his wife, Sophia, had barely spoken for a decade. Sophia had borne the Commodore twelve children, and when they were young and poor she worked at Bellona Hall, their Staten Island boardinghouse, with one child at her breast and several others trailing behind.
Two decades earlier, when Sophia mustered the courage to object to his pursuit of the children's governess and refused to go along with his plan to move to a New York City town house, the Commodore established complete domination over her by having her committed to Dr. McDonald's insane asylum in Flushing. His eldest son, the obese William Henry, arranged for his mother's internment, and when the governess fled after being driven frantic by the Commodore's sexual advances, William Henry found another young girl "to content" his father, saying, "The old man is bound to have his way and it is useless to oppose him."
If not for the intercession of Vanderbilt's mother, Sophia would have remained in the insane asylum while the Commodore played his beloved whist, raced his magnificent trotters, conducted business, and pursued women. Phebe Hand Van der Bilt summoned her son to Staten Island and told him he'd better fetch his wife back if he knew what was good for him. (Sophia was not only his wife but his first cousin, the daughter of his father's sister, Eleanor.) "I would never cross that woman," he said of his mother. From that time on, Sophia had been compliant, uncomplaining, and aloof.
The summer after the Commodore met Victoria and Tennessee, following his usual routine, he repaired to Saratoga for the racing season and to take the spa water. Sophia, having little place in her husband's life, remained behind and moved into the home of her daughter and son-in-law, Mary Louise and Horace Clark. On August 17, 1868, Commodore Vander-bilt was sitting on the porch of the United States Hotel, drinking beer and smoking a black cigar when word came that Sophia had died of an apoplectic attack. Six hours later, he arrived in New York in his private railroad car, _Duchess_. Two days after that, Sophia was interred in the Vanderbilt Mausoleum at the Moravian Cemetery on Staten Island. Horace Greeley, who had been one of the pallbearers, wrote in the _Tribune_ that Mrs. Vanderbilt had "lived nearly seventy-four years without incurring a reproach or provoking an enmity."
Soon Victoria and Tennessee became daily fixtures in the Commodore's household, for they answered his considerable needs: Although he was still slim and spry enough to race his horses and spend hours at his one-desk, one-secretary office on Fourth Street near Broadway, he was slowing down. His hearing was failing, and he suffered from heart trouble, a hernia, kidney stones, constipation, and an enlarged prostate. At meals he had been known to consume pâté de foie gras, woodcock, Spanish mackerel, and saddle of venison, accompanied by Burgundy or Veuve Clicquot or beer. Tennessee took over: She babied, cajoled, and disciplined Vanderbilt, removing the rich foods from his diet and insisting that he walk as well as drive his horses. She tried to get him to stop smoking, but he said, "When I have to give up smoking, you may give _me_ up." Tennessee's treatments consisted of clystering (a high enema), manipulating the prostate, and magnetic healing. With her left hand acting as a negative magnet, the right as a positive, she claimed to reverse the polarity of his body and to expel negative energy.
Almost everyone was terrified of the Commodore, but not Tennessee. Their relationship was light and affectionate. She called him "old boy" and "the old goat." He called her "my little sparrow." Four servants were later to recall that they often found her occupying the Commodore's bed in the morning. "Ample"—that was Vanderbilt's word for Tennessee—ample breasts, ample hips, inviting. She was what she was and didn't care what anyone thought. Her jealous sister Utica said that when Tennessee asked the Commodore how many women he'd had, he replied, "A thousand," to which Tennessee laughingly responded, "Then I am only half as bad as you are, for I have had but five hundred." Within a few months of Sophia's death, in the fall of 1868, the "old goat" asked Tennessee to become the next Mrs. Vanderbilt.
If Tennessee ministered to the Commodore's body, Victoria ministered to his soul and eventually to his purse as well. Sitting in the darkened parlor under Phebe Hand Van der Bilt's portrait, Victoria transmitted messages to him from his mother, who had died fifteen years earlier. When Victoria was in the room, the Commodore said he could even smell his mother's presence—a combination of strong soap and lavender. His mother was the only woman he had truly loved. He liked his eight daughters well enough but told people, "After all, they're not Vanderbilts." His two living sons, William Henry and Cornelius Jeremiah, he called a "blatherskite" and a "sucker," respectively. William Henry was hardworking and dull and, what was more irritating to the Commodore, prudent and obedient: He'd bring round his sons, Willie K. and Cornelius, for hymn singing and money-grubbing, but the Commodore said there was no life in any of them. The Commodore refused to see Cornelius Jeremiah, an epileptic, because he was unable to admit that he could produce a less-than-perfect human being. He also felt that Cornelius Jeremiah was God's reprimand to him for having married his first cousin. This poor son nevertheless loved his father deeply and would stand for hours on the back porch hoping to catch a glimpse of the Commodore. Horace Greeley occasionally lent Cornelius Jeremiah money. The Commodore told Greeley that he would never pay him back for the loans to his son. "Who the devil asked you?" Greeley shot back.
The Commodore had once despised his father as well, but he softened his attitude in his old age. Convinced that Victoria Woodhull truly was able to relay messages from his beloved mother, Vanderbilt offered her $100,000 if she would go into a trance and conjure up his father, then describe the old man well enough for an artist to paint a portrait. Prudently, she declined, but soon Victoria was to make a great deal more money than that by conjuring up some numbers for the Commodore.
In February 1868, Commodore Vanderbilt engaged in a battle with Jim Fisk and Fisk's partner, Jay Gould, for control of the Erie Railroad. Fisk and Gould had won because at a secret meeting in a suite at the American Club Hotel at Broadway and Seventeenth Street, the Erie directors decided to print more than one hundred fifty thousand new shares of Erie stock. The suite was the residence of Josie Mansfield, Fisk's newly acquired mistress and a friend of Victoria Woodhull's from their acting days in San Francisco. As Fisk remarked, "If this printing press don't break down, I'll be damned if I don't give the old hog all he wants of Erie." Finally, on March 10, having bought the one hundred fifty thousand shares without gaining control of the Erie, the Commodore caught on. The humiliation was as painful as the swindle. With Victoria Woodhull's help, it would not happen again.
The world of high finance and the low life of prostitutes seemed totally separate, but in fact they converged in the elegant brothels Victoria visited. At these establishments, women entertained the richest and most powerful men of the day in their beds, and yet they were considered insensate and invisible. Some madams, including Annie Wood, undoubtedly made the most of the opportunity this provided. As Wall Street traders, city officials, businessmen, and politicians gathered in her parlor, Annie listened carefully to what they said. She also trained her "girls" to encourage the men to boast of their financial maneuverings, instructing these women both in the art of extracting information and in seeming ignorant of what they had heard.
Victoria was an intimate of Annie Wood and knew many of the prostitutes who worked in her house. It was there that Vicky became reacquainted with her friend from San Francisco, Josie Mansfield. Josie and her husband, Frank Lawlor, had come to the city in 1864 in the hope of finding employment in the theater. They were unsuccessful and soon Lawlor, who admitted that Josie had married him only to escape the sexual abuse inflicted upon her by her stepfather, found that his wife was "going astray" and divorced her. According to Annie Wood, Josie became so impoverished that she had but one dress and could not pay her rent. It was then that she began to frequent Annie Wood's house on Thirty-fourth Street, where she set her sights on the overblown Jim Fisk, a regular patron.
Josie knew of Fisk's wild behavior, his quiet wife in Boston, and his free spending: He would give $100 bills to any pretty prostitute who caught his eye. In November 1867, Fisk arrived at the bordello for an evening's entertainment and was immediately taken with the buxom Miss Mansfield. Although Josie professed to find Fisk intelligent and manly, she refused his money and rebuffed his advances. For three months Josie skillfully withheld her favors, thereby inflating her worth. Then she allowed Fisk to pay the overdue rent on her tiny room on Lexington Avenue, after which he installed her in the American Club Hotel suite. Fisk underwent a remarkable change: He trimmed his unkempt red mustache and waxed the ends to handlebar perfection, he wore French cologne and kept his boots shined. This besotted lover bought Josie a roomful of dresses and gave her $50,000 in cash and about five times that in Tiffany emeralds.
Josie became a lady of fashion. Daily the hairdresser called, teasing her hair into a variety of puffs, curls, and frizzes. Once every two weeks the enamelist painted her face, shoulders, neck, and arms with a compound of bismuth and arsenic that gave her skin a much-desired deathlike pallor. To be a woman of fashion was a full-time occupation carried to an absurd degree. George Ellington described the requirements:
The _elite_ do not wear the same dress twice... she has two new dresses of some sort for every day in the year, or seven hundred and twenty.... She must have one or two velvet dresses which cannot cost less than five hundred dollars each; she must possess thousands of dollars' worth of laces, in the shape of flounces, to loop up over the skirts of dresses as occasion shall require. Walking-dresses cost from fifty to three hundred dollars; ball-dresses are frequently imported from Paris at a cost of from five hundred to a thousand dollars.... Nice white llama jackets can be had for sixty dollars; _robes princesse_ , or overskirts of lace, are worth from sixty to two hundred dollars. Then there are... dresses for all possible occasions. A lady going to the Springs takes from twenty to sixty dresses and fills an enormous number of Saratoga trunks.
_Josie (Helen Josephine) Mansfield, 1871_
_Jim Fisk. Hoping for money, the impoverished Ninth Regiment of the New York State National Guard elected him colonel_. (Illustration Credit 14.1)
Fisk complained that in one year he spent $30,000 to equip Josie ($5,000 more, he noted, than the president's salary). Her casket of jewels contained her fabulous emeralds, a set (necklace, earrings, tiara) of diamonds, a set of pearls, a set of corals, a medallion set, and twenty-five finger rings. Within the year Fisk had given Josie a house in her own name at 359 West Twenty-third Street, only half a block west of Fisk's Opera House. He also supplied her with several maids, a cook, butler, and coachman. Yet Josie's demands on Fisk escalated: He said that she was more temperamental than an opera diva. Once when she asked Fisk for an extra $30,000, he sent Boss Tweed himself to mediate. Tweed asked her what it was she really wanted. "I want more," she replied.
One of Fisk's business associates was a startlingly handsome playboy with jet-black hair and classic features named Edward (Ned) Stiles Stokes. Stokes was married to the daughter of a furniture tycoon, and both his father and father-in-law paid his debts and supplied him with cash to disport himself at well-known restaurants and Broadway gambling houses. Fisk, taking a liking to Stokes, lent him some money for an oil venture, named one of his 125 canaries after him, and introduced him to Josie.
Several times a day, Fisk sent a messenger to Josie's house with notes outlining his plans. Woodhull was to observe of Mansfield that "she obtained not only Fisk's money but she also participated in his business secrets. He concealed nothing... of his business plans and aspirations from her." Although both women denied any business association, there is little doubt that, for a price, Josie Mansfield shared these plans with Victoria Woodhull. And on the client register of Woodhull, Claflin & Co. (a Wall Street investment firm formed in the winter of 1869) there was listed an H. J. Mansfield. Josie Mansfield's given name was Helen Josephine.
By the end of 1868, Vanderbilt was giving Victoria and Tennessee a percentage of the profits from his business transactions. In a trance state, Victoria offered him uncannily accurate financial advice, and the Commodore was shrewd enough not to question whether her sources were from this world or another. In December, Vanderbilt made the boldest move of his career. He declared an 80 percent stock dividend on the Central Pacific Railroad (issuing four extra shares for every five now owned). On Christmas Day, Victoria visited the Commodore and offered advice from the spirits. That evening, he told a young widow to place all her savings in Central stock. "It's bound to go up.... Mrs. Woodhull said so in a trance," he declared. On the Monday after Christmas, the Central stock opened at $134. By the time the exchange closed, it had bounded up to $165. It was rumored that the Commodore had allocated the profits on three thousand shares, or $93,000, to Victoria and Tennessee. When the Commodore was asked how he made such astute financial decisions, he laughed and replied, "Do as I do. Consult the spirits."
THROUGH THE COMMODORE, Victoria and Tennessee were gaining money and influence. Victoria would later say that this was part of the spirits' preparation for a great mission. But for now, the nature of her work was yet to be revealed and seemed confined to self-aggrandizement. Meanwhile, across the river in Brooklyn, seemingly unrelated events were about to bring into sharp relief the issues with which the new society struggled, ones that would eventually clarify for Victoria that her spirit-guided mission would be to launch "a social revolution."
The relationship between Lib Tilton, her husband, Theodore, and Henry Ward Beecher was becoming increasingly complex. Like Victoria Woodhull and Josie Mansfield, Theodore Tilton had also entered a privileged world of great wealth. Women of all classes had joined the woman's rights movement and the previous summer there had been a convention in one of America's most fashionable resorts, Newport, Rhode Island. The exquisite Paulina Wright Davis invited Tilton to stay at her mansion. Mrs. Davis had been involved in woman's rights since 1850. Widowed at thirty-two, she married Thomas Davis three years later. Davis, a man of wealth and social position, served in the Rhode Island legislature and for two decades had unstintingly supported his wife's efforts.
Never had Tilton seen such "great luxury" and display as in Newport. "Here one sees at the same glance the ocean and its waves, and silk dresses and scented handkerchiefs. I don't relish the comparison... I am more than ever content with my own moderate resources," he wrote Lib. Paulina was comfortable in this opulent milieu, though Lucy Stone criticized her for wearing magnificent French dresses, made on a mannequin in Paris. Stone called Davis "fickle," but Paulina defended herself, saying that she was "determined to do my utmost to remove the idea that all woman's rights women are horrid old frights with beards and mustaches who want to smoke and swear."
Davis knew that many women did not understand their own bodies, and she determined to do something about it. Her own couture mannequin inspired her to have an anatomically correct _femme modele_ constructed, which she imported to America and used in demonstrations before women's groups. Paulina had been a Calvinist until the church's indifference to slavery and its repression of women drove her from the church and into abolition and woman's rights. "I was not a happy woman until in mature life I outgrew my early religious faith and felt free to think and act from my own convictions," she asserted. As early as 1848, Davis had been instrumental in the drafting of the Woman's Property Act, which enabled Stanton and other women to own houses in their own names. Ironically, this act was sponsored by rich men who did not wish their sons-in-law to inherit property they had left to their daughters.
At dinner one night in Newport, Tilton was seated next to Laura Curtis Bullard, an heiress whose considerable fortune derived from a patent medicine, Winslow's Soothing Syrup, created by her father. Frank Curtis had this one child only, and he doted on her, educating her in Switzerland and supporting her lavish lifestyle in Paris. For the past two years she had been a widow. Having spent much time in Europe, she had adopted the French upper-class morality and the free love views of Charles Fourier, John Humphrey Noyes, and Stephen Pearl Andrews. Bullard was attracted to Tilton and suggested that an "affinity" could be pursued as long as it didn't interfere with their lives. Soon, she was to envision herself as Theodore Tilton's "Spiritual Wife." Soon, he was to embark on what he termed "a great friendship" with Mrs. Bullard.
While Tilton was away from Brooklyn, Beecher spent more and more time at the Tiltons' home, where, seated at Greeley's desk, he worked on the final installments of _Norwood_. As a token of his growing affection, he gave Lib a small watercolor of a trailing arbutus and symbolically wrote of it in his novel: "It is like the breath of love. The pure white and pink blossoms in sweet clusters lie hidden under leaves or grass, and often under untimely snows. Blessings on thee! Thou art the fairest, most modest, and sweetest-breathed of all our flowers!" When Susan B. Anthony visited, Lib showed her the painting. "My pastor brought that to me," she said with pride.
In _Norwood_ the love remained secret: Beecher called it "nest hiding." He wrote of Rose Wentworth, his heroine: "It would seem as if, while her whole life centered upon his love, she would hide the precious secret by flinging over it vines and flowers, by mirth and raillery, as a bird hides its nest under tufts of grass, and behind leaves and vines, as a fence against prying eyes." Seven years later, Lib wrote Beecher of her efforts to protect him from scandal: "My endeavor was entirely to help you, keep you from all suffering, to bear myself alone. My weapons were love, a large untiring generosity, and _nest hiding_!"
With the end of his novel in sight, Beecher once again went to see his friend John Tasker Howard, the influential publisher. Howard, in turn, approached a Jay Cooke employee, Samuel Wilkeson (who happened to be Elizabeth Cady Stanton's brother-in-law). Wilkeson was known for bringing excellent manuscripts to the J. B. Ford syndicate. During the recent war he had been both a correspondent for the _New York Times_ and a promoter of the bonds that Cooke was selling to help finance the Union armies. Wilkeson arranged for the J. B. Ford syndicate to advance Beecher $20,000 for a new book, a two-volume life of Christ. Wilkeson himself was so enthusiastic about its potential profits that he invested $10,000 of his own money in the project. Beecher began work immediately on _The Life of Jesus the Christ_ , and when he finished the first chapter he read it privately to Mrs. Tilton. Lib was pregnant once again, which no doubt kept carnal thoughts at bay. When Beecher asked how the Virgin Mary might have felt with the Christ child in her womb, Lib described her own feelings as she touched her swelling stomach. Then, she later recalled, she went upstairs and returned with "a little sketch of one of my Bethel lessons on 'Mary, the Mother of Jesus,' as an example of a woman's faith."
In Henry Ward Beecher, Lib Tilton found the affection she had so long craved. One Saturday he arrived at Livingston Street with his arms so full of yellow chrysanthemums that he had to pound on the door with his foot. "I am hungry to see your children," he declared.
"Are you really? Then come up directly and see them," she answered. That day Lib had laid out on the bed of Florry's room a profusion of white lace, dimity, and calico dolls' dresses. Lib and Beecher sat on the floor with the three children, laughing and chatting as they dressed each doll. Six-year-old Alice carried her doll to him. "What will you name her?" Beecher asked. When Alice did not reply, he suggested, "Rose Wentworth," the heroine of _Norwood_. "Yes, yes, Rose," agreed Alice, and clapped her hands in delight.
As Lib's confidence grew, the woman's rights people who congregated about her husband and who met at her house no longer seemed intimidating. She became corresponding secretary of the New York branch of the American Equal Rights Association and informed Theodore after a meeting, "You will be amused to know that Susan [B. Anthony] made me Chairman and said afterwards that 'I did as well as Theodore himself.' I always want to represent you well." For the first time she felt herself competent; when Anthony began to publish _The Revolution_ , Lib became its poetry editor.
Paul Tilton was born at the end of January 1868, and the following week, when Theodore returned home, Lib presented him with his new son. On the night his son was born, Tilton had been lecturing in Akron, Ohio, after which he retired to the Empire House for a passionate encounter, presumably with Laura Curtis Bullard. The next morning he experienced "a terrible wrestling with my soul" and decided that his own heart was "a dark place to look into." Tilton vowed that for Paul and for his own soul, he would try to make a "rebeginning." In an agony of self-flagellation he told his wife that he was prepared to break off his "great friendship" with Mrs. Bullard. He vowed to be a "better man," a "faithful husband," and swore that if Lib would help him, they would form a unit against the world. They fell on their knees and prayed to God "to perfect our union."
For her part, Lib Tilton, having fallen in love with her preacher, had struggled to keep that love "pure." She no longer had the protection of an advanced pregnancy, and Beecher was becoming more explicit in his sexual advances. He kissed her on the lips in what he later termed "a paroxysmal kiss." He told her, "I have a greater love for you than any woman in my life und wish to find in you the solace of life which had been denied to me by an unaffectionate marriage at home."
Pressured by both her ardent preacher and her repentant husband, an experiment in faithfulness seemed a safe choice. And so the Tiltons embarked on this quest, at first with high hopes. He wrote, "I regard my last evening spent with you at home as the most memorable point in my whole life. You opened for me, that night, the gate of Heaven, which had so long seemed shut." She wrote, "I am so safe and strong and glad in your love that I am conscious of an entire change toward every one, an independence like maidens feel when they decide upon one of their choice.... I am waking to a new life."
But, in Theodore's absence, high hopes gradually turned to grim determination. The pressure began to build as Lib came to question her husband's faithfulness and her own ability to resist her preacher. Two months later, Theodore again declared his fidelity: "My vows I shall keep, and not break. With God's help, and with yours, I shall be the faithfulest man in the world." Lib answered, "Theodore, do you _know_ what you have written?... That it might be so, I earnestly wish. That it is so, _I cannot believe_." His waffling reply was, "Innocence or guilt, in most of us, depends more on the measure of our temptation than on the measure of our virtue.... How morally sublime was Christ's word to the woman, 'Neither do I condemn thee.' "
Finally, in despair, Lib wrote to her husband complaining of his protracted absences and neglect of her. And, as if warning him, she ended the letter with, "My brain has run wild since 4 o'clock—sleep forsook me. The love of man or two men is not restful, while the love given to God is peaceful beyond expression." Theodore, otherwise occupied, replied,
Your last note reminded me that I had not written you from Washington as warmly as I wrote from the West. I felt that you wronged me by not reflecting on the difference between my situation here and my situation there. Here every minute of my day is busy, there I did nothing all day but sit in a car or a hotel and wait for the evening.
In Washington, Theodore Tilton was indeed involved in momentous events. Even after the Republicans gained control of Congress in 1866, President Johnson continued to oppose them. Throughout the year 1867 a series of Reconstruction acts had been passed by Congress over Johnson's repeated vetoes. The president cut back on the military and fired more than sixteen hundred officeholders he deemed sympathetic to the radical Republicans and replaced them with men who would strengthen the local civil governments he had previously installed. On March 2,1867, Congress retaliated by passing an act restricting his authority as commander in chief of the armed forces. The Tenure of Office Act, passed the same day, prohibited the president from removing any civil officeholder until his successor had been approved by the Senate.
That summer, when Congress was not in session, Johnson removed his bitter enemy, Edwin M. Stanton, as secretary of war. When Congress reconvened, Stanton was reinstated, and in December furious radical Republican members of the House introduced a resolution to impeach the president. The resolution was rejected 108 to 57. Two months later, Johnson declared the Tenure of Office Act unconstitutional and once again dismissed Stanton. On February 21, 1868, two years after Theodore Tilton had first called for Andrew Johnson's impeachment in the _Independent_ , public and political opinion had become so negative that the House by a vote of 126 to 47 resolved that the president "be impeached of high crimes and misdemeanors in office." Congress was now compelled to find evidence to justify its action. A seven-man committee was appointed to prepare the articles of impeachment.
During this period, Theodore Tilton spent many months in Washington writing article after article for the _Independent_ supporting Radical Reconstruction and criticizing Johnson for his "vulgarity and traitorous behavior." Tilton was among the most influential writers in America in turning the public against Johnson, but it was Johnson himself who, by opposing the Republicans and Radical Reconstruction, had lost all chance of being nominated for president. Even Tilton quickly saw that the real reason for removing the president—his bigotry and pro-South sympathies—could not be grounds for impeachment. Benjamin Butler, newly elected to the House of Representatives, and a member of the seven-man committee, was carried away with anger and wanted to charge Johnson with numerous crimes. Although Tilton knew that these charges were false, he objected only to Butler's accusation that Johnson had been complicitous in the assassination of Lincoln. As Tilton yielded to political expedience, he began to question his own character. "I have less faith in my moral integrity now than at any former period in my life," he wrote Lib. "I feel myself scarred, spotted, miserable."
Finally, on March 3, eleven articles of impeachment were adopted. The first eight related to Johnson's attempt to remove Stanton; the others related to his "attempt to bring into disgrace, ridicule, hatred, contempt, and reproach the Congress of the United States." On the instructions of his defense counsel, William Evarts, the president did not appear before the Senate. Evarts, a member of Plymouth Church, hated Theodore Tilton for first suggesting that Johnson should be impeached. He vowed that one day he would destroy Tilton as Johnson was now being destroyed. When the trial began, Evarts cleverly steered the testimony to the narrow legal issue of the meaning of the Tenure of Office Act. Within two weeks Tilton wrote that in his view the president would be acquitted, and even Butler admitted to Evarts that he wished he were on the other side.
Behind the scenes, President Johnson and Evarts maneuvered to compromise with the moderates in the Senate. When the Senate voted on May 16, the vote fell one short of the required two-thirds majority needed to impeach Johnson. Amid the pandemonium that ensued, a dying Thaddeus Stevens, who had spent his life fighting against slavery, was carried into the chamber by two attendants. He called out, "What was the verdict?" When he heard it, Stevens threw his arms into the air and shouted, "The country is going to the devil." Benjamin Butler called a press conference to announce that he was launching an investigation of certain senators who had taken bribes.
The next day, Tilton rode the cars to Chicago for the Republican National Convention where General U. S. Grant (his given name was Hiram Ulysses Grant, but U. S. Grant had a ring to it) received the presidential nomination. The Democratic National Convention followed two months later. On the twenty-second ballot, Horatio Seymour was chosen to run against Grant. Susan B. Anthony had applied to be a delegate to the convention and to present a petition that woman suffrage be made a plank of the Democratic platform. Although too ill to attend the convention, Mary Cheney Greeley was one of the four signers of this petition, once again lending her support in a manner guaranteed to antagonize her husband. Anthony was rejected as a delegate, but a clerk duly read her demand on the floor of the convention, at the conclusion of which the women in the gallery cheered and then the delegates returned to business as usual. The woman suffrage issue was not dealt with during the campaign.
In August 1868, while Theodore Tilton was once again in Newport, his eight-month-old son, Paul, whom Tilton hardly knew, died of cholera. A distraught Lib was on her knees, praying in the sitting room in which Paul's body lay when, according to her account, the stained-glass dome overhead became suffused with light. The room glowed as two angels descended and, each taking a hand, flew with Paul through the dome and disappeared into the clouds.
_The committee that drafted the Articles of Impeachment against President Andrew Johnson. Benjamin Butler appears at the lower left_. (Illustration Credit 14.2)
Beecher came down from Peekskill to officiate at the funeral. And a most unusual funeral it was: With the death of her baby, a dramatic new system of belief entered Lib Tilton's life. Like her husband before her, Lib Tilton now turned to Spiritualism. Susan B. Anthony, who attended the funeral, was surprised to find the house "filled with sunshine" and Lib not in the traditional black of mourning but dressed all in white in the Spiritualist tradition. To Anthony, Lib seemed gladdened by the thought that her "babe" was safe in heaven.
Theodore Tilton quickly returned home, but his reserved nature made it impossible for him to say anything that would comfort his wife and she was later to say that she resented him for this. He did, however, have sexual intercourse with her, and when he left in September she once again found herself pregnant. She also said that her heart yearned for her lost baby and she felt guilty about his death. Lib asked Beecher if perhaps she could have done something to save Paul. He reassured her that there was nothing. Her preacher comforted her and filled the void. Almost a decade later, Theodore Tilton was asked in civil court, "Do you believe that your wife loved Beecher more than you?" To which he replied,
No, but he was the sympathetic, ever-present friend. In my family life I could not give her the novelty of sympathy. Our loss was common.... Beecher was her God. I was her husband.
Beecher was all self love. She was merely convenable. In one of Mrs. Beecher Stowe's novels the remark is made that a seduced woman thinks herself purest when she is nearest to the point of being debauched. Had the date of this novel, "The Minister's Wooing," been nearer our time I should have thought Mrs. Stowe drew that point from my wife.
On October 9, 1868, the Brooklyn Academy of Music was filled to overflowing for an election rally for U. S. Grant. On a curtain behind Henry Ward Beecher there appeared a greatly enlarged Thomas Nast cartoon showing Grant being welcomed at the front door of the White House by Miss Columbia, Nast's symbol for America. Nast had included in the cartoon a dejected Horatio Seymour who crouched next to a lamppost from which hung the corpse of a lynched Negro.
Lib Tilton, in the tenth row, was enthralled. Beecher's endorsement of Grant was unqualified and addressed the issues that troubled voters the most.
It is said that General Grant is a drunkard. I do not believe a word of it. But I had rather have General Grant a drunkard than Horatio Seymour sober! It is said that General Grant knows nothing of civil affairs. Certainly Seymour knows nothing of military affairs. It is said that Grant is not a statesman. I have always said that what we meant to make a President of, was a great deal of common sense.
The applause was thunderous. Lib later said that she tried to fight her way down the aisle to congratulate her preacher but the crowd was too dense. Therefore, on the following afternoon, October 10, 1868, she went to Beecher's house in Columbia Heights. Beecher's family was still in Peekskill. Lib found him in his writing room on the second floor at the rear of the house, with an unparalleled view of New York Bay, the southern tip of Manhattan, and the East River. That evening a glowing Lib wrote in her diary, "A Day Memorable."
_Henry Ward Beecher's writing room: "A Day Memorable_ "
Henry Ward Beecher had a way of speaking of his personal concerns through his sermons. A few weeks later from the Plymouth Church pulpit he intoned,
The man who has been wallowing in lust, the man who has been on fire in his passions, and who by God's great goodness has been brought to an hour and a moment when... his monstrous wickedness stands disclosed in him—that man ought not to wait so long as the drawing of his breath.... He will stand up and say, "Here I am, a sinner, and I confess my sin and I call on God to witness my determination from this hour to turn away from it." That is the wise course, and you would think so, if it was anybody but yourself.
# CHAPTER FIFTEEN
WE ARE READY. WE ARE PREPARED.
ON CHRISTMAS Day 1868, as Victoria Woodhull advised the Commodore that his Central Pacific Railroad stock was "bound to go up," in a Philadelphia garret an incident was about to take place that would bring into sharp focus the injustice toward women in the postwar society.
The German immigrant landlady spoke little English. She held up two fingers to indicate $2 and said the word "Now." Twenty-year-old Hester Vaughn, desperate for a place to stay, reached into the pocket of her loose calico dress and put the money in her palm. The third-story garret was unheated and as a blizzard raged, Hester, who had been without food for three days, began a violent labor. She crawled to the door to call for help. Twenty-four hours passed before the landlady heard Hester's low moans. When she opened the door, Hester lay semiconscious on the rough wood floor in a pool of blood. Her dead baby was still attached to a crudely tied umbilical cord. The constables arrived an hour later and, since she could not walk, they carried her to prison. The newborn's body was so frozen that a portion of the skin stuck to the floor and was torn away as the corpse was lifted. Hester Vaughn was charged with infanticide. She was tried without a lawyer and convicted.
To bring Hester into the courtroom, where she would be sentenced to the gallows, required the services of a Philadelphia physician, Dr. Susan A. Smith. She found Vaughn lying on a bare cot in Moyamensing prison, staring blankly at a rat crouched in the corner. As she examined the prisoner, she elicited her story: Hester Vaughn had come from Gloucestershire to marry her fiancé, only to find him already married to another. Alone and deserted, she became a scullery maid in Jenkintown, then a dairymaid at a nearby farm. One night as she lay in bed, her employer raped her. She was too dispirited to run away. Four months later when her pregnancy began to show, this man gave her $40 and told her to clear out by dusk. She went to Philadelphia, worked as a seamstress, slept in doorways and in an almshouse, and finally rented the garret where, alone and unassisted, she had given birth. She told Dr. Smith that she had knotted the cord in an effort to save herself and her child from bleeding to death. Susan Smith left the prison and contacted Anna Dickinson, who lived a few houses away from Dr. Smith on Locust Street.
Anna was a natural choice to intercede on Hester's behalf, for she had become one of the most famous women in America, "the Queen of the Lyceum." Across the nation, she had wooed her audiences to the cause of abolition, and when the war ended, lectured on woman's rights so passionately that she was often referred to as the "Joan of Arc" of the movement. With few professions open to women, the Lyceum lecture circuit provided a desirable way of life—a means to earn money and fame, to travel, to have one's opinions heard. Agencies such as Redpath's, Fall's, the Boston Lyceum Bureau, and the American Literary Bureau in New York arranged these tours. Though not as successful as Anna Dickinson, Anthony, Stanton, and other woman's rights advocates also traversed the country, lecturing in town after town and resting only on Sundays. On one such tour, Anthony noted in her diary, "The year's work full 13,000 miles travel—170 meetings."
Other speakers on the Lyceum circuit included Spiritualists, trance mediums, political orators, and medical practitioners. In the quiet world of provincial America, their appearance was eagerly anticipated. From October to May thousands of people across the country were willing to pay from 50 cents to $5 for the privilege of listening to these visitors, who earned anywhere from $75 to $1,000 a night (less expenses and agents' fees).
These lecturers adopted unique mannerisms, presentation, and dress, to distinguish them from all the rest. Dickinson, who had once worn a plain gray Quaker dress, now wore a Paris gown and expensive jewelry. She bobbed her curly brown hair and wore a jaunty Alpine cap. Anna became the idol of repressed women: She earned her own money; she dressed beautifully; she traveled wherever she wished; she owned a horse and scandalized people when she raced the animal astride.
Dickinson's popularity and political power were especially appealing to three influential men. Although an aura of intimacy pervaded these relationships, had it been publicly perceived that she was actually sexually involved with any of the three, it would have been her ruin. The first two were married, and adultery brought ostracism from traditional society and, in several states, a prison sentence. The third, a bachelor, had vowed he would marry a virgin.
Wendell Phillips was Anna's first mentor. At fifty-seven, he retained his wiry energy and craggy good looks; a full head of dark hair, close-cropped burnsides, a pronounced cleft in his chin. His aristocratic mien and devotion to Radical Reconstruction seemed to put him above all personal concerns. By 1868, this great orator had been married for thirty-three years to an invalid who was confined to her room and often to her bed. They had no children.
The second of Anna's admirers was Benjamin Butler. He was physically unattractive, small and plump, with a disproportionately large head and sunken eyes engulfed in bags of flesh. A drooping eyelid added to his sinister appearance. Yet Butler had such vitality and power that many women were attracted to him. His wife, Sarah, a former actress, was ill with thyroid cancer, but she joined her husband and their only daughter, Blanche, in the campaign for woman's rights.
Butler was drawn to Anna and gave her political advice. It was rumored that he also wrote many of her speeches. He became close to Anna's entire family and secured a job for her brother Edwin in the Washington pensions bureau. From there Edwin wrote to his sister that Butler had dropped in and asked for her address, "So I suppose thee'll have a love letter the day before or after this reaches thee."
And finally, there was Whitelaw Reid, who came to interview Anna in 1863, when he was a correspondent for the _Cincinnati Gazette_. He was then twenty-six, and she twenty-one. Reid was immediately taken with her. Although from a humble background, Reid envisioned for himself the life of an aristocrat. In 1864, Reid became the London correspondent for Horace Greeley's _Tribune_ and lived in splendor on Chester Square, giving dinners that reportedly "outshone those at the American Embassy." When peace came the following year, he tried his hand at plantation farming but soon returned to the editorial staff of the _Tribune_.
Of Anna's three devotees, Whitelaw Reid was the only bachelor and newspapers frequently carried the rumor that he and Anna would marry. Reid wrote Anna's sister Susan that he'd received a letter from Schuyler Colfax congratulating him on "my impending marriage to 'that brilliant girl, Miss D.,' besides any number of other letters and newspaper paragraphs without number." He ended the letter by praising Anna's courage in bearing "it as stoically as I do."
Anna, when she learned of Hester Vaughn's plight from Dr. Smith, left immediately for New York, where she went at once to the offices of _The Revolution_. As always, Susan B. Anthony was delighted to see "her love," "her darling," "her chick a dee dee." Anna had totally captured Susan's heart. Anthony wrote to her on tear-stained paper, "My soul goes out to you in real mother yearnings—I don't believe you have believed the depths thereof." Susan yearned to give Anna "one awful long squeeze," and no matter how busy she was, she would travel anywhere to see her. "I do so long for the scolding and pinched ears and everything I know awaits.... What worlds of experience since I last snuggled the wee child in my long arms."
_Whitelaw Reid, 1863. At twenty-six he was a correspondent for the_ Cincinnati Gazette.
Anthony was horrified, but not surprised, to hear of Hester Vaughn's death sentence. But times were changing: Now she had her own journal, and article after article appeared in _The Revolution_ condemning the injustice done to Hester. The Workingwomen's Association organized a protest at Apollo Hall, where Anna Dickinson brought tears to the eyes of hardened reporters. The association demanded a new trial or an unconditional pardon. Elizabeth Cady Stanton visited Governor John W. Geary of Pennsylvania to ask for the pardon. She argued that Vaughn was a victim of a cruel male society and that Hester's seducer was the one who should be tried for this crime. Geary refused to help, explaining to Stanton, "You have no idea how rapidly the crime of infanticide is increasing. Some women must be made an example of. It is for the establishment of a principle, Ma'am."
"Establishment of a principle, indeed! Yes, I suggest you inaugurate this good rule by hanging a few women," Stanton replied.
For a year _The Revolution_ continued to pursue Vaughn's story, running many articles on Hester's failing health. Stanton wrote that women should be judged by a jury that included women and that capital punishment was cruel and useless and should be abolished. The New York branch of the American Equal Rights Association circulated petitions throughout the country protesting Vaughn's imprisonment. Members of Congress were solicited, and George W. Julian and Samuel C. Pomeroy wrote letters to Governor Geary. After a year of mounting protest, Vaughn was pardoned. The first branch of the Workingwomen's Association, whose members earned scarcely enough to feed and clothe themselves, helped raise $300 to send Hester Vaughn back to England.
The Workingwomen's Association had been Susan B. Anthony's idea; she believed that women should be self-supporting and must organize to achieve this goal. With this in mind she set up New York's Number One Branch of the Workingwomen's Association to benefit female typesetters and clerks, and a few weeks later created a second branch in a boarding-house tenement housing factory girls, most of whom were seamstresses. These young girls worked from ten to fourteen hours a day and generally earned $4 to $8 a week. Jammed into the boardinghouse dining room for the initial meeting were more than a hundred weary women in plain calico dresses. When the chairs ran out they sat or sprawled on the floor. Anthony told them she wanted to know about their working conditions so that she might help them. At first the women remained silent but then one spoke up: "I get two dollars apiece for making ladies cloth cloaks."
"How long does it take to make one?" asked Anthony.
"Less than a day. It's partly machine and partly handwork."
There was a ripple of dissent in the room—this seemed far too easy. One chalk-faced girl said, "Me—I make lace collars for twenty-two cents a dozen. In a day, twelve hours work, that's sixty-six cents."
A woman held up her raw, blistered hands. "I am a carpet-sewer. I work nearly three days and sew fifty yards of carpet, and when I asked the boss to pay me for sewing the borders, which is additional work, he laughed and said it was 'chucked in.' "
Another woman explained that after she had made a coat she was required to dampen it in starch water and iron it without extra pay. No coat that was not glossy and well pressed was accepted. "I took in two dozen last Tuesday," said the bleary-eyed woman, "and had a neighbor in to help. But both of us working steady all day and half the night, just as hard as we could, finished the lot by Friday evening. It was a hard earning of a dollar and a half, Ma'am, to divide between two, and both of us widders, but we're thankful to get it for all that. There's many a one wanting work that can't get any."
Susan told these dispirited women, "You must not work for these stingy prices any longer. Have a spirit of independence... and you will get better wages for yourself. Get together and discuss, and meet again and again. I will come and talk to you."
At this point talk was all Anthony had to offer but she inspired women with her quiet determination not to accept oppression. And there was no doubt that _The Revolution_ bolstered her effectiveness. Its columns advocated such measures as fair wages, an eight-hour workday, and a tax on foreign products. Its Wall Street news revealed the manipulations of Fisk and Gould and their fellow schemers. Accounts of their misconduct were often picked up by the popular press. Elizabeth Cady Stanton used _The Revolution_ to attack preachers, politicians, and the insensitive wealthy.
Women are starving today and every day in the streets of New York but what cares the pulpit about it? Such a question need not be asked the politician. Still less need we stop the gilded coaches in Central Park to ask of beauty, wealth and fashion a question so impolite, so impertinent. And yet... these three classes... enact every civil statute, establish and regulate every civil law and custom, frame fashion and enforce all the religious faiths. The statute book benefits the rich only.
Stanton's position, however radical, was understandable, but her blind rage against the male establishment, combined with her feelings of class superiority, weakened her argument and further alienated those she most needed to accomplish her purpose. "The male element is a destructive force, stern, selfish, aggrandizing, loving war, violence, conquest, acquisition, breeding in the material and moral world alike discord, disorder, disease, and death.... The idea strengthens at every step, that woman was created for no higher purpose than to gratify the lust of man.... Society as organized today," she concluded, "is one grand rape of womanhood under the man power."
Members of the conservative Boston branch of the American Equal Rights Association were appalled by this increasing radicalism. The New England society tried to distinguish itself from the New York society by drawing up its own agenda, emphasizing that woman suffrage was to be "a clear cut separate single question," not mixed with "side issues" such as the problems of working women, the present laws of marriage and divorce, and the protest against the male establishment. As a practical matter, Lucy Stone on behalf of the New England society encouraged the support of men, which she considered essential to the cause. This group voted unanimously that "prominent and able men, particularly ministers" should join with them. Henry Ward Beecher promptly became a member.
The New England group made it clear that there would be no reconciliation with New York until all connection was severed with the intolerant George Francis Train, financier of _The Revolution_ , who used its pages to attack "niggers" and "that infamous Fifteenth Amendment." The most vocal adversary of Train was Mary Livermore, who decided that Train must be wiped out of the picture as effectively as she had wiped out the disease of scurvy in the Union troops during the Civil War. "We are now in the midst of a serious quarrel with Miss Anthony and Mrs. Stanton and the Train admixture," wrote Lucy Stone.
Anthony knew that a fractured woman's movement would have little power, and so she acceded to Stone and Livermore's demand to drop Train's name before the next American Equal Rights Association convention. Train, realizing that he could no longer use _The Revolution_ to promote his views, told Anthony, "Omit my name from your journal—and let me work out my destiny alone." He withdrew not only his writing but his monetary support, and on May 10, 1869, Train left New York on the first trip by rail to California and then attended the ceremony where the Union Pacific and the Central Pacific Railroads were connected at Promontory Point near Ogden in Utah Territory. Here spikes of gold and silver were driven into the tracks that now extended from the Atlantic to the Pacific oceans.
A reconciliation between the two groups seemed possible. Mary Livermore, seemingly placated, wrote Anthony, "I have written to the New England friends to let bygones be bygones and come to the May meeting. It seems to me personal feelings should be laid aside and women should all pull together." On the morning of May 9, the American Equal Rights Association convention met in the handsome Steinway Hall on Fourteenth Street between Union Square and Irving Place. On the platform sat Vice Presidents Elizabeth Cady Stanton, Frederick Douglass, and Henry Ward Beecher. Lib Tilton, with the secret flowering of her relationship with Beecher and her new feelings of confidence, had joined the Executive Committee. She stood at the rear of the hall with her daughter Florry, collecting tickets.
_A Union Pacific train in rugged Utah territory in 1869, when the Union and Central Pacific Railroads finally connected_ (Illustration Credit 15.1)
Lucretia Mott, the association's president, was home with a spring flu. In her place Stanton, as first vice president, assumed the chair. The Reverend O. B. Frothingham stood and, as if trying to head off trouble, announced that he was sure a "generous, sweet atmosphere will prevail" at this meeting. No sooner had he finished speaking than Stephen Foster, a New England farmer who had been an early abolitionist, sprang to his feet and declared, "I object to certain persons who have prevented harmony and destroyed the efficiency of this organization." Then looking at Stanton he announced, "One of these is our presiding officer."
Stanton answered, "I would like you to state in what respect." In reply, Foster mentioned Train and his ridicule of the Negro and opposition to the Fifteenth Amendment. Foster demanded that _The Revolution_ repudiate Train "because of his course in respect to the negro's right." When Anthony declined, Foster insisted that both Stanton and Anthony "retire and leave us to nominate officers who can receive the respect of both parties." His ultimatum drew applause, after which there was a heated exchange concerning Anthony's bookkeeping and renewed charges that she had misused $1,000 of American Equal Rights Association funds to cover the cost of Train's speaking tour in Kansas the previous year.
"She never kept any books or account of the expenditures," declared Foster.
"That is false!" Anthony exclaimed. "Every dollar ever received by me and every dollar expended, item by item, was presented to the trust fund committee of Boston, of which this gentleman is a member." Then, in a desperate effort to justify herself, Anthony added, "They voted me a check of $1,000 to balance the account."
This in fact was not true, for Anthony had received no such check, and Stephen Foster, hoping to discredit her by making more of this minor distortion than was warranted, replied, "I would be glad to believe Miss Anthony, but her statement is not reliable."
Stanton rapped down her gavel. "You are out of order, Mr. Foster," she declared. Stanton was indignant that this matter was being publicly aired to humiliate Anthony before so large an audience. Lucy Stone's husband, Henry Blackwell, interceded.
The facts of the case are these.... Miss Anthony brought in a statement of her expenditures to the society. No one doubts that all the expenditures were actually made as she reported. Her statement made due to herself from the society $1,000. Miss Anthony, for the sake of harmony and the good of the cause has given up her claim for this $1,000. In regard to this we have to say that we are entirely satisfied with the settlement thus made. I think this question might well have been kept back.
Then Stanton called for a vote of confidence from the convention, which was carried, but the atmosphere had turned adversarial. The tension increased when Stanton proceeded to attack the male establishment and the Negro. Bringing up the same old threats and playing on the same old fears used time and again, she warned,
The Republican Party today congratulates itself on having carried the Fifteenth Amendment of the Constitution, thus securing "manhood suffrage" and establishing an aristocracy of sex on this continent.... The lower orders of men... the slaves of yesterday are the law-makers of today.... The legislation of the ignorant African... in whose eyes woman is simply the being of man's lust... must culminate in fearful outrages on womanhood, especially in the Southern states.
Think of Patrick and Sambo and Hans and Yung Tung, who do not know the difference between a monarchy and a republic, who cannot read the Declaration of Independence or Webster's spelling book, making laws for... the daughters of Adams and Jefferson... women of wealth and education.... Shall American statesmen, claiming to be liberal, so amend their constitutions as to make their wives and mothers the political inferiors of unlettered and unwashed ditch-diggers, bootblacks, butchers and barbers, fresh from the slave plantations of the South... to establish an aristocracy based on sex alone?
As soon as Stanton finished, Frederick Douglass rose and moved toward the lectern. He had come to the convention to observe, not to speak, but now he could not contain himself. In the past he had declared his loyalty to Stanton, recalling, "When there were few houses in which the black man could have put his head, this woolly head of mine found refuge in the house of Mrs. Elizabeth Cady Stanton, and if I had been blacker than sixteen midnights without a single star, it would have been the same." But Stanton's speech had deeply wounded him. She had joined the horde of oppressors. Controlling his emotions, he told the convention:
There is no name greater than that of Elizabeth Cady Stanton in the matter of woman's rights and equal rights, but... the employment of certain names, such as Sambo, and the gardener, and the bootblack, and the daughters of Jefferson and Adams and all the rest that I cannot coincide with....
Douglass's own cause burned with a white-hot flame.
I must say that I do not see how anyone can pretend that there is the same urgency in giving the ballot to woman as to the negro. With us, the matter is a question of life and death, at least in fifteen states of the Union. When women, because they are women, are hunted down through the cities of New York and New Orleans; when they are dragged from their houses and hung upon lampposts; when their children are torn from their arms and their brains dashed out upon the pavement; when they are objects of insult and outrage at every turn; when they are in danger of having their homes burnt down over their heads; when their children are not allowed to enter schools, then they will have an urgency to obtain the ballot equal to our own.
Lucy Stone also felt that Stanton's bigoted arguments were holding women back. If women lent their support to powerful men and to the cause of Negro enfranchisement, the favor, she believed, would soon be returned. In this she represented the vast majority of woman suffragists. The audience was quiet as Stone spoke. Even Anthony conceded that Stone "made _the speech_ of _all the women_ —and quite outdid her old self."
The gentleman who addressed you claimed that the negroes had the first right to the suffrage.... But I want to remind the audience that when he says what the Ku-Kluxes did all over the South, the Ku-Kluxes here in the North in the shape of men, take away the children from the mother and separate them as completely as if done on the block of the auctioneer. Over in New Jersey they have a law which says that _any_ father—he might be the most brutal man that ever existed—may by his last will and testament dispose of the custody of his child, born or to be born, and that the mother may not recover her child. And that law modified in form exists over every state in the Union, except in Kansas.
Woman has an ocean of wrongs too deep for any plummet, and the negro, too, has an ocean of wrongs that can not be fathomed. There are two great oceans; in the one is the black man, and in the other is the woman. But I thank God for that Fifteenth Amendment, and hope that it will be adopted in every state. I will be thankful in my soul if _anybody_ can get out of the terrible pit.
At the evening session, the Stanton and Anthony faction were overwhelmed by the powerful men and New England women who dominated the convention. A series of resolutions supporting the ratification of the Fifteenth Amendment and urging a separate amendment for woman suffrage was presented and passed over the objections of the New York group. Henry Blackwell then read out a resolution clearly aimed at disavowing their association with Stanton's more radical views.
_Resolved:_ That in seeking to remove the legal disabilities which now oppress woman as wife and mother, the friends of woman suffrage are not seeking to undermine or destroy the sanctity of the marriage relation, but to ennoble marriage, making the obligations and responsibilities of the contract mutual and equal for husband and wife.
No sooner had Blackwell read the resolution than Mary Livermore stepped forward. This forty-eight-year-old woman who weighed nearly two hundred pounds was as commanding as she was direct and blunt. She said that originally she had submitted a stronger resolution, and although she'd agreed to the Stone-Blackwell version, having slept on it, she had changed her mind. Then she read,
_Resolved:_ That while we recognize the disabilities which the legal marriage imposes upon woman as wife and mother, and while we pledge ourselves to seek their removal by putting her on equal terms with man, we abhorrently repudiate Free Loveism as horrible and mischievous to society, and disown any sympathy with it.
There it was— _free love_ —an epithet hurled straight at Mrs. Stanton and members of the New York branch of the American Equal Rights Association. Anthony was the first to respond, saying, "This howl comes from those men who know that when women get their rights they will be able to live honestly and no longer be compelled to sell themselves for bread, either in or out of marriage." Free love was an easy target for opponents of the New York group. In fact, they did not advocate free love; their agenda was ambiguous on this issue. But Mary Livermore seized upon free love to denounce their radicalism in general and reduced Stanton's entire effort to reform the laws of marriage and divorce to this simple assault.
At this point, Victoria Woodhull seemed to have no common ground with the New York women who sought suffrage. But the concept of free love was broad and murky, and could encompass everything from amending the Constitution of the nation regarding women's enfranchisement to allowing one to indulge in sexual relations with whomever one chose for however long one chose. Free love meant different things to different people.
Lucy Stone was outraged by her friend Mary Livermore, because they had thoroughly discussed keeping the explosive issue of free love out of the convention. Stone declared,
I am unwilling that it should be suggested that this great, sacred cause of ours means anything but what we have said it does. If anyone says to me, "Oh, I know what you mean, you mean Free Love by this agitation," let the lie stick in his throat. You may talk about Free Love, if you please, but we are to have the right to vote. Today we are fined, imprisoned, and hanged without a jury trial by our peers. You shall not cheat us by getting us off to talk about something else.
Lucy Stone had a powerful personal reason for avoiding the subject of free love. It was common knowledge that her husband and fellow equal rights advocate Henry Blackwell was passionately pursuing Abby Hutchinson Patton, the wife of a prominent stockbroker, Ludlow Patton. Abby was also a member of "The Hutchinson Family," a singing group that campaigned for woman's rights. She had been on the campaign trail with Blackwell, three years earlier, at a time when he had felt jealous of his wife's burgeoning career and fame. Abby Patton fed Henry's ego and called him a "grand and secret love."
Henry's sister, the physician Elizabeth Blackwell, wrote of his infatuation, "I can say nothing about Harry [as the family called Henry]. The _thing_ grieves and disgusts me. Poor fellow!" and added the directive so frequently seen in such letters, "Burn this at once!" Indicating that this subject was bandied about, Emily Blackwell, another sister, wrote that she'd heard through a third party that "Mrs. P." might wish to break off her "relations" with Henry and she hoped that he did too, but "I shall not feel sure... how their good resolutions hold out... nor do I know what the result will be between H & L for I have not seen them."
Like Lib Tilton, Lucy Stone blamed her husband's straying on her own sexual inadequacies. His sisters agreed, pointing out that in her devotion to woman's rights Lucy had neglected her husband. At this very meeting Abby Patton sat scarcely ten feet from Lucy Stone, and few could miss the difference between the two. Abby Patton—petite, dimpled, with a voice "like a songbird"—wore a lavender velvet dress with a large garnet breast-pin and green satin ribbons in her hair. Lucy Stone—small, plain, and plump, a dark mole prominent above her upper lip—wore a black silk dress with a delicate white lace collar; her hair, parted in the middle, was severely pulled back and unadorned. The tension in the air was palpable.
Month after month Henry had vacillated between his wife and Mrs. Patton. A brother-in-law noted that he had a strong "butterfly streak." Lucy, who had begun to suffer from blinding headaches, wrote Henry, angrily demanding the income from her own properties, which she wanted to separate from his. Lucy wanted her husband back but could not bear the humiliation of demanding his return. Shortly before this convention she gave him an ultimatum: She was putting their daughter Alice in boarding school and moving to Boston. She would try to finance a newspaper that they could edit together if he wished. He could follow or not. As she prepared to move to Boston she wrote, "I feel crushed, and torn, and homeless.... I shall try and work through the paper for the future and quit this lecturing field altogether... and shall certainly not continue my mode of work—tho it is my natural way."
By the final day, the convention had clearly split apart on the issues. Ernestine Rose, a member of the New York group, pointed to the powerful men who offered no help. She said that when Horace Greeley was asked what "impartial suffrage" meant, he replied, "Why man, of course. The man and the brother." "They do not speak of women and sisters," said Rose. A frustrated Stanton's form of address indicated to the gathering just who had the power: " _Gentlemen_ and Ladies, I take this as quite an insult to me. It is as if you were invited to dine with me and you turned up your nose at everything that was set on the table."
The imposing Mary Livermore answered, "It certainly requires a great amount of nerve to talk before you, for you have such a frankness in expressing yourself that I am afraid of you." The audience laughed at this, for Livermore had her own sharp claws. "If you do not like the dish, you turn up your nose at it and say, 'Take it away, take it away.' " She made a dismissive gesture with her left hand. "Throughout the day the men who have attended our convention have been turbulent.... I say it frankly, that the behavior of the majority of men has not been sympathetic to you!" There were cheers in the hall.
On leaving Steinway Hall, Livermore halted in front of Susan B. Anthony. "The American Equal Rights Association is an awful humbug. I would not have come... nor would any of us, if we had known what it was. We supposed we were coming to a woman suffrage convention."
The following night at a meeting of the Brooklyn branch of the AERA, Stone and Blackwell heard a rumor that Stanton and Anthony were so upset that they were about to form a national society of their own. On Saturday morning, Livermore asked Anthony if there was any such plan. She denied it. That night, after Stone, Blackwell, and Livermore had left the city, Stanton and Anthony called a meeting at the Women's Bureau. As soon as the parlors, halls, stairway, and offices were filled, Elizabeth Cady Stanton announced a so-called impromptu business meeting. That night a new group was formed—the National Woman Suffrage Association. Stanton wanted only women to join the new organization, a proposition her husband Henry Stanton heartily endorsed: "Having been drilled for twenty years privately, I am convinced that women could do it better alone." But Tilton and Anthony disagreed, and an all-female organization was voted down. New goals were set forth: to oppose the Fifteenth Amendment because it did not include women (an almost hopeless task, since the amendment already had been ratified by twenty of the thirty-six states); to secure a Sixteenth Amendment specifically enfranchising women; to fight for every aspect of woman suffrage, including women's right to an eight-hour workday and pay comparable to men; and to demand more rights in marriage and more lenient divorce laws so that women would not be regarded as their husbands' "property." Declared Elizabeth Cady Stanton, "We are ready. We are prepared."
# CHAPTER SIXTEEN
SOUP FOR THREE
VICTORIA WOODHULL was about to fulfill the prediction that one day she would possess $700,000. Her good fortune would come about through the worst financial debacle in the history of America: the collapse of the gold market, September 24, 1869—Black Friday.
At the end of the Civil War, American paper currency had become so weak that it took $241 of paper "greenbacks" to buy $100 worth of gold. Four years later, as prosperity returned, $100 in gold could be bought for $135 in greenbacks, but the price was unstable, and the gamblers on Wall Street speculated in currency as they did in stocks. In the spring of 1869, gold fell to $131, and Jim Fisk and Jay Gould bought $7 million worth. For bold operators such as these it seemed possible to manipulate the price of gold and even corner the market. But the U.S. Treasury had nearly $100 million in gold in its vaults and would, on occasion, sell a few million dollars' worth when it was needed for trade. Only if these vast reserves were kept off the market could the price of gold be manipulated.
The previous year, directly after Abel Rathbone Corbin had married President Grant's middle-aged sister Jenny, Jim Fisk and Jay Gould began to cultivate Corbin as a friend. In the summer of 1869, Gould bought $1.5 million in gold and earmarked it for the Corbins. He told them that if the price of gold rose, the profits would be theirs. Gould then offered a similar arrangement to Assistant U.S. Treasurer Daniel Butterfield and to General Horace Porter, Grant's personal secretary and trusted friend. Though Butterfield accepted the bribe, Porter sent Gould's offer back with a sharp note: "I have not authorized any purchase of gold and request that none be made on my account. I am unable to enter into any speculation whatever." Finally, in a bold stroke, Gould purchased $1.5 million in gold earmarked for President Grant's wife, Julia. Within the week Gould forwarded to Abel Corbin two checks for $25,000 each, representing profits, made out to Jenny Grant Corbin and to her sister-in-law, the first lady of the nation.
On September 2, 1869, the president arrived in New York on his way to Saratoga and had breakfast with the Corbins. Abel Corbin somehow persuaded Grant to write a secret letter instructing Treasury Secretary George Boutwell not to sell gold until he personally gave the order. When Boutwell came to New York from Washington two weeks later, several prominent businessmen told him how the gold market was being manipulated. Fisk, worried that Boutwell might crack under the pressure, arranged to have William O. Chapin, his trusted employee, deliver a letter to the White House urging the president to stand fast on his decision not to sell gold.
On September 18, 1869, Chapin took the eight a.m. Pennsylvania Central train and arrived in Baltimore that evening. He hired a horse at a local stable and rode all night, forty-one miles at a breakneck pace, arriving at the White House the morning of September 19 to find President Ulysses S. Grant and General Horace Porter playing croquet on the lawn. Chapin dismounted and gave the president Fisk's letter. Grant opened the envelope, read the message, put it in his pocket, and said, "There will be no reply." Chapin proceeded to the nearest telegraph office and wired his employer, "Letter delivered all right." The telegraph clerk tapped out the message, which arrived at Fisk's office. But it read, "Letter delivered. All right." One tiny dot, one tiny error—but when Fisk received the reply that seemed to say "All right," he believed that the corner on the gold market was safe.
The week after Grant wrote Boutwell the secret letter advising him not to sell gold, Victoria Woodhull visited Commodore Vanderbilt and in a trance advised him to buy gold at $132. Vanderbilt committed his entire reserves and, using several brokers, acquired $9.5 million at that price. The sale of gold was not conducted in the stock exchange itself but in an adjacent Gold Room on Broad Street and Exchange Place. The Gold Room resembled a Greek amphitheater, in the center of which was a fountain decorated with a bronze cupid and dolphins spouting water. Two mechanical indicators, one inside, another outside the building, told the current price of gold. On Wednesday, September 22, gold rose to $137 and then to $141.
At lunchtime, Jenny Corbin received a letter from her sister-in-law, Julia Dent Grant. The final line read, "Tell Mr. Corbin that the President is very much distressed by your speculations and you must close them as quick as you can." Abel Corbin summoned Jay Gould. Both men were in a panic. Corbin insisted Gould buy him out. Gould refused but said he would guarantee that the price of gold held. Early Thursday morning Gould met with Fisk at the home of Josie Mansfield to plan a strategy. They decided to drive the price of gold up to $150 and then, with others still eager to buy, they would sell out their holdings. To effect this plan, they placed anonymous sell orders with several brokers. Then Fisk returned to the Gold Room and bought up all the gold in sight, thereby stimulating others to do the same. When the exchange closed, gold stood at $144.
That evening, Victoria Woodhull once again visited the Commodore and advised him to sell his gold at exactly $150. She said that she had seen a vision of a gold bubble bursting and the number 151. Perhaps the figure of $150, the price at which Fisk and Gould planned to unload their gold, came from the spirits, though it is more likely that Victoria had heard what the plans were from Josie Mansfield. In any case, the Commodore promised Victoria that if her prophecy came true, she would receive half of his profits.
Friday morning, September 24, was cloudless and temperate. When trading opened at ten o'clock the indicator read $150, and the Commodore, heeding Vickie's advice, sold his $9.5 million worth of gold. By eleven-thirty gold was worth $160. As the price rose further, hundreds of screaming brokers and speculators crowded the floor of the Gold Room, waving scraps of paper. Men rushed to the fountain in the center of the room and splashed their fevered faces; some thrust their heads under the spouting dolphins. The National Guard was sent from Brooklyn to keep order in the street. The outdoor indicator steadily rose up and up. By noon gold stood at $162.
On the corner of Broad Street and Exchange Place, Jim Fisk, accompanied by Josie Mansfield, watched from his carriage as the outside monitor touched $163. Close by in her own carriage, Victoria Woodhull sat next to Colonel Blood. None of the parties seemed perturbed that gold had risen past $150. The _Sun_ later noted that "amid the pandemonium Mrs. Woodhull sat unmoving, a half-smile on her lips." There is no doubt that once the indicator moved past $150, Victoria knew that she had fulfilled her mother's prophecy that evermore she would ride around in her own carriage.
For days brokers had been pleading with the president to avert a national collapse by selling gold. At his New York office, Assistant Treasurer Daniel Butterfield, who had been counting the profits on the $500,000 in gold Fisk and Gould had bought for him, understood that he could delay no longer. He sent a wire to Treasury Secretary George Boutwell in Washington which read, "Much feeling and accusation of government complicity."
Boutwell and Grant conferred. At one o'clock, Boutwell wired Butterfield, "Sell four million gold tomorrow and buy four million bonds." Butterfield secretly sent a messenger to Fisk and Gould informing them of what was about to happen. Then he publicly announced the government's plan to sell gold the following day. The impact was instantaneous; within thirty minutes gold plummeted from $164 to $132.
Investment houses that had been the bulwarks of Wall Street went bankrupt in the riptide. By midnight twenty-five people had committed suicide. It took months to straighten out the debacle of Black Friday. Dozens of clerks spent eight days simply compiling the flood of transactions. A congressional committee eventually established that Fisk and Gould had indeed sent the first lady $25,000 in profits, but Abel Corbin had cashed the check, so there was no way to verify that the money had reached Mrs. Grant. The committee decided that the president himself probably knew nothing of the scheme. Daniel Butterfield was asked to resign but was not further penalized. Commodore Vanderbilt made a profit of $1.3 million, of which he was said to have given Victoria Woodhull half. That, added to her earlier profit on the Central Pacific, made her a very rich woman.
_The panic in the Gold Room on Black Friday, September 24, 1869_ (Illustration Credit 16.1)
SOME SAID IT had started as the Commodore's joke: He'd boasted that he could set up a monkey on Wall Street, and when the public heard he was behind the critter, it would make millions. Two beautiful female stockbrokers would be sure to attract attention in a world geared to celebrity. In the welter of publicity that followed, no one mentioned Vanderbilt's bold acumen in backing an enterprise that attracted more than four thousand visitors on opening day and found an untapped source of female money, to say nothing of information, unavailable to other Wall Street firms.
In the winter of 1869, Victoria and Tennessee rented two parlors at the genteel Hoffman House and set up the brokerage firm of Woodhull, Claflin & Co. On January 19, 1870, calling cards were sent from the firm. "Mrs. Victoria C. Woodhull," read one; the other, "Mrs. Tennie C. Claflin" (she no longer wished to be called Tennessee). Each parlor was decorated with oil paintings, statuary, green velvet sofas, and a piano, making it look, as a reporter observed, like "a ladies' drawing room." A portrait of their patron, Commodore Vanderbilt, was displayed next to a painting of two plump angels, their pink arms wrapped around a gold cross, under the inscription Simply to Thy Cross I Cling.
The trial balloon floated giddily away: Within a month Woodhull, Claflin & Co. had become so successful that Victoria and Tennie C. were forced to move to larger quarters at 44 Broad Street. The desks in the new offices were carved with a Greek scroll-and-key design in honor of Victoria's spiritual guide, Demosthenes. In the front office a telegraphic stock indicator clattered steadily. Facing the indicator, with a Marvin safe at his back, Colonel Blood sat at a massive desk. Blood had demonstrated his business acumen in St. Louis and now took the lead in running the firm. The colonel enlisted his brother George, a former newspaperman, as chief book-keeper. The records were meticulously kept.
The crowd attending the official February 14 opening included Commodore Vanderbilt, Boss Tweed, and Jim Fisk accompanied by the elegantly dressed Josie Mansfield. By noon so many people crowded Broad Street that policemen, one hundred strong, were sent in to keep order. Thereafter, they all came—celebrities, customers, other brokers, the curious, and, of course, women. Woodhull, Claflin & Co. differed from other Wall Street firms in that there was a spacious private back office, completely cut off from the front offices by a richly carved walnut partition topped with ornamental glass. This office, accessible by a rear entrance, was restricted to women.
Society women and heiresses, small-business owners, writers, teachers, and housewives who had saved modest amounts hidden from allowances supplied by their husbands flocked to the rear entrance. So did women like Josie Mansfield, to invest the money they had earned as actresses or at houses like Annie Wood's and Molly de Ford's. (Indeed, Madame de Ford later said that the firm had earned her a profit of $30,000 in one year.) These women brought information and money to Victoria, who knew what to do with both.
_A cartoon of Victoria and Tennessee driving the bulls and the bears of Wall Street, in the_ Evening Telegraph, _February 18, 1870_ (Illustration Credit 16.1)
The press welcomed the flashy novelty of Victoria and Tennie C., two beautiful women who made good copy and courted publicity, riding to work in their open chariot, tucking solid-gold pens behind their ears, wearing identical empress-blue velvet gowns. So many men crowded the offices that they were forced to put a sign in the window: "All gentlemen will state their business and then retire at once." The press called them "The Bewitching Brokers," "The Queens of Finance," "The Sensation of New York." Many afternoons the sisters could be found in the women's office, where they offered female clients champagne and strawberries dipped in chocolate as they sat behind their massive desks. They left the actual work of the brokerage to the men up front while they entertained clients or stuffed their scrapbooks with clippings.
Commodore Vanderbilt appeared at the Woodhull, Claflin & Co. offices almost daily, and he once remarked to a _Sun_ reporter that many women were buying the Central certificates sold through the firm because they had his picture on them. The renamed Tennie C. remained close to the Commodore even though six months previously he had reneged on his offer of marriage. It seems that William Henry, the Commodore's canny son, aware that he and his wife, Maria, would never be accepted into society if his crude father married a Spiritualist fortune-teller, had arranged for the visit of a distant cousin from Mobile, Alabama, a nurse who was widowed during the war. Mrs. Crawford arrived with her nineteen-year-old daughter, Frank. On August 21, 1869, the Commodore eloped to Canada with Miss Frank Crawford. Later, when he was asked why he had married the daughter instead of her more suitable mother, he replied, "If I'da married her, Frank would have gone off and married someone else. Now I have them both."
In any case, the Commodore's marriage had no effect on his appetites, and his young wife offered no visible objection when on Tennie C.'s next visit to Washington Place he kissed her full on the mouth and exclaimed, "You might have been Mrs. Vanderbilt."
"Didn't you say you'd marry me?" asked Tennie.
"I intended to have done so, but the family made other plans."
The rumor persisted that Tennie C. was still sexually intimate with the Commodore. When asked by a reporter from the _Brooklyn Eagle_ if she found the role of Vanderbilt's "protégée" awkward, she replied, "Were I to notice what is said by what they call society, I could never leave my apartment except in fantastic walking-dress and ballroom costume. I despise what those squealing, crying girls or powdered, counter-jumping dandies say of me."
Susan B. Anthony, intrigued by these sisters, visited them. As she made her way down Broad Street, she lifted the skirt of her black silk dress to avoid the mud, garbage, and tobacco juice. When Anthony arrived at Woodhull, Claflin & Co., she was directed to the rear women's entrance, where Victoria welcomed her enthusiastically and praised her for her work. Although this was their first meeting, Anthony, always on the lookout for people to aid her cause, asked Victoria if she would join the suffrage movement. Victoria replied, "Just wait until we get ourselves firmly established in our business and we will show you what we will do for the rights of our sex." At this point, a waiter entered with a lunch tray, which he placed on an empty desk. In an article she wrote for _The Revolution_ , Anthony expressed her pleasure that these businesswomen were waited upon just like men. She ended the article with, "I found two bright, vivacious creatures, full of energy, perseverance, intellect, and pluck, and I said to myself, here are the elements of success."
Although the sisters' escapades undoubtedly were designed to attract publicity, they also violated the feminine taboos. Tennie C. once called a reporter into her office to show him the man's banking suit she was wearing. The reporter gaped at the trousers that ended three inches above her ankle and said, "If you wear that out on the street, there'll be a riot worse than the draft riot." Tennie C. nevertheless wore the man's jacket and waistcoat but added a long black broadcloth skirt, and Victoria soon joined her in wearing this attire.
Clothes were a political statement. Women who dressed like men threatened the entire structure of male domination. An eminent physician, reinforcing this belief, stated that women who wore men's clothes manifested an aggressiveness unbecoming to their sex. Earlier when Elizabeth Cady Stanton's cousin Elizabeth (Libby) Smith Miller first wore the "bloomers" that she had designed, woman's rights advocates, including Stanton herself, recommended this as an unconstricting, healthful costume. However, Stanton was so criticized for this mode of dress that after several months she abandoned it, feeling that women were spending too much time defending their attire and not enough on the issues.
During the war, women such as Mary Livermore and other members of the Woman's Sanitary Commission wore bloomers as they nursed the wounded and waded through the muck and gore of the battlefield. But with the war over, women were pressured into wearing whalebone fashions that squeezed their waists so tightly that they were often short of breath, their internal organs were displaced, and the pressure on their kidneys and bladder increased. Skirts used twenty yards of material, and bustles of iron wire often weighed as much as twenty pounds. Fashionable women went through life like hobbled horses.
The fashions of the day also reflected the current sexual standards. Pure, passionless wives were prized, but to procreate, a man must be sexually aroused, and therein lay the dilemma. Women solved this problem by concealing themselves from head to toe but with breasts upthrust and buttocks padded. Tennie C. and Victoria chose men's attire not only for publicity but also to protest the fact that women were kept in their place by clothes.
The sisters conspicuously flouted the accepted conventions of the day in other ways as well. It was an unwritten rule that women were not permitted in restaurants in the evening unless accompanied by a man. Delmonico's, at the corner of Broadway and Chambers Street, was among the grandest of these establishments. One night at seven, when Victoria and Tennie C. arrived at Delmonico's and ordered tomato soup for two, the startled waiter ran to fetch the owner. Lorenzo Delmonico leaned over the table and explained, with seeming solicitude, the error that had been made in seating the ladies: "We assumed a gentleman was joining you. Just pretend to be talking to me and I'll walk out the door with you. Then people will think you just came to speak to me. That will make it look all right."
"Make what look all right?" Victoria asked.
_Tennie C. Claflin, in men's clothing, c. 1870_
"I can't let you eat here without a man. It would start an awful precedent," Delmonico said nervously.
"Don't let us embarrass you," Tennie C. remarked, then stood and left the restaurant only to return a moment later with their coachman in scarlet coat and leather boots. She sat him uncomfortably between herself and Victoria. "Now, waiter," directed Victoria, "you may bring us tomato soup for three."
# CHAPTER SEVENTEEN
A HARD PLACE
BY THE fall of 1869, Victoria Woodhull was flourishing, but Elizabeth Cady Stanton was foundering. Her personal life had broken apart. With her father's legacy, Stanton bought a house in Tenafly, New Jersey. Her husband, Henry, remained in New York, concentrating on his newspaper career and the company of his political cronies. Except for the youngest, the children were away at college. Though Henry visited from time to time, the marriage was in effect over. To soothe her shattered ego, she joined the Lyceum circuit and traveled throughout the West. She told Anthony that her views on marriage and divorce were warmly received and that women flocked to her "with their sorrows."
As Stanton retired from the fray, Lyman Beecher's youngest daughter, Isabella Beecher Hooker, was making herself known within the woman's rights movement. Mrs. Hooker lived with her husband, John, at Nook Farm, a prosperous community just outside Hartford, Connecticut. Her distinguished neighbors included Isabella's half-sister, the author Harriet Beecher Stowe, and her husband, Calvin; the Charles Dudley Warners (Warner was the editor of the highly respected _Hartford Courant_ ); and new arrivals Samuel Clemens and his wife, the former Olivia Langdon. Livy was the girl who in 1860 had slipped on the ice and lay paralyzed until a Spiritualist healer cured her. Clemens, using the name Mark Twain, had just published _Innocents Abroad_ , whose great success launched him on his illustrious career. Of the inhabitants of Nook Farm, Isabella, with her Beecher assurance and preacher predilections, was clearly the social leader, and invitations to her Sunday night dinners were coveted.
For years Isabella had tried to "measure up" to the rest of the Beechers, especially Harriet and Catharine, and to her famous preacher brothers, especially Henry. She wondered when she would find something for herself, something that would bring her the recognition they had achieved. Belle, as she was called, was seventeen when she met John Hooker, a law clerk in her brother-in-law's office. When he proposed, she looked at the women around her, at their repeated pregnancies and domestic servitude, and observed that the role of wife might mark an end to her life before it began. "I do regret that so many fine women are made unhappy by firstly, being petted and spoiled by attention and admiration while young ladies and then learning all at once... they are to have _no_ will of their own except so far as it coincides with their husband's." After two years of vacillation, however, Isabella married John and fell into the life she had feared, isolated domesticity and quiet anger.
Of her four children, one died in infancy. In her diary she wrote, "I should enjoy reading [a scientific book] through, but it requires close attention and this it is almost impossible for me to give with Mary at my elbow and with my brain half asleep from want of fresh air and company and exercise." Isabella blamed her inadequacies on her unequal schooling. "At sixteen and a half, just when my brothers began their mental education, mine was finished.... Father, poor minister as he was, could send them to College and Seminary, all _six_ —cost what it might—but never a daughter cost him a hundred dollars a year, after she was sixteen."
In 1852, when she was thirty, Isabella visited Harriet, a few months after the publication of _Uncle Tom's Cabin_ , to find her sister celebrated throughout the land. If only Isabella too could become successful. "Oh my _soul_ —if you would only teach me how to earn money," she mused, "but there's no use in hoping—I can't write a book nor draw pictures—nor do any other productive work." To her children, Isabella was as stern a disciplinarian as her mother had been. Once, when eight-year-old Mary annoyed her, she talked to her "gently and persuasively," but when the child ignored her, Isabella flew into a rage, grabbed a long-handled brush, and savagely beat her daughter, who struggled violently. "She was frightened horribly and tho half choked" looked over at her little sister and shouted, " 'I want Alice to go out.' " Although Mary behaved after this beating, her mother was mortified at her own loss of control.
Undeniably, Isabella was a woman of passions, some uncontrollable. Later she admitted that her first feelings of power developed during a secret flirtation in the early years of her marriage. "You _can have_ no idea of the pleasure of being admired and loved after having been shut out from the world as I have been... and filled with care and anxiety and labor.... The fact is I was engaged so young that I had little time to know my power until after my destiny was sealed."
In 1860, Isabella wrote her first article for publication, a timid essay called "Shall Women Vote? A Matrimonial Dialogue." It concluded, "If absolute power were in my hands, I would not open the polls to women today—no, nor next year, nor ever, unless public opinion demanded it." She signed herself Mrs. John Hooker. But by 1868 forty-six-year-old Isabella, still energetic and unfocused, found herself at Nook Farm with a near-empty nest. Only her fourteen-year-old son, Ned, was still at home. It was then that she turned to the woman suffrage movement. In the November and December issues of _Putnam's Magazine_ , she published a second literary effort, printed anonymously, "A Mother's Letters to a Daughter on Woman Suffrage." She wrote that the woman's movement should address itself to prostitution, divorce, education, "and the whole system of insane asylums, poor houses, jails and many other institutions of modern civilization." Shortly thereafter she became president of the Connecticut Woman Suffrage Association.
In the summer of 1869, Isabella spent convention week in Newport with Paulina Wright Davis—a week that would change her life. There she met many members of the newly formed National Woman Suffrage Association, including Anthony and Stanton. For three days, Isabella literally sat at the feet of these two women and became convinced of their sincerity. She wrote, "Mrs. Stanton... is a noble woman, a magnificent woman... my prejudices against her... certainly go down under the influence of her presence and conversation.... I love her as well as I do Miss Anthony.... Sometimes she fails in judgment... but in right intentions never."
After leaving Newport, Isabella was summoned by Harriet, who wanted her editorial help. For Stowe, accustomed to adulation, criticism was something new, but then she had never before written about sex. In the _Atlantic Monthly_ , she had published "The True Story of Lady Byron's Life" in response to a recent biography of Byron that portrayed his wife as a harridan whose frigid nature had driven the poet to despair and death. Harriet, a friend of Lady Byron's, refuted this, writing that Byron had banished his wife and infant daughter for the sake of an incestuous relationship with his half-sister, Augusta Leigh.
Stowe's article was denounced by a public that refused to acknowledge the possibility of incest. The _Independent_ found the article "revolting, obscene garbage... barren in proof, inaccurate in dates, infelicitous in style and altogether ill advised." With Isabella's help, Stowe boldly turned her article into a book. It was ignored by the press, however, and became her only literary failure. Thereafter, she observed that public censure was something to be avoided.
While helping Harriet to edit her Lady Byron book, Isabella took it upon herself to organize a woman's rights convention to take place in Hartford. Her husband, John, sisters Harriet and Catharine, and brother Henry agreed to attend. Imperiously, she advised the New York delegation on "dress, manners, and... all the Christian graces." She instructed them to moderate their militant rhetoric in the cause of unity. Stanton replied in a letter to Hooker that the instructions amused her, but she indicated that she was annoyed at Isabella's officiousness. She promised to wear a dress "worthy of a place in the opera house of Hartford... give my hair an extra curl and wreath my face in its sweetest smile," but she added that when she had "demanded my right of suffrage twenty-five years ago, all my friends were shocked and grieved.... Your anxiety reminds me of the unhappy hen that butchered a bunch of ducks that danced in the waves and flapped their wings, and seemed to enjoy an element where the good hen feared to go."
_Isabella Beecher Hooker, the half sister of Henry Ward, came late to the woman suffrage movement_.
_John Hooker, Isabella's husband, reluctantly supported his wife's views_.
_Paulina Wright Davis. She taught women to understand their own bodies_. (Illustration Credit 17.3)
Stanton accommodated Mrs. Hooker, against her better judgment but the Hartford convention did little to unify the New England and New York groups. One month later Stanton wrote,
I did my best to obey orders, and appeared in a black velvet dress with real lace and the most inoffensive speech I could produce. All those passages that would shock the most conservative were ruled out, while pathetic and aesthetic passages were substituted in their place. From what my friends said, I believe I succeeded in charming everyone but myself and Susan who said it was the weakest speech I ever made. I told her that was what it was intended to be.
Lucy Stone, who led the New England delegation at the Hartford convention, was under great stress. Newly settled in Boston, she was trying to cope with her husband's affair with Abby Patton while simultaneously raising money for a newspaper. In October 1869, Emily Blackwell reported that her brother was once again with Mrs. Patton and that he "injuriously and foolishly keeps up Lucy's discomfort and distress by refusing utterly to terminate" the affair. Stone also brooded that Stanton and Anthony had formed the National Woman Suffrage Association without her knowledge. "Underhanded," she called it, and concluded that the solution was to turn the New England society into a national association to be called the American Woman Suffrage Association. Henry Ward Beecher agreed to be president, and the first AWSA convention was scheduled for late November 1869, in Cleveland.
Lucy Stone rationalized that the new organization was based on different convictions, not rivalry, and mailed a copy of the "call" to Anthony at _The Revolution_ with a cordial note: "I _hope_ that you will see it as I do that with two societies... we shall secure the hearty active cooperation of _all_ the friends of the cause better than either could do alone.... So far as I have influence this society shall never be an enemy or antagonist of yours in any way." But when Stone heard that Stanton and Anthony had been invited to the AWSA Cleveland convention, she wrote a friend that she hoped they would stay away, since it would be "so dreadful an incubus to take them up again!" And Henry Blackwell made a special trip to New York to ask Theodore Tilton to dissuade them from attending.
When Stanton heard of this, she responded by writing in _The Revolution_ that the Boston group were "hypocrites" and that "the names of persons are appended [to the call] who have been sedulously and malignantly working for two years to undermine certain officers in the National Association." Stanton refused to attend the convention, but Anthony, putting personal feelings aside, went, hoping to gain support for a Sixteenth Amendment granting the vote to women. She was not invited to sit on the podium and was appalled to find that the delegates, rather than supporting a Sixteenth Amendment, merely criticized the direction in which the assertive National Woman Suffrage Association was leading the movement and attacked Stanton's views on marriage and divorce. Finally, Anthony felt impelled to speak:
So help me, Heaven! I care not what may come out of this convention, so that this great cause shall go forward to its consummation! And though this convention by its action shall nullify the [NWSA] of which I am a member, and though it shall tread its heel upon _The Revolution_ , to carry on which I have struggled as never mortal woman or mortal man struggled for any cause... still, if you will do the work in Washington so that this [Sixteenth] Amendment will be proposed, and will go with me to the several legislatures and _compel_ them to adopt it, I will thank God for this convention as long as I have the breath of life.
On the day after the Cleveland convention, Horace Greeley attacked Stanton and Anthony's NWSA, writing in the _Tribune_ that it mistook "rashness for courage, folly for smartness, cunning for sagacity, badinage for wit, unscrupulousness for fidelity, extravagance for devotion, effrontery for heroism, lunacy for genius, or an incongruous mélange for a simple palatable dish." But an even larger controversy had arisen in New York, one that was about to rip asunder the last shred of unity in the woman's movement.
_Lucy Stone—the eloquent New England woman's rights champion_
_Henry Blackwell—Lucy's straying husband, who blamed his sexual infidelity on his wife's career_ (Illustration Credit 17.3)
AT 5:30 IN the afternoon on Wednesday, November 25, 1869, the famous journalist Albert Richardson entered the _Tribune_ offices, strode across the newsroom, and stopped to read his mail. From behind a pillar Daniel McFarland took aim and fired. The ball, discharged from scarcely three feet away, entered Richardson's abdomen an inch below the navel. McFarland took off at a run, turning up Printing House Square and running down William Street. He was apprehended a few hours later in his room at the Westmoreland Hotel.
Albert Richardson had returned from the war a broken man. The horrors he had seen as correspondent for Greeley's _Tribune_ had been followed by two years in the hands of the rebels in Salisbury prison, where he learned that his wife and child had died of cholera. Devastated, he returned to the _Tribune_ and wrote a successful book on the generals of the Union army but until he met Abby Sage McFarland he felt that life was merely to be endured until death came as the rescuer.
Abby Sage had been living in Madison, Wisconsin, when she met Daniel McFarland at a reading she was giving for the benefit of the local hospital. She was seventeen when she married McFarland, who, impressed by her sweet voice, trained her to be an actress. Convinced that Abby could support him, he sold their furnishings, pawned her jewelry, and moved to New York City. When Abby could not find employment, McFarland began to drink heavily. He beat his wife when she became pregnant, then abandoned her. Abby's parents, asking no questions, welcomed her back to their small farm in Manchester, Vermont. Eighteen months later McFarland begged for a reconciliation. Abby returned, but so did Daniel's drinking and abuse. Abby was later to say,
He would lock himself in the room with me and give way to such terrible furies that only extreme pride and self-control prevented me from making my misery known. He brought home what he professed was prussic acid and threatened to force me to take it. He would snatch my scissors from my workbasket and bearing open his breast he would brandish them about, swearing he would let out his heart's blood before me. He told me that he kept loaded pistols with which he would at any moment shoot me.
In the winter of 1866, Lucia P. Calhoun, the _Tribune's_ society editor, for whom Abby had written a few theater notes, told two friends of her impoverished state. The first, Horace Greeley's sister, Mrs. Sinclair, befriended Abby and gave her money to buy clothes for herself and her son, Percy. The second, Albert Richardson, also tried to help her by prevailing upon Greeley to recommend Daniel McFarland for a patronage job at the Custom House. McFarland, often too drunk to perform his duties, was fired. Richardson intervened again and secured a position for him with the Internal Revenue Service.
One morning Abby arrived at the _Tribune_ offices with her face swollen and a black eye concealed with makeup. Richardson took her to lunch, and she told him that her husband had savagely beaten her over the years but there was nowhere to turn, no one to help. The next day Richardson rented a room next to the McFarlands' at a boardinghouse at 72 Amity Street. He pushed his bed against the wall between his bedroom and theirs and vowed that if he heard one blow or one cry, he would enter the room and rescue Abby. Although he had never spoken of it, he had fallen in love with her.
At 11:30 p.m. on Wednesday, March 14, 1867, Albert met Abby at the stage door of the Winter Garden theater, where she was playing the small role of Nerissa in _The Merchant of Venice_. As they walked arm in arm, Daniel McFarland emerged from a dark side street carrying a fourteen-inch four-barreled revolver. Richardson felt the gun at his back and instinctively spun round, striking McFarland. The gun discharged three times, wounding Richardson in the hip, but he held McFarland until two policemen arrested him. In the presence of witnesses, Daniel McFarland agreed to a separation. Some three weeks later, however, he intercepted a letter from Richardson in which he addressed Abby as "My Dearest Wife" and said that, should she become legally free, he wanted her to be just that. He declared his love and wrote that he would do whatever he could to make her life rich and beautiful.
With money from Richardson, Abby went to Indiana to establish the year's residency required for a divorce. (Indiana had the most liberal divorce laws in the country except for Wisconsin, where incompatibility was grounds for divorce.) She returned to New York the week of November 20, 1869. Daniel McFarland learned that Richardson had purchased a farm in New Jersey and after his marriage expected to live there with Abby. Live there with _his property!_ Five days later, McFarland shot Richardson.
When he was shot, Richardson slowly pulled himself up the narrow stairs of the _Tribune_ to the office of Whitelaw Reid, where he collapsed into a chair and announced, "I am in need of a surgeon." He was so calm that Reid didn't realize that Richardson was wounded, but when he saw blood seeping through his waistcoat, Reid quickly summoned a doctor. Richardson was carried to the Astor House. He asked the doctor, "Do you think I will live?" The doctor answered, "If you live forty-eight hours you will be out of danger, but your life is in extreme peril."
When it became increasingly obvious that Richardson would die, as a last wish he requested that Henry Ward Beecher marry him to Abby. The preacher, caught up in the ecstasy of his own relationship with Lib Tilton, was only too happy to comply. He performed the ceremony with Horace Greeley as witness. Richardson died the following morning.
Almost immediately the news circulated that Abby was still legally married to Daniel McFarland—an Indiana divorce was not recognized in New York. Beecher was severely criticized for performing the wedding ceremony. The press took up the attack. William A. Bartlett, a prominent lawyer, wrote in the _Sun_ ,
Consider, married men of New York! Husbands and fathers! By what frail and brittle tenure your homes... your wife... are yours.... Reverend Henry Ward Beecher stands ready to marry her to the first libertine... who comes boldly and even proudly forward, holding by the hand and leading Lust to her in triumph over religion.
_The deathbed marriage of Abby Sage McFarland and Albert Richardson. The Reverend Henry Ward Beecher officiates_. (Illustration Credit 17.1)
Members of the clergy also joined in the criticism of Beecher. The Reverend George B. Cheever wrote, "The tendency of these times towards excessive self-enjoyment and loose views of religion and the marital bonds led to this tragedy as a natural result." The Reverend W. A. Scott added, "The deathbed marriage was a blasphemous act and was more sensational than the tragedy itself."
Henry Ward Beecher found himself more and more threatened. In the _Sun_ , he defended himself: "I took every statement of every kind respecting the affair... without time to investigate. The man was dying. Was that a time for sifting evidence?"
Unlike Beecher, members of the National Woman Suffrage Association did not run for cover. To a crowded audience at Steinway Hall, Anna Dickinson said of Richardson, "Once in a while God blesses the world with a great lover. This man was such a one." When she heard hisses from the gallery, she addressed the men in the audience directly: "Love signifies companionship, friendship, understanding, some similarity of sentiment, of pursuits, of interests.... I stand here to warn you, sirs, that the woman of the present is not the woman of the past, that she will mete to you as you mete to her."
Stanton's response was still more radical. On behalf of the NWSA, she called for a revision of the divorce laws and wrote,
Marriage today is in no way viewed as an equal partnership, intended for the equal advantage and happiness of both parties. Nearly every man feels that his wife is his property, whose first duty, under all circumstances, is to gratify his passions, without the least reference to her own health and happiness, or the welfare of their offspring; and so enfeebled is woman's judgment and moral sense from long abuse that she believes so too and quotes from the Bible to prove her own degradation.
Lucy Stone alleged that members of the NWSA, through their stand on the McFarland-Richardson case, wished to undermine the laws of marriage. She accused them of being infiltrated with "loose women" and "free lovers" who intended to make "easy divorce" a part of their platform. She appealed to Beecher, as president of the AWSA, to persuade the women of Plymouth Church to join her organization, not Stanton's. Perhaps in her effort to preserve her own precarious marriage Stone feared anyone who would make the laws of divorce more lenient. Lucy was trying desperately to keep her husband. Stoic but anguished, she begged her sister-in-law Emily to influence her brother to give up Mrs. Patton. "It is not good for him to... take up the old snare," she wrote.
On January 8, 1870, the second anniversary of _The Revolution_ , the first issue of Stone's _Woman's Journal_ appeared. The pointedness of the date was, of course, intentional. _Woman's Journal_ , an immediate success, was backed by six prominent Boston businessmen who invested $10,000 each. Mary Livermore, a powerful adversary of the NWSA, was appointed editor. Harriet Beecher Stowe, Catharine Beecher, and Henry Ward Beecher became contributors.
Susan Howard—the wife of Beecher's friend and benefactor John Tasker Howard, and the mother of _Sun_ editor Joe Howard—was perplexed by the controversy. She wrote Isabella Beecher Hooker that Lucy Stone had accused members of the NWSA, including Mrs. Stanton, "of holding free love doctrines." Isabella replied that she could not understand why Lucy was now making such a fuss over Stanton's views, since "it is certain that Mrs. S. entertains no new views, none that Mrs. Lucy has not heard from her years ago." Isabella explained that Stanton "has come to look upon easy divorce as a blessing and a necessity. But to say that she ever advocated this as a means of _personal gratification_ to woman or man is simply to insult her."
To reassure Mrs. Howard that the goal of the NWSA was "purely suffrage," Isabella enclosed a copy of its constitution but then, in an odd gesture, checked off the names of the women who might be pointed out by adversaries as "loose women or free lovers." Isabella wrote that Lucy Stone had made the same free love accusations to her brother Henry and ended her letter with an appeal to her friend to think for herself.
Susie, you are in a hard place and I am not urging you to try to defend anybody.... I only wish my dear brother [Henry] could be persuaded to hear some of these things and not because I wish to wean him from his new alliance or friendships but because he could be much more useful to them and to the whole cause by understanding the countercurrents... if he does not allow himself to be warped and set against other workers.... Oh Susie, _few men_ know what this battle means—but many women, wives and mothers, know and feel it all, feel it for their sisters if not for themselves. I am one of these and I stagger under the weight of my load.
Evidently, Susie Howard was not allowed the privilege of thinking for herself. Isabella's letter was answered by Mrs. Howard's son Joe, who wrote not to Isabella but to her husband, John Hooker, stating that Lucy Stone's allegations were widely accepted in Brooklyn. John Hooker, who had unwillingly been dragged into the controversy, replied of Mrs. Stanton, "From what little I have seen of her I do not fancy her at all.... She is a bold and strong thinker but she has not a refined nature. She is however a pure woman, and seeking to promote purity in others. There is not a particle of the sensuality that we generally regard as incident to 'free-loveism' about her. Her notions are very far from pleasing me, but they are notions, not feelings, a groping mind, not a wantoning heart."
Lucy Stone and her compatriots could level their accusations at Mrs. Stanton because the doctrine of free love and woman's rights had much in common: The concept of equal relations between the sexes was identical and led to the fight for many reforms, including the vote and the liberalization of marriage and divorce laws. But free love was abhorrent to many women as well as men because it openly acknowledged equal female sexuality. Free love presupposed female sexual satisfaction, which was scorned as "personal gratification" and therefore "vile." It granted women the right to refuse sexual intercourse, freeing them from male sexual coercion. In this it violated both legal strictures and the teachings of the church, which mandated sexual submission. Also, while free love did not advocate promiscuity, its tenets were liberal enough to encompass flagrant sexual behavior that shocked many open-minded women. It was one thing to believe in the theory of free love and quite another to practice it.
# CHAPTER EIGHTEEN
THE EVANGEL
RICH AT LAST! Scarcely four months after Black Friday, in January 1870, Victoria and Tennie C. leased an elegant brownstone at 15 East Thirty-eighth Street, just off Fifth Avenue in Murray Hill. After a lifetime of wandering, of rented dwellings and few personal possessions, the sisters could finally achieve their fantasy of success and luxury. The next weeks were filled with shopping sprees for Oriental rugs, oil paintings, and Venetian glass chandeliers. The downstairs rooms were hung throughout with purple velvet. And everywhere there were mirrors, even on the ceiling of the mahogany-paneled library. In the main salon, incense burners hung from the gilded ceiling. According to the _Sun_ , this room featured a "wondrous dome... a flood of light beaming through a circular sheet of glass painted in the most exquisite colors [depicting] the loves of Venus in delicate lines." Victoria's bedroom was lined in green velvet with a matching green velvet and gold fringed bedspread and gilt chairs, while Tennie C.'s bedroom was draped in deep purple velvet and lilac-patterned silk imported from France. The _Star_ called their house a "Modern Palace Beautiful."
Even before the house was furnished fifteen family members had moved in. In addition to Victoria, Tennie C., and Colonel Blood, there was Victoria's son, Byron, a gangly sixteen-year-old with the mind of an infant, and her devoted nine-year-old daughter, Zulu Maud. Buck and Roxy arrived, followed by Utica, Margaret Ann and her three children, and Polly and her daughter, Rosa Burns, as well as Polly's new husband, Dr. Sparr, a magnetic healer. Many afternoons Roxy and Utica raided Tennie C.'s and Victoria's closets, playing dress-up like two children; then they would sit in the parlor drinking together. When there was no other alcohol Utica drank drugstore bay rum. Mama Roxy took to pawning Tennie C.'s jewelry, which led to screaming arguments ending with Mama's tears and promises to reform. Buck charged a new wardrobe to his daughters and, decked out like a proper Wall Street broker, began visiting the offices of Woodhull, Claflin & Co., offering advice to Colonel Blood. When a runner arrived with $3,000 worth of bearer bonds, Buck cashed them and pocketed the proceeds.
One evening there was a knock on the door, and on the doorstep stood Dr. Canning Woodhull, stick-thin and shaking like a wet spaniel from delirium tremens. "Oh, Doc, you poor fellow!" exclaimed Victoria. Dr. Woodhull joined the Claflin ménage and, when he was not fogged with morphine and alcohol, helped look after his son, Byron. Victoria told a _Herald_ reporter that with her staff of twelve and all her family, "it costs us over twenty-five thousand dollars a month to live."
At this rate, within a short time Victoria's funds would be exhausted, but for the moment she was free of debt and ready to make her mark. Later she claimed that in February, Demosthenes came to her and told her, "Your work is about to begin." That work, she said, was to lead a domestic and social revolution to rectify the injustices toward women. Although the transformation from self-aggrandizement to selflessness would take some time, it had begun. Colonel Blood, who had read Karl Marx and believed in a world free of cruelty and corruption, was undoubtedly a major influence. The greatest influence on Victoria Woodhull, however, was the "New Age" philosopher Stephen Pearl Andrews, who sought her out the month she moved into 15 East Thirty-eighth Street.
With his flowing gray hair and beard and his abstracted air, Andrews cut a striking figure. For the past four years he had lived with his second wife, Esther, a Spiritualist, magnetic healer, and trance speaker. Their boarding-house on the corner of Fourteenth Street near the Academy of Music had become the center for a group of radical Spiritualists who believed in his vision of a utopian society, which he called the Pantarchy. Andrews, as leader, called himself the Pantarch. He declared that the Pantarchy would "replace the corrupt system that was evolving to afford plunderers' profits to the few, while demeaning the dignity of labor" and predicted "a grand domestic revolution" whose keystone was his long-held belief in free love.
Andrews's problem was money. He and his group were just scraping along until he met Victoria Woodhull. So impressed was Victoria with his ideas, which coincided with some of her own practices, that she immediately backed him. Her spacious parlors were put at Andrews's disposal for meeting rooms and he filled them with members of the Pantarchy and other radical Spiritualists. His wife had just completed a course at the Eclectic Medical College of the City of New York, "for the free and untrammeled investigation and practice of medicine" and had become an expert at herbal healing as well as magnetic healing. Esther Andrews and Victoria became friends at once. In a trance, Esther reportedly suggested successful cures for maladies that had eluded doctors. To cure Colonel Blood's arthritis, she warned him to avoid nightshade vegetables. "Abandon your carriage and walk.... Eat what flies or grows above the earth, and you will improve," she instructed. At the séance table, Esther warned Victoria that a spirit had said, "Beware the Judas kiss," and told her that she would be betrayed by a member of her own family.
Despite the presence of her crude kin, Victoria soon established herself as a notable hostess, and her house attracted not only members of the Pantarchy but also advanced thinkers and celebrities of the day, including the famous Spiritualist Laura De Force Gordon, a trance medium and defender of woman's rights who was greatly admired by Susan B. Anthony. Among Victoria's other guests was Elizabeth Stuart Phelps, author of the Spiritualist book _The Gates Ajar_ , in which she assured her readers that the living would be reunited with the dead once one passed through "the gates" separating this world from heaven. Reflecting one of the main appeals of Spiritualism, she maintained that she had written _The Gates Ajar_ to comfort "the bereaved wife, mother, sister, and widowed girl." Though vigorously attacked by the clergy as "heresy," it sold eighty-one thousand copies in the United States and one hundred thousand copies abroad in less than a year.
Victoria also attracted powerful business, political, and military figures, mostly of a freethinking kind, among them Benjamin Butler, who was intrigued by the beautiful Victoria and her unconventional entourage. When Butler first met Victoria in 1870, his wife, Sarah, was in Germany, seeking a cure for her thyroid cancer. The rumor was afloat that Anna Dickinson would not marry Whitelaw Reid, as predicted, but would marry Butler as soon as Sarah died. Butler was indeed pursuing Dickinson and writing her passionate love letters while she herself seemed to be making plans for married life with a secret lover. Anna announced that this would be her last lecture season and that a personal involvement would cause her to retire from public life. The _Cincinnati Enquirer_ noted, "The name of the gentleman was not ascertained nor when it is to come off." Reid enclosed the clipping and wrote defensively, "Are my congratulations in order? And shall I send congratulation or condolence to the gentleman from Boston? Have you seen the latest about _me_? My physicians have forbidden me to marry and so I am making the Tribune my bride!" Whitelaw Reid's and Dickinson's biographers both assumed Butler was "the gentleman from Boston," but as future events were to reveal, they were mistaken as to the identity of Anna's secret lover.
In any case, by the summer of 1870, Butler had become a frequent visitor at Woodhull's house, and their relationship was close. Victoria spoke of Butler's intimate habits, his compulsive snacks of doughnuts washed down with whiskey, the way his unlit cigar traveled round and round in his mouth when he tried to stop smoking. Whether Butler's relationship with Victoria was sexual as well as political is unclear, although she readily admitted that she visited Butler "at night" to ask him if he would open the Judiciary Committee's mind to a Sixteenth Amendment to give women the vote. There was even a published rumor that Butler had offered to help Victoria Woodhull in the cause of woman suffrage if she would allow him "an opportunity to feast his eyes upon her naked person." When Butler was confronted with this report he remarked coyly, "Half-truths kill."
Benjamin Butler was a strong supporter of woman's rights and was known to encourage controversy in Congress. In Victoria Woodhull he undoubtedly saw a way to further his ideas. Thus far Woodhull had limited knowledge about either of the woman's rights organizations and was unaware of the split between the New York and Boston groups or of the opprobrium heaped on Henry Ward Beecher for the deathbed marriage of Albert Richardson and Abby Sage McFarland, but now she was rich and longed for a mission. The canny Butler decided to exploit this ambition, and he was supported by Andrews and Blood, who also saw the political use to which this beautiful, charismatic woman could be put. While Stanton and Stone were squabbling, and Henry Ward Beecher was defending himself against accusations of free love, Victoria Woodhull, propelled by the men behind her, boldly asserted herself. In April 1870, a notice was placed in the _Herald:_
While others of my sex devoted themselves to a crusade against the laws that shackle the women of the country, I asserted my individual independence.... While others sought to show that there was no valid reason why woman should be treated... as a being inferior to man, I boldly entered the arena of politics and business and exercised the rights I already possessed. I therefore claim the right to speak for the unenfranchised woman of the country and... I now announce myself as a candidate for the Presidency.
Since the presidential election was still two years away, Victoria's announcement belonged more to the world of publicity than to politics. But Butler, Blood, and Andrews even then were building a constituency and a platform for their ideas of social reform. They made their first substantial move the following month, on May 14, 1870, with the first issue of _Woodhull & Claflin's Weekly_. Victoria Woodhull and Tennie C. Claflin were listed as editors and proprietors and J. H. Blood as managing editor. Ten cents bought a single copy and four dollars a year's subscription. The first issue contained a mélange of radical theorizing, fashion, sports, a serial by the female author Amandine Dupin Dudevant (who had adopted the nom de plume George Sand), and several columns on Spiritualism, trance speakers, and magnetic healing, prompting one observer to say that the journal was "edited in one world and published in another." The _Weekly_ was as eccentric and eclectic as Victoria herself, but its purpose was straightforward: to promote Victoria Woodhull as a leading advocate of the political reforms of the day.
_The first page of the_ Weekly _endorsing Victoria's bid for the presidency_ (Illustration Credit 18.2)
Victoria and Tennie C. used their celebrity status and the interest in Spiritualism to draw attention to themselves. In the third issue, Victoria published the first of a series of political writings under her own signature. She explained that they were written on the two or three evenings a week when, in a trance state, she sat either in her upstairs parlor in winter or on the roof in summer. As Victoria spoke, Colonel Blood, sitting nearby with pen and paper, took down her words. Then a few hours later, when she emerged from her trance, Blood would read back what she had said, and these thoughts would become her articles. Victoria declared that she did not recall what she had said in her trances.
It is unlikely that the intricate political arguments and legal knowledge that marked these articles originated with her. They echo the writings of Andrews, Blood, and Butler. Interspersed with these technical arguments, however, were illustrations drawn from Victoria's own life. What Victoria Woodhull declared in her various trances, what was her personal inspiration, and what was written for her, no one can say. Some said Victoria was the instrument of powerful men; she herself contended she was the instrument of the spirits. While others no doubt used her as a flamboyant parrot, she herself came increasingly to believe that the spirits had chosen her to become "the ruler of the whole world." And she went still further to declare, "I am the evangel."
AT THE DISASTROUS American Equal Rights Association convention in May 1869, when the woman's movement split apart, Elizabeth Cady Stanton ended her final speech by declaring, "The need of this hour is not territory, gold mines, railroads or specie payments, but a new evangel of womanhood, to exalt power, virtue, morality, true religion, to lift man up into the higher realms of love, purity and thought." But as yet she had no idea where that "evangel of womanhood" might be, and meanwhile her organization was in desperate trouble. The delegates who were about to meet at the National Woman Suffrage Association convention in Washington in January 1870 were a considerably weakened group. Lucy Stone's rival American Woman Suffrage Association had attracted many powerful women and men, including Henry Ward Beecher, Wendell Phillips, and William Lloyd Garrison. Stanton and Anthony's ranks, on the other hand, were filled with inexperienced young women venturing out on their own for the first time. Most had to "beg at the pockets" of husbands or fathers for the money to go to Washington to protest male domination. The NWSA delegation was received by members of Congress in a committee room of the Capitol. The most prominent women sat at a table in the center while behind them the new members settled into their seats like a covey of doves. The world of politics was strange and incomprehensible to most of them. Instead, there was much talk of dress and proper deportment, intermingled with a giddy feeling of escape. Amid this chatter, Stanton called the meeting to order.
The portly Reverend Samuel May opened with a prayer and then called for every woman in the room who desired the ballot to raise her hand. Only a few timid hands rose, accompanied by much giggling. Stanton was astounded by the reaction of her raw recruits. "The Reverend May has adopted a very bad manner in submitting the question," she said. Then Stanton directed any woman opposed to a Sixteenth Amendment to enfranchise women to stand. "Those in favor are to remain seated," she said emphatically. To Stanton's dismay, as the laughter continued, an overwhelming majority of women from her delegation rose to their feet.
_William Lloyd Garrison with his daughter Fanny. The great abolitionist favored the vote for black men before women_. (Illustration Credit 18.1)
Stanton and Anthony were paying a heavy price for their opposition to enfranchising black men unless women also got the vote. At an antislavery festival in New York two weeks later, Stanton approached Wendell Phillips, but he turned on her "like a shark with sarcasm and ridicule." Stanton took this as another demonstration of male oppression and wrote,
To have Wendell Phillips withdraw his velvet paw as if one were unworthy to touch the hem of his garment—this is enough to rouse one's blood to the white heat of rebellion against every "white male" on the continent. When I think of all the wrongs that have been heaped upon womankind, I am ashamed that I am not forever in a condition of chronic wrath, stark mad, skin and bone, my eyes a fountain of tears, my lips overflowing with curses, and my hand against every man and brother! Ah, how I do repent me of the male faces I have washed, the mittens I have knit, the trousers mended, the cut fingers and broken toes I have bound up!
It was in this atmosphere of animosity that Daniel McFarland was brought to trial for the murder of Albert Richardson. He sat in the courtroom in a tattered black frock coat, conversing earnestly with the palm of his hand, held in front of his face. The all-male jury found McFarland not guilty by virtue of insanity. But the "insane" Daniel McFarland—not Abby, the child's mother—was granted sole custody of their son, Percy. He walked from the courtroom holding the weeping boy's hand.
On May 17, 1870, the National Woman Suffrage Association organized a protest against the McFarland verdict, to be attended by women only. Three thousand assembled at Apollo Hall, where Stanton argued that "the husband's right of property in his wife" was the central problem in marriage. She said the jury had found McFarland not guilty because "neither woman nor slave can testify against their supposed masters." By declaring Abby's divorce illegal, the court permitted any "bloated drunkard or diseased libertine" to possess and coerce a woman sexually. It was time therefore for "an entire revision of the laws of New York on marriage and divorce." Marriage as it existed, said Stanton, was "nothing more or less than legalized prostitution.... I rejoice over every slave that escapes from a discordant marriage."
On the same day as the NWSA protest, the opposite view of marriage and divorce appeared in the _World_ and the _Tribune_ written by the mighty Catharine Beecher, the domestic high priestess of family life. Addressing the case of Abby McFarland, she wrote, "A woman may separate from her husband for abuse or drunkenness and not violate law, but neither party can marry again without practically saying, 'I do not recognize Jesus Christ as the true teacher of morals and religion.' " In this she represented the popular view that marriage was a sacred institution, entered into for life, no matter what the consequences. In fact, remarriage for widows was widely considered immoral as well. Even Anthony herself said, "I hate the whole doctrine of 'variety,' of 'promiscuity.' I am not even a believer in second marriages after one of the parties is dead, so sacred and binding do I consider the marriage relation." But Anthony was able to separate her beliefs from what she felt was the greater good of womankind.
Lib Tilton was distressed by the friction between the NWSA, in which her husband was so active, and the rival AWSA, headed by her beloved preacher. As she saw her friends at Plymouth Church joining Beecher's organization she asked her husband to intervene and unite the two associations, saying that the chance for enfranchisement would fall by the wayside unless they could reconcile their differences. Theodore agreed and advised Anthony that, in spite of all that had happened, reconciliation would be a wise course of action. "No, no," Anthony replied, "self-respect, self-justice forbid it. I stand plaintiff, not defendant, before that woman [Lucy Stone] and the Boston clique." Nonetheless, Tilton determined to try for an amalgamation of the two organizations. Lucy Stone was as opposed as Susan B. Anthony, but she agreed to a meeting. When they met and Tilton failed to persuade Stone to unite with her rivals, he lost his temper and threatened, "I will form a new society.... I can carry it, for I have a paper, I have the platform, and I have a pen."
The parent American Equal Rights Association, whose membership was now split between the rival organizations, had virtually ceased to function, but some branches still had not affiliated with either of the two new organizations. In a clever but deceptive maneuver, Tilton plotted with Stanton to change the name of the NWSA to the Union Society and thus absorb as many branches of the old AERA as possible. Stanton admitted that she had "grand times" doing this but wrote, "Boston is _awful_ sore." Having grabbed off whatever AERA membership it could, the Union Society then dissolved and resumed its old name. Lucy Stone's group called this action "unconstitutional" and retaliated by announcing that the AWSA would hold its 1870 convention in New York City. The abyss yawned wide. There was no way left to bridge it.
And so, as the NWSA held its convention at Apollo Hall in New York early in May, the AWSA held a simultaneous meeting in the same city at Steinway Hall. The New England visitors attracted large audiences and a favorable press; the New York group was ill attended and largely ignored. The NWSA meeting had barely convened on the second day when a resolution was put forth that, since the AWSA had chosen Beecher as president, Stanton and Anthony should resign their offices and a "popular man" should lead them. The two women had no choice but to agree. Stanton, who blamed men for her subjection, yielded control to a man. Theodore Tilton was elected president. Henry Ward Beecher and Theodore Tilton were now presidents of rival organizations. Tilton wrote Beecher a congratulatory letter and requested "mutual co-operation in the cause of woman's enfranchisement." Beecher responded with a cordial but noncommittal note. Both letters were a sham.
_The Revolution_ too had lost ground to _Woman's Journal_ and was beset by financial problems. Anthony spent much of her time trying to raise money, but she was overwhelmed. She hoped that Anna Dickinson, whose huge lecture fees had made her rich, would take over the editorship of _The Revolution_. "If Anna Dickinson will be sole editor, I say Glory to God!.... Tell our glorious little Anna if she only will nail her colors to that mast and make the dear old proprietor free once more, I will sing her praises till the end of time." But Anna, on the road lecturing with Wendell Phillips, declined. Finally, Theodore Tilton arranged for _The Revolution_ to be sold for a token payment of $1 to the heiress Laura Curtis Bullard, with whom he had carried on a long-term love affair and who remained completely under his spell. Anthony wrote in her diary that giving up her paper "was like signing my own death-warrant" and added, "I feel a great calm sadness like that of a mother binding out a dear child that she could not support." Laura Curtis Bullard inaugurated her editorship with a lavish reception at the Brooklyn house she had purchased to be near Tilton. And Tilton wrote to Anna Dickinson, "The Revolution is (as perhaps you have heard) in what is equivalent to my own hands, 'To have and to hold.' "
Although Stanton railed against the AWSA, calling its members "sick unto death with propriety," and Anthony declared, "What an iceberg is that Boston, God help them to their live senses—nothing human can," there was no doubt that the AWSA was triumphant. It now signed up men and women in twenty-two states and territories. In the first six months of 1870, thirteen state suffrage associations affiliated themselves with the AWSA, and only a branch in Wisconsin went to the NWSA.
William Lloyd Garrison went so far as to write that Stanton had become a "female demagogue" and that she and Anthony were both "untruthful, unscrupulous and selfishly ambitious." Their "old friends" had deserted them. Years later, Stanton was to write that only a few "stood firmly together under a steady fire of ridicule and reproach even from their lifelong friends... and most of the liberals in the press. The position of the women seemed so untenable to the majority that at times a sense of utter loneliness and desertion made the bravest doubt the possibility of maintaining the struggle of making themselves fairly understood." But Stanton had said what was needed in these desperate times was "a new evangel of womanhood" and, as if the spirits themselves had intervened, Victoria Woodhull had announced, "I am the evangel."
# CHAPTER NINETEEN
YOUR CHILD IS NOT MY CHILD
JULY 4, 1870: The wiry Bowen and the plump Beecher were encased to the waist in the burlap potato sack. "One, two, three—jump," shouted Beecher as the two former adversaries hopped across the Woodstock Common. The crowd cheered as the mismatched pair crossed the chalk line to win the sack race. John Tasker Howard ran up to hug both men. These new allies were warmed by the sun, by the adulation of the crowd, and by the presence of their illustrious guest, President Ulysses S. Grant—in short, by their burgeoning power.
EVER SINCE HENRY Bowen had tried to drive Henry Ward Beecher from Plymouth Church four years earlier and Beecher had retaliated by cutting off Bowen's political patronage, they had been bitter enemies. Though Bowen missed no opportunity to slander Beecher, he had done little to hurt him. Six months before this Independence Day celebration, Bowen had tried to strengthen his hand by buying control of the _Brooklyn Daily Union_ , a small-circulation Republican newspaper, and appointing Tilton editor. With this paper added to the popular _Independent_ , Bowen hoped not only to influence the people of Brooklyn, but to enrich himself further by supporting candidates who could repay him with patronage. But in a surprise move Samuel Wilkeson, on behalf of Jay Cooke, teamed up with John Tasker Howard, and purchased a newspaper for Henry Ward Beecher. The paper, named the _Church Union_ , was promptly renamed the _Christian Union_ to broaden its scope. Henry enlisted his famous sisters, Harriet and Catharine, as well as his brothers, Edward and William, to write for his paper and began successfully competing with Bowen and Tilton for subscribers.
Beecher was clearly regaining his power, and it was his Republican faction, not Bowen's, that was fast becoming Brooklyn's dominant political force. Until 1868, the federal government and the state legislature had controlled the patronage in the city of Brooklyn. The latter had passed a bill to transform Fourth Avenue into a boulevard and another to improve Third Street, work that had cost $600,000 in taxpayer money over the previous eight years and was still far from finished. In the election of 1868, by campaigning against the corrupt state legislature, the Democrats won every office by a vast majority—in fact, 110 percent of the registered voters. District Attorney Morris, a Democrat, was called in to conduct an investigation into illegal voting. Though few anticipated that he would find wrongdoing in his own party, Morris obtained sixty indictments. The Democratic Party then released papers showing that Morris himself was drawing the salary of a nonexistent clerk in his office. The indictments were dropped, but the people of Brooklyn had become disillusioned with Democratic rule. The Brooklyn Republicans won the next election by a wide margin, and Plymouth Church provided a powerful political base for them.
In the previous three years, a group of Republicans had joined Plymouth Church and were now so close to their preacher that they were known as "Beecher's boys." These were not religious parishioners but ambitious politicians. They had been able to join the church because Beecher had nullified the requirement that applicants must subscribe to the Articles of Faith. Newcomers merely had to express a "desire" to become a "Christian" for admission to his flock. Among this group was Benjamin Tracy, a pugnacious lawyer who was appointed United States district attorney in 1870, as a reward for his loyalty to the Republican machine. Tracy was well suited to replace Tilton as Beecher's acolyte. He believed that personal loyalty and friendship superseded all moral principles and declared, "Lying in defense of a good friend is justified." Tracy organized the Republican members of Plymouth Church into a tight, effective group.
While the Beecher-Tracy contingent was popular with the federal government, Bowen was not. The problem was Theodore Tilton. Bowen had promised Grant his full cooperation and yet, under Tilton's leadership, his newspapers had become hostile to the president. Tilton had praised Grant's inaugural address but soon began to sour on the new administration, attacking its corruption both in the _Independent_ and the _Brooklyn Daily Union_. He wrote that Grant dispensed his vast political patronage to "second rate people" and accepted money and favors in return. Tilton also censured the president for failing to bring peace to the South and for his tardiness in suppressing the activities of the Ku Klux Klan. He wrote that the Republican Party was indifferent to the Negro question. "If we do not educate the Negro into the fabric of our society, we will be paying the penalty for generations to come," Tilton warned. In December 1869, he even went so far as to recommend impeaching the Supreme Court for failing to uphold the Civil Rights Act.
Henry Bowen suffered on another front as well: Samuel Wilkeson was not only Henry Ward Beecher's friend and supporter but also the chief agent for Jay Cooke's Northern Pacific Railroad. In 1869, with the granting of the railroad's charter, 47 million acres of land had fallen into Cooke's hands. Wilkeson, sent on a reconnaissance trip to the West, had written Cooke, "There is no end to the possibilities of wealth here.... Jay, we have got the biggest thing on Earth. Our enterprise is an inexhaustible gold mine." With the generous government grants and subsidies, and the army taking care of the "Indian question," the Northern Pacific promised vast rewards.
Wilkeson understood that the merits of the great railroad must be sold to the American people for the government grants to continue. Cooke gave Wilkeson virtual carte blanche to bribe influential members of the press with free Northern Pacific stock. Beecher was given $15,000 worth of stock for the purpose of "influencing the public mind to favor the new railroad" in his newspaper, and Horace Greeley received $20,000 worth on the same premise. However, when Wilkeson approached Theodore Tilton, he flatly refused the gift, saying, "I am a poor man, but my principles are not for sale." When Bowen made an appointment to see Sam Wilkeson and asked that he too be given Northern Pacific stock, Wilkeson answered that Tilton had refused to promote Northern Pacific stock in the _Independent_ and had said he might even investigate whether the railroad bonds were worth the price they were selling for. Since Bowen could not control his own editor, said Wilkeson, how could he expect to be given stock?
Adding to his dilemma, Bowen had married a young woman who complained bitterly that her social life had been ruined because Henry Ward Beecher clearly avoided the Bowens and other church members had followed suit. Without the benevolence of Plymouth's pastor, the Bowens had become political and social pariahs.
It was then that Bowen saw a way to weaken Tilton's position. He volunteered to pay all of Tilton's expenses as long as he remained on the road, lecturing or writing for Bowen's newspapers. Then, with Tilton sidetracked, Bowen made a bold move to recapture the patronage that had once enriched him. He sent his nephew Henry A. Bowen, a major Grant supporter, to Washington to see the president and tell him that his uncle was now in complete control of both the _Independent_ and the _Brooklyn Daily Union_ , and that Theodore Tilton was merely an employee. Young Bowen told the president that both newspapers would fully support him, and he invited Grant to spend the Fourth of July weekend at his uncle's estate in Woodstock, New York. Grant accepted.
Once this was accomplished, Bowen sent a message to Beecher and his backers, Sam Wilkeson and John Howard, saying that they were all men of influence and adding that the president was shortly to arrive as his guest. If they could be reunited, he would be pleased to invite them as guests as well. A meeting followed, and when Bowen entered, he walked over to Beecher and embraced him, saying, "My friend, the time has come for us to forget all our differences." "Yes," Beecher replied, his eyes misting with tears, "we must be friends. There must be no break between us. It would kill me." Bowen then declared that if the breach caused by Tilton could be mended, together they might control the entire Republican Party in Brooklyn. After the meeting Bowen, Beecher, and Howard walked the streets of Brooklyn, their arms linked as they recalled how twenty-three years before they had conceived the blessed Plymouth Church.
And so by noon on July 4, 1870, as the threatening skies cleared, thousands gathered on the Woodstock Common awaiting a glimpse of Ulysses S. Grant, the first presidential visitor since George Washington. One by one the dignitaries took their places on a platform wrapped with red, white, and blue bunting. One speaker followed another, and it was almost two o'clock when General Stewart L. Woodford began to speak. By this time, no matter how momentous the occasion, the crowd was hungry and restless. Woodford's voice was so low that few could hear him, and most only saw him wildly waving his arms. Occasionally a shrill phrase, "on this our glorious anniversary," "the dear old flag," or "Fathers of the Republic," would float forward. President Grant, the _Times_ noted, sat "like a mummy in a little chair and seemed in profound thought." The crowd began talking and jeering and kept it up until Woodford sat down. Beecher stood and called out, "Hey. See here. I want you boys to quiet down. It isn't a square thing to make all that noise." The audience fell silent and then began to chant, "Beecher, Beecher, Beecher." When the preacher spoke, there was no movement in the crowd, and when Beecher referred to Grant as "our greatest warrior, our greatest president," Grant "brightened and smiled."
The Russian minister Baron De Catacazy spoke last, after which Henry Bowen called for three cheers for the emperor of Imperial Russia. The ceremony concluded with the singing of "The Star Spangled Banner." Then four hundred dignitaries and their wives repaired to Bowen's mansion for a late lunch, after which the ladies played croquet and the men conducted various competitions, including the sack race. United with Beecher, Bowen had won not only the sack race but also a prominent position in the race for political power and patronage.
THE MATCHED GRAY horses charged through the black night, but no night was as black as the darkness in Lib Tilton's soul. When the carriage stopped at 174 Livingston Street, Brooklyn, she alighted and stood for a minute clutching her silk shawl about her shoulders and gazing up at the room on the second floor in which baby Paul had died. Perhaps if he had lived she would not be facing this night and the darkness within her. Another son, Ralph, had his first birthday on June 20, just two weeks before, but even this baby had not erased the pain of Paul's loss.
With a heavy tread, she mounted the stairs to the room where her husband sat at his desk under the gaslight, absorbed in his writing. Theodore Tilton, whose controversial views were well known, had not been invited to Woodstock. He imagined Lib to be, as usual, in Schoharie with the children to avoid the July heat. What impelled her sudden return was not immediately apparent. Lib bent forward to kiss his cheek. Noting her husband's surprise, she explained, "I have come to tell you a secret I have long kept in my heart, a secret I resolved to tell you but lacked the courage."
Then sinking to her knees before her husband she implored, "Before I speak I wish your solemn word to do no harm to the person concerning whom this secret is to be told." Bewildered, Theodore Tilton agreed. Lib wrapped her arms around her knees and, swaying back and forth, told the details of "a passionate fondness" for Henry Ward Beecher that had become a full-blown sexual affair. Theodore questioned his wife closely: Why had she never told him of this affair? Why was she telling him now? Lib replied that she was consumed with guilt and remorse and that in the past few weeks the burden had become too much to bear. What she did not mention was that she was pregnant again and that her husband would realize that this time the child could not be his. Theodore Tilton had been away on a four-month Lyceum tour when her "love babe," as she called it, was conceived.
When Lib's confession was over, Tilton grabbed his hat and coat and stormed out of the house feeling "just blasted." He went to his office at the _Brooklyn Daily Union_ and remained there for three days of soul-searching. He was later to say that after that "purging" he emerged "light-headed—ready to forgive." Unable to change his vision of his wife as pure, Tilton rationalized that her affair must have been prompted by "high religious love. I think she sinned as one in a trance. I don't think she was a free agent." He determined to make only one demand: that Lib tell her seducer that her husband knew about the affair but would keep it secret. Tilton was so struck by his own "magnanimity" that later he said, "For the following two weeks I lived a kind of ecstasy."
After those two weeks, Tilton decided that the open marriage he enjoyed should extend to his wife as well. They would be "loving friends," both free to pursue their "affinities" should any arise. With that Theodore piled his wife into a carriage and took her around to meet Laura Curtis Bullard, so that Laura could hear of the new arrangement from Lib's own lips. Soon after, Mrs. Bullard became a frequent visitor to the Tilton household and Lib's closest friend. Theodore told associates that his wife was "as advanced in her social theories as myself."
Even though Tilton and Beecher were presidents of rival woman suffrage organizations and politically at odds, their personal relationship seemed restored. Once again the Reverend Beecher was welcomed into the Tiltons' household. Theodore encouraged his wife to take an "affinity... to save her health and life." One day when Beecher visited, after a few minutes of conversation he asked Lib to accompany him on a carriage ride. When she declined, her husband "playfully reproached her," declaring that she should go, that the air would do her good and certainly she was entitled to some "private time." And so she went. But this exercise in free love lasted only three months. One evening in October, as Lib ascended the staircase her husband looked at her thickening body with a sudden realization. "It is his!" he shouted. His wife trembled but did not reply.
The next day Stanton and Anthony came to Brooklyn to meet with Tilton. In the afternoon Stanton accompanied him to the new offices of _The Revolution_ , which he and Laura Curtis Bullard now edited, and then to Mrs. Bullard's house for dinner. As the meal progressed, Tilton, wineglass in hand, became increasingly agitated and finally exploded: "Henry Ward Beecher, that abominable man, has ruined my life. Oh, that the damned lecherous scoundrel should have defiled my bed and at the same time professed to be my best friend." Tilton said that he could accept free love, that he had confessed his own infidelities to his wife and there was no jealousy. Was not Mrs. Bullard now his wife's best friend? Under these circumstances he expected that Lib would confess her infidelities to him. But her unexpected pregnancy struck him with full force. "Had he come to me like a man and confessed his guilt, I could perhaps have ignored it, but to have him creep like a snake into my house and I so blind as not to see.... Oh, it is too much." Stanton recalled that she had never seen "such a manifestation of mental agony. He seemed upon the very verge of insanity."
It was eleven-thirty when Tilton returned home to find his wife and Susan B. Anthony waiting for him in the parlor. Lib jumped to her feet and dashed across the room, planting her scant five feet before his six-foot-three-inch frame, as if to block his entrance. She berated him for not picking them up for dinner as he had promised. Tilton looked genuinely surprised. "Was I to take you?" he asked.
"Liar!" screamed Lib.
"Your child is not my child. I don't know who my children belong to," Tilton cried out in an anguished reply. Then with a swift gesture he struck Lib full across the face.
Lib dashed up the stairs and into her bedroom with Anthony following behind. No sooner had Anthony locked the door than Tilton turned the knob, pounding and shouting, "Let me in, let me in."
"You're insane!" screamed Lib.
"Open up, or I'll break it down."
Anthony moved toward the door. "I will not turn this key," she announced. At that moment she felt that Tilton might kill his wife.
"No woman shall stand between me and my wife," yelled Tilton.
"If you enter this room it will be over my dead body!" Anthony shouted back.
There was silence, and then they heard Theodore's footsteps retreating. Frightened by what might happen next, Lib and Susan pushed the bed against the door in case he tried to break in. After an hour or so the two women were so tired that they climbed into bed together and Lib, snuggling close to Susan, poured out the details of her love affair with her "God and preacher," Henry Ward Beecher. She said it was her husband who had set her on the path to free love. Theodore had promised to forgive and forget, but her pregnancy had shattered that hope. He'd taken money from the mouths of their children to pay for an abortion at Madame Restell's for some girl he'd gotten pregnant. Now he wanted her to abort Beecher's baby. She knew that her "love babe" was in danger.
"Did the Reverend Beecher use force in having you yield to his advances?" asked Susan, trying to comfort this distraught and fragile woman.
"No. He treated me with the kindness one would a child. I resolved many times to yield no more, but as often my good resolutions failed."
"Is Beecher aware that your husband knows of this affair?"
"No," replied Lib. She told Anthony that her husband's one request was that she tell Beecher but that she couldn't bring herself to do it. She admitted that she had invented excuses because it was "so difficult to say." First he had been away on vacation and then when he came back Theodore had welcomed him into the household, and things seemed calm once again. But now she could not hide her pregnancy and she did not know what to do. She told Susan that were it not for the children and the fact that she had no means to support herself, she would leave Theodore forever. She wanted to run away, but where could she go? Then, abruptly, as if she realized there was no escape, she sat up and reached for her robe. "I think I'd better go to him and apply cool water to his head," she said. "It will soothe him."
"No," directed Anthony, "you are not to leave this room. It is not safe."
In the morning, the three assembled for breakfast. Theodore spoke not a word through the entire meal. When it was over, he looked at Anthony and said in a voice brimming with hatred, "Never enter my house again."
"I shall enter whenever I choose," Anthony shot back. But she never came again. And, she confided to her diary: "It is almost an impossibility for a man and a woman to have a close, sympathetic friendship without the tendrils of one soul being fastened around the other, with the result of infinite pain and anguish."
Elizabeth Cady Stanton had spent the night at Laura Bullard's. When she returned to her home in Tenafly, New Jersey, Stanton found Anthony sitting on her doorstep. Susan scolded her for forgetting that Mrs. Bullard's dinner invitation had included herself and Mrs. Tilton and then told her about the terrible events of the previous night. "Fresh with astonishment," they compared notes. Stanton said, "Theodore related a very strange story to Mrs. Bullard and me last evening." After she had heard the details, Anthony replied, "Mrs. Tilton told essentially the same thing to me."
Under the circumstances, it was no wonder that the Tiltons wanted privacy, but it was just at this time that Lib's mother, the volatile Mrs. Morse, asked to live with them. The Honorable Nathaniel B. Morse had withstood his wife's violent temper long enough. After she came at him one night brandishing a carving knife, he determined to divorce her. When Tilton told Mrs. Morse that she could not live with them, she turned on her son-in-law with rage, calling him "a scoundrel" and saying that his debts piled up while her poor daughter, humiliated by her "skinflint of a husband," scrimped along mending clothes.
As Mrs. Morse's fury mounted, she mentioned what Tilton later described as "my liaison with Mrs. Tilton's most intimate and honored friend," and accused him of spending time with that "vile woman" and of helping her with _The Revolution_ , which paid him not a penny. "All these woman's rights people, these free lovers, you put upon my child."
"Madam," warned Tilton, "speak respectfully of the master of this house and of his guests or leave now. As for good behavior, in this respect I hold your daughter responsible."
An incredulous Mrs. Morse turned to Lib. "Why? What have _you_ done, my child? Have _you_ been doing wrong?"
The abruptness of the question rendered Lib mute. Without answering, she left the room. Mrs. Morse ran after her, shouting, "Is it Beecher? Is it Beecher? Is it Beecher?" over and over. Lib stopped still, then slowly turned toward her mother, hung her head, and nodded.
"Oh, my God!" cried Mrs. Morse. Flying to her daughter's defense, she turned on Tilton: "If you had gone for your family instead of looking after woman's rights meetings, you would not be obliged to look up your lost trunk!"
"Leave my house!" bellowed Tilton in return.
Mrs. Morse then struck him repeatedly with her parasol, shouting, "Woman's rights have killed you!"
The following morning, Lib Tilton packed her bags and without a word fled to Marietta, Ohio, to the home of friends. From there she wrote,
Oh Theodore, Theodore!
What shall I say to you? My tongue and pen are dumb and powerless but I must force my aching heart to protest against your cruelty.... Do you not know that you are fulfilling your threat that "I shall no longer be considered the saint"? Do you not know, also, that when in any circle you blacken Mr. B's name—and soon after couple mine with it—you blacken mine as well?
Theodore, _your past_ is safe with me, rolled up, put away never to be opened—though it is big with stains of various hue—unless you force me for the sake of my children and friends to discover it, in self-defense or their defense.... Would _you_ suffer were I to cast a shadow on any lady whom you love?.... Once again I implore you for your children's sake, to whom you have a duty in this matter, that _my past_ be buried—left with me and my God. He is merciful. Will you, His son, be like Him?
I feel that you are not in the condition of mind to lead the woman suffrage movement, and I implore you to break away from it and from your friends Susan, Mrs. Stanton and every one and every thing that helps to make a conflict with your responsibilities as husband and father.
Tilton was at his office at the _Brooklyn Daily Union_ when his wife's letter arrived. He read it twice over and then burned it. It did not occur to him that she had made a copy and had sent it to her mother. Lib returned to 174 Livingston Street on November 9, to find that Theodore's rage had not abated. When she entered the library, he said, "So the harlot has returned!" Pointing to the red velvet chaise longue by the fireplace, he exclaimed, "This has been consecrated to sexual intercourse between you and Reverend Beecher. Our daughter Florry knows. She asked me about it. I can stay here no longer." He promptly left for Frank Moulton's house.
Tilton next resigned from Plymouth Church. Jealousy aside, his views had become too radical for Beecher's church. He now espoused Stanton's views of marriage and carried them one better. In view of his own experience, he wrote an article in the _Independent_ titled "Love, Marriage and Divorce."
Marriage without love is a sin against God—a sin which, like other sins, is to be repented of, ceased from and put away. No matter with what solemn ceremony the twain may have been made one, yet, when love departs, then marriage ceases and divorce begins. This is the essence of Christ's idea. To say that He granted divorce only for a gross and fleshly crime is to forget that He called the eye a paramour and the heart of wanton's bed.
On December 3, 1870, Laura Curtis Bullard put her pen to paper as well and asserted that woman suffrage was dead as a political movement and that the right to divorce was much more important.
Women know their own wants, and they know that they do not want suffrage a thousandth part as keenly as they want a reform of the marriage and divorce laws, and a general readjustment of the family relations.... What a woman wants is freedom to marry and to be mistress of herself after marriage; freedom to freely sunder a yoke she has freely bound.
Lucy Stone and many other women were shocked that Tilton and Bullard, in their espousal of free love, had abandoned the cause of woman's rights. Moreover, Mrs. Morse was now convinced that Tilton planned to divorce her daughter and elope to Europe with Mrs. Bullard. Without her daughter's knowledge she consulted her own divorce lawyer to see what settlement could be made in such an event. She reasoned that Henry Ward Beecher was the most powerful of men, and if he was as truly devoted to her daughter as he professed, could he not become her discreet companion and give her "the comfort of a paramour"?
Bessie Turner, a fifteen-year-old who had come to the Tiltons as a servant but had become Lib's ward, was dispatched to the Beechers' home to implore the preacher to visit Mrs. Tilton. As instructed by Mrs. Morse, Bessie recited a litany of Theodore Tilton's cruelties toward his wife and added that when Lib was gone, Theodore had tried to seduce her but that Bessie had rebuffed him.
Beecher appeared the next morning. He sat next to Lib. Mrs. Morse, who did most of the talking, sat facing them. "You are about to hear of cruelties beyond imagining," she began and then matter-of-factly told Beecher that Tilton came home drunk at night and frequently knocked his wife to the floor with his fist and kicked her. He was subject to uncontrollable rages. When on the road he held orgies with "strange women." "My daughter has lived a life of great unhappiness and has been subject to great cruelty and deprivation. Her life with Mr. Tilton has become intolerable."
What Beecher said next filled Lib with dread: "This is a case in which I feel a man cannot give the best counsel. It is a case, it seems to me, where a woman is needed. If you will allow me, I shall be glad to bring my wife and let her hear, for I think much of her judgment about such things."
Mrs. Morse sprang to her feet. "I will bless her if she will come. Bless her as long as I live."
As Beecher was leaving, Mrs. Morse explained in a hurried whisper Lib's so-called embarrassing condition. In an odd turnabout reflecting the morality of the day, she blamed her daughter for Beecher's having seduced her. "She is mourning for _her sin_ ," Mrs. Morse intoned. The preacher put his arm around Mrs. Morse's shoulder and said, "You must call me 'Son,' and I will call you 'Mother.' " And then he confirmed Lib's fear that her baby might be snatched from her by assuring Mrs. Morse that he personally would provide the money to see that her "darling" received the proper medical attention.
Beecher returned the next day with his wife, Eunice, who listened to Mrs. Morse's complaints of Theodore's inhumanity to his wife. Of course, there was no mention that Lib was pregnant with the child of Eunice's husband and that this was the reason for Theodore Tilton's behavior. When Mrs. Morse was through, Eunice, who was greatly touched, turned to her and said of Lib, "I shall adopt your poor daughter." The following morning, as the Beechers sat across from each other at the breakfast table, Eunice told her husband that she had been up most of the night thinking about poor Mrs. Tilton's suffering. She must separate herself from that irresponsible villain.
Hardly looking up from his newspaper, Henry commented, "It is a drastic step."
"Were she my daughter, I would say, 'Separate,' " insisted Eunice. "He will drag her down and disgrace her." Then Henry casually reached for a scrap of paper, scribbled some words, and pushed them across the table to his wife. The note read, "I incline to think that your view is right and that a separation and settlement of support will be wisest—Henry Ward Beecher."
"Is this all right?" he asked. And when Eunice said yes, he instructed, "Bring it round for me then."
NOW BEGAN Lib Tilton's bleak December days, liquid gray days that ran together with no beginning or end. She was more than five months pregnant and it was becoming more difficult to conceal the life within her. She sent Florry, Alice, Carroll, and baby Ralph to her mother's. She sent the servants away. The house fell silent. There was no fire in the grate, no food in the larder. Her unborn child, the proof of her infidelity, had caused "all this torture and embarrassment and humiliation," yet she wanted this "love babe." At night when no one could see her shame, Lib, feeling "that there was no place for my head in that doomed house," would wander the streets of Brooklyn wearing a hooded cloak. She returned at dawn "to creep into the basement and lie down anywhere, feeling utterly wretched."
When Mrs. Beecher brought the note from her husband, Lib Tilton knew full well that the life she carried within her was about to be expunged. No one wanted it—not her husband, or her preacher, or his wife, that beetle-browed woman who called her "daughter." One morning she felt an uncontrollable urge to flee. She dressed quickly and ran down the steep steps of the Livingston Street house, turning her back on that place with its memories of infidelity and death. For hours Lib Tilton wandered the streets. How could she run away? Where would she go? Later, in a stumbling, lame explanation she would say, "I thought I would never come back, but then, you see, I found that I had left my purse at home and so of course I had to return. You see I had to."
On the eighth, or perhaps the tenth day of her isolation, for she had lost track of the days, she saw that the sky was clear and a light snow had descended. Once more she donned her cloak and pulled the hood over her head to conceal her face. Now Lib knew where she was going. A man driving a wagonload of wood saw her trudging along the muddy road. She shook her head "no" to his offer of a ride. At noon she arrived at the Greenwood Cemetery and made her way to the place where her children, Paul and Mattie, were buried in the frozen earth. "Such tiny graves," she thought, and sank to her knees, then spread her body across the cold earth as if to warm the graves of her children. She lay there "and felt peace." She knew not how long it was before she felt a rough hand grasp her shoulder. "Get up, girl," commanded the groundskeeper. "There's no place for you here." Lib did not move. "Get up, do you hear me!" he said, and began to shake her.
Lib Tilton stood. "If there is one spot on earth that is mine, it is these two graves," she said. The groundskeeper regarded her soberly, then removed his hat and bowed his head. "I did not know they were yours," he said. He turned and left. Lib recalled, "I stayed there lying on the little graves all the rest of the day."
_The entrance to Greenwood Cemetery_ (Illustration Credit 19.1)
# CHAPTER TWENTY
THE YAWNING EDGE OF HELL
HENRY BOWEN THREW a copy of the _Brooklyn Daily Union_ on the desk and demanded of his editor, "What is the meaning of this?"
"Of what?" Tilton asked.
"Of this line, 'Let General Grant when he is smoking his cigar think over this matter.' I never smoked a cigar in my life, sir, and if any one of my sons were to smoke I would disown him."
"Everybody knows that General Grant smokes," replied Tilton wearily.
"That may be," shouted Bowen, "but I will not have them know it through the _Union!_ "
Bowen was furious with Tilton. Ever since the Fourth of July celebration when he had entertained Grant, Bowen had tried to no avail to persuade Tilton to support the president in the _Brooklyn Daily Union_. Making him the editor had been a miscalculation. "If anybody in King's County expects to see the _Union_ consenting to be chained like a coach dog to the Republican or any other party, he is woefully mistaken," Tilton wrote, and pledged himself to expose corruption at all levels of government. His paper advocated the nomination in both parties of more able men who would not tolerate such scandals as Black Friday and the Tweed ring, which he called "the curse of two great cities."
Bowen was a partner in several warehousing operations with Erastus D. Webster, who was in charge of the Brooklyn Navy Yard and its various activities. When Webster received the regular Republican nomination to fill a vacant seat in Congress in an interim election for the Third District of Brooklyn, Bowen promised him the support of his two newspapers. However, in a meeting in Bowen's office, Tilton told Webster that he had no intention of supporting him. Webster turned to Bowen: "I understand what Mr. Tilton is saying, but with your assistance I think..."
_Boss Tweed's New York by Thomas Nast_ (Illustration Credit 20.1)
"Address yourself to me, sir," Tilton interrupted. "Mr. Bowen has no more charge over this newspaper's opinions than you have."
Tilton attacked Webster in print as "a corrupt dispenser of federal patronage." Across the East River, New York City seethed with corruption, and Tilton, fearing that the same thing was about to happen in Brooklyn, joined with a group of young idealists who called themselves liberal Republicans. They nominated their own candidate, W. A. Fowler, to oppose Webster, and Tilton agreed to support Fowler in the pages of the _Brooklyn Daily Union_. For Bowen this was the last straw. He called Tilton into his office and told him, "I will not mince words. In the future the _Independent_ and the _Brooklyn Daily Union_ must support my candidate."
"I will not agree to that," Tilton replied.
"Then I surmise the only way for me to obtain the backing I want would be to dismiss my editor."
"Yes, but that is the only way," said Tilton.
That would not be easy: In the nine years since Tilton had taken over as editor of the _Independent_ , the annual revenues from subscriptions and advertising had almost tripled, from $91,733 to $300,305. It had become the most popular and influential religious weekly in the nation, claiming more than half a million readers. On December 19, 1870, Henry Bowen found a way around the problem: He told Tilton that he was taking over the editorship of the _Independent_ himself but would pay Tilton $5,000 a year as a contributor. Tilton would continue as editor of the _Brooklyn Daily Union_ at a yearly salary of $7,000. Tilton also was guaranteed a minimum of $3,000 a year in bonuses. Tilton's former annual salary of $10,000 would now be increased by half. Implicit in this new contract, at least in Bowen's mind, was that the increase would buy loyalty.
A meeting of the Republican general committee was scheduled for the evening of December 21. That morning the _Independent_ carried two notices: one that Henry Bowen had become its editor and another stating that the _Independent_ was endorsing Erastus D. Webster for Congress. At seven o'clock that evening Bowen marched into the meeting confident that he would recapture his former patronage. But as the meeting progressed, Bowen was astounded to find that the Beecher contingent, headed by United States District Attorney Benjamin Tracy, would not support him and instead had allied themselves with Tilton's liberal Republicans and the Democrats. Together they were supporting Fowler to crush Webster and gain control of Brooklyn.
A motion was put before the general committee to endorse Fowler. When Bowen objected, he was shouted down. Then Benjamin Tracy read a report he had prepared asserting that when Henry Bowen was tax collector, he had placed Brooklyn's funds in a personal bank account for months at a time, defrauding the city of the interest. Colonel Morton, another member of Plymouth Church, moved that Bowen be brought up on charges of tax fraud.
A furious Bowen countercharged that Tracy was making this accusation so that his own group would "control patronage," thereby admitting that he himself sought that control. Oliver Worth, a blacksmith on Bowen's payroll, argued with Colonel Morton, who suddenly "bolted across the room and struck Worth a tremendous blow in the face." A melee ensued—chairs were thrown into the crowd, and four enraged men seized Bowen and began beating him. "Fists flew and heavy knocks fell thick and fast.... Men struggled and clenched and confusion prevailed until the lights were put out and the whole business brought to a close by the intervention of the police," read a report in the _Brooklyn Eagle_. A bruised Bowen went home to bed. The following day he received written notification from the general committee that it had expelled him. Once again, Henry Bowen had tumbled from power.
ON THE MORNING of December 23, according to a later account in the _Sun_ , Mrs. Theodore Tilton, escorted by her mother and Bessie Turner, was seen entering and leaving the Fifth Avenue office of the abortionist Madame Restell. The article further alleged that money for an abortion had been provided by Henry Ward Beecher. When Emma Moulton arrived that afternoon to drop off some Christmas presents for the children, she found her friend Lib lying in bed, the sheets red with blood. Lib told her that she'd had a miscarriage as a result of the "anxiety night and day" created by her mother.
In the next room, Mrs. Morse sat at Greeley's desk, her fury flowing out through her pen. To Tilton she wrote,
You infernal villain! This night you should be in jail. Your slimy, polluted, brawny hand curses everything you touch.... I will publish you from Dan to Beersheba.... I can with one stroke of my pen bring you to your knees and brand you for life. The world will be better for the riddance of such a villain and think no more of putting you aside than killing the meanest cur which runs in the street.
Next she wrote to Henry Ward Beecher, whom she addressed as "My Dear Son," reminding him that he had told her to call him by that name. "I told _darling_ [Lib] I felt if you could, in safety to yourself and all concerned, you would be to me all this endearing name. Am I mistaken? Mother."
Her last letter was to her son-in-law's employer, Henry Bowen. In it she accused Theodore of being a free lover and informed Bowen that he was about to divorce her daughter and move to Europe with Mrs. Bullard. She concluded, "Of Theodore Tilton's infidelities Mrs. Beecher can enumerate them far better than I."
Bowen received Mrs. Morse's letter the day after Christmas. It gave him the excuse he had been seeking to get rid of Tilton. He summoned Theodore to his home on Willow Street and began to dress him down as a heathen, libertine, and drunkard. Bowen referred to the essay Tilton had written on "Love, Marriage and Divorce," saying that such blasphemy had no place in a religious publication. Bowen was clearly preparing to dismiss his editor. "I note, sir," he said, "that you have not attended church much lately."
Tilton replied, "Never again will I cross the threshold of Plymouth Church. And _you_ better than any man living know why." Then he told Bowen how the Reverend Beecher had seduced Lib just as he had seduced Bowen's late wife, Lucy. Both men had been cuckolded.
Bowen once had hoped his knowledge of Beecher's adultery would give him power over the preacher, but then he saw that there was no way to destroy Beecher without injuring himself, his ten children, and their mother's memory. For eight long years he had watched helplessly as Beecher's wealth and popularity grew. In an instant Bowen changed direction: Perhaps Theodore Tilton would be the God-given instrument that finally would destroy Beecher, or failing that, these men might destroy each other, thus ridding him of two problems. "I cannot stand it any longer," Bowen declared. "You and I owe a duty to society in this matter. That man ought not to stay another week in his pulpit. It is not safe for our families to have him in this city. I challenge you as a matter of public duty to write an open letter."
Tilton agreed, and the two began to draw up a statement. "Sir, I demand you immediately cease from the ministry of Plymouth Church," Tilton wrote.
"And that he leave Brooklyn and his influence here behind," Bowen said.
"And that you quit the city of Brooklyn as a residence," Tilton wrote.
"Good, good," said Bowen. Then, pressing his advantage, he dictated, "And that you withdraw from the _Christian Union_ and never again write for that publication nor have your sermons printed."
Tilton paused for a moment. He sensed something wrong. "No," he said, "I won't write that."
Bowen backed off, but suggested, "We must add the phrase 'for reasons which you explicitly understand.' " Tilton added that phrase and wrote the note anew.
Sir,
I demand that for reasons which you explicitly understand you immediately cease from the ministry of Plymouth Church and that you quit the city of Brooklyn as a residence.
When it came to signing the letter, Bowen told Tilton that it should bear his signature alone. Again Tilton felt uneasy. Would such a demand be honored with only his signature? Bowen reassured him by saying that he would personally carry the letter to Beecher, sustain the charges, and produce the necessary documentation if called upon. And so Theodore Tilton signed the letter and gave it to Henry Bowen.
When Tilton returned home, he was surprised to find Frank Moulton, whose wife had come to relieve Lib's nurse, in the parlor. When Tilton told his friend what had just occurred, Moulton instantly understood that Bowen was using Tilton to accomplish his own ends. "You made your demand all alone?" he asked incredulously.
"Bowen gave me his word he would sustain my charges."
"Oh, Theodore, you have left him a chance to play you a trick. You are a ruined man!"
Lib Tilton lay in bed, listlessly watching as her husband paced up and down, but she sat up abruptly when he told her of his ultimatum to Beecher. "If Mr. Bowen makes war on Mr. Beecher and if you join in it and if Mr. Beecher retires from his pulpit, as he must under such an attack, sooner or later everybody will know the reason why and that will be to my shame and to the children's shame. I cannot endure it!" To these men, infidelity was a pawn in the game of power and wealth but to Lib it was her life. "I have always been treated as a nonentity—a plaything—to be used or let alone at will. But it has always seemed to me I was a party not a little concerned," she later reflected. "Make peace," she implored her husband, "if not for me then for the love of your children."
Seeing her so desperate and knowing he had blundered, Theodore softened toward his wife: "Then you must write him telling him I know everything." Elizabeth propped herself up in bed and, calling for pen and paper, wrote Beecher what she had never been able to tell him, that her husband knew everything about their relationship. She ended her letter with a prayer for reconciliation between her preacher and her husband.
BOWEN HAD GONE immediately to Beecher's study at Plymouth Church with Tilton's letter demanding the preacher's resignation. Beecher read the demand and then sat silently, drumming his fingers on the arm of his chair. Finally, he looked at Bowen and said, "This is sheer insanity. The man is crazy." The demand did not refer to Beecher's affair with Mrs. Tilton, and Bowen realized that Beecher had no inkling Tilton knew of it.
"Have you any answer?" inquired Bowen.
Once again there was a long silence; then Beecher asked, "Where do you stand in this?"
_Frank (Francis De Pau) Moulton—the mediator in this matter_
Knowing that his ploy to remove Beecher had failed, Bowen said, "Mr. Tilton has become your enemy, but I will be your friend."
"You are friendly toward me?" asked a skeptical Beecher.
"We have settled our differences," replied Bowen.
Bowen was ready to join the powers he could not defeat but wanted as much as possible in return. In a veiled manner, the negotiations began. Bowen assured Beecher that he had already removed Tilton as editor of the _Independent_ and was ready to dismiss him altogether. "In my opinion you are correct," said Beecher. "A man as tainted as Tilton cannot properly be retained on the _Independent_ without doing it damage." Beecher, whose own newspaper was threatened by Tilton's success, went even further: "With respect to Mr. Tilton's continued employment on the _Union_ , I think that as the editor of the Republican organ in Brooklyn he will get the paper into trouble." Bowen nodded in assent.
The bargaining continued. On a small note card Bowen mapped out the price of his "friendship": His _Independent_ would once again have the right to publish Beecher's sermons and lecture-room talks. Beecher would publicly announce at a Friday-night prayer meeting what an excellent man Bowen was, thereby restoring his social status. He would express his friendship to Bowen in a letter, which could be shown to President Grant if necessary. Beecher would retract what he had said to injure Bowen politically, and the tax fraud charge would be dropped. He would instruct Wilkeson and Howard to grant Bowen a free interest in the J. B. Ford syndicate, which owned Beecher's two-volume _The Life of Jesus the Christ_ as well as a new edition of his sermons and hymns, thereby giving Bowen a financial incentive to protect Beecher's reputation.
And finally came the big reward: Beecher would have Wilkeson arrange to retain Bowen as an agent to sell Jay Cooke's new issue of Northern Pacific railroad bonds. That night Bowen left with his note card initialed by Beecher. There had been no mention of Beecher's affair with Mrs. Tilton. The interview had been "strictly business."
On December 29, 1870, Bowen received the following letter:
Dear Mr. Bowen,
The understanding between us whereby you fully and permanently identify yourself, your paper... and your nephew with the interest of the Northern Pacific Railroad Company is as follows: On your part you give... the exclusive interest and influence of your money columns and your editorial columns to the enterprise and bonds of the Northern Pacific Railroad Company.... On our part... we will advance you on January 2, 1871, $15,000 on the Northern Pacific bonds... and each month thereafter a sum of $10,000 is to be credited until the sum of $60,000 is thus earned.
Jay Cooke & Company
Tilton was at his desk at the _Brooklyn Daily Union_ when he looked up to see Bowen. "I have determined to meet Mr. Beecher face to face—," he began.
With the passion of a man caught out, Bowen interrupted. "If you should inform Mr. Beecher of the allegations I have made concerning his adultery... or tell him I had anything to do with that letter, you will be cashiered from both my papers."
"I'll do just as I think best," Tilton answered coldly.
"Then you are finished," bellowed Bowen. "If you intimate I had any connection with this, I shall have you ejected from these offices by force. I shall cast you into the street."
Scarcely an hour later, a letter was delivered to Tilton's home canceling his contract both as a contributor to the _Independent_ and as editor of the _Brooklyn Daily Union_.
FRIDAY, DECEMBER 30, 1870: The wind howled across Columbia Heights as parishioners alighted from their carriages and hurried into Plymouth Church for the weekly prayer meeting. Once inside, top hats and bonnets were removed and shaken; boots were stamped free of snow. Women's coats with ermine or fox trim were hung next to men's lined with beaver. Skates were hung on pegs by parishioners who had come from a nearby pond. Despite the weather the crowd was more than a hundred strong, for these Friday-night meetings were popular events. Henry Ward Beecher would sit in a commodious green-velvet armchair in the downstairs lecture room and converse with his parishioners about matters of the heart and soul. He improvised as he spoke, intoxicated by his own rhetoric, and he encouraged others to cross-examine him on any subject. The pastor cherished these meetings, for it was here that he came in personal contact with members of his flock. One of them described Beecher as "a sort of evangelical Bacchus."
Beecher had started down the stairs to the lecture room when Frank Moulton stepped forward and grasped his arm. "Mr. Beecher, Theodore Tilton is at my house and wishes to see you immediately."
Beecher pulled away, shaking off Moulton's hand. "This is Friday night. This is a prayer meeting night. Tell him I cannot go."
"He wants to see you in regard to a letter he sent you through Mr. Bowen."
Beecher seemed unimpressed. "Not now."
Moulton persisted. "There is a letter written by Mrs. Tilton in regard to your relations with her. Mr. Tilton, I believe, has his wife's full confession. I think you had better go."
Moulton watched Beecher's face flush. As Benjamin Tracy passed the two men on the stairs, the preacher called out to him, "Take the prayer meeting for me please, Ben," and silently followed Moulton from the church. Whirling snow surrounded them as they trudged, heads bent low, fighting the wind. "This is a terrible night. There is an appropriateness in this storm," muttered Beecher. "Has he truly got that confession?"
"He has, for I have seen it," answered Moulton.
"Oh, my God," said Beecher. "Where can I go? What can I do?"
"I don't know. I am not a Christian, I am a heathen, but I will try to help you."
Together they moved silently through the snow to Moulton's house. Tilton was waiting for Beecher in the upstairs parlor. As Beecher entered, Tilton locked the door, put the key in his pocket, and gestured to Beecher to sit down. Tilton remained standing. "I presume, sir, that you received from me, through Mr. Bowen, a letter demanding your retirement from your pulpit and from the city of Brooklyn," he said formally.
"Yes, I have received it," Beecher answered with equal formality.
"I called you here tonight in order to say you may consider that letter unwritten, unsent, blotted out!"
A look of relief passed over Beecher's face. "Thank you. Thank you, dear Theodore."
"Your thanks should not go to me but to Elizabeth. This meeting is not for your sake, nor for my sake, but for her sake. We must protect one another and make peace. But you must know that as for me not only have you injured me in my business relations but you have corrupted my wife, teaching her to lie, to deceive." He withdrew from his pocket Elizabeth's confession and began reading it aloud. Beecher's face and neck turned bloodred. "Theodore, I am in a dream—this is Dante's Inferno," he blurted out. "You have a right to kill me. I have forfeited my life to you. Take it."
"No," answered Tilton, "but you shall not walk serenely over this town while I go to work with sorrow and a sore heart every day.... Know that you are discovered."
Henry implored Theodore to give him permission to see Lib "for the last time," and Tilton acquiesced, telling him to come to his house later that evening but warning him, "See to it, sir, that you do not chide her for the confession she has made. For if you smite her with a word, I will smite you in ten-fold degree." With that, Tilton unlocked the door.
As Beecher staggered down the stairs clutching the banister, Frank Moulton, who was standing below, heard him mutter to himself, "This will kill me." When the dazed preacher walked through the front door without his coat, Moulton dashed out and covered him. Beecher was later to say that the news of Lib's confession "fell like a thunderbolt on me... I went forth like a sleepwalker while clouds were flying in the sky. The winds were out and whistling through the leafless trees, but all this was peace compared to my mood within."
When Beecher arrived at the Tiltons' house at eleven that evening, Bessie Turner led him to Lib's bedroom, where he found her lying "white as marble with closed eyes as in a trance, her hands upon her bosom, palm to palm as one in prayer." He pulled up a chair next to her bedside, reached out and took her hand in his, and then began to cry. Through his tears he observed the faintest quiver run through Lib's body. It seemed "as if she was going to die," he later said, yet his first thought was that he might be found responsible and "left by her death with this terrible assertion hanging over me."
"You have slain me. You have ended my usefulness," Beecher whispered in her ear.
Lib, who had been unable to resist her preacher while strong, was now without will. His words of persuasion took but little time. Then he gave her pen and paper, and as he held the inkwell for her, she wrote what he dictated. When Beecher left her chamber, the note was tucked firmly in his breast pocket:
December 30, 1870
Wearied with importunity and weakened by sickness I gave a letter implicating my friend Henry Ward Beecher under the assurances that that would remove all difficulties between me and my husband. That letter I now revoke. I was persuaded to it—almost forced—when I was in a weakened state of mind. I regret it and recall all its statements.
E. R. Tilton
At midnight, the embers in the fireplace still glowed. The nurse, Lucy Mitchell, finally satisfied to hear Mrs. Tilton's even breathing, dozed off. She was awakened by a "buzzing sound." As Lib Tilton cowered in a corner of the bed, a frightened bird in a nest of rumpled sheets, her husband loomed above her. "This will never do, Mr. Tilton," said Lucy, rousing herself. "Mrs. Tilton is very ill, seriously ill, and must not be disturbed."
"Get out!" commanded Tilton. Lucy, seeing the look in his eyes, scrambled to obey. Outside the door she was unable to hear the exact words, but later said that Mr. Tilton was shouting and his wife was answering in a tone of entreaty. After some time, Theodore jerked open the door, and Lucy almost fell inside as he stormed past her. Lib Tilton was shaking uncontrollably. Lucy put a pillow under her to elevate her hips. She stroked her patient's head and repeated again and again, "There, there, dear!"
December 30, 1870—Midnight
My Dear Husband,
I desire to leave with you, before going to bed, a statement that Mr. Henry Ward Beecher called upon me this evening, and asked me if I would defend him against any accusation in a _Council of Ministers_ and I replied solemnly that I would in case the accuser was any other person than my husband. He dictated a letter, which I copied as my own... designed to vindicate Mr. Beecher against all other persons save only yourself.... You and I are pledged to do our best to avoid publicity. God grant a speedy end to all further anxieties.
Affectionately, Elizabeth
_Beecher directs Lib Tilton to write a letter retracting her confession of adultery_. (Illustration Credit 20.2)
The following day Mrs. Tilton summoned Frank Moulton to her bedside. She implored him to retrieve the letters she had written under duress the previous night, first for Beecher and then for her husband. "Burn them!" she begged.
ON NEW YEAR'S EVE, a gun tucked under his jacket for protection, Frank Moulton checked the warehouses of Woodruff & Robinson to make sure they were secured against holiday revelers. He then proceeded to Beecher's home to retrieve Mrs. Tilton's letter vindicating the preacher. He found him in his bedroom, staring out of the window at the river view. A weary Moulton removed his overcoat and placed his gun on a nearby bureau. Moulton explained that Theodore Tilton had given him a letter from Mrs. Tilton renouncing the letter Beecher had dictated to her. The only way this matter could be kept quiet was if all the letters were destroyed. He must hand over Mrs. Tilton's letter of retraction. Beecher later said he agreed to comply because he felt both endangered and conscience-stricken: "The case was strongly against me. My old fellow worker Tilton had been deprived of his eminent place and influence, and I had counseled it. His family had well nigh been broken up, and I had advised it. His wife had long been sick and broken in health and body, and I had been the cause of all this wreck." With the usual flood of tears, the preacher turned to Moulton: "I call upon you to save me. My life is ended." Then he walked to his desk, opened the drawer, and handed Moulton Lib's letter.
Beecher and Moulton had identical recollections of this conversation, but Beecher denied that he had also justified his behavior by saying to Moulton, "The red lounge on which we consummated our love was to me an almost sacred object.... My acts of intercourse with that woman were as natural and sincere an expression of my love for her as the words of endearment which I addressed to her. There seemed to be nothing in what we did together that I could not justify to myself on the ground of our love for each other and I think God will not blame me for my acts with her. I know that at present it would be utterly impossible for me to justify myself before man."
Moulton was to say that "this was impressed upon my mind because it was the first annunciation of the justification of the doctrine of free love that I had ever heard."
WITH A CLEAR SKY and the Brooklyn temperature at forty degrees, the year 1871 began. As Tilton sat at his desk writing a letter to Bowen denouncing his treachery, a few blocks away at 124 Columbia Heights, Henry Ward Beecher stood before a ten-foot Christmas tree. The Sunday-school children had decorated it with garlands of pine cones and holly and swags of brightly colored crocheted balls studded with cloves. The white lace tablecloths were heaped with pies, cookies, cakes iced in red and green, preserved fruits, and other sweetmeats. The piano tinkled out a series of hymns and popular tunes. For the first time in seven years, Henry Bowen and his young wife were invited to the reception. While his wife mingled happily with her friends as they compared Christmas gifts, Bowen drew Beecher aside and proudly told him he had followed the preacher's advice and fired Tilton from both of his newspapers. Beecher felt "a sudden stab." Events were becoming more and more complicated. Only two days before, this action would have been welcomed, but now it might precipitate a crisis.
Beecher hastily wrote a note to Frank Moulton asking him to come by as soon as possible. When Moulton arrived at seven, Beecher excused himself from the festivities and the two men went upstairs. What would happen now that Bowen had fired Tilton? Beecher asked. Would an enraged Theodore tell all? According to Moulton, he "buried his face in his hands and wept," saying "he had meditated suicide and could not live to face exposure.... He had wakened as from a sleep and likened himself to one sitting dizzy and distracted on the yawning edge of Hell." Beecher told Moulton that if the affair were to be revealed, he wanted "notice in advance so that he might take some measure, either by death or flight, to hide himself from the world's gaze." For the first time Beecher shifted the blame for all that had happened. "I don't understand how _she_ could have done this to me," he complained.
"To avert the storm," Moulton suggested that Beecher write him a letter saying that he would resign as president of the AWSA, thereby eliminating one area of friction with Tilton, president of the NWSA. He told Beecher, "I know that Theodore loves his wife and will hesitate to bring his own family down." If Beecher apologized for his conduct, Moulton would carry his letter to Tilton, whose pride would be saved, and everything might be set aright. Too agitated to write, Beecher dictated the letter to Moulton and then signed it.
Brooklyn, January 1, 1871
My Dear Friend Moulton,
I ask through you Theodore Tilton's forgiveness, and I humble myself before him as I do before my God. He would have been a better man in my circumstances than I have been. I can ask nothing except that he will remember all the other breasts that would ache. I will not plead for myself. _I even wish that I were dead_....
H. W. Beecher
# CHAPTER TWENTY-ONE
THE WOODHULL MEMORIAL
"ISHALL GO TO the Washington convention to work, not to speak. Tilton should be secured by all means—his wife too. Our parlor needs her demure, motherly, angelic sweetness as much as our platform needs him. These little, quiet, domestic women are trump cards nowadays. I wish I had a whole pack of them," wrote Isabella Beecher Hooker to Susan B. Anthony in December 1870. Isabella, like almost all the Beechers, was an expert at telling other people how to behave. She was convinced that she could do a better job of running the National Woman Suffrage Association convention, scheduled for January 1871, than Stanton and Anthony, who she felt had become too controversial. When Isabella first proposed "taking the Washington convention into my own hands," a dispirited Stanton wrote to Anthony that she was ready to let Isabella do just that. "Let us exalt Mrs. Hooker, who thinks she could manage the cause more discreetly and more genteelly than we do.... I am ready to rest and see the salvation of the Lord."
In the preceding few months Stanton had withdrawn. "You know when I drop anything, I drop it absolutely," she wrote to Anthony. "You cannot imagine what a deep gulf lies between me and the past." She refused to cancel her lecture engagements in order to attend the Washington convention, but she did send Isabella $100 as a substitute for her presence. Mrs. Hooker was delighted, but Anthony "looked disgusted" and told her that Stanton "must appear before the congressional committees, at any rate." Isabella tactlessly wrote back to Stanton, "I don't know what to say. You will talk more forcibly than any one else and in committee you are invaluable. Still, I want your money, and I could do without you on the platform."
Isabella's letter was "a slap in the face... quite blunt and egotistic and somehow it hurts my self-respect," Stanton wrote. She decided to boycott Hooker's convention and wrote Anthony urging her to do the same, but Anthony replied,
Mrs. Hooker's attitude is not in the least surprising. She is precisely like every new convert in every reform.... The only thing surprising in this case is that you, the pioneer, should say to each of these converts, "Yes, you may manage. I grant your knowledge, judgment, taste, culture, are all superior to mine. I resign the good old craft to you altogether."
To my mind there never was such suicidal letting go as has been yours these last two years. I am now tee-totally discouraged.... How you can excuse yourself is more than I can understand.
When the women Hooker had assembled gathered for a preliminary meeting in a committee room of the Capitol, Anthony congratulated George Julian of Indiana for introducing a Sixteenth Amendment to grant women the vote, but she pressed the congressmen to move the stalled bill out of the House Judiciary Committee. When she finished, one of the senators scolded her as if she were a schoolchild. "Miss Anthony," he said, "Congress has no time to attend to such questions, even if they were worthy of consideration." Anthony answered, " _If_ we had the _votes_ , gentlemen, the time would be forthcoming and even at hand."
After the meeting, Hooker and Anthony returned to the home of Senator Samuel Pomeroy, with whom they had been staying. Picking up the _Daily Patriot_ , Isabella read an announcement that one Victoria Woodhull, a self-proclaimed candidate for president, had been granted a hearing before the Judiciary Committee of the Joint Houses of Congress, to present a memorial on the question of woman suffrage. This unprecedented honor was to occur on the same morning that the NWSA convention was officially scheduled to open.
Isabella expressed her astonishment to Senator Pomeroy and his wife. How did this Victoria Woodhull secure a hearing? Who was she? Who were her antecedents? Pomeroy chastised her: "She is for your cause, is she not?... Men never could work in a political party if they stopped to investigate each member's antecedents and associates. If you are going into a fight, you must accept every help that offers."
Susan then told Isabella about a document known as _The Woodhull Memorial_ , which Benjamin Butler was said to have written. She had procured a copy and felt that it might provide the argument they needed for women to achieve enfranchisement. The Washington press had been alerted to the hearing and Anthony was sure that Victoria Woodhull would make news. She had met Woodhull briefly when she interviewed her for _The Revolution_ and was impressed both with her beauty and her wealth. Their own delegation had received scant attention and only one newspaper notice, which said the delegates were "beautiful enough to win the hearts of Congress," hardly a recommendation for political action. What they must do, Susan insisted, was to postpone the opening session of their own convention and appear the following morning in the committee room in a show of solidarity before Congress and the press.
ALL THIS HAD come to pass through Benjamin Butler's efforts. During the winter of 1870, the pragmatic Butler had informed Victoria that a Sixteenth Amendment to give women the vote would go no further than the Judiciary Committee. To put a bill in committee was a classic way to kill it with study or neglect. It was a common political practice that would never change, he told her. On November 19, 1870, under Victoria's signature, _Woodhull & Claflin's Weekly_ published an article asserting that the Sixteenth Amendment was dead in committee but that this was irrelevant because the Constitution of the United States defined women as "citizens," and citizens were entitled to vote. This was followed two weeks later by an erudite document, complete with legal and constitutional precedents. Victoria would later say that when she awakened from a trance one winter afternoon, Colonel Blood gave her this document, which had been dictated through her but conceived by the spirits. Skeptics noted, however, that it featured the blunt language and piercing arguments for which Butler was known.
In December, at Butler's behest, Woodhull appeared in Washington with _The Memorial of Victoria C. Woodhull_ in hand and petitioned Congress to pass a declaratory act asserting women's right to vote. Two days later, she delivered a copy of the _Memorial_ to the Senate, and George Julian requested that it be presented to the Judiciary Committee. Woodhull spent Christmas at Butler's house in Washington while his family was away. No sooner had Victoria returned to New York than she received a message from him: She had been invited, on Thursday, January 11, 1871, to present _The Woodhull Memorial_ to the combined Judiciary Committee of both houses of Congress—in person.
The woman who went to Washington was actually an amalgam of others—her views of enfranchisement, marriage, divorce, and the relations between the sexes were largely Elizabeth Cady Stanton's. She wore the same stylish black velvet dress and Alpine hat as Anna Dickinson. She pulled her hair back in a bun like Lucy Stone, and her only adornment was a single white rose at her throat. Yet there was one trait that was truly her own—her passion.
Because of her stage training and her extraordinary visual memory, Victoria could render the most complicated legal arguments convincingly, without notes. She needed these skills to keep her composure, for she was later to admit that, although she knew the spirits were her allies, she was terrified. In the corridor of the Capitol building, Benjamin Butler stood next to the trembling newcomer. The meeting was to be held in a spacious marble-walled conference room, but smoke from a malfunctioning stove forced its occupants to leave. As they waited for a new room, Butler grasped Victoria's arm and whispered, "Be strong." Her defense of free love, he knew, made her the radical of radicals, but he told her to ignore any slights or snubs from the other suffragists, especially from their leader, Isabella Beecher Hooker. A congressman standing nearby overheard Butler and remarked, "It would ill become these women, especially a Beecher, to talk of antecedents or cast any smirch upon Mrs. Woodhull, for I am reliably assured that Henry Ward Beecher preaches to at least twenty of his mistresses every Sunday." The nervous Victoria was only dimly aware of the remark. It was the first she had heard of the rumor, but she noted that it seemed "to have a subduing effect instead of arousing indignation" among those standing nearby.
A few minutes later, Butler left to inspect the new room. He seemed completely in charge of the proceedings, ordering the windows opened to ventilate the cramped quarters. A large rectangular mahogany table dominated the room. Chairs were carried in and placed around this table and in rows on both sides. Isabella Beecher Hooker and Susan B. Anthony sat in the second row of chairs, side by side. Newspaper reporters leaned against the bookshelves lining the back wall. Butler checked to make sure that all the committee members were seated around the table, disappeared, and reentered leading Victoria Woodhull into the room. She held tightly to Butler with one hand while with the other she clutched Tennie C., who was engulfed in a ruffled green taffeta shawl that set off her copper hair. As she was about to sit down, Victoria turned and stared Isabella full in the face, and Isabella felt herself mesmerized by those blue-gray eyes.
It was ten in the morning when Butler called the meeting to order. John A. Bingham of Ohio, the chairman, introduced Victoria Woodhull. As she stood, her face drained of color; she leaned forward and grasped the edge of the table. Isabella was sure she was about to faint. She began in an almost inaudible whisper with long pauses between words. Then she stopped. Her lips moved silently as if in prayer. Suddenly, a change came over Victoria: Red patches glowed on her cheeks as she began reading her _Memorial_ in a mellifluous, low voice that gained authority with every phrase. Isabella noted that her voice had become infused with "fire and freedom."
Woodhull maintained that women already had the right to vote and proposed that Congress merely pass a declaratory act to that effect. She noted that the first section of the Fourteenth Amendment states, "All persons born or naturalized in the United States and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside." Furthermore, though the Constitution gave the states the right to "regulate suffrage," the wording did not grant them the right or power to prohibit it.
The women of the Wyoming territory had been granted the vote the previous year, and Woodhull asserted that they, and every other woman in the nation, were entitled to vote under Article Four, Section Two of the Constitution, which states, "The citizens of each state shall be entitled to the privileges and immunities of citizens in the several states." She pointed out that the Founding Fathers had used the word "persons," specifying neither male nor female, but that
after the adoption of the Fourteenth Amendment, it was found that still more legislation was required to secure the right to vote to all who were declared to be citizens. And the Fifteenth Amendment was passed by Congress and ratified by the states.... We must regard this amendment as though there had been no negroes requiring enfranchisement. We must divorce our minds from the negro and look at the Constitution _as it is_... the basis of equality is constructed _by all_ and _for all_ and from which all partake of _equal_ rights, privileges and immunities.
The sovereign will of the people is expressed in our written Constitution, which is the supreme law of the land. The Constitution makes no distinction of sex. The Constitution defines a woman born or naturalized in the United States, and subject to the jurisdiction thereof, to be a citizen. It recognizes the right of citizens to vote.... All people of both sexes have the right to vote, unless prohibited by special limiting terms.... No such limiting terms exist in the Constitution.
Isabella was fascinated by Woodhull and could see that for the first time men were listening to "the sheer logic of a woman's argument." And there was more: This woman had succeeded in a man's world. She had the power to change things. When Woodhull ended her speech, she sat down abruptly, limp as a rag doll. The physical transformation was noticeable to the reporters who recorded the scene. Tennie C. reached over and held her sister's hand. Mrs. Hooker observed that she regarded her sister "with a look of prayerful sympathy... one of the most beautiful expressions I have ever seen."
After the meeting Anthony and Hooker followed Julian to his office, where they found Victoria and Tennie already seated on his couch. When Woodhull was introduced to Mrs. Hooker, she sprang to her feet, clasped both of Isabella's hands in her own, looked at her intently, and remarked, "We have met before." And Isabella knew it was true. She felt the intense current of Woodhull's personal magnetism. Here at last was the evangel they had so long awaited. To Isabella this woman was the instrument of "the Heavenly Father.... It impressed me profoundly and in a manner I could never describe with the conviction that she was Heaven sent for the rescue of woman from the pit of subjection."
Victoria later told Isabella that in a trance she had seen "a great vision.... I see the near approach of the grandest revolution the world has yet known and for the part you shall play in it, thousands will rise up and call you blessed. It was not for nothing that you and I met so singularly."
A few days after Woodhull presented her _Memorial_ , as Mrs. Hooker passed Charles Sumner, the senior Republican senator from Massachusetts, he drew her aside and said, "I was at a dinner party of twenty senators and each was asked in turn whether Woodhull's arguments could be answered, and not one suggested an answer.... Mrs. Hooker, this is the first time in my life that I realized that disenfranchisement means the same to you that it would to me."
Isabella believed that women would soon have their freedom. The previous year they had faced a "dead Congress." Now they were being noticed, talked about, helped. Four committee rooms at the Capitol were made available to them. Congressman Albert Gallatin Riddle arranged for _The Memorial of Victoria C. Woodhull_ to be set in electrotype. Thousands of copies were sent out, as well as five thousand letters asking for petition signatures from women demanding the vote. Instead of the usual mailing costs of 25 cents for a single sheet and 50 cents for a double one, these materials were sent free of charge, saving thousands of dollars, because of congressional franking privileges supplied by Benjamin Butler.
_Isabella Beecher Hooker. Now a complete convert to Spiritualism, she felt Victoria Woodhull was "Heaven sent for the rescue of women._ " (Illustration Credit 21.1)
The second week in February, Victoria Woodhull received an invitation to visit President Grant at the White House. He greeted her warmly and then gestured to the presidential chair. "Someday you will occupy that chair," he said, according to her recollection of the event. No doubt this was flattery, but the impressionable Woodhull believed him. She also believed that President Grant had been directed by a higher power to deliver this message to her. In fact, he may have been directed by his wife, Julia, who had been impressed by _The Woodhull Memorial_ and had written Victoria indicating her support.
Woodhull was bringing a new strength and optimism to the movement. Stanton and Anthony—desperate to further their cause—embraced her eagerly and asked no questions about her "antecedents," following Senator Pomeroy's suggestion not to pry into her past. Whatever she had done or been was none of their business. Stanton declared that Woodhull's arguments constituted a "new departure," and as a result the entire "manner of agitation is radically changed." She wrote to Victoria,
In declaring that women are already citizens and pointing the short way to freedom, you have inspired the strongest of us with new hope and enthusiasm. In securing a hearing before the Judiciary Committee of Congress... you have lifted the debate on woman suffrage from the low ground of expediency, where ordinary men insist on holding it, into the higher realm of constitutional law. You have attacked too the last stronghold of the enemy—the social subordination of women.
And Anthony wrote,
Bravo! My Dear Woodhull!
I feel new life—new hope that our battle is to be short, sharp and decisive.... I have never in the whole twenty years' _good fight_ felt so full of life and hope. Go ahead! bright, glorious, young and strong spirit, and believe in the best love and hope and faith of
Susan B. Anthony
Anthony knew that in addition to leadership Woodhull had enough money to subsidize a small delegation of women to remain in Washington until spring. One of those who stayed was Isabella Beecher Hooker, who within three months had compiled the names of eighty thousand women who demanded the vote through a declaratory act. This list was presented to Congress, and a buoyant Isabella wrote, "Early next winter we _shall_ be declared voting citizens."
Only Stanton seemed discouraged when the Majority Report on the Memorial denied the "petitioner's prayer" for a declarative act that women as citizens could vote. She wrote to Anthony that it was "the crowning insult to us from the Republican party." Benjamin Butler and William Loughridge wrote a favorable Minority Report and copies were sent out to twenty thousand influential individuals.
Isabella began to believe that Victoria might be "His [God's] own instrument for working out the deliverance from bondage.... He has opened the Red Sea before us who were willing to pass through under her leadership." Seeing Victoria day after day, Isabella felt it her duty to groom this unconventional woman for the exalted position she was to hold. As she had once instructed Mrs. Stanton on proper deportment and behavior, she now set about molding Mrs. Woodhull into an upper-middle-class matron. On receiving a letter from Victoria scrawled on four sheets of wrapping paper, Isabella responded:
Burn this as soon as sent! I want you to use nice notepaper hereafter—and send in envelopes. These are too rough... a dreadful eyesore to me. Now if you are to be our accepted standard bearer—be perfect, be exquisite in neatness, elegance and decorousness.
But as Isabella was trying to impose her vision on Victoria, the latter in turn was imposing her own ecstatic spiritual vision on Mrs. Hooker's mind and heart. "I have seldom been so drawn to any woman," wrote Isabella. "I know she has visions and is inspired by _spiritual_ influences but her inspiration seems very like my own, a simple reliance on a Heavenly Father."
When Victoria confided to Isabella her conviction that she was "to be President next time and thus _ruler of the whole world_ —our country being destined to rule all others," Isabella cautioned her never to disclose this. But she wrote her husband, John, of it and added, "I dare not tell you all I see in the future.... God knows it and that is enough."
On Thursday, February 16, 1871, Victoria Woodhull, Isabella Beecher Hooker, and Benjamin Butler were asked to lecture at Lincoln Hall in Washington on the subject of constitutional equality. An elated Isabella wrote to Victoria, "I have this moment read your letter of acceptance _My Darling Queen_ , and it is all I could ask. You are fitted for political strife and a pure leadership.... I give you my blessing and deepest sympathy and warmest prayers."
When Isabella confessed that she had never spoken to such an audience and "did not dare see that I could," Victoria told her to draw upon her other powers. With the help of the spirits Isabella would not speak for herself but would transmit words of immortal truth. "As I am about to speak, I call upon the spirits. They surround me and protect me. I sense them hovering about me in the air... and the light beaming through. I am doing their bidding." This convinced Isabella that the spirits were inspiring and protecting her as well. When Isabella rose to speak, she, who had so long ago written to her husband, "I can't write a book nor draw pictures—nor do any other productive work," found a rapt audience being influenced by her. Isabella later wrote that she had given a powerful speech, "The finest address I ever delivered.... It flowed out of my inner consciousness as if it were part of my very being." Long after, that speech seemed to her a "veritable inspiration," and from that moment on she felt that evermore, "my strength shall be equal to the day."
Woodhull's own speech, which followed, incorporated passages from the Butler and Loughridge Minority Report. A newspaper report called it "a triumph," and Hooker noted, "I verily believe not one who heard her great speech (for it is truly great—I never saw an audience of 1400, many standing, listen as this did to a bare legal argument in my life) has wished to utter a word against her or has been able to do it."
But Isabella had not counted on the American Woman Suffrage Association and its _Woman's Journal_ , which stated that the alliance of the National Woman Suffrage Association with a woman like Victoria Woodhull was far worse than its association with the bigoted George Francis Train. Rumors of Victoria's past had begun to catch up with her. Mary Livermore was a Chicago resident, and she recalled that there had been unsavory newspaper stories about the Claflin clan. Livermore was quoted as saying that Woodhull was unfit to be involved in woman's rights: "Her hands are unclean." And once again members of the AWSA accused the women of the NWSA of embracing free love. Anthony had her fill of these accusations. She wrote to the Spiritualist Laura De Force Gordon,
If I had time or space I could tell you... the lie to it.... Our fastidious Boston friends can't _see nor_ hear the roar of the Woodhull shot... because it isn't of the _true blue Boston ring_ —I verily believe those _men_ and their _echoes_ , the women—would rather women should grovel in the mire of disfranchisement another whole century than that they shall be lifted out by what they term _"unclean hands._ "
At Harriet Beecher Stowe's behest, Henry Ward Beecher wrote a letter to their sister Isabella asking her to drop her association with this "pseudobanker" whose opinions were "unusual and dangerous." Catharine Beecher and Mary Perkins also reacted violently to their sister's association with this creature of dubious reputation. Finally, Harriet wrote Isabella at length that a parishioner of Plymouth Church had observed Mrs. Woodhull in the company of a "certain senator" while traveling at night on the train to Washington. The parishioner related how Woodhull had invited him to visit her hotel room later that night, but he declined. In the morning, he met the senator and told him that Woodhull was "not a proper woman for you to be with," to which the senator replied that he needed her because she had promised to "bring over" a congressional opponent.
Much disturbed, Isabella wrote to this member of her brother's church to confirm Harriet's account. But even when he replied that those indeed "were the facts," Isabella concluded that the facts were not damning, only the inferences drawn from them. It had become impossible for Isabella to envision Victoria Woodhull, this woman whom she had quickly come to love, as anything but pure. "It is a great comfort to me thus to believe in her innocence," she confessed.
However, as more rumors of Victoria and Tennie C.'s exploits drifted back to Isabella, she wrote to Anthony for advice.
Dear Friend Susan,
Now as to Mrs. Woodhull—we are in a quandary... my sisters, all three... have nearly crazed me with letters imploring me to have nothing to do with her.... I was never so perplexed to form a judgment in my life and my prevailing belief is in her innocence and purity.... I shall always love her and in private shall work for her redemption if she is ensnared.... I can't guess who is really behind her nor what will become of herself and paper and party if she has any. You will send this I hope to Mrs. Stanton and... she will be able to make some examination of this mysterious family.
Of Tennie C. I hear the most dreadful stories—yet she has the face of a sweet innocent child.
Undoubtedly, Isabella thought that women who had led the life attributed to Tennie would appear dissolute. The idea that sin was reflected in one's countenance was a common misconception. After Victoria visited Lucretia Mott and her sister, Martha Coffin Wright, in Philadelphia, the latter wrote to Anthony: "No one can be with her [Woodhull] without believing in her goodness and her purity but with her past we have nothing to do. She said to me, 'All that I am I have become through sorrow' and I had no desire to lift the veil from that." Anthony answered Isabella's query about looking into Woodhull's background in an irate, direct manner.
When we begin to search _records, past_ or _present_ —of those who bring brains or cash to our work for enfranchising women—it shall be with those of _the men—not the women_ , and _not a woman—not Mrs. Woodhull_ —until every insinuation of gossip of Beecher, Pomeroy, Butler, Carpenter shall be _fully investigated_ —and each of them shall have proven to _your_ and our satisfaction—that he never flirted, trifled with, or desecrated any specimen of Womanhood.
No! No!! Mrs. Hooker—it won't do to begin that— _are we more than Christ himself_ —that we shall thrust from us the _woman accused_ —Christ repeats to you and to me and to all women— _"Let her who never sinned in spirit_ or in act cast the first stone"—not until we _chastise and refuse men_ will I consent to _question women_ —and it is _only that Mrs. Woodhull is a woman_ —and that _we are women—all of an enslaved class_ —that we ever _dream_ of such a thing....
You see the _theory_ you propose for the _Woodhull Scandal_ applied to _men_ , living or dead is simply ridiculous.... I know you will see at once that the moment we begin the work of making _one woman account to us—our turn_ may come next and no end to the _persecution_ that would follow.
Anthony's admonition was exactly what Isabella needed to hear. When next Mary Livermore wrote her, disparaging Woodhull's antecedents and accusing her of "licentiousness," Isabella replied,
Woman cannot be injured by woman—only men are in danger—to them a vile woman is indeed the spark to a tinder, to us she is a sister for whom God has made us accountable.... Indeed the reason this is any campaign at all is that men perceive at last that the little white symbol of power is coming into our hands and begin to realize what we shall do with it.... The lips of women are being unsealed... and the day of judgment and purity draws nigh.
# CHAPTER TWENTY-TWO
SILENCE, TIME, AND PATIENCE
ABITTERLY cold winter had settled over Brooklyn. "There came scarcely one human being of all the church people and not a friend to call on me," lamented Lib Tilton. So quickly had the rumors spread across Brooklyn and into the heart of Plymouth Church that all shunned her. She lived, she said, in "a marked house"—marked with the sign of Cain, for had Theodore Tilton, her husband, and Henry Ward Beecher, her God, not been as brothers?
On January 11, 1871, in a snowstorm, Lib heard a knocking at her door. It was the elegantly dressed Paulina Wright Davis. "Oh, you have come to see me! For weeks I have been shut up from the world. I thought no one ever would come," Lib said, and burst into tears. Paulina entered the house to find it freezing. No fire had been lit in the hearth. At once, Lib obsessively related the events of the past months, starting with her husband's rage, triggered by the confession that she was enceinte by Mr. Beecher. Lib seemed surprised that Mrs. Davis had no knowledge of her affair and told her that the facts were "known and discussed in a considerable circle" and were "impossible to conceal."
Mrs. Davis was struck by Lib's desperation. She seemed pathetic, even deranged. Paulina left that evening for Washington, where she was introduced to Victoria Woodhull. Like Isabella, Paulina was much taken with the fiery Spiritualist. One evening she told Victoria about Mrs. Tilton's affair and the abortion, and said with passion, "I came away from that house, my soul bowed down with grief at the heart-broken condition of that poor woman, and I felt that I ought not to leave Brooklyn until I had stripped the mask from that infamous, hypocritical scoundrel Beecher."
For all his protestations of forgiveness Theodore Tilton too thought of Beecher as a villain who had cost him his marriage, his profession, and his power. As he brooded on this, he drank heavily and contemplated suicide. He could hardly look at his wife and took most of his meals with the Moultons. One afternoon Theodore unexpectedly returned home at four o'clock and found his wife on her knees in prayer beside the red velvet chaise longue, the scene of her infidelity. He pulled Lib to her feet, wrenched the wedding ring from her finger, and forced her to accompany him to Greenwood Cemetery. There he ground the ring beneath his heel into the dirt that covered their children's graves.
The enraged Tilton did not keep his promise of silence about his wife's affair. Lib's mother, Mrs. Morse, wrote to Beecher, "I know of twelve persons he has told and they in turn have told others.... He swears as soon as the breath leaves her body, he will make the whole thing public and this prospect is the one thing which keeps her alive."
Frank Moulton had not destroyed Lib's incriminating letters as he had promised. He was later to say that he had kept them in case Beecher once again damaged Tilton. In any case, Tilton wrote out several copies of Beecher's letter of apology and distributed them at whim. On her next visit he gave a copy to Elizabeth Cady Stanton. He pointed to the phrase in the letter of apology that read, "I humble myself before him," and boasted, "We have Plymouth Church at our feet."
Stanton tried to restore calm. Having been tempted to run away with Edward Bayard, she understood Beecher's dilemma. She urged Theodore to build a house for the family near her in New Jersey, where they could "enjoy a new peace and happiness." She wrote, "Let the dead past go, waste no energies in regret, but garner up the wisdom that comes from experience for future worth. You do not know how much interest and sympathy I feel for you and your little wife."
It did not take long for Sam Wilkeson to discover that his sister-in-law had a copy of Beecher's self-incriminating letter, and he arranged a meeting with her. She arrived with Susan B. Anthony, and both women related all that Lib and Theodore had told them separately that evening in Brooklyn. Stanton then told Wilkeson that women in her organization as well as many men "heard and repeated the scandal." (She was later to write, "That Henry Ward Beecher was deeply infected with these [free love] doctrines was no secret at all.... Mr. Beecher held a sexual theory which he believed to be in advance of the present constitution of society.") Wilkeson mopped his ample brow and said, "If this gets out, it will knock _The Life of Christ_ higher than a kite!"
Wilkeson beseeched these women not to make their discoveries public. He explained that Henry Ward Beecher must be supported at all costs: The _Christian Union_ , Plymouth Church, and the J. P. Ford syndicate, publishers of _The Life of Jesus the Christ_ , had too much at stake to see Beecher sacrificed. Stanton realized that "the destruction of Mr. Beecher would be the destruction of all these." As Beecher himself was later to say, "I have been the centre of three distinct circles... the _great_ church... the _newspaper_... the _book_.... The _sensitiveness_ of so many of my people would have made any appearance of trouble... an occasion of alarm and notice, and have excited, when it was important that rumors should die and everything be quieted."
_Samuel Wilkeson, who tried to suppress the scandal. Portrait by Mathew Brady, 1863_
Stanton put it more bluntly: "It was a matter of money. The church property is not taxed, its bonds in the hands of the wealthy men of that organization are valuable, and the bondholders, alive to their financial interests, stand around Mr. Beecher, a faithful, protecting band—not loving truth and justice less, but their own pockets more." Though Elizabeth Cady Stanton was his wife's sister, Wilkeson threatened her. He said that with all this money at stake the "lie," as he called it, would be "saddled upon us women," wrote Stanton. Beecher must not fall. Sam Wilkeson assumed that most problems could be solved with money. Using his position as chief representative of Jay Cooke's Northern Pacific bond syndicate, Wilkeson wrote Tilton that he was prepared to send him, as he had Henry Bowen, an immediate payment of $15,000 and added,
You are in trouble. I come to you with a letter just mailed to Jay Cooke, advising him to secure your services as a platform speaker to turn New England, old England, or the great West upside down about our Northern Pacific.
Pluck up heart! You shan't be trampled down. Keep quiet. Don't talk. _Don't publish_. Abide your time and it will be a very good time. Take my word for it.
Sam Wilkeson
Tilton was hard up for cash but, unlike Bowen, he was not a man to take a bribe. He did not reply to Wilkeson's letter.
It fell to Frank Moulton to suppress the scandal and make peace among the parties. With Beecher's encouragement he went to Tilton and offered him a trip abroad with his family to be financed by Wilkeson and his associates. As Moulton might have predicted, Tilton angrily refused. This was not the way to placate an honorable man. Moulton knew that much of Tilton's bitterness came from the loss of his job and thus his public platform. Laura Curtis Bullard's ownership of _The Revolution_ had in effect made it Theodore Tilton's publication, but Laura had tired of the financial and emotional drain. Anthony wrote to her beloved Anna that Mrs. Bullard wanted her to " _take back The Revolution_ —she is sick & tired of the work & the new Co. of sinking the cash—& they have spent it _so foolishly_." Anthony worried that her rivals in Boston would acquire the paper, " _thus putting_ in _Lucy Stone's hands the prestige_ , the results of all _my hardest 20 years_." But after owning _The Revolution_ for eighteen months, Mrs. Bullard simply abandoned it.
Moulton told Tilton that he would be able to secure funds from people who admired him to finance a new publication, the _Golden Age_ , which would be solely owned and edited by Tilton. What Moulton did not tell Tilton was that Henry Ward Beecher would be his major financial backer. To raise his share of the money, Beecher persuaded his wife, Eunice, to sign a mortgage for $5,000 on their Columbia Heights home. Also, unbeknownst to Tilton, Moulton arranged for Beecher to pay the tuition at a private boarding school in the West for Bessie Turner, Lib's ward, who had been privy to the entire scandal.
But no sooner had one problem been solved than another arose. Mrs. Morse had counted on Beecher's becoming her "Son"—in other words, on his supporting Lib as his permanent mistress. But Beecher, in her eyes, had simply abandoned them. She wrote to the preacher asking him for money. "Can you help?... With a rent of $1,500 and an income of $1,000 the consequence is with other expenses I shall be by the first of the month terribly behind." When Beecher ignored her letter she wrote again: "You have not seen fit to pay any attention to the request I left at your house now over two weeks since.... Do you know when I hear you cracking your jokes from Sunday to Sunday and I think of the misery you have brought us, I think with the Psalmist there is no God." Finally, she threatened to reveal "the secret of her [daughter's] life."
Beecher prevailed upon Moulton to win Lib Tilton's confidence and urge her to stop her mother's threats. He assured Moulton that his love affair with Mrs. Tilton was ended forever and he "would do all he could to restore her love" to her husband. But all communication between Beecher and Lib had been forbidden by Tilton. Frank Moulton suggested that the preacher write to Mrs. Tilton to convince her to trust Moulton as the peacemaker while Frank himself would secure Theodore's permission to deliver the letter. Beecher wrote,
When I saw you last I did not expect ever to see you again or to be alive many days. God was kinder to me than were my own thoughts. Mr. Moulton the friend whom God sent to me has proved above all friends that ever I had, able and willing to help me in this terrible emergency of my life. His hand tied up the storm that was ready to burst upon our head. May not this friend stand as a priest in the new sanctuary of reconciliation, and mediate and bless you, Theodore, and my most unhappy self? He will be a true friend to your honor and happiness. In him we have a common ground.
With Moulton as go-between, Mrs. Tilton stopped her mother's threats, and the Tilton household once again took on a semblance of normality. Feeling that at last all was well, Lib wrote Laura Curtis Bullard about the events of the past months. Only Laura could fully understand the great social experiment on which they had embarked, an experiment that had so disastrously failed.
Dear Friend and Sister:
You, like me, have loved and been loved.... I find in you an element to which I respond... I cannot reason—only feel. Memories of you fill me with admiration and delight. I have caught up your card picture and kissed it again and again, praying with tears for God's blessing to follow you and to perfect in us three the beautiful promise of our nature. But my sweet and dear I realize in these months of our acquaintance... how almost impossible it is to _bring out_ these blossoms of our heart's growth—God's gifts to us—to human eyes. Our pearls and flowers are caught up literally by vulgar and base minds that surround us on every side, and so destroyed or abused that we know them no longer as our own, and thus God is made our only hope.
My husband has suffered much with me in a cruel conspiracy made by my poor suffering mother, with an energy worthy of a better cause, to divorce us by saying that I was seeking it because of Theodore's infidelity, making _her_ feeling _mine_....
All might have been well had not Theodore decided to summon Beecher to his house to determine the paternity of Ralph, who had been born on June 20, 1869. Beecher arrived while the Tiltons were finishing breakfast, and Lib was asked to confirm the date that their affair began. She checked her diary and said it was on October 10, 1868, that she had made the entry "A Day Memorable." After making a quick calculation, Tilton, satisfied that Ralph was his child, announced that he would change the boy's name to Frank, in honor of Frank Moulton. But Lib, having seen her preacher again, could not resist the urge to correspond with him clandestinely. "My future either for life or death would be happier could I but feel you _forgive_ while you forget me," she wrote. And a week later: "Does your heart bound _towards all_ as it used? So does mine! I am myself again... of course I should like to share with you my joy but I can wait for the Beyond!" To which Beecher replied, "Your note broke like spring upon winter and gave me an inward rebound to life. No one can ever know, none but God, through what a dreary wilderness I have passed."
VICTORIA WOODHULL HAD gone to Washington to find fame but stayed to work for woman's rights and soon came to believe she possessed the power to alleviate the burdens of women like herself who had been abused and exploited and had been powerless in a man's world. She often heard spirit voices telling her what she must do. She vowed that no longer would she use her powers frivolously or for personal gain but to transform society and lead women to the millennium.
On the night she returned from Washington, Victoria, "as a talisman against any temptation toward untruthfulness," stitched into the sleeve of each of her dresses the second verse of the 120th Psalm: "Deliver my soul Oh Lord from lying lips and from a deceitful tongue." To Isabella Beecher Hooker she poured out her innermost thoughts.
My Dear, dear friend,
I am often compelled to do things from which my sensitive soul shrinks and for which I endure the censure of most of my friends. But I obey a Power which knows better than they or I can know and which has never left me stranded and without hope. I should be a faithless servant indeed were I to falter now when required to do what I cannot fully understand, yet in the issue of which I have full faith.
None of the scenes in which I have enacted a part were what I would have selfishly chosen for my own happiness. I love my home, my children, my husband, and could live a sanctified life with them and never desire contact with the wide world. But such is not to be my mission. I know what is to come, though I cannot yet divulge it. My daily prayer is that Heaven may vouchsafe me strength to meet everything which I know must be encountered and overcome....
On March 28, 1871, the very day that Victoria Woodhull returned from Washington, Paris workers supported by elements of the bourgeoisie, who were embittered by France's loss to Prussia in the War of 1870, and offended by the new laws to protect creditors and landowners, rebelled. They formed their own municipal council and issued the proclamation of the Paris Commune. Two days later the national government and the communards were at war. Immediately, Stephen Pearl Andrews became a supporter of the Paris Commune and convinced Victoria Woodhull that the communards were honoring the rights of women by guaranteeing them economic, social, and political equality with men. In Paris, he said, women were living her dream of breaking away from their traditional roles.
Victoria Woodhull, following Andrews's lead, was determined to follow in the footsteps of her French sisters and allied herself with radical elements who hoped to transform American society. The American Labor Reform League, a group composed of anarchists, socialists, and other freethinkers, had scheduled its first meeting for May 8 at Cooper Institute, and at Andrews's suggestion, Woodhull presented the keynote speech. It had been composed by Andrews, although at the time she claimed to have written it. She read "The Great Social Problem of Labor and Capital" without passion and stumbled over a few of the longer words, nonetheless she added the support of radical labor reformers to the Spiritualist and woman's rights groups who endorsed her for the presidency.
For the first time Victoria took an active part in the editorial policies of _Woodhull & Claflin's Weekly_. She changed the motto from "Onward and Upward" to "Progress! Free Thought! Untrammeled Lives." Early in 1871, while she was in Washington, articles appeared in the publication favoring the legalization of prostitution. Now editorials by Woodhull herself explained that those articles, written in her absence, were contrary to the actual views of the _Weekly_.
WHEN ISABELLA BEECHER Hooker returned to her Nook Farm home in Connecticut, she set about changing the minds of her sisters about Victoria Woodhull, particularly Catharine, who had been living nearby with their sister Harriet Beecher Stowe, with whom she had co-authored _The American Woman's Home_ , a guide to proper deportment, domestic duties, and child care. Catharine had recently completed _Woman Suffrage and Woman's Profession_ , in which she praised domesticity as woman's total fulfillment and insisted that women had no place in politics. For thousands of women Catharine Beecher spoke with authority on maternal and marital felicity, neither of which she had experienced.
Catharine was the eldest of Lyman Beecher's eleven surviving children, born at a time when her father, a great Calvinist preacher, was young and full of fire. Catharine was the center of his attention until she was two and his first son was born. She had been a rambunctious girl whose father was constantly trying to control her, both mind and body. When she was eight, Lyman Beecher held her head in a bucket of water until, sputtering, drowning, her willfulness gone, she acquiesced to his complete authority. Equality vanished in the water that swept into her mouth and nose. Fourteen years later, sexuality vanished in the water that swept her fiancé to a fate where her father insisted he had gone unredeemed. At twenty-two, Catharine had met Alexander Fisher, an acclaimed mathematician and, by several accounts, a gentle soul. "I could not ask for more delicacy and tenderness," she wrote a friend. Shortly after their engagement, Fisher sailed for Europe to study advanced mathematics but his ship foundered and he was lost at sea.
At the time of Fisher's death, Lyman Beecher was forcing upon his daughter the act of conversion, the complete submission and obliteration of self in the name of his Calvinist faith. Before Fisher's departure Catharine had attempted to achieve conversion, but with his death she refused her father's harsh religion with its commitment to predestination, which allowed the possibility that Alexander might burn in hell. Yet in her dreams she saw the water engulf him and heard her father's voice call out the dread phrase, "No salvation!"
Catharine rejected Calvinism and chose a life whose rules she created for herself. Teaching would be her religion, and guiding a generation of women, her mission. She believed women should use gentle wiles to persuade the men who dominated them, on the theory that they would not deny their dutiful wives and children. Women might thus gain power by influencing husbands and sons toward greater benevolence. She feared men for their abstract power, but most of all Catharine feared sex, equating it with violent possession. "As things now are, men have the physical power that can force obedience; in most cases they have the power of the purse and in all cases they have civil power." "Force obedience" the way her father, Lyman Beecher, had when he held her head in that bucket of water.
Catharine was particularly fond of her half-sister, Isabella, who had been born just two months before Fisher was lost at sea. At fourteen, Isabella was taken into the school Catharine had founded, the Western Female Institute in Cincinnati, Ohio, and given the only education she was to receive. Catharine tried to keep an open mind concerning her sister's attachment to Mrs. Woodhull but she strongly believed that Victoria and women like her were introducing an explosive element into politics, demonstrating a power that they should not possess, much less wield. She wrote, "Many intelligent and benevolent persons imagine that the grand remedy for the heavy evils that oppress our sex is to introduce woman to political power and office, to make her a party... in the scramble and fight for political offices, thus bringing into this dangerous melee the distinctive tempting power of her sex. Who can look at this new danger without dismay?"
If only her sisters could meet Victoria Woodhull, thought Isabella, they would be "just as much in love with her as the rest of us." She explained to Harriet and Catharine that Woodhull had been selected as God's instrument for righting social injustice. Although she did not understand all of Woodhull's theories, Isabella told Catharine that she was convinced that this woman was a visionary and that Isabella herself was destined to play a great part in her divine plan. Woodhull had at her command spirit forces that would bolster her. She had penetrated the curtain between heaven and earth so that the spirits could cross over. As Victoria had demonstrated to her, Isabella made a ladder with her fingers showing her sisters the way one climbed to higher and higher planes until they passed over effortlessly from earthly life to another world.
Though Catharine was disparaging, Harriet Beecher Stowe was sympathetic to Spiritualism. After her son, Henry, drowned while swimming in the Connecticut River in New Hampshire, Harriet turned to Spiritualism to make contact with him. She said she could see her dead son's spirit and other spirits hovering nearby. Her husband, Calvin, also had visions. Once Calvin left their house to catch a train but, remembering he had forgotten his hat, returned to get it. Harriet was sitting at the desk writing. He passed her by without a word. Finally, she asked, "Why will you not speak to me?" Calvin replied, "Oh, I thought you were one of my visions."
Harriet believed that Spiritualism was "a reaction from the intense materialism of the present age"; nonetheless she told Isabella that she disapproved of any of the Beechers associating with this "vile woman." Perhaps because Harriet so disapproved, Catharine gave in to Isabella's persuasion and agreed to meet with her sister's "Darling Queen." Catharine felt that "Mrs. Woodhull is a pure woman holding a wrong social theory and ought to be treated with kindness if we wish to win her to the truth."
As Catharine stood under the immense crystal chandelier in the Woodhull living room, Victoria greeted her cordially, extending both her hands so that Catharine could feel her magnetism. Since it was a mild spring day Victoria suggested a ride in Central Park. At three o'clock carriages arrived for their afternoon promenade. Victoria and Catharine settled into the crimson velvet seat of Woodhull's brougham, which was equipped with parasol holders, a small vanity, and a call bell for summoning the driver. Catharine Beecher felt obligated to show this uneducated woman how misguided she was: No doubt Woodhull's feelings for woman's oppression were sincere but her methods were destroying any hope of progress. Catharine opposed almost everything Victoria was for. "Disastrous influences—the teachings of Free Love, the baneful influence of Spiritualism... the fascinations of the _demimonde_ , the poverty of thousands of women who, but for desperate temptations, would be pure—all these malign influences are sapping the foundations of the family state," she had written.
Catharine, accustomed to the deference of other women, began in her didactic way to lecture Victoria who, once or twice, tried to answer but found it impossible to interrupt this determined woman. Catharine explained that it was terribly dangerous for a woman to try to establish herself as the equal of man. Only at great peril did she try to challenge man's power directly. She told Victoria that a woman of breeding—one with proper antecedents—could gently persuade men and that this moral power was the only proper one to wield.
_Harriet Beecher Stowe and her husband, Calvin Stowe. President Lincoln said to Harriet, "So you're the little woman who started this great war._ " (Illustration Credit 22.1)
_"Antecedents!_ " Would Victoria never escape these women who felt superior to her, who regarded her simply as one of the shiftless Claflins? _Antecedents_ —how dare she!
Catharine droned on, preaching her domestic gospel: A woman's sexual acquiescence was a gentle gift that found its proper place in procreation. Any other manifestation was evil. Marriage was a sacred, lifelong bond. Easy divorce and free love were "the same as asking civilized human beings to return to the lustful instinctiveness of animals." If Victoria adhered to this dangerous doctrine she would soon be alone in her satanic beliefs, which, Catharine cautioned, "must come from a misunderstanding of women's role or else you are in the possession of some powerful, malignant spirits."
Anger welled up in Victoria; suddenly around Catharine's head she saw a band of devils with rat-tails. Later Victoria wrote of this occasion, "Catharine Beecher imagined the malignant spirits manifested by herself were mine instead of hers. She was drunk of them and as drunkards invariably do, thought that they were confronting instead of being possessed by her."
"You are misguided," Victoria finally said. "Many great people have already accepted and are living my theories of social freedom though they are not ready to become its avowed advocates, as I am. You speak of Free Love with derision while your own brother, Henry Ward Beecher, the most powerful preacher in America, openly practices it. I do not condemn him, I applaud him. Would that he had the courage to join me in preaching what he practices."
"Evil!" Catharine exclaimed. It seemed to Victoria that Catharine suddenly wilted, drawing herself into the corner of the carriage. "I know my brother is unhappy but he is a true husband. I will vouch for my brother's faithfulness to his marriage vows as though he were myself."
"But you have no positive knowledge that would justify your doing so," said Woodhull.
"No... no positive—," stammered Catharine. "I know he is unhappy. Mrs. Beecher is a virago, a constitutional liar and a terrible woman altogether, so terrible his friends and family seldom visit. But unfaithful—no. I will hear no more of it."
"You will hear," said Victoria. "In concubinage with his parishioner's wife—it is common knowledge. And if you were a proper person to judge, which I grant you are not, you should see that the facts are fatal to your theories."
Catharine was now livid. Wisps of hair as gray as her complexion shook as if on tiny springs. "Victoria Woodhull, I will strike you for this. I will strike you dead!" she cried out.
"Strike as much and as hard as you please. Only don't do it in the dark so I cannot know who is my enemy."
"Stop!" commanded Catharine, who then clambered out of the carriage, and holding her voluminous skirt moved rapidly across the Greensward of Central Park. Victoria watched her go, the devils with rat-tails sailing off through the sky, a fluttering covey of malignancy like a dark cloud above her head.
From Hartford, Isabella noted, "Sister Catharine returned last night. Saw Victoria and attacked her on the marriage question, got such a black eye as filled her with horror and amazement. I had to laugh inwardly at her relation of the interview and am now waiting for her to cool down!" Catharine Beecher never denied Woodhull's account of what had passed between them during their Central Park carriage ride, but she did not tell Harriet or Isabella the specifics of their conversation or mention their brother Henry. From then on, however, Isabella was under increased pressure from her sisters to abandon her new friend. Catharine referred to Victoria and Tennie C. as "those two prostitutes." Harriet Beecher Stowe and Mary Beecher Perkins wrote a joint letter to John Hooker, imploring him to bring Isabella "back to God and away from that harlot."
Isabella fought back. "Do you believe I could have taken her to my heart as I have done, if I did not believe her true and pure?" she wrote her sisters, and reminded Harriet how she had stood by her during the Lady Byron affair. Isabella bluntly added, "You and Mrs. Perkins each have a drinking, licentious son, and yet you know that you would be glad for them to marry the purest young girl and have tried every means to effect it hoping it might redeem them, and yet you judge Mrs. Woodhull."
When Susan B. Anthony was asked by a _Tribune_ editor if she knew about "Mrs. Woodhull's antecedents," she answered that she "did not care any more for them than I do about those of the members of Congress." Victoria Woodhull, though she professed indifference, was stung by the accusations against her. It was at this point that she began what was to become a lifelong attempt to manufacture an idealized genteel background for herself. Eventually, this was to extend to her changing the names and professions of various members of her family. She offered one such sanitized history to Martha Coffin Wright, one of the most powerful members of the woman suffrage movement, who then relayed it to her daughter, Eliza: "She and her five sisters, all married but one, have always lived together in their father's house—one of the largest on Fifth Avenue.... She scorns the innuendoes against her character and challenges any one to point to a single act in her life that would not bear scrutiny. She says 'let them float,' they do not hurt her and she has not time for any notice of them." But, in truth, notice them Victoria Woodhull did. And how they rankled. Victoria wrote Isabella,
Under all the curses and imprecations which are being heaped upon me, strong though I feel, I need some little sustaining presence from those who I believe comprehend me.... I must confess to not a little surprise that whatever I have done or may do is at once denounced as imprudent, unwise and the endeavor made to stigmatize me as a very improper person.
I shall not change my course because those who assume to be better than I desire it.... It grieves me that there should be anything to interfere with obtaining justice at the earliest possible moment. Some say they would rather never obtain it than that it should come from such a source. Why should the... _Woman's Journal insinuate_ without stating. I thought that it was a question of _Right_ under the _Constitution_ , I did not know that it was a question of _Antecedents!_
# CHAPTER TWENTY-THREE
THE WORST GANG
TWENTY-THREE years had passed since Seneca Falls and the first glimmering of woman's rights. To celebrate that anniversary, the National Woman Suffrage Association scheduled a weeklong convention to begin at New York's Apollo Hall on May 11, 1871. And once again the American Woman Suffrage Association announced a simultaneous convention at Steinway Hall. The previous year, the NWSA had been virtually ignored or dismissed while the AWSA received glowing press coverage. This time, however, their organization would not be overlooked, for Anthony, Stanton, and Hooker had aligned themselves with Victoria Woodhull, a woman no one could ignore.
For the forces behind Victoria Woodhull, the NWSA convention provided an opportunity to promulgate their views and develop a greater constituency for her bid for the presidency. Stephen Pearl Andrews drew up twenty-five articles to be presented to the convention, representing his own vision of a utopian world. The first article dealt with woman's rights, but the rest advocated such wide-sweeping reforms as a revised civil and criminal code, government control of public enterprises, public ownership of mines and waterways, distribution of public land to settlers, high taxes on income and property. As Andrews was completing this list, his wife, Esther, died. Stricken, he locked himself in an upstairs bedroom of Woodhull's house and said he would remain there until Esther informed him that she had "safely made the journey to Summerland." Elizabeth Cady Stanton stepped into the breach and stayed at Victoria's house to help complete the final arrangements for the convention.
Ever since her arrival in New York, Mama Roxy had been openly hostile to visitors and "that damn scoundrel Blood," who had stolen the daughters who were her livelihood. Her daughter Polly's husband, Dr. Benjamin Sparr, had moved into the house and soon became Roxy's ally in trying to extort money from her daughters' acquaintances. The final rift came when Roxy confided to Tennie that, just as they used to do, a letter had been written to "a certain gentleman" that should yield them lots of money. When it arrived they could go back to Ohio to begin life anew. Tennie bristled. "What have you done?" she demanded of her mother. "Never, never will I go back to that life." It was one of those blackmail letters—she knew it. And worse, she made her mother admit that it had gone to Commodore Vanderbilt. Then and there Tennie knew that she had lost the Commodore as a patron. Vanderbilt didn't care what people said about him, but he had trusted Victoria and Tennie. Now that trust had been irrevocably broken and Tennie knew that no matter what she did or said he would never forgive them.
Tennie, as always, was unable to cope with this woman who had emotionally enslaved her. Instead, she turned her rage on Dr. Sparr. This was his doing. Three days before the convention, Roxy and Buck Claflin and the Sparrs were banished from the house on East Thirty-eighth Street, but Tennie paid the rent for her parents at a comfortable boardinghouse. As Tennie had predicted, the Commodore severed all connection with them. Soon after, when a reporter asked Vanderbilt how his Wall Street protégées were faring, he answered, "They hain't no friends of mine. From what I hear, you shouldn't be associating with such folks."
There was not a vacant seat in Apollo Hall: Members of the NWSA, Spiritualists, working women, businessmen, politicians, and members of Section 12 of the International Workingmen's Association—a socialist organization that represented labor in its Marxian opposition to capitalism—were in attendance. In 1869, Stephen Pearl Andrews had received permission from the General Council of the International in London to form two American branches. Victoria was named honorary president of Section 12, the more visible of the two. Some NWSA members refused to sit next to Victoria, lest the contagion of her reputation infect them. Susan B. Anthony noted, "great trouble allowing Mrs. Woodhull on our platform." Stanton insisted that Woodhull be placed between herself and the venerable Lucretia Mott "to give her respectability." When Mott hesitated, Stanton told her that Victoria had the courage to advocate openly what people like Henry Ward Beecher practiced secretly. Woodhull was duly seated.
A reporter noted that Victoria sat perfectly still: "... a small splinter of the indestructible.... If her veins were opened they would be found to contain ice." But her keynote address was electrifying. She was obviously in the grip of an ecstatic vision: Women could change the world if only they dared change it. Her rhetoric was now militant.
Why do I war upon marriage... because it is, I verily believe, the most terrible curse from which humanity now suffers, entailing more misery, sickness, and premature death than all other causes combined.... Sanctioned and defended by marriage, night after night there are thousands of rapes committed, under cover of this accursed license. I know where of I speak—millions of poor, heartbroken, suffering wives are compelled to minister to the lechery of insatiable husbands when every instinct of body and sentiment of soul revolt in loathing and disgust.... Prate of the abolition of slavery, there was never servitude in the world like this one of marriage.
I have asked for equality nothing more.... Sexual freedom means the abolition of prostitution both in and out of marriage, means the emancipation of woman from sexual slavery and her coming into ownership and control of her own body, means the end of her pecuniary dependence upon man... means the abrogation of forced pregnancy, of antenatal murder of undesired children and the birth of love children only.
Woodhull preached a new gospel. She directed women to
rise and declare... yourself free. Women are entirely unaware of their power. Like an elephant led by a string they are subordinated by... just those who are most interested in holding them in slavery. If the very next Congress refuses women all the legitimate results of citizenship... we shall proceed to call another convention expressly to frame a new constitution and to erect a new government.... We mean treason, we mean secession, and on a thousand times grander scale than was that of the South. We are plotting revolution! We will overthrow this bogus Republic and plant a government of righteousness in its stead....
Cheers filled the hall, and women waved their handkerchiefs to show approval. When Woodhull returned to her seat, Lucretia Mott, tears rolling down her cheeks, reached over and clasped Victoria to her breast.
The applause had scarcely quieted when Paulina Wright Davis came forward and read out a series of resolutions that had been written by Stephen Pearl Andrews the previous month. One of them, though it did not mention Victoria Woodhull by name, was clearly written with her in mind:
The inquisitional impertinence of an investigation into the personal characters of women who are able and willing to cooperate in the movement... shall be completely and definitely set aside and ended. All laws shall be repealed which are made use of by Government to interfere with the rights of adult individuals to pursue happiness as they may choose.... This will place the intercourse of persons with each other upon their individual honor.
This resolution was, of course, an endorsement of free love. The women of the rival AWSA were shocked, and Mary Livermore countered with opposing resolutions. Anthony wrote in her diary, "AWSA meeting passed resolutions saying that they were not Free Lovers. Why not one saying that men are not thieves and murderers?"
Martha Coffin Wright felt that the press, and especially Horace Greeley's _Tribune_ , had falsely reported that the NWSA had adopted Stephen Pearl Andrews's "ridiculous string of resolutions," which Mrs. Davis had read "in the innocence of her heart." In fact, she noted, "they fell deader than Casca, and no action at all was taken on them." Then, reflecting her reservations about Andrews's influence, Wright wrote to Anthony, "It seems to me we have pretty squally times ahead. I wish there was an oyster big enough to hold that _Pearl_ of great price, or that he might share the fate of his namesake _Stephen_. His foolish mistimed resolutions have done harm in giving the philistines a chance to rejoice." Anthony agreed: " _Professor Purlo's hypotheses_ are too ridiculous—his moonshine impracticalities—oh dear dear what terribly _rough seas_ we do have—but the _calm_ must be near." The worst, however, was yet to come.
On the morning that Victoria Woodhull left for Apollo Hall to make her revolutionary speech, Colonel Blood had hidden the _Times_ , which carried the news that Roxanna Claflin, the mother of the self-appointed presidential candidate Victoria Woodhull, had appeared before one Justice Ledwith and sworn out a warrant for the arrest of "James H. Blood, alias Dr. J. Harvey," on charges that he "had succeeded in corrupting her daughters Victoria and Tennessee" and had "entirely weaned them from their affectionate and never to be consoled mother." Mrs. Claflin also charged that she had "often heard Blood insist that Tennessee should make efforts to secure the attentions of different married gentlemen of wealth in order that they might make money out of them. Such men as were secured, she charged, were blackmailed by Blood." Mrs. Claflin alleged that the house on Thirty-eighth Street was overrun with woman's rights people, communists, and free lovers brought into the household by Blood and Andrews.
On the afternoon of May 15, 1871, the Essex Market police court was jammed with spectators. Roxy Claflin, wizened in a worn black silk-faille dress and a black Spanish-fringed shawl, told the judge, "My daughters were good daughters until they got in with that man Blood. He threatened my life several times.... He said he would not go to bed until he had washed his hands in my blood. He is one of those who has no bottoms in his pockets. You can keep stuffing in all the money in New York but they never fill up. If my daughters would just send this man away, as I have told them to do, they might be millionairesses."
Roxanna Claflin drew a breath and then clasped a Bible to her breast. "I say here and I call heaven to witness that there is the worst gang of communists and free lovers in that house that ever lived—Stephen Pearl Andrews and Dr. Woodhull and lots more such trash and—"
Roxy's own lawyer, Steven Townsend, interrupted. "Keep quiet, old lady," he commanded. But she continued. "I want to tell the judge what these people are—they're trash.... I was threatened in that house. If it had not been for my son-in-law Dr. Sparr they would have put me on Blackwell's Island—so help me God. They would have driven me to an insane asylum. They've got Stephen Pearl Andrews, that old free lover who is the worst man on earth. Until that free love and communist business came into my family there was peace and quietness. Stephen Pearl Andrews, that old free lover who asked, 'What is Jesus Christ more than any other man?' " In the course of her testimony Roxy Claflin spoke of communism and free love over and over again. After Roxy was excused, Polly Sparr took the stand and stated that she had not seen Blood use violence toward her mother but she had seen Tennie and Vickie invoke the spirit of Demosthenes, who, "speaking through Vickie, threatened my mother that she would be taken to an insane asylum."
Victoria was the next witness. She testified that she had married Dr. Woodhull at fifteen years of age and had subsequently divorced him. She said she had married, divorced, and remarried Colonel Blood and had lived with him for eleven years. Victoria added that the proceeds from her brokerage house supported as many as twenty family members. At various times, there had been ten children residing at Thirty-eighth Street. She asserted that her mother had "abused Colonel Blood without any cause whatsoever. The whole trouble was that my mother wanted to get Tennie back, wanted her going around the country telling fortunes. That is the cause of this action."
Then Colonel Blood testified. Townsend questioned him on his marital status. Colonel Blood was evasive. He said he had married Mrs. Woodhull in 1866.
"Was Mrs. Woodhull divorced when you married her?" asked Mr. Townsend.
"I don't know."
"Were you not afterwards divorced from Mrs. Woodhull?"
"Yes, in Chicago in 1868."
"How long were you separated from her?"
"We were never separated. We continued to live together."
"When did you remarry?"
"I'm not sure."
"Have you seen Dr. Woodhull?"
"I see him every day. We are living in the same house."
"Do you and Mrs. Woodhull and Dr. Woodhull occupy the same room?"
Colonel Blood glared at Roxy's lawyer but made no reply. His own lawyer, John Reymert, said, "Please tell the court that Dr. Woodhull lives in the same house and who supports him."
"The firm of Woodhull, Claflin and Company has supported the whole of them," Colonel Blood shot back. "Mrs. Woodhull's boy received a fall when he was young. He needs his father's medical care and treatment."
There was a stir in the courtroom as these irregular arrangements came to light. Blood had couched his explanation in conventional terms, for he was not about to testify to his true belief in "the abolishment of the marriage statute as it is now constituted." Both Blood and Woodhull openly espoused free love theoretically but maintained that their personal conduct was a strictly private matter. They were aware that throughout history, sexual promiscuity had been used as a weapon against those one wished to destroy.
Tennessee Celeste Claflin followed Blood to the stand. She turned to smile at the reporters in the front row, looked up imploringly at the judge, turned to stare at Townsend, and then kissed the Bible. She began to testify as if she were speaking to friends. "Mr. and Mrs. Sparr, they lived with me until I had to dump them out. I have always supported that man and his wife." Not waiting for a question from Townsend, she added, "I never knew Colonel Blood to use violence toward my mother. He only treated her too kind. In fact, I don't see how he stood it. My mother and I always got along together until Sparr came into the house. Benjamin Sparr has been trying to blackmail people through my mother."
Judge Ledwith interrupted: "This is altogether irrelevant, if it is objected to, I will rule it out."
Townsend said, "I have objected, but I can't stop her from talking." From her purse Tennie removed a sheaf of letters and began waving them. "I have been accused of being a blackmailer. The _Tribune_ said that I humbugged Commodore Vanderbilt. Other papers have said that I blackmail men for money. I want it ventilated. I have a lot of letters here supposed to be written by my mother for the purpose of blackmailing different persons in this city. My mother can't read or write. They were written by this man Sparr."
Again Judge Ledwith cut in, "Those letters are ruled out."
Tennie disregarded him. "I came here to sustain my character and I'm going to sustain it. Mrs. Woodhull has a protector but it's been announced to the world that I am a blackmailer. They're trying to make me out a bad woman."
Townsend tried to clarify the matter. "What was the reason that your mother quarreled with Colonel Blood?" he asked.
"My mother was against my sister Vickie and Colonel Blood because they have taken me from the life of a traveling clairvoyant and fortuneteller. They are the best friends I ever had outside of my mother, and they rescued me. I am a Spiritualist, and I have great power, but it was a hard life, and I could not endure it. My mother has said that she would ruin me and bring me back to my old life. But she was influenced by this man Sparr and his wife."
Reymert leaned forward and said softly, "Madam, just try to answer the questions." But Tennie was overwrought. "I have always been the martyr of this family. Yes, since I was eleven years old I used to tell fortunes with her, and since I was fourteen years old I have kept thirty or thirty-five deadheads. I am a clairvoyant. I am a Spiritualist. I know my power!
"Many of the best men in the street know it too. Commodore Vanderbilt knows my power. I have humbugged a great many rich people, I admit it. But I did it to make money to keep all these deadheads. I have led a terrible life, and because I will not return to it and leave my present position, I am persecuted."
The judge turned to Reymert and asked, "Can't you keep her from this irrelevant testimony?" Then he directed both lawyers to approach the bench. While they conversed, Tennie sprang from the witness chair, ran to the mother who had caused all this damage and trouble, and clasped her in her arms. "Come with me, come with me, let us leave here," Tennie implored. Polly Sparr pulled at her mother, trying to separate the two women. Colonel Blood moved forward, patted Tennie on the cheek, and whispered softly, "Retire, my dear. Do retire. You are only making yourself conspicuous." Reymert then asked for a recess while Tennie was led into an inner chamber. Shortly thereafter, it was announced that Tennie C. Claflin would not resume the stand.
At last reporters began to look carefully into the history of Victoria's family. All during anniversary week, accounts of "the Woodhull Convention" were accompanied by scandalous revelations. The _St. Louis Times_ divulged much of the Claflins' tawdry background, including Buck's pilfering of the post office mail in Homer, Ohio, and the suspicion that he had burned down his own gristmill to collect the insurance. The paper then turned to Victoria herself, stating that she had a scandalous "antecedent history." After she had operated "a _house_ in Chicago in a grand and _peculiar_ style," it reported, "she removed to the sedate city of St. Louis... where she appeared as the proprietress of an eclectic institution located on Sixth Street next door to the present City Dispensary."
The _Cleveland Leader_ reported, "One unfortunate fact... is that Mrs. Woodhull has made herself a prominent figure in the Woman Suffrage Movement... and now her shameful life has been exposed, it will follow that the enemies of female suffrage will point to her as a fair representative of the movement.... In Cincinnati years ago, she was the same brazen, snaky adventuress that she now is."
Henry Bowen in the _Independent_ excoriated both Woodhull and her publication.
No subject discussed during anniversary week excited so much attention as the question of suffrage for women.... The excitement was not a little fanned by the sudden revelation of facts in the private life of Mrs. Woodhull.... Mrs. Stanton, Miss Anthony and Mrs. Hooker had been foolish to have given a prominent place to Mrs. Woodhull.... _Woodhull & Claflin's Weekly_ with its coarse treatment of all the sacred things of human life is enough to condemn anyone whose name is associated with it.
In a letter to Mrs. Stanton written during anniversary week, Martha Coffin Wright complained about Horace Greeley's "persistent determination to fasten the Free Love question" on the NWSA. Now Greeley attacked with even more ferocity. The _Tribune_ headline read, "Free Love Is Free Lust." Of Woodhull, Greeley wrote that she was "one who has two husbands after a sort, and lives in the same house with them both, sharing the couch of one, but bearing the name of the other (to indicate her impartiality perhaps)." He demanded that the NWSA explain why it had made a leader of someone who followed this doctrine.
Horace Greeley might be a popular hero, but to Victoria Woodhull he was a villain. She had met Mrs. Greeley through Stanton and knew that Mary's rage against her husband was the only thing that kept her going. Here was a man who had destroyed his wife's physical health with repeated pregnancies and her mental condition with his gross insensitivity. She struck back in _Woodhull & Claflin's Weekly_. Victoria wrote what no one else was willing to say aloud.
Mr. Greeley's home has always been a sort of domestic hell. I do not mean that Mr. Greeley has proved an unfaithful husband.... On the contrary he has been held up... as a model husband in that particular and for that reason the fault and opprobrium of domestic discord has been heaped on Mrs. Greeley. Who has ever troubled himself to inquire how much... Greeley has had to do with souring the temper, unstringing the nerves, and completely disorganizing the machinery of a delicate woman's organization.
_A portrait used during Greeley's presidential campaign, 1872_. LEFT TO RIGHT: _Horace Greeley, daughters, Gabrielle and Ida, Mary Cheney Greeley_ (Illustration Credit 23.1)
Free love, two husbands, the destruction of marriage laws—even some of the most forward-thinking members of the woman's movement balked at having Woodhull associated with their cause. Sara Burger Stearns expressed the opinion of the majority: "I would like to have the name of Victoria C. Woodhull as connected with Woman Suffrage as soon _forgotten_ as possible. We can better afford to wait for years... than to attempt the question of 'free love'.... The woman who declares 'marriage' the 'greatest evil of the Age' must not be the principal one to be heard."
But Isabella Beecher Hooker refused to denounce her beloved Victoria "at the bidding of a hostile press." She admitted, "I do not understand all her views—and I have had no time to ask her concerning them—nor to study these new social theories." In answer to Sara Burger Stearns's critical letter, Isabella explained that she was sure that Victoria Woodhull used the term _free love_
with a meaning of her own different from this hateful one, as she will someday explain, I hope.... I know that she is striving to put down lust and exalt love, that her motives are exalted and her life pure and her whole nature spiritual in an uncommon degree.... The _Tribune_ knows and so does every editor and reporter and reader, that neither you nor I, nor anyone of the prominent workers for suffrage believes in Free Love, according to the popular meaning of that word, which is free lust... but they know as well that if they can frighten us into disavowing any sympathy with such a powerful woman as Mrs. W. is proving, both because of her own brain and heart and because of her command of money (a thing we never had in our ranks before) and because of the influence of her paper among _Spiritualists_ (a very large and increasingly influential class) and among businessmen and politicians, then they have dealt a severe blow at the whole suffrage movement and set it back years.
Elizabeth Cady Stanton was infuriated by the press and the women of the AWSA, who eagerly joined in the attack on Woodhull. How dare these women accuse Woodhull when their own lives were rife with scandal! And so it began, the rending and tearing of all they had accomplished. Stanton told Woodhull about such matters as Henry Blackwell's affair with Abby Patton. There had been talk about other leaders of the AWSA: The Reverend Phebe Hanaford had just moved to New Haven without her husband and had declared that she intended to separate from him, and there was some question whether Mary Livermore would follow her example. Furthermore, Stanton told Victoria the names of two "kept women" who sat on the Boston platform. Finally, in her fury she disclosed the reason that Henry Ward Beecher had resigned as president of the AWSA and described the behind-the-scenes efforts to quell the scandal.
Neither Stanton nor Anthony had any illusions about Woodhull's personal conduct, nor did they feel the need to justify it. Their relationship was strictly for the sake of a greater purpose. Anthony wrote, "I will take by the hand every prostitute I can find who seeks to escape the inequalities of that law which places all womanhood at the mercy of manhood." Stanton too dealt with the larger issue:
In regard to the gossip about Mrs. Woodhull I have one answer to give all my gentlemen friends. When the men who make laws for us in Washington can stand forth and declare themselves pure and unspotted from all the sins mentioned in the Decalogue then we will demand that every woman who makes a constitutional argument on our platform shall be as chaste as Diana....
We have had women enough sacrificed to this sentimental, hypocritical prating about purity. This is one of man's most effective engines for our division and subjugation. He creates the public sentiment, builds the gallows, and then makes us hangmen for our sex.... If Victoria Woodhull must be crucified, let men drive the spikes and plait the crown of thorns.
It was in a mood of indignation over Woodhull's persecution by hypocrites who in truth were practicing free lovers that Stanton left to meet Anthony on a western tour. Amelia Bloomer, president of the Iowa Woman Suffrage Society, was later to say, "Mrs. Stanton whispered the scandal to [me] and said 'the Woodhull knows all about it.' " As Stanton and Anthony traveled across the West, the story of the Beecher-Tilton affair spread in woman suffrage circles. A miasma of scandal engulfed the cause.
Had the woman suffrage movement been a cohesive force, undoubtedly it would have had a strong voice in American politics. This was a time when the radical Republicans were sympathetic to the rights of women. They needed women's support, as they did that of black men. But the woman's movement was now shattered by personal attacks and counterattacks, while the greater issue of enfranchisement languished amid gossip and recrimination. The window of opportunity was closing, and almost fifty years would pass before women were granted the vote.
On the day Victoria's trial ended in the Essex Market courthouse, France's government forces and the Paris communards clashed in a bloody battle. Victoria said later that upon returning from the courtroom she fell into an exhausted sleep only to be awakened by two of her spirit guides, Bonaparte and Josephine. At their behest, Victoria left her body and journeyed to a hill high above the Town Hall of Paris, where she could see that the Palais-Royal, the Ministry of Finance, the Rue de Rivoli, and the Rue Royale were all aflame. As the fire spread, the Porte Saint-Martin, the Hotel de Ville, the Bastille, and Bercy "were belching forth blood-red columns of fire."
As a final act of defiance, the communards set fire to the Tuileries Palace and, on the Left Bank, the government buildings of the Quai d'Orsay. When the last barricades fell on May 28, 1871, much of Paris lay in ruins. The government forces lost one thousand men and the communards twenty thousand. Forty thousand communards were arrested during the next month.
Mama Roxy's accusations of communism and the destruction of the Paris Commune coincided. The threat of communism haunted the United States. Though only two of the ninety seats in the Paris Commune had been occupied by Marxist communists—the commune consisted largely of working-class socialists—the American press ignored the distinction. The _Telegram_ stated that the New York "communists... given the opportunity of the Parisians would be the same repulsive monsters," and the _World_ called Section 12 of the International Workingmen's Association and its honorary president, Victoria Woodhull, "a menace" and "a foul contagion." In St. Louis, the liberal reformer Carl Schurz wrote that the International had originated in Sicily and was linked to the Mafia.
On the morning that the revolt of the Paris Commune reached its climax, Colonel Blood staggered out of Victoria's house, blood streaming down his face. Utica had struck him on the head with a chair. Twenty minutes later, Mama Roxy appeared at the police station and lodged yet another complaint against Blood. When reporters arrived at the Thirty-eighth Street house, Mama Roxy was sitting on the stoop awaiting them. "You know what a hell-hound that Blood is," she protested. At that moment Tennie walked up and surveyed the scene. "Come on, come on," she said to the reporters with resignation, and beckoned them inside, where she sank into a chair and began a tirade of her own. "All these people," she gestured wildly. "Leeches! For years they have lived this contemptible life and now useless like so many bloated corpses they utter the screams of the infirm."
Victoria, hearing the commotion, descended the stairs and said quietly, "I believe Tennie ought to use the gift God has given her not in the mercenary way. She was forced... to prostitute her powers.... We don't want any scandal."
But scandal there was, and it followed them everywhere. One Miss Achsah Truman, known as "a woman of the town," lodged a suit against Victoria saying that she had given her $800 to invest and that was the last she had seen of the money. Utica Brooker was arrested three times on charges of drunkenness and soliciting men. At the end of May, Dr. Benjamin Sparr's naked body was discovered slumped over a desk in a room at French's Hotel. He had died of an apoplectic attack. Clutched in his hand was an unfinished letter beginning, "Vickie, Colonel and Tennie included..."
As further scandals surfaced, the women of the AWSA tried to disassociate their organization from any views expressed by "the Woodhull." Mary Livermore accused Victoria of falsely claiming "to lead the movement" while "advocating and practicing... lawless licentiousness." Lucy Stone's daughter wrote, "No woman is regarded with so much abhorrence by almost all decent people."
_Paris burns: The communards set fire to the Tuileries Palace and, on the opposite bank, the government buildings on the Quai d'Orsay_. (Illustration Credit 23.2)
Anthony wrote of the AWSA, "What effort more than Herculean to save themselves from contamination from the touch even from the hem of the Woodhull garments. It is too sick! When will they begin the _washing_ of their men champions and workers? It is high time." And Martha Coffin Wright, who had seen the ups and downs of the movement since its inception at Seneca Falls, deplored the situation: "The division is so senseless... that I have no patience with it."
Victoria Woodhull retaliated in the way she knew best—the Claflin way. In the _Weekly_ appeared an editorial:
Without pretending to a perfect knowledge of, or caring a fig about, the history of the _personnel_ of either branch of the woman movement, we are led to suspect that this over-pious, over sensitive, Boston wing have much more to conceal than their more outspoken sisters of Apollo Hall.... Mrs. Livermore, it is a rather delicate thing for... those "who live in glass houses to throw stones," and you very well know that most people do live in these brittle tenements.
Anthony knew the risks that members of the American Woman Suffrage Association were taking by provoking Victoria Woodhull, who was privy to scandal concerning Livermore and other people in this group. Anthony commented,
This fresh howl will soon be over—I learn the Boston folks begin to feel the kick of their Free Love gun firing quite painfully—they must be awfully ashamed of themselves—if there be any such possibilities left unto them.
Mrs. Hooker is in Boston today preaching... Christ and the Woman at the Well—"Thou hast had five husbands and the one thou now livest with is _not_ thy husband—go preach my gospel"—has Woodhull a more _numerous_ record of husbands that are _not_ husbands? Well, we are fallen upon strange times but there is no way but to live them... judging not, lest we be also judged.
Woodhull needed support and needed it quickly. On May 22, 1871, she wrote a "card" to the editors of the _World_ and the _Times_.
Because I am a woman and because I conscientiously hold opinions somewhat different from the self-elected orthodoxy which men find their profit in supporting... self-elected orthodoxy assails me, vilifies me and endeavors to cover my life with ridicule and dishonor....
One of the charges made against me is that I live in the same house with my former husband, Dr. Woodhull, and my present husband, Colonel Blood. The fact is a fact. Dr. Woodhull, being sick, ailing and incapable of self-support, I felt it my duty to myself and to human nature that he should be cared for.... My present husband, Colonel Blood, not only approves of this charity but cooperates in it. I esteem it one of the most virtuous acts of my life.
Victoria issued a warning:
I do not intend to be made the scapegoat of sacrifice to be offered up as a victim to society by those who cover over the foulness of their lives and the feculence of their thoughts with a hypocritical mantle of fair professions, and by diverting public attention from their own iniquity in pointing the finger at me.... I believe in Spiritualism. I advocate Free Love in its highest, purest sense, as the only cure for the immorality... by which men corrupt and disfigure God's most holy institution of sexual relation.
Then in a thinly disguised reference to the Beecher-Tilton affair she added,
My judges preach against Free Love openly, practice it secretly. For example, I know of one man in a neighboring city, a public teacher of eminence, who lives in concubinage with the wife of another public teacher of almost equal eminence. All three concur in denouncing offenses against morality. "Hypocrisy is the tribute paid by vice to virtue." So be it. But I decline to stand up as the "frightful example." I shall make it my business to analyze some of these lives and will take my chance in the matter of libel suits.
# CHAPTER TWENTY-FOUR
THIS GIRL IS A TRAMP
THEODORE TILTON appeared at the Woodhull residence the following morning, the _World_ tucked under his arm. "Who do you mean in this card?" he asked with feigned innocence.
"I mean you and Mr. Beecher," Victoria replied. Tilton, of course, already knew that. The previous evening, in a three-hour conference with Moulton and Beecher, he had been designated to endear himself to Woodhull so that "some influence might be brought to bear to induce this sudden enemy to suppress the dangerous tale." Tilton implored Victoria, "Don't take any steps now.... I can be of service to you.... Let me take you over to Mrs. Tilton's, and you will find her in no condition to be dragged before the public."
Victoria agreed. Lib greeted her cordially and asked her to sit and talk while she sewed a checked-silk dress for Florry. They spoke quietly about the joys and burdens of motherhood. When Victoria was about to leave, Mrs. Tilton rose and presented her with a book of poetry, which she inscribed, "To my friend Victoria Woodhull."
Woodhull soon became a frequent visitor in the Tilton home, where the conversation revolved around woman's rights, Spiritualism, and free love. According to Victoria, she told Theodore that the trouble was not his wife's affair but "the false social institutions under which we still live." She ridiculed the "dreadful suzz, maudlin sentiment, and mock heroics" he was exhibiting over a matter she found to be "the most natural in the world" and commented that "our sickly religious literature, Sunday school morality, and pulpit phariseeism humbugged him into the belief he ought to feel and act in this harlequin and absurd way." And she told Lib, "Be kind to and sympathize with the new attraction rather than be waspish and indignant.... I have learned that the first great error most married people commit is in endeavoring to hide from each other the little irregularities into which all are liable to fall. Nothing is so conducive to continuing happiness as mutual confidence."
Frank Moulton also was enlisted to befriend Mrs. Woodhull. He was struck both by Victoria's intelligence and her "unexpected modesty." Because he considered his wife his full partner in life, he said to Emma, "I want you to take a look at that woman and tell me what you think of her." Emma, as described by her husband, was "a broad-minded, self-possessed woman able to take care of herself," one who accepted "life on the theory of human imperfection." She found Victoria to be high-strung, magnetic, and dedicated to the idea that she had the power to eliminate women's suffering. Emma too opened her home to Victoria.
To ingratiate himself further, Tilton offered to put his pen at Woodhull's service to legitimize her and her theories. Anna Dickinson was then the most sought-after woman lecturer in America, and Victoria was determined to outdo her. In the _Golden Age_ , Tilton bestowed Anna's title upon Victoria, writing, "If the woman's movement has a Joan of Arc, it is this gentle but fiery genius.... Little understood by the public, she is denounced in the most outrageous manner by people who do not appreciate her moral worth." Because Dickinson was admired for the physical freedom that few women were able to enjoy, Tilton wrote of Woodhull, "She can ride a horse like an Indian, and climb a tree like an athlete; she can swim, row a boat, play billiards, and dance. Moreover, as the crown of her physical virtues, she can walk all day like an Englishwoman."
Tilton rewrote _The Woodhull Memorial_ so that it would have a more popular ring when she presented it on the Lyceum circuit. But his most ambitious task was a biography that Woodhull insisted he write based on rough notes from Colonel Blood. Woodhull and Tilton were in the difficult position of having to explain how her mother had turned on her and accused her of scandalous acts in the recent court action. In fact, it was hard to describe to an outsider the tremendous hold both parents had established through their psychological and physical domination of their children. Tilton's explanation was clever if convoluted. Describing the Claflin family, he wrote,
Such another family-circle of cats and kits, with soft fur and sharp claws, purring at one moment and fighting the next, never before filled one house with their clamors since Babel began. They love and hate—they do good and evil—they bless and smite each other.
Being daughters of the horse-leech, they cry "give." The whole brood are of the same feather—except Victoria and Tennie.... Victoria is a green leaf, and her legion of relatives are caterpillars who devour her. Their sin is that they return no thanks after meat; they curse the hand that feeds them. They are what my friend Mr. Greeley calls "a bad crowd."
But it was Victoria's other powers that proved the most thorny: In the first draft, Tilton made no mention of her spiritual gifts. Victoria objected. "You have left out the most important parts!" she admonished him. Spiritualists were among her strongest supporters, and it was vital to Woodhull that she prove herself to this group. Tilton was told to include her communication with the other world, her success as a clairvoyant and trance speaker, and even the incident where, like Christ himself raising Lazarus, she had brought her son, Byron, back from death.
"You want me to say that you have called a dead child to life?" Tilton asked incredulously.
"Yes, to do otherwise would be as if you were writing Hamlet and decided to leave out his father's ghost." So Tilton wrote,
I must now let out a secret. She acquired her studies, performed her work, and lived her life by the help (as she believes) of heavenly spirits. From her childhood till now (having reached her thirty-third year) her anticipation of the other world has been more vivid than her realizations of this. She has entertained angels, and not unawares. These gracious guests have been her constant companions. They abide with her night and day. They dictate her life with daily revelations and like St. Paul, she is "not disobedient to the heavenly vision."... Every characteristic utterance which she gives to the world is dictated while under spirit influence, and most often in a totally unconscious state. The words that fall from her lips are garnered by the swift pen of her husband and published almost verbatim.
Tilton went all out in his service to Mrs. Woodhull. His biography was so florid that _Harper's Weekly_ wrote of it, "If apples are wormy this year, and grapes mildew, and ducks' eggs addle, and bladed corn be lodged, it may all be ascribed to the unhallowed influence of Mr. Tilton's life of Victoria Woodhull." Lucretia Mott's niece Marianna wrote of the biography, "Can anything but infatuation or aberration explain its absurdities?" And Julia Ward Howe, author of the "Battle Hymn of the Republic," made the dire prediction, "Such a book is a tomb from which no author again rises.... It would have sunk any man's reputation anywhere for common sense."
Tilton, however, seemed unembarrassed and believed what he had written. When interviewed by a reporter from the _Sun_ , he said of Victoria, "Her sincerity, truthfulness, nobility and uprightness of character rank her in my mind, as a pious Catholic would rank Saint Teresa."
"You astonish me," said the reporter.
"Not more than I myself was astonished at the singular revelation of her character to me, as one of the most upright, truthful, religious, unsullied souls I have ever met."
There is no doubt that Tilton's involvement with Woodhull, which began for practical purposes, had now become infatuation. During the summer of 1871 he spent many nights at the Woodhull house and often slept with her on the cool rooftop. They swam at Coney Island and rowed in Central Park. Woodhull inculcated Tilton with her free love doctrine, which he was more than ready to assimilate. She told him, "A popular objection against Free Love is that it breaks up families. My answer to this indictment is that a family which falls in pieces when Free Love strikes it is already broken up and waiting for a loophole out of which to escape."
While appearances suggest that they were lovers, Tilton never commented on the subject. Three years later, however, after Tilton had disassociated himself both from free love and from Victoria herself, she told a reporter from the _Chicago Times_ , "Mr. Tilton was my devoted lover for more than half a year.... He slept every night for three months, in my arms." But when a New York journalist later asked her to verify this statement, she replied, "I am not a fool. A woman who is before the world as I am would not make such a flagrant statement, even if it were true." Perhaps she had never said it. Perhaps it was said by a Victoria possessed by the spirits, who was another woman altogether.
Of all the allies Woodhull sought, Henry Ward Beecher was the most valuable. She insisted on meeting him to woo him to her cause. Although Moulton and Tilton prevailed upon Beecher to accede to Woodhull's wishes, he flatly refused. Three weeks after her "card" the following statement appeared in _Woodhull & Claflin's Weekly:_
Civilization is festering to the bursting point in our great cities and notably in Brooklyn.... At this very moment, awful and herculean efforts are being made to suppress the most terrific scandal which has ever astounded and convulsed any community.... We have the inventory of discarded husbands and wives and lovers, with dates, circumstances and establishments.... Confidences which are no confidences abound.
Beecher changed his mind.
My Dear Victoria,
I have arranged with Frank that you will see Mr. Beecher at my house on Friday night. He will attend a meeting at the church till ten o'clock and we will give you the rest of the evening as late as you desire. You may consider this fixed. Meanwhile on the sunshiny day I salute you with a good morning—peace be with you.
Yours, Theodore Tilton
They met in the late evening, these two charismatic people who had much in common: Both could persuade and provoke, both were inspired by the adulation of their followers, both derived their strength from higher powers. In an effort to exert his considerable charm, Beecher, according to Woodhull, confided that he kept a spirit table at Plymouth Church and informed her of his early use of mesmerism. In this and in two subsequent meetings, Beecher convinced her that they thought alike. He told her that he too advocated a reform of the marriage laws. "Marriage is the grave of love," Beecher avowed. "I have never married a couple that I did not feel condemned."
"Why then do you not preach that conviction?" asked Woodhull.
"If I were to do so I should preach to empty seats." Beecher said he preached the truth as fast as he thought people could accept it. "Milk for babies, meat for strong men."
Victoria "held society to be upside down." She felt she could persuade Beecher to join her in her holy quest to achieve the idyllic world that the spirits had revealed to her. "Beecher's life was Woodhull's faith," Frank Moulton was to observe. "She hoped from hearing him talk in his lofty way... that she might wheedle or compel him to come forward and thus make her theory of the reconstruction of society respectable."
Convinced that her new allies would stand by her, Victoria wrote Isabella that she was sure her brother Henry would provide the introduction for a speech she was to present at Steinway Hall in November on "The Principles of Social Freedom." Isabella well remembered the letter she had received from her brother criticizing Woodhull, so she questioned the truth of Victoria's assertion and wrote her "a most indignant and rebuking letter" saying that her brother would never become involved. Woodhull's reply astonished Isabella "by its calm assertion that she considered [Henry] as true a friend to her as I myself." She wondered what hold her friend Victoria had over her brother and sent her letter to Theodore Tilton, asking him to "write me what it all meant." Tilton did not reply.
Isabella was totally dedicated to Victoria—a disciple of this "strange prophet." Nothing could sway her. When a friend, Anna Savery, sent her Woodhull's "card" and the threatening statement that had appeared in the _Weekly_ mentioning "the inventory of discarded husbands and wives and lovers" and asked her for an explanation, Isabella replied:
As to what Mrs. Woodhull means by that threat I do not know and I have no time to ask. I have never talked with her on the social question fifteen minutes and I will not. My ground is that she alone of all the women in the U. States succeeded in getting a hearing and a report out of a dead Congress.... She has... appeared to me as... a prophetess full of visions and messages to the people which it would be woe unto her to refrain from proclaiming even though martyrdom was sure to follow....
She is a mystery to me but so is every _forerunner_ to the people of his or her day and for one I am determined to keep my eyes and ears and heart open to any woman who thinks she has a word from above to deliver unto the nations.... I verily believe that the _hour_ of woman approaches and is even upon us.... If the spirit world is what she conceives it to be and is eagerly waiting to pour down upon us in a purifying flood—let it come... my heart will exult with joy....
The feeling on this Woodhull matter has nearly killed me.... I am driven to death.... Just now my sister Catharine is attacking Mrs. W's private character _infamously_ so as to keep people from going out to hear her. The result is she will have a jam next time and last week had a good house on a stormy night. It is dreadful having foes in your own household.
When Catharine Beecher heard that Victoria Woodhull was to lecture in Hartford she attacked her in the _Courant_. Under the pseudonym of "a Lady of Connecticut," she asked, "Can anyone support a woman who maintained that when a wife and mother loved elsewhere, she should forsake her family for that love, who maintained also that a woman who fulfilled conjugal duties after that specific love was gone was guilty of prostitution?" Catharine then went on to conclude that only Victoria's sister Tennie C. "exceeds her in indecencies."
Stephen Pearl Andrews countered with an article for the _Weekly_ reminding the readers of the _Courant_ that Mrs. Woodhull had spoken to an audience of four thousand in Cleveland and had said nothing offensive. To bolster Woodhull's reputation, he published the letter from Isabella Beecher Hooker in which she had addressed Victoria as "My Darling Queen." As a result of all this publicity, as Isabella had predicted, the Hartford Opera House was filled to capacity.
This brought on another barrage of criticism. The Reverend Phebe Hanaford rescinded her invitation to Isabella to speak at her First Universalist Church of New Haven because of her "unholy alliance" with "the notorious mistress of Colonel Blood's affection." Referring to the "Darling Queen" letter, she wrote to Isabella, "Every drop of Beecher blood in your veins ought to cry out against her life of shameful disregard of propriety.... A repentant Magdalen I can accept as a co-laborer—but a woman who 'glories in her shame' never!"
Hard put to defend herself, Isabella blamed Andrews in her reply: "That private note of mine was the outpouring of an admiration I could no longer repress.... I said this woman is a Born Queen and I owe her the allegiance of my heart—so I wrote just as I felt and it was printed by a _man_ without her knowledge."
While Isabella had become Woodhull's greatest defender, Horace Greeley and his editor Whitelaw Reid, who was in charge of the day-to-day operations of the _Tribune_ , led the attacks on Mrs. Woodhull. Presumably, Isabella felt that Anna Dickinson could influence Reid, for she repeatedly wrote to her complaining of his treatment of Woodhull. The previous spring, when Woodhull first threatened to expose the private lives of those who opposed her, she wanted to use the _Tribune_ as an outlet. Greeley was away on a lecture tour so Woodhull sent an urgent message to Whitelaw Reid asking him to visit her. According to his account, when he arrived he found Victoria and Theodore together. "Mrs. Woodhull received me cordially and at once began detailing her stories about Mr. Beecher to the effect that he was a practical free lover who ought to be honored in his practice but denounced for his hypocritical professions—who kept a regular harem in Brooklyn and had more illegitimate children than any other man in town—with much more equally revolting and incredible." She then went on to denounce other prominent people in Brooklyn, New York, and Boston. When she had finished, Tilton begged Reid to dissuade Mrs. Woodhull from trying to publish her accusations in his or any other newspaper.
After a second visit, when Reid was chastised by Victoria both for failing to publish her story and for not trying to stop the critical coverage of her in the _Tribune_ , she warned him that she possessed certain information about Anna Dickinson that would prove disastrous were it made public. Less than a fortnight after Reid's second visit to Woodhull, he received a frantic summons from Anna that implied that she was being threatened. He answered, "I shall break all engagements, and go over Saturday night—leaving here at midnight, and reaching the Continental about five in the morning. I'll go up to your house immediately after breakfast—say by nine to ten o'clock." After Reid's visit he wrote to Anna, consoling her and advising her how to handle threats and gossip. "Take Beecher for a model," he instructed. "He works with ease to himself and bears the misfortunes that harrow the minds of all... in a way that strengthens rather than weakens him."
As Isabella continued to write to Reid complaining about the _Tribune's_ treatment of Mrs. Woodhull, he began to weary of the situation. He visited Isabella at Nook Farm in August hoping to placate her, but to no avail. Finally, an exasperated Reid wrote to her explaining that neither he nor Horace Greeley had written anything about Victoria Woodhull that was not true. He added with sarcasm, "I must and do always treat her as I treat all women, demi-rep, slanderers or ladies, with needful courtesy where I meet them."
Harriet Beecher Stowe, no longer able to ignore the situation, visited Henry at Twin Mountain House in New Hampshire and confronted him with the rumors. He vigorously denied them, after which she announced that Woodhull was "a snake and should be given a good clip with a shovel." Using the powerful weapon of her pen Harriet began a serial, "My Wife and I," which appeared in installments in the _Christian Union_. Stowe felt that humor would accomplish what a diatribe might fail to do. Victoria Woodhull appeared in a thinly veiled portrait as the shameless adventuress and newspaper owner Audacia Dangyereyes. The scenes depicted were perilously close to those that had recently occurred.
"I have just come from the Police Court, where there's a precious row. Our friend 'Dacia Dangyereyes is up for blackmailing and swindling and there's a terrible wash of dirty linen going on."
"How horribly disagreeable," said Eva, "to have such women around. It makes one ashamed of one's sex."
In a play on words, Isabella appeared as Mrs. Cerulean, since "her head was in the blue." Pointing to the scandal in the courtroom and all that had followed, Harriet hit upon Isabella's trait of loyalty to the underdog. As Isabella had admitted, "The truth is, the more people are mistaken and going in a wrong way... the more I am inclined to keep company with them in the hope of using with them what influence I may have, toward a right way." Now Harriet wrote of her sister's loyalty to Victoria Woodhull, "You will see, Mrs. Cerulean will adhere all the closer for this. It's persecution, and virtue in all ages has been persecuted. Therefore, all who are persecuted are virtuous. Don't you see the logical consistency? And then, don't the Bible say, 'Blessed are ye when men persecute you, and say all manner of evil against you'?"
_Victoria Woodhull as the seductress Audacia Dangyereyes, in Harriet Beecher Stowe's "My Wife and I_ " (Illustration Credit 24.1)
Harriet also satirized Isabella's spiritual vision and painted her sister as a gullible fool.
'Dacia... conducted herself in a most sweet and winning manner and cast herself at her feet for patronage; and Mrs. Cerulean, regarding her through those glory spectacles which she usually wears, took her up immediately as a promising candidate for the latter day. Mrs. Cerulean don't see anything in 'Dacia's paper that, properly interpreted, need make any trouble; because, you see, as she says, _everything ought to be love_.... Then the infinite will come down into the finite, and the finite will overflow into the infinite, and in short, Miss 'Dacia's cock's-feathers will sail right straight up into Heaven, and we shall see her cheek by jowl with the angel Gabriel, promenading the streets of the new Jerusalem. That's the programme.
Meanwhile, 'Dacia's delighted. She hadn't the remotest idea of being an angel, or anything of the sort, but since good judges have told her she is, she takes it all very contentedly. Mrs. Cerulean is a respectable woman, of respectable family, and this girl is a tramp, that's what she is, and it is absolutely impossible that Mrs. Cerulean can know what she is about.
The final draft of Woodhull's speech, "The Principles of Social Freedom," to be presented at Steinway Hall on November 20, was carefully crafted by Stephen Pearl Andrews to be "as soft" as possible so that Henry Ward Beecher could introduce Victoria without embarrassment. The bulk of the speech traced the changing attitudes toward social freedom from the sixteenth century to the present. Victoria herself carried it to Brooklyn, met Beecher in his study at Plymouth Church, and presented him with the document. She asked that he read it carefully and suggest revisions before introducing her at Steinway Hall. But as the days grew short, she had no word from Beecher, and Tilton told her that Beecher was afraid that if he introduced her he might be "tarred and feathered" with her belief in free love. On the day before she was to speak, the following letter was delivered to Beecher.
Dear Sir:
For reasons in which you are deeply interested as well as myself and the cause of truth, I desire to have an interview with you, without fail, at some hour tomorrow. Two of your sisters have gone out of their way to assail my character and purposes... and thus to defeat the political ends at which I aim.
You doubtless know that it is in my power to strike back and in ways more disastrous than anything that can come to me but I do not desire to do this. I simply desire justice from those from whom I have a right to expect it and a reasonable course on your part will assist me to it. I speak guardedly, but I think you will understand me.
I repeat that I must have an interview tomorrow, since I am to speak tomorrow evening at Steinway Hall and what I shall or shall not say will depend largely on the result of the interview....
They met at the Moultons' house, in the fading afternoon light. Beecher was fully aware of the Damoclean sword Woodhull held over his head, but still he resisted. When all of her arguments seemed to fail she told Beecher that his attachment to Mrs. Tilton would surely be revealed to the public. "The only safety you have is in coming out as soon as possible as an advocate of social freedom and thus palliate, if you cannot completely justify your practices, by founding them at least on principle. Your introduction of me would bridge the way."
"I cannot. I cannot!" Beecher cried out. "I should sink through the floor. I'm a moral coward on this subject and you know it. I am not fit to stand by you, who go there to speak what you know to be the truth. I should stand there a living lie."
Beecher knelt on the sofa beside her and clasped her face between his hands. He began to weep and to beg her, "Oh, let me off. Let me off."
These histrionics had no doubt been effective in the past, for Beecher had used them often, but Victoria was not impressed with this "maudlin display." "Mr. Beecher, if I am compelled to go upon that platform alone, I shall begin by telling the audience why I am alone and why you are not with me."
Beecher replied, "I cannot face this thing! I can never endure such a terror. Oh! if it must come, let me know of it... in advance that I may take my own life."
# CHAPTER TWENTY-FIVE
YES!
I AM A FREE LOVER!
IN FRONT of Steinway Hall, ten-foot-long streamers, bright red with gold letters, read Freedom! Freedom! Freedom! They were drenched with rain, and several had whipped free and twisted in the wild wind. At the entry door, seven-foot-high posters proclaimed Victoria C. Woodhull... The Principles of Sexual Freedom... Free Love, Marriage, Divorce and Prostitution.
At seven o'clock, a full hour before the lecture, three thousand people, a capacity audience, jammed into the hall and crowded the stairs and corridors. The raffish and the respectable jostled each other in eager anticipation. On the street, a "lady of the evening," her hair a bright hennaed red, called out to no one, "I hope by God I haven't come here in all this rain for nothing!" Nearby, as hundreds of people were turned away, fistfights broke out.
In a small anteroom behind the stage, Victoria and Tennie C. found themselves surrounded by reporters. Victoria wore a black silk dress with her signature white rose at her throat. She whispered to a _Herald_ reporter, "Your paper has never misrepresented me, I know it won't now," then reached into a vase nearby and handed him a white rose. Tennie, who was talking animatedly to the reporters, suddenly lifted her skirt to reveal her ankles encased in gaiters.
From time to time Victoria glanced toward the entrance to the anteroom. At seven forty-five Tilton and Moulton elbowed their way through the crowd to her side. After Woodhull left Brooklyn that afternoon, they had tried to persuade Beecher to introduce her. Tilton suggested that Beecher say he disagreed with her views but that she had a right to express them. Finally, Beecher said he would be there, though he was not sure if he "could bear the ordeal." But at ten minutes after eight Henry Ward Beecher still had not appeared. An impatient audience began to clap and pound their feet. Then a chant began: "Woodhull, Woodhull, Woodhull!"
Victoria, close to tears, clutched Theodore's arm. "There isn't one brave man in the circle of two cities to preside at my meeting." Then resolutely she walked down the long dim corridor toward the stage. When she was about ten paces ahead of him, Tilton moved quickly after her. "Are you going to introduce her?" Moulton called out.
"Yes, by Heaven, since no one else has the pluck to do it."
Victoria paused in the wings as the clapping and stomping roared in her ears. Tennie, noting her sister's pallor and glazed eyes, saw the danger that might lie ahead if the spirits intervened. If that happened Vickie would become another person—ecstatic, swept away, engulfed by her visions—and who knew what she might say or do? "Vickie, be calm," Tennie cautioned. As Victoria moved reluctantly onto the stage, the audience rose as one. People ran forward to crowd around the platform, leaning perilously close to the gas footlights. Utica—still beautiful and sweet-faced but in her usual state of partial inebriation—adjusted the panniers of her pink _peau de soie_ skirt and settled next to a group of her friends in a front box that partially jutted out onto the stage. Tennie repaired to an identical box opposite.
Amid the uproar, Victoria stood frozen in place, blinking rapidly and staring out at the audience. At that moment Tilton strode on stage. Taking Victoria by the arm, he led her forward and raised his arms to silence the crowd.
Ladies and gentlemen, happening to have an unoccupied night, which is an unusual thing for me in the lecture season, I came to this meeting actuated by curiosity to know what my friend would have to say in regard to the great question which has occupied her so many years of her life. Five minutes ago, I did not expect to appear here, but several gentlemen have declined to introduce our speaker, one after another for various reasons, chief among them being objections to this lady's character. I know it, and I believe in it, and I vouch for it. As to her views, she will give them to you herself and you may judge for yourself.
Tilton's voice became infused with passion. "It may be that she is a fanatic. It may be that I am a fool. But before high Heaven I would rather be both fanatic and fool in one, than be such a coward as would deny this woman the sacred right of free speech."
From her perch in the stage box Utica Brooker leaned forward to study her sister as Vickie began reading her speech. In the wings, Stephen Pearl Andrews ran his finger across the page, following the words in his copy as she spoke them. In the context of a historical review of social freedom, Woodhull examined male and female relationships in several societies. The audience sat quietly but this was not what they had come to hear. It was not until she reached the present day that Victoria Woodhull turned to the controversial issue of woman's rights.
The basis of society is the relation of the sexes.... There is no escaping the fact that the principle by which the _male_ citizens of these United States assume to rule the _female_ citizens is _not_ that of self-government but that of despotism.... Our government is based upon the proposition that all men and women are born free and equal and entitled to certain inalienable rights, among which are life, liberty, and the _pursuit_ of happiness. What we, who demand social freedom ask is simply that the government of this country shall be administered in accordance with the spirit of this proposition. _Nothing more, nothing less_.
Warming to the subject of sexual equality and free love, she looked out at an audience who now eagerly awaited her words. A flush colored Victoria's pale cheeks as once again she created an invisible connection with the audience. Abandoning Andrews's written text, she now peppered her speech with language of her own.
The wiseacres stop and tell us that _everybody_ must _not_ pursue happiness in his or her own way.... I say they're wrong. They're just humbugging you. What I believe to be the truth I endeavor to practice and in advocating it permit me to say I shall _speak_ so _plainly_ that _none_ may complain that I did not make myself understood.
Marriages have endeavored to... hold the people in subjection to what has been considered a standard of moral purity.... But let us inquire into this matter.... Is marriage where two meet and realize that the love elements of their natures are harmonious... or is it where a _soulless form_ is pronounced over two who know _no_ commingling of life's hopes?... The courts hold if the law solemnly pronounce two married that _they are married_ , whether love is present or not.... It is a stupid law which can find no analogies in nature.
Marriage must consist of love or of law.... People may be married by _law_... and they may also be married by _love_ and lack all sanction of law.... Law cannot compel two to love.... This is a matter that concerns _these two_ and _no_ other living soul has _any human_ right to say aye, yes or no, since it is a matter which is none of their business.... Where there is _no_ love as a basis of marriage there should be _no_ marriage!
Victoria could feel the spirits all about her. It was these spirits for whom she spoke, all those suffering souls whose burden she carried on her frail shoulders. As she continued to depart from Stephen Pearl Andrews's speech, she seemed unaware of the rapt audience drinking in her radical doctrine or of the force of her words.
I do not care where it is that sexual commerce results from the dominant power of _one sex_ over _the other_ , compelling him or her to submission against the _instincts of love_. And where hate or disgust is present, whether it be in the gilded palaces of Fifth Avenue or in the lowliest purlieus of Greene Street, _there_ is prostitution, and _all_ the law that a _thousand_ State assemblies may pass cannot make it otherwise.
Victoria's passion lifted the audience to a spiritual frenzy as an almost sexual ecstasy gripped the crowd. Caught in her spell, they burst into wild applause and cries of "Hurrah!" But as the roar subsided, from the stage box where Utica sat came the sound of hissing, which led to more applause and still more hissing. Then the jealous Utica stood. Like her mother, she was determined to bring her sister down. Her envy, fueled by laudanum and alcohol, knew no bounds. At thirty, Utica had been drinking steadily for half her life. A beatific smile lit her face as she called out to her sister, "How would you like to come into this world without knowing who your father was:
Utica instinctively knew that the philosophical doctrine of free love collapsed in the real world. The theories of the desiccated old Stephen Pearl Andrews had little emotional power, but Victoria was a desirable, magnetic woman, and from her, free love became explicitly sexual. And if one actually practiced free love, the result would be illegitimate children, often with no one to raise them. Victoria stumbled as she attempted to answer her sister. "There are thousands of noble men and women in the world today who never knew who their fathers were," she replied. "Every person is... entitled to pursue _happiness_ in whatever direction he or she may choose."
But Victoria's spirit spell had been shattered. She had lost control of her audience. The hissing escalated. Utica sat down with a smile of victory as Victoria tried to continue: "In all contracts, people have the protection of the government to contract for an hour, a day, a week... least of all does the government require that any of these contracts be entered into for life. Why should the social relations of the sexes be made subject to a different theory?" Without specifically mentioning Beecher she said, "When Christian ministers are no longer afraid or ashamed to be Christians they will embrace this doctrine. Free Love will be an integral part of the religion of the future."
From Utica's box a woman's voice screamed out, "Yah, yah, yah—we've had enough of you." Once again, Vickie struggled on.
They assert that the murder of Richardson by McFarland was the legitimate result of _Free Love_ , but I deny it. McFarland murdered Richardson because he believed that the law had sold Abby Sage's _soul_ and _body_ to him, and, consequently, that _he owned her_ and that no other person had any right to her favor and that she had no right to bestow her love upon any other person.... The murder of Richardson, then, is not chargeable to his love or her love, but to the fact of the supposed ownership, which the law of marriage conferred on McFarland.
I have a better right to speak, as one having authority in this matter, than most of you have, since it has been my province to study _Free Love_ in all its various lights and shades. When I practiced clairvoyance, hundreds, aye thousands, of desolate, heart-broken men, as well as women, came to me for advice.... The tales of horror, of wrongs inflicted and endured, which were poured into my ears, first awakened me to a realization of the hollowness and the _rottenness of society_ and compelled me... to ask the question whether it were not better to let the bound go free. In time, I was fully convinced that marriage laws were productive of precisely the reverse of that for which they are supposed to have been framed.
If our sisters who inhabit Greene Street and other filthy localities choose to remain in debauch, and if our brothers choose to visit them there, they are only exercising the same right that we exercise in remaining away.... I can see no moral difference between a woman who marries and lives with a man because he can provide for her wants and the woman who is not married but who is provided for at the same price.... The sexual relation must be rescued from this _insidious_ form of _slavery_. Women must rise from their position as _ministers_ to the _passions of men_ to be their equals. Their entire system of education must be changed. They must be trained like men, [to be] independent individuals, and not mere appendages or adjuncts of men, forming but one member of society. They must be the companions of men from choice, never from necessity.
Again, Utica stood and waved her white silk handkerchief rapidly to attract attention. She knew her sister was now totally in the thrall of the spirits. Cheers and hisses filled the hall. "Sit down," yelled a voice. "It's her sister," screamed out another. Vickie could no longer be heard above the tumult. She walked to the right side of the stage, leaned over the rail of Tennie's box, and asked her sister, "Can't you get Utie to go out?" At that moment a policeman appeared in the box on the opposite side of the stage, but as he caught hold of Utica's arm people cried out, "Leave her alone." The policeman sheepishly left the box. Tilton walked forward and shouted, "This meeting was called for the lady who was speaking when this interruption occurred...."
"I am her sister," Utica called out.
"I am in the chair," responded Tilton.
"Mrs. Woodhull, are _you_ a free lover?" a voice shouted.
"The love I cannot command is not mine," she replied. "Let me not disturb myself about it nor attempt to filch it from its rightful owner—"
"You have not answered the question," interrupted Utica.
Victoria flung her speech to the floor. She tore the white rose from the neck of her dress, and her words tumbled out: " _Yes! I am a free lover!_ I have an inalienable, constitutional, and natural right to love whom I may, to love for as long or as short a period as I can, to change that love every day if I please! And with that right neither you nor any law have any right to interfere. I mean just that and—"
But the noise was deafening, and few heard the end of Victoria's speech. What Utica had sought to do, she had done. Victoria's opponents were now in the ascendancy. The newspapers reported the most inflammatory aspects of her Steinway Hall speech. "Free Lover Lectures on Free Love," announced the _Argus_ headline. "Died of Free Love... The Woman Suffrage Movement," proclaimed the _Gazette_. Even Tilton ran from the scandal. The _Tribune_ quoted him as saying, "It was not the printed speech that did the damage, it was the interjected remarks.... She said violent things!"
Victoria Woodhull's problem was passion. When she openly admitted, "To preach the doctrine you must live the life!" she was censured as "a vile woman," one who possessed the carnal power to captivate and destroy men.
Martha Coffin Wright said that all over the nation women lined up against Woodhull and joined "the Free Love panic." She wrote,
I cannot understand the nice distinctions on the Free Love question and I don't mean to try.... It is hardly worthwhile to ostracize those who believe they have found a remedy for it all in some refined and incomprehensible idea of Free Love. The outcry, in most cases, proceeds I think, either from the timid, who are afraid of their own, or other people's shadow, or from the hypocritical, who join in the cry of "stop thief!" or from those who have always held aloof, as an excuse for their indifference, people who cannot trust in the eventual victory of truth over error.
Woodhull's supporters were now limited to members of the NWSA, who saw in her beliefs their last chance for enfranchisement; to Spiritualists, who accepted her as one of their own and had elected her president of their national association; to members of Section 12 of the International Workingmen's Association, of which she was honorary president.
To solidify the support of labor, Woodhull spoke at a meeting of the Reform Labor League on "The Impending Revolution." The content was vintage Andrews: a condemnation of the railroad barons, of corporate monopolies and of the great land grab that had appropriated property belonging to the Indians. After Roxy's clumsy attempt to blackmail Commodore Vanderbilt, Victoria and Tennie C. no longer had access to his world of money and power. With nothing left to lose, Victoria read Andrews's attack on the Commodore and his son William Henry.
A _Vanderbilt_ may sit in his office and manipulate stocks or make dividends by which in a few years he amasses fifty million dollars from the industries of the country and he is one of the remarkable men of the age. But if a poor, half-starved child were to take a loaf of bread from his cupboard to prevent starvation, she would be sent first to the Tombs and thence to Blackwell's Island.... It is a crime for a single person to steal a dollar, but a corporation may steal millions of dollars and be canonized as saints.
The _Times_ ridiculed the speech saying that Woodhull was "capable of mischief in inflaming the unthinking hostility of the poor to the rich." Undeterred, Andrews published the _Communist Manifesto_ in the _Weekly_ , the first appearance in America of Marx and Engels's 1848 document that called for the abolition of private property, child labor, and all rights of inheritance as well as calling for free education and a heavy progressive income tax.
On Sunday, December 17, 1871, a parade was organized to protest the French government's execution of Louis-Nathaniel Rossel the previous month for his participation in the Paris Commune. (Ironically, Rossel had been a member of the commune for only nine days, after which he'd resigned in protest over its military inefficiency.) Andrews selected Victoria to lead the parade. On the appointed day, approximately ten thousand people assembled at Cooper Institute and marched to the muffled beat of a drum corps composed of black Civil War veterans. First there was a catafalque drawn by six gray horses draped in black, behind which walked Tennie C. and Victoria carrying the flag of the Paris Commune. They were followed by members of Section 12 of the International, holding a banner that proclaimed, "Honor to the Martyrs of the Universal Republic." Cuban revolutionists marched with flags of blue and white, French refugees with red banners. Irish, German, and Italian groups were joined by members of the Printer's Society and the Workingwomen's Association. The procession moved like a funeral cortege down the Bowery. Police guarded their way as crowds gathered to mock and jeer the marchers.
Five years later, when criticism of her radicalism had reached its height, Victoria explained that she had reluctantly agreed to Andrews's request because the spirit of Napoleon Bonaparte had appeared before her and had warned her that "there would be a riot of the most fearful character" if she did not appear at the head of the procession. After Napoleon left Victoria said that she had experienced a vision in which she saw herself leading the parade past the door of her former residence, 17 Great Jones Street. Victoria then described how Bonaparte's prophecy had proved true: As the parade moved down the Bowery it was blocked by a wagon upset by a runaway team. The marchers were unable to move and the crowd moved menacingly toward them. Victoria, noticing that they were at the intersection of Great Jones Street, turned east on to the nearly empty street and led the parade directly past her old house. The police told her afterward that her quick action was what had saved the crowd "from being massacred."
This explanation was typical of the convoluted manner in which Woodhull used her clairvoyance to justify her actions. By the time she gave this explanation in 1876, Victoria realized that her support of the Marxist International had been misguided. To justify her error she claimed that the decision had not been hers but the spirits'. In retrospect, she wrote that although this parade "was the beginning of our financial ruin (as those who were supporting us could not understand why we should connect ourselves with the parties of whom that movement was made up) we never faltered in our allegiance to the spirits whom we served, but literally obeyed their every command."
The Rossel demonstration, which was meant to coalesce Woodhull's supporters, did just the opposite. Not only were conservatives offended but hard-line trade unionists and Marxists denounced it as a shameless publicity stunt. A German branch of the International noted that the Americans were more interested in Woodhull's "dazzling eyes and free love" than in establishing an eight-hour workday. After reading a subcommittee report that free love had "perverted the aims of the Association," Karl Marx himself suspended Section 12 until the next General Congress, scheduled for the following spring, with a strong recommendation that it be expelled at that time.
It was becoming increasingly risky for the women of the NWSA to place their hopes on Victoria Woodhull, but Elizabeth Cady Stanton saw no other way to achieve their goal. She wrote,
Some people carp at the "National" organization because it endorses Mrs. Woodhull.... Those of us who were convinced by her unanswerable arguments that her positions were sound had no choice but to follow. What if foul-mouthed scandal, with its many tongues, seeks to defile her? Shall we ignore a champion like this?... Admit for the sake of argument that... she has been or is a courtesan in sentiment and practice. When a woman of this class shall suddenly devote herself to the study of the grave problems of life... shall we not welcome her to the better place she desires to hold? Victoria C. Woodhull stands before us today a grand, brave woman, radical alike in political, religious, and social principles. Her face and form indicate the complete triumph in her nature of the spiritual over the sensuous. The processes of her education are little to us, the grand result is everything.
ON JANUARY 8, 1872, Victoria Woodhull and Colonel Blood took a train to Brattleboro, Vermont, to attend the funeral of Jim Fisk, who had died violently two days earlier. The scandalous death was attributable to his mistress, Josie Mansfield. Josie had grown discontented with Fisk and decided to leave him for Ned Stokes, the married dandy who had caught her eye. To support Stokes, Josie tried to blackmail Fisk for $200,000. Unless the money was forthcoming, Josie threatened to tell how he had bilked the public through the Erie Railway Company. "It is only four years ago since you revealed to me your scheme for stealing the Erie books," she wrote. She also threatened to reveal how he had rigged the gold market. "You surely recollect the fated Black Friday," she wrote with sarcasm, and asserted that he was mistaken in his assumption that there had been "no witnesses" to his "transactions." She said that she knew the value of this information, and it could be measured in dollars. Finally, she said that unless money was promptly paid, she was prepared to make their affair public by giving the _Herald_ some thirty-nine letters Fisk had written to her.
But Fisk was not a man to be blackmailed. He told his wife about his relationship with Josie. Lucy Fisk had come to terms with her errant husband and was not alarmed. Fisk had bought her a house in Boston, where she lived a quiet life with a female friend, attended church, played the piano, tended her garden. He lavished on her every luxury and seldom bothered her with his presence. She regarded him more as an attentive son than as a husband. Lucy had all the material possessions she wanted, a life with no cigar smoke, no alcohol, no sexual demands—a more felicitous arrangement than was the lot of many of her sex.
Josie also had misjudged the reaction of the public. Fisk was an acknowledged knave and buffoon, but in these rapacious times, success counted more than virtue. When he was not forthcoming with the cash, Stokes removed $200,000 from an oil account that, unbeknownst to Fisk, still bore Stokes's name as a cosigner. Fisk retaliated by filing an embezzlement suit against Stokes, after which Josie filed what was perhaps the first palimony suit in America.
Fisk marshaled his forces against his former mistress and her lover. Madame Annie Wood, in a sworn affidavit, asserted that shortly after the affair with Stokes began, Josie had asked Annie to provide a room at her Thirty-fourth Street "house" for trysts with him. Josie's servants also testified about her several lovers, including Stokes. Then on November 28, 1871, Ned Stokes was indicted for embezzlement. He was taken to the Ludlow Street jail, where he was allowed to bring his Oriental carpet, whiskey, and bottled water, and to have his dinner sent in from Delmonico's. On Saturday, January 6, 1872, Stokes was released on bail. He proceeded to the Yorkville police court to attend the second hearing of Mansfield v. Fisk. Throughout the morning session, Stokes heard himself depicted as a dissolute wastrel and a thief. That afternoon he hailed a carriage and arrived at the Grand Opera House just in time to see Fisk getting into his carriage and overheard him direct his driver to the Grand Central Hotel. Stokes took a shortcut through the ladies' entrance of the hotel and stationed himself at the top of the stairs. Fisk was on the fourth step when Stokes produced a pistol from under his double-breasted gray overcoat and fired twice. His porcine target was impossible to miss. A doctor probed for the bullets, but they were deep in Fisk's bowels. "When you were a boy did you ever run away from school and fill yourself with green apples?" Fisk asked the doctor. "I feel just as I used to.... I've got a bellyache." He died the following morning.
_Ned (Edward Stiles) Stokes—Josie Mansfield's vain, playboy lover_ (Illustration Credit 25.1)
_Ned Stokes shoots Jim Fisk_. (Illustration Credit 25.2)
After Fisk's death, Josie Mansfield enjoyed a certain celebrity. She sued his widow for the $200,000 she claimed Fisk owed her, but lost the case. Subsequently, she married an alcoholic playboy and moved to Paris. Stokes, who was tried three times for Fisk's murder, spent a scant six months in Sing Sing. Commodore Vanderbilt said that after Jim Fisk's death he was able to elicit his late rival's financial advice through a trance speaker, Mrs. Harris. He said that Fisk was now both truthful and helpful.
# CHAPTER TWENTY-SIX
A HEAVY LOAD
December 29, 1871
Dear Mrs. Woodhull,
Will you ask Demosthenes if there is any new argument not yet made on the 14th and 15th Amendments that he will bring out through some of us at the coming convention.... I want Frank Moulton to make a speech and his pretty wife to be there to adorn the platform. _I have written T.T. and will never forgive him if he fails us at this time_. We must not please Boston by having our convention a failure. Could we get _Beecher to speak one evening all by himself?_ He ought to do that much for us having spoken and presided for Boston. Try and bring that to pass. I think Moulton and Theodore could. See them about it.
Ask the spirits of Rachel, Elizabeth Barrett Browning, Malibran and Hemans to send down some fine woman's suffrage songs. I know they are all interested in our struggle and see as clearly as I do that we must sing as well as argue ourselves into the political kingdom. Can you manage to get our _call_ in all the New York papers? It should be flying all over the country by this time.... Have you invited some of the best Labor Reform speakers and some of the best Spiritualists? Let us have a real old fashioned protracted meeting full of enthusiasm. I would rather make a few blunders from a superabundance of life, than to have all the proprieties of a well embalmed mummy. Let me hear from you.
With Kind Regards, Elizabeth Cady Stanton
In this remarkable letter Stanton, with no discernible hesitation, asks Woodhull to contact some of the most famous women of the ages, all dead: the biblical Rachel; the poet Elizabeth Barrett Browning; Maria Felicita Malibran, a world-renowned singer who died in 1836, at the age of twenty-eight; and Felicia Dorothea Hemans, a Welsh poet whose body lay in a crypt beneath St. Anne's church in Dublin.
The second week in January 1872, Woodhull appeared on the platform of Lincoln Hall in Washington bracketed by Stanton and Anthony, who had traveled from Nevada without breaking her journey. After Woodhull's admission that she was a practicing free lover, further attempts to defend her personal conduct were abandoned. Even Isabella had changed tactics: She read a poem to the convention whose meaning was clear. It was about a stained little hand that might save all of mankind and womankind because it did not fear to work where a pure white hand would never venture.
At Stanton's request, Woodhull had written Beecher asking him to speak at the convention. He answered that his lecture schedule did not permit him to come, but before mailing the letter sent it to Moulton for approval and noted, "I do not mean to speak on the platform of _either_ of the two suffrage societies."
Moulton and Tilton had taken all responsibility for placating Woodhull. Prudently, Beecher, whose career was now at its zenith, had kept his distance. The first volume of _The Life of Jesus the Christ_ brought Beecher accolades, and a series of lectures delivered at Yale, published as _The New Haven Lectures_ , confirmed him as America's leading liturgical thinker. Tilton, on the other hand, thanks to his radicalism and his public alliance with Woodhull, found himself sinking—lectures canceled, money scarce, fame extinguished. Moulton asked Beecher to put a good word in the _Christian Union_ about Tilton's publication, the _Golden Age_ , but Beecher wrote merely that the _Golden Age_ was a small publication unattached to any church views. A dismayed Moulton wrote, "I am ashamed of it and would rather you had written nothing."
Tilton was stung by this latest insult, and when he drank he told anyone who would listen that Beecher had seduced his wife. Sam Wilkeson said of it, "This whole affair [is] the subject of conversation in the clubs." Lib Tilton, continuing her forbidden correspondence with Beecher, warned him of her husband's rage. A fearful Beecher turned Lib's letter over to Moulton and wrote to him,
It seems that a change has come to T.... Ever since he has felt more intensely the force of feeling in society and the humiliations which environ his enterprise, he has growingly felt that I had a power to help which I did not develop....
Nothing can possibly be so bad as the horror of great darkness in which I spend much of my life. I look upon death as sweeter-faced than any friend I have in the world.... To live on the sharp and _ragged edge_ of anxiety, remorse, fear, despair, and yet put on all the appearance of serenity and happiness, cannot be endured much longer.
The pressure on Beecher was unrelenting. At this time the Northern Pacific syndicate was so short of cash that workers were being paid in scrip, but when Sam Wilkeson told Henry Bowen that he would not be able to continue his commission payments, Bowen too threatened to expose Beecher. Frank Moulton decided it was time to settle matters once and for all. He went to Wilkeson and pointed out how much money would be lost if Beecher's reputation was destroyed. An enlightened Wilkeson agreed to bribe Tilton and Bowen into silence. Bowen was given a year's commission from the strapped Cooke syndicate and an extra $7,000, with which he paid off Tilton, who had lodged a suit against Bowen for violating his newspaper contracts. Subsequently, Tilton wrote Anna Dickinson that with this payment, "The whole case has terminated most happily."
Having made a major investment, Sam Wilkeson insisted that a written document, a "tripartite agreement," be executed in which Bowen, Tilton, and Beecher agreed to put aside all their past grievances and accusations "real or imagined" and make no further threats. When the agreement was signed, Wilkeson wrote to Moulton and advised, as a "closing act of justice and duty," that he place all the damaging correspondence in "the flames of the friendly fire in your room of reconciliation." Frank Moulton did not burn the correspondence.
In the coming presidential election, Tilton saw a chance to regain his power. Grant's first term had been so corrupt that the word "Grantism" became a synonym for graft, greed, and nepotism. The laissez-faire president had surrounded himself with relatives and politicians who dispensed patronage for gargantuan personal gain. Disaffected Republicans, including Theodore Tilton and many other newspapermen, calling themselves liberal Republicans, advocated "an uprising of honest citizens."
Since the night at Steinway Hall when Victoria had admitted to practicing free love, Tilton had seen that Beecher was right in distancing himself from her, and he had begun to do the same. Now came the final break. Tilton told Victoria that he was going to the liberal Republican convention with Whitelaw Reid and would report it for the _Tribune_. Since that paper had been particularly critical of Woodhull, clearly he could no longer be associated with her politically. "You are a liar!" Woodhull told him. "Why not be truthful and admit you are going to Cincinnati to nominate Mr. Greeley for president?" Later Victoria was to say that at that moment she experienced a vision of Tilton walking along, followed by a coffin containing the body of Horace Greeley. She mentioned this vision to Tilton and cautioned, "You will be responsible for putting him in that coffin... responsible for his death."
In her otherworldly way, Woodhull had sensed the change in Tilton. His own abject failure and Beecher's success had made a cynic of this once-idealistic firebrand. Now, through the liberal Republican Party, Tilton saw a chance to become more powerful than Beecher, and he was about to sacrifice his principles for personal gain.
On May 1, 1872, the liberal Republican convention convened. Whitelaw Reid and Theodore Tilton lobbied to put Greeley's name among the presidential candidates. On the sixth ballot, the convention nominated Greeley for president and chose Governor Benjamin Gratz Brown of Missouri for vice president. Many newspapermen supported the nomination of a fellow journalist. (One exception was E. L. Godkin, the editor of _The Nation_ , who wrote that this was the biggest disaster since the news of the first battle of Bull Run.) The majority of experienced political reformers, however, were disappointed to have a candidate with no previous political experience.
Susan B. Anthony attended the convention hoping to gain support for woman's enfranchisement, but was shocked to hear Tilton declare that woman's rights was not to be an issue in the campaign. An irate Anthony observed, "None but the liberals deride us now and Theodore Tilton stands at their head in light and scurrilous treatment." At the regular Republican convention in Philadelphia a month later, Grant was nominated by acclamation for a second term. Three weeks after that, the Democrats in Baltimore joined the Liberal Republicans in nominating Horace Greeley, an odd marriage to say the least: Men who had spent their lives as Republicans were now allied with Democrats who urged a "soft peace" and an end to Radical Reconstruction.
When Susan B. Anthony arrived at the Democratic convention she stated her position in the _World:_ "Baltimore Warned—Anthony says the Democrats Must Endorse Woman Suffrage or the Strong Minded Will Work for U. S. Grant." The warning did little good. At none of the conventions had Anthony been able to obtain support for women. Finally, Henry Blackwell managed to persuade the regular Republicans to issue a statement that "the honest demands of any class of citizens for equal rights should be treated with respectful consideration." Stanton called this weak statement the "Philadelphia splinter" in the party's platform and said that the Republicans would soon cut it down to a "toothpick." But Anthony felt that this was their only alternative. "Baltimore will not recognize our claim so of course we must make as _big a noise_ over the 'splinter'... as possible."
In the coming election, women faced a dilemma, for it was obvious that both parties intended to give only token support to woman's enfranchisement. Only Victoria Woodhull unequivocally supported women. Martha Coffin Wright wrote her daughter Ellen Garrison that Woodhull maintained that "both parties were utterly corrupt and a new one, labor and woman suffrage, should be formed. She was full of enthusiasm and faith in the coalition." By early spring Hooker and Stanton had become so convinced that Woodhull alone represented their needs that they seriously considered creating this political party to endorse her for president. Anthony, who knew that this would amount to a mere political protest with no chance of convincing either Republicans or Democrats, was becoming increasingly wary of lending her name in support of a woman who was controlled by men like Blood and Andrews. She wrote,
We have no element out of which to make a political party, because there is not a man who would vote a woman suffrage ticket... and all our time and words in that direction are simply thrown away. My name must not be used to call any such meeting....
I tell you I feel utterly disheartened.... Mrs. Woodhull has the advantage of us because she has the newspaper and she persistently means to run our craft into her port and none other. If she were influenced by _women_ spirits, either in the body or out of it, in the direction she steers, I might consent to be a mere sail-hoister for her. But as it is, she is wholly owned and dominated by _men_ spirits.
The 1872 National Woman Suffrage Association convention was scheduled for the first week in May at Steinway Hall. Isabella Beecher Hooker was now a complete convert to Victoria's belief in spirit guidance. She wrote, "Oh, if I were only _omnipresent_ wouldn't I do this business up handsomely. It makes me frantic, if I would allow myself to be anything but calm and hopeful, to think how much I could do if _my soul could work outside my body_."
By now Isabella was also convinced that her brother was guilty of adultery. Victoria had been the first to tell her of her brother's free love practices and assured Isabella that they would soon be revealed and that she would become a heroine if she could persuade Henry to come forward with his views on the subject. Victoria told Isabella, "I know that Mr. Beecher... need not stand alone for an hour, but that an army of glorious and emancipated spirits will gather spontaneously and instantaneously around him and that the new social republic will have been forever established." Stanton and Anthony also confirmed that the story was true, although Anthony would provide no details.
Isabella later wrote Henry,
Mrs. Stanton told me precisely what Mr. Tilton had said to her when in the rage of discovery he fled to the house of Mrs. B. [Bullard] and before them both narrated the story of his own infidelities as confessed to his wife and hers as confessed to him.... The only reply I made to Mrs. Stanton was that if true you had a philosophy of the relation of the sexes so far ahead of the times that you dared not announce it, though you consented to live by it.
Isabella now asked Henry if there was anything she could say at the next NWSA convention that would pave the way for more liberal sexual attitudes. Henry was appalled by the letter and decided to play on his sister's emotions.
April 25, 1872
My Dear Belle,
I do not intend to make _any_ speeches on any topic during Anniversary Week. Indeed, I shall be out of town. I do not want you to _take any ground this year except upon suffrage_. You know my sympathy with you. Probably you and I are nearer together than any of our family. I cannot give reason now.... Of some things _I neither talk nor will I be talked with_. For love and sympathy I am deeply thankful. The only help that can be grateful to me or useful is _silence_ and a silencing influence on all others. A day may come for converse. It is not now. _Living or dead, my dear sister Belle, love me_....
Isabella's immediate reaction was fear that her brother would kill himself once the truth came out. She later recalled, "My mind flew back to the sentence which suggested suicide to me the moment I read it.... And I believed even that." Her fears seemed to be confirmed when Stanton showed her the copy she had been given of Beecher's letter of apology in which he'd written, "I even wish I were dead." An overwrought Isabella determined that she must find a way to save her brother's life.
Isabella warned her husband, John, that it was only a question of time until the scandal became public, and that Henry might end his life. John reacted by telling his wife that his own nervous condition was such he was sure he would die from this added strain. Sensing catastrophe, Isabella arranged for her husband to go to Europe. He begged her to come with him, but, worried about Henry, she sent her daughter Mary in her place. A trip to the Alps did not improve John's health as Isabella had hoped, but at least at this distance the anxieties of one did not exacerbate those of the other.
With her husband abroad, Isabella occupied herself by taking control of the arrangements for the NWSA meeting in May. Both Stanton and Anthony were away on Lyceum lecture tours and knew little of the plans. In early May, a weary Susan B. Anthony was sitting in a railroad station in Illinois, waiting for the train that would take her to her next speaking engagement, when she saw a man reading _Woodhull & Claflin's Weekly_. When he left she picked up the discarded paper and, leafing through it, saw that a "People's Party" was to be inaugurated at the NWSA convention to support Victoria Woodhull for president. The "call" read in part, "We believe the time has come for the formation of a new political party whose principles shall meet the issues of the hour and represent equal rights for all." The sponsors of this new party were listed as "Isabella Beecher Hooker, Elizabeth Cady Stanton... Susan B. Anthony." Anthony jumped to her feet, dashed to the wireless office, and sent sharp messages to Stanton and Hooker demanding the withdrawal of her name. "I do not believe in any of us women, the majority of whom do not even own our bodies to say nothing of our purses—forming a _political party—slaves_ and paupers—do we see the farce."
Three days before the start of the convention, Anthony returned to New York and called on her friend Elizabeth Phelps of the Women's Bureau. Mrs. Phelps was well known for her involvement in woman's rights and was frequently featured in the popular press. She was said to be an heiress, yet no one could determine where her money came from or if indeed she had any. There was said to be a mysterious benefactor but again no one could be sure. To the tabloids' delight, in the spring of 1871 this socalled "millionairess" was arrested for shoplifting 22 cents' worth of sweet sticks at Macy's. She was released the same afternoon, but this notoriety exposed Elizabeth Phelps to gossip, speculation, and blackmail.
Victoria Woodhull seemingly had come to Mrs. Phelps's defense in the _Weekly_ , where she described how Mr. Macy's underpaid waiter-girls were known to pocket the money for an item that had been legitimately sold to a customer. Victoria suggested that merchants such as Macy and A. T. Stewart could stop the stealing by paying their employees a livable wage. But was Victoria Woodhull really Elizabeth Phelps's defender? Soon after Anthony's arrival at the Women's Bureau, a much disturbed Mrs. Phelps confided to her that Victoria had visited the previous week and demanded $500 or her name would be included in an article describing "the sexual liaisons and free love practices of some of the best-known women in the reformatory movement." Mrs. Woodhull, said Phelps, not only specifically accused her but also Laura Curtis Bullard, Mary Livermore, and Phebe Hanaford among others, and remarked, "If these women practice what I preach and yet play the hypocrite, I am prepared to broadcast this news all over New York City." Mrs. Phelps said she had not given Victoria any money because she didn't have it to give. Anthony wrote in her diary, "Called on Mrs. Phelps. Heard Woodhull's move to blackmail the women."
Desperate to recruit support for her presidential bid, Victoria had indeed compiled a set of "slips" detailing the sexual behavior of various individuals in the suffrage movement who she felt were maligning her. Much of this information came from Stanton, Hooker, and from the more scandalous daily tabloids. These "slips" were sent out to the women named, accompanied by a note stating that unless the accusations against her ceased and a payment to support her campaign was forthcoming they would see themselves in an article on free love in the _Weekly_ titled "Tit for Tat." When Tilton heard this, he told Victoria that if she destroyed the "slip" on Laura Curtis Bullard he would help her "kill the rest." Sometime later, when what she had done came to light, Woodhull tried to justify her actions by writing,
I concluded to shut the mouths of a clique of loose and loud-tongued women who were continually stabbing me... and making me a fiend incarnate in the eyes of the people. I grouped the clique together in an article of which I sent each member a printed _slip_.... The filthy fountains suddenly ceased to vomit forth their slime and I have had no occasion to publish the article but if it still arise I shall not hesitate to do so.
Paulina Wright Davis, who had retreated to Europe for her health, received a letter from Victoria enclosing a proof of "Tit for Tat." Mistakenly thinking it had been published, Mrs. Davis wrote,
My Dear Victoria,
Driven to bay at last you have turned, poor hunted child, and dealt a cruel blow.... Every one of these women you name has been hounded by men and now it suits them to make cat's-paws of them to hunt you.... Dear child, I wish you had let them pass and had taken hold of these men whose souls are black with crimes and who set up to be the censors of morality.... The first time I ever saw Mrs. Phelps, I was told by a _man_ that she was a woman of damaged reputation. Men are the chief scandalmongers of the age. It is they who report all the vile scandals of New York here and so make society detestable. You are not fooled by them, hence you must be crucified.
Anthony had no such sympathy for Victoria and was outraged by her blackmail schemes. After leaving Mrs. Phelps she traveled to Tenafly to confront Stanton about Woodhull and the People's Party. Anthony said that she had paid for Steinway Hall to be used by the NWSA, and she insisted that any other group could rent its own hall and meet somewhere else. Stanton replied that Anthony was being "narrow minded and domineering."
At a business committee meeting of the NWSA, held the day before the convention was to begin, Victoria Woodhull announced that this was to be a joint convention of her People's Party and the NWSA. Anthony responded that her group alone could use Steinway Hall. Stanton was now so angry with Anthony that she resigned as president and refused to preside at the convention. Anthony was elected in her place. However, Stanton did attend the first session and, defying Anthony, asserted in the keynote speech that women should vote in the coming election as members of Woodhull's People's Party. Then, according to Anthony's diary, Woodhull's supporters "claimed right to possess the meeting. Evident they were bound to try their strength and strategy." Anthony controlled the situation by refusing to recognize Mrs. Woodhull, but later wrote, "There was never such a foolish _muddle_ all come of Mrs. S. [Stanton] consulting _with_ and conceding _to_ Woodhull."
On the following evening, however, as the meeting was about to end, Victoria Woodhull stepped forward and announced in a clear voice, "I move that this convention adjourn and meet jointly with the People's Party tomorrow morning at Apollo Hall!" Anthony, sensing the audience was with Victoria, refused to put the motion. But Victoria once again called out the motion, and a majority of voices cried, "Aye, aye, aye." In a desperate effort to stanch the flow to Woodhull, Anthony announced, "You are not members of our organization. You cannot vote. This meeting is adjourned. We will meet in this same place tomorrow." She was ignored. Victoria Woodhull stood before the audience and spoke as if possessed by the spirits. She told them of her vision of a world free of corruption and oppression. But as the words tumbled out, Anthony left the platform, located the janitor, and ordered him to extinguish the footlights and then every other gaslight in the hall. In the darkness, bewildered women groped their way to the exits.
May 10, 1872: "Friday—National Convention... small audience. The fiasco's perfect—from calling People's Convention—never did Mrs. Stanton do as foolish a thing—all cause near being lost," Anthony wrote in her diary. "Saturday: I was never so hurt with this folly of Mrs. Stanton. Sunday: Pleasant day—but sad to me. Our movement as such is so demoralized by the letting go of the helm of the ship to Woodhull—though we rescued it—it was by a hair breadth escape."
Anthony waited apprehensively to see how the _Weekly_ would depict the newest schism in the woman's movement. "I never before came so near to losing all knowledge of myself as at New York those days—the Woodhull paper is not yet here." Days later, she wrote in her diary with some relief, "Woodhull paper comes freighted with glorious triumph. She subsumes everything into service, no denunciating, simply calls too slow. But the job a tee-total failure..."
_Susan B. Anthony's diary. The entries from June 11 to June 22 have been ripped out_. (Illustration Credit 26.1)
AT THIS MOMENT Victoria Woodhull was in her glory. At the Saturday Apollo Hall meeting the name People's Party was quickly changed to the Equal Rights Party and Victoria Woodhull was officially nominated as its candidate for president of the United States. She chose as her running mate Frederick Douglass, who later was said to remark, "I never heard of this," but he never formally requested that his name be removed from the ticket.
There were now three woman's rights organizations: the conservative Boston AWSA, the NWSA under Anthony's sole control, and Woodhull's Equal Rights Party, supported by Stanton and Hooker. The woman's rights movement was hopelessly fragmented, a disaster that Isabella Beecher Hooker failed to grasp. She wrote to Stanton,
_Apollo Hall was a success_ and through it the suffrage army moves in _three_ columns instead of _two_ —and each wing is a host. I have never realized the situation before and now I see the whole battlefield as plainly as though in the Heavens looking down upon it from the side of the Great-God Himself commanding the host. Do not smile at my enthusiasm. You know it is not _we_ who have brought ourselves out of recent perils.... Though Victoria has been a heavy load... now by the absolutely deferential tone of the Press toward Apollo and by the red flags and Communistic mottoes there displayed we must recognize the powerful aid the new party brings to Suffrage.
Isabella then wrote to Anthony praising Woodhull and asking if she was not "overpowered by the sublimity of this hour." Anthony, feeling exhausted and maligned, answered,
_Yes, I am overpowered_ with the sublimity of this hour—but no more so— _not_ so much so—as to have been at the sublimity of every hour and day for the past _twenty years_.... But I am too much in the condition of tee-total collapse of heart and soul... to be able to see or say the word or work of the hour.
The following week Victoria Woodhull held a rally to ratify her nomination. She was introduced by Isabella, who sat next to her on the platform. Using the pseudonym "Broadway," Isabella wrote about this event in the _Weekly_.
Mrs. Woodhull, the nominee for the Presidency, passed into an ante-room, where her friends crowded to congratulate her. She was in ecstacy.... The ladies kissed her and embraced her, kissed each other and kissed her again. I never before saw so much kissing and hugging in public, nor, for that matter, in private either. Men were not afraid to pass hands round women who were not their wives, and women indulged in political osculation till they were tired.
Anthony wrote Stanton, "What a ridiculous letter that is that Mrs. Hooker has published. It is too bad this kissing and hugging and putting away old men and getting new ones to hug and kiss emblazoned in print constantly of _our leading_ W.S. women—it's simply sickening—what can be Mrs. Hooker's object."
Susan B. Anthony had ample reason to distrust and dislike anyone or anything connected with Victoria Woodhull. In losing Stanton, Anthony had lost her other half. For almost a quarter of a century these two women had worked together, and the strength of the NWSA lay in this unity. Woodhull, in addition to being a blackmailer, had stolen away both her closest friend and the membership of her group, thereby jeopardizing the chances for enfranchisement. On Thursday, May 23, Anthony noted in her diary that she had stopped at the Rochester home of Mary Hallowell, an old friend, and had stayed for dinner with her, Amy Post, and a Mrs. Millis and her daughter. The evening was free of constraints, and the conversation revolved around Victoria and her sister Tennie C. Anthony was convinced that Woodhull would stop at nothing in order to force people to support her candidacy. She spoke of the "slips" against Phelps, Livermore, and Hanaford and told the others that Victoria was "resorting to blackmail intentionally and intended to succeed by that and other methods equally objectionable." What else was said—or allegedly said—that night was to have repercussions, for someone mentioned it to a Dr. Orvis who lived nearby.
The gossip spread with lightning speed. Fifteen days later, on the afternoon of June 7, Anthony arrived in Philadelphia and headed straight for the house of "my darling" Anna Dickinson. That night Anthony wrote in her diary, "Surprise found her home—not the old time greeting but cold, cold. She charges me with betraying her confidence—Livermore and Hanaford." Three days later, Anthony visited a friend and learned from him that Anna's sister Susan had told people that Anthony " _fabricated_ terrible charges against T. C. [Tennessee Claflin]."
The entries from the next day, June 11, until June 22 and again from July 23 to July 28 were ripped from Anthony's diary and burned. However, correspondence with others during this period describes the nature of the charges she allegedly made. Martha Coffin Wright wrote to Anthony, "We do not believe the charge of Orvis that you made the public charge against Tennie Claflin that he asserts you did, as there was a law passed by our Legislature a year or two ago making the charge of unchastity an indictable offense. On that account, if no other, you would be careful." Adultery could bring a jail sentence.
In a burst of letter writing, Anthony denied that she had accused Tennie C. of adultery. To Isabella she wrote, "That letter of Dr. A. Orvis was a _perfect fabrication_ of _course_. Mrs. Stanton gave me a similar one he had sent to Mrs. Woodhull. He's a _jack_ —you may supply the blank or not as you please. None but an _evil-doer_ could report me thus but how _little faith_ we have even _in those we know best_." And five days later she wrote to Isabella again concerning Woodhull: "I shall welcome her work, every bit of it, for women's freedom—though I do not choose to follow her into a party with these _men_. I have never on paper or with tongue said to anyone aught else. I did not reply or explain that _Dr. Orvis'_ letter from Rochester to you because I never dreamed of your counting it other than a _lie_ as it was."
In a subsequent article in the _Weekly_ , Victoria Woodhull stated that it had come to her attention that Anthony was assassinating the character of her sister Tennie and spreading rumors that Victoria was a blackmailer. She wrote, "The charges... as made by Miss Anthony... were either entirely without foundation or most outrageously perverted." Anthony again defended herself, writing that Woodhull had made her the "marplot." But, indicating that, in truth, she did speak out against the sisters, Anthony confided to her diary, "Well thanks to the good fates that somebody was enough to save us from that _swindle bound system_ —it is too humiliating."
Perhaps Dickinson broke off her relationship with her dearest friend for fear that Woodhull might retaliate against her for Anthony's insults. Woodhull could destroy Anna's career. Later, when Dickinson was asked why she had abruptly terminated her relationship with Anthony, she replied that Susan had grown "too old"—hardly a convincing reason. In any case, for the next quarter of a century not one word passed between these two women.
There was no doubt that scandal swirled around Anna's head and that she was frightened by it. Several friends knew that she practiced free love, and she had confided to them the details of a sexual relationship with none other than that great abolitionist Wendell Phillips. Phillips, a married man, was Anna's "secret lover." Whitelaw Reid wrote her concerning Phillips, "By the way I have been hearing a precious mess about you." A letter from Isabella Beecher Hooker to her husband written at the same time indicates that knowledge of the Dickinson-Phillips affair was widespread. John Hooker's reply anticipates the devastating effect this revelation and other sexual scandals might have on the woman's movement.
What you write us is shocking and overwhelming beyond expression. The matter of Anna D. was wholly new to me. I had heard a discreditable story about Phillips, but never a hint of her being the other party. It is horrible, I cannot conceive how she, at her ripe age, and with her good sense, and her comprehension of the whole case could submit herself to such a desecration.
But it is very strange that it's got to get out—or get so generally known that, as you say, it has got to come out. I do not wonder that in her distress, she should have divulged it to one or two confidential friends but it is inconceivable that they should have told it to others. If they did they ought to be shot.
But the case of H. [Henry] is more horrible in every way... it is greatly aggravated by the Woodhull matter mixing in with it.... When all this comes out—the H. story and that of Anna D. and Phillips, all woman suffragists, it will give that cause a heavy blow.... The Woodhull matter is a heavy load for it to carry and I do not wonder if all this superadded (and really illustrating Woodhullism) should create a reaction in the public mind that will send the struggle down to another generation and postpone its triumph for thirty years.
# CHAPTER TWENTY-SEVEN
MRS. SATAN
Whoever is set up to be President of the United States is just set up to have his character torn off from his back in shreds and to be mauled, pummeled, and covered with dirt by every filthy paper all over the country. And no woman that was not willing to be dragged through every kennel, and slopped into every dirty pail of water like an old mop, would ever consent to run as a candidate. Why, it's an ordeal that kills a man. It killed General Harrison and it killed old Zach [Taylor]. And what sort of brazen tramp of a woman would it be that could stand it and come out of it without being killed?
SO COMMENTED Harriet Beecher Stowe when Victoria Woodhull announced her bid for the presidency. Horace Greeley might have heeded this advice as well, for the campaign of 1872 deteriorated into one of the most vicious in American history. The woman's rights advocates were divided among Grant, Greeley, and Woodhull. Anthony was enraged by the damage that Victoria Woodhull had done to her personal life and her cause. She wrote to Martha Coffin Wright, " _Victoria_ and her _People's Party_ seceded from us and we are left alone. She having shaken the very dust of us from off her feet. Next to Horace Greeley, she is the most taken off her feet by the honors conferred upon her. Horace is surely a second child, if he ever was anything else and Tilton's _fawning_ around him is utterly disgusting." Moving forward with her remnant of the NWSA, Anthony and her group campaigned for Grant. "We must now all pull together to kill Greeley.... Let us hold up a looking glass before his eyes that will make him quake in his boots."
Beecher, too, supported the popular incumbent, President Grant. Tilton, seeing in Greeley his chance to redeem himself, dropped all other obligations to become Greeley's campaign manager. Since Tilton and Beecher were once again politically opposed, Frank Moulton examined every speech by both men to make sure there would not be another confrontation to upset the peace they had vowed to uphold in the "tripartite agreement."
Greeley put Reid in charge of the _Tribune_ and began a vigorous campaign, attacking the corrupt Grant administration. But soon Zach Chandler, chief of Grant's campaign committee, went to the railroad barons, stock manipulators, and promoters and warned them that if Greeley were elected he would put an end to the federal grants that netted them untold millions. Jay Cooke, who controlled the Northern Pacific Railroad stock syndicate, counted on federal aid. He'd given Greeley $20,000 worth of stock to promote his railroad, and now he felt this ungrateful editor had turned on him. Cooke promptly contributed $50,000 to Grant's campaign, and other powerful businessmen followed suit. Within weeks a realistic Greeley wrote, "The Grant folks are full of money and are using it with effect."
With business interests solidly behind Grant, Tilton and others advised Greeley to cultivate the Southern Democrats, who were reclaiming their states from the "carpetbag" administrations that had functioned mainly for their own profit. Tilton, who had said that Negro troops should be sent to the South as "terriers to watch such rats," now advocated the withdrawal of all federal troops, saying that their presence would only harden Southern hatred for the Negro. This man who had found Grant's Reconstruction policies "too soft" now abandoned all attempts at reform, writing,
We believe that the anti-slavery battle has been fought out. Slavery is abolished, and the Thirteenth Amendment makes its re-enactment impossible. The negro has been invested with the right of suffrage, and the Fourteenth and Fifteenth Amendments make his disenfranchisement impossible. Legally the negro stands exactly where the white man does. Socially whatever stigma rests upon him is far more oppressive in the North than in the South.
Tilton, the former idealist, had become as self-serving as the times in which he was living. He turned his back on his old friends from abolition days. When Benjamin Butler tried to enlist his help to create a civil rights act that would contain a provision mandating integrated education, he refused to become involved. Similarly, Tilton declined when Frederick Douglass asked him to intervene in the case of James W. Smith, the first black cadet at West Point, who had endured three years of severe persecution only to be drummed out of the academy for supposedly failing a test—one given to no other cadet.
Tilton's barely veiled policy of placating the South in order to win Greeley's election soon angered Northern voters and especially newspapermen. The cartoonist Thomas Nast, for example, turned the powerful weapon of his pen on Horace Greeley in _Harper's Weekly_. In a savage parody of Greeley's presidential acceptance letter, in which he appealed for the North and the South to "clasp hands across the bloody chasm," Nast drew a series of cartoons, including pictures of Greeley shaking hands with a Confederate who had just shot a Union soldier, stretching out his hand to John Wilkes Booth across Lincoln's open grave, and assisting a Ku Klux Klan member who had just lynched a black man and stabbed a black mother and child. In the cartoon "What I Know About Bailing Out," Nast depicted Greeley clutching the bail that he'd put up in 1867 for the former president of the Confederacy, Jefferson Davis. Although Tilton had broken with Woodhull, the tar of free love stuck to him and his associates, and Nast depicted the straight-laced Greeley surrounded by embracing free lovers.
To a public always susceptible to the instantaneous vivid image, the politically inept Greeley came to be seen as a malevolent panderer to the South, a floundering old buffoon. But Susan B. Anthony, who not long before had said she wanted to "kill Greeley," now wrote that though working for the "Republican Party and all women... I shall not join with the Republicans... and the liberals... in hounding Greeley.... The fact that old pro-slavery men proposed to vote for him, no more makes him proslavery than that drunkards or rum-sellers vote for him makes him a friend and advocate of the liquor traffic. My sense of justice and truth is outraged by the _Harper's_ cartoons."
With her break from Anthony, Anna Dickinson came more under the influence of Whitelaw Reid, who prevailed upon her to campaign for Greeley. Anna vacillated between Greeley and Grant. She deplored the excesses and corruption of the Grant administration, but she wanted proper payment to stump for Greeley. A courtship began: Tilton wrote her inviting her to spend the weekend with Greeley at his Chappaqua farm. But that same week, Grant offered her $20,000 to make six speeches for him. Finally Reid, realizing that Greeley needed a strong woman's rights speaker, offered Anna $10,000, to be delivered at some future date, to speak for Greeley. To further her career, Anna needed Reid's friendship and that of Greeley's _Tribune_. Under pressure, she agreed to Reid's proposal.
_The Whited Sepulchre. Greeley tries to cover a monument to his infamy with his trademark white hat and coat, in a Nast cartoon, 1872_. (Illustration Credit 27.1)
Anna had begun her career as an abolitionist and she too had a difficult time justifying Greeley's friendship with the South. Ultimately, she defended her decision to support him by maintaining that a vindictive policy would embitter an entirely new postwar generation. When Dickinson finally spoke at Cooper Institute in New York, she accused Grant of putting "thieves" in the internal revenue service, the Post Office and "other departments of the government." She said that if he had disqualified those who gave him gifts, "the state of New York would be officerless." Grant, Anna alleged, was directly responsible for stealing millions of the people's money, "and the half not told!"
With Dickinson campaigning for Greeley, and Hooker campaigning for Woodhull's Equal Rights Party, Anthony found herself in the odd position of being allied with the Boston AWSA against her dearest friends and compatriots. The _Woman's Journal_ stated of the NWSA, "We hope they have got rid of the 'Free Love' incubus which has done incalculable harm to the cause of woman suffrage. Women, like men, are known by the company they keep. The withdrawal of Mrs. Woodhull and her so-called 'radical reformers' will result we hope in bringing to the front of the New York society, women whose intellect and character command public respect." Lucy Stone wrote that she regarded the "Woodhull & Claflin tribe" as a "real curse."
Soon Henry Blackwell, Stone's wandering husband, added his advice:
And now, dear Miss Anthony, let me give you a few words of counsel. You work with a somewhat different class of women from ours, and will have to exert an influence with some of them, or they will undo all I have done politically.... Don't let them try to drive, bully, or threaten the party. Women can _persuade_ men, can reason with them, can appeal to their sense of _justice_ and _chivalry_. They cannot scold them into compliance.... Above all keep the Victoria Woodhulls... at a distance. For God's sake do not let that destructive element come into your meetings.
Not only did Anthony and Blackwell alike shun Woodhull, but the newspapers too gave her scant coverage. "Isn't it amusing to see how the papers ignore the Equal Rights Party and candidate or the _Tribune_ at least for I seldom see the _World_ or other papers," observed Martha Coffin Wright. Victoria Woodhull and her associates felt that a deliberate campaign was being mounted to suppress her ideas. Greeley's _Tribune_ decided not to report on Woodhull's political efforts, and the other newspapers followed suit.
Just as he had attacked Horace Greeley's credibility, Nast now demonized Victoria Woodhull for her demand for sexual equality. In _Harper's Weekly_ he depicted her as Mrs. Satan. There she stood, a winged female devil, her countenance beautiful and sinister. A set of horns sprouted through her gleaming dark hair. She held a sign that read Be Saved By Free Love. Behind her stood an emaciated woman with two ragged children, and on her back she carried a disheveled man clutching a bottle of whiskey. The caption read, "Get thee behind me, Mrs. Satan."
_Thomas Nast demonizes Victoria Woodhull as Mrs. Satan_.
This was a mortal blow: Woodhull was no longer considered interesting or impassioned but was perceived as someone to be avoided. As quickly as her fame had come, so quickly did it vanish. In city after city her speaking engagements were canceled. Without Vanderbilt's support, the Wall Street firm of Woodhull, Claflin & Co. foundered. Victoria was nearly out of money and could not meet the rent on the Thirty-eighth Street house. The lease was canceled. Down came the mirrors, the crystal chandeliers, the yards of velvet and damask, the Lawrence painting _Haying at Dorking_ , the cherubs and smiling nymphs.
Once again, the Claflins were on the move. They rented a less expensive house on Twenty-third Street and supplemented their income by holding twice-monthly Psyche Club meetings where Spiritualism, free love, and utopian world government were discussed. At these meetings Utica openly solicited the attentions of men for money and, as in the old days, there were rumors that the upstairs rooms were used for sex. The newspapers soon carried the story of how Francisco de Martín, the chancellor to the counsel general of Spain in New York, spotted a girl of about fifteen standing quietly in the hall at one Psyche Club meeting. Seeing the tears streaming down her face, he approached her and said, "My child, this is no place for you. They're discussing things here which you shouldn't listen to." The girl, Rosa Burns, was so happy to find a sympathetic friend that she confided to de Martín that she was pregnant and had been abandoned.
Rosa was the daughter of Victoria's sister Polly Sparr and her first husband, Ross Burns, whom Polly had pursued so many years before, holding the body of their dead child in her arms. Burns had become lieutenant governor of Kansas and knew nothing of his daughter's whereabouts. De Martín determined to remove Rosa from the Claflins' home and with no objection from her mother took her to live at his house with his own daughter, whose name also was Rosa. Despite de Martín's attentions, Rosa Burns's baby was born dead. The following day Polly Sparr showed up, accused the Spanish official of being the baby's father, and demanded money from him. De Martín denied the accusation and said he would pay for the burial, but that was all.
Polly then went to the collector of the port and accused de Martín of smuggling. A case was brought against him but he was proved innocent. Not long after, he returned to his house one day to find that Polly had stolen all of his furniture. De Martín then wrote Ross Burns himself (since Rosa Burns was illiterate) telling him of all that had transpired, and Burns immediately sent for his daughter to come live with him in Kansas. Even with the best of care, Rosa died several months later from complications of childbirth. In her will, she left Rosa de Martín what little money she had. Polly Sparr sued to recover her daughter's money, but lost the case.
Scandal heaped upon scandal: In an effort to break his morphine addiction, Canning Woodhull's doctor cut his dose. As a result, he died. Utica, vowing to "take revenge on the family," rushed to the coroner's office and informed him that Dr. Woodhull had died "under suspicious circumstances." The coroner arrived at the house and claimed the body. Though an autopsy revealed no foul play, the newspapers carried the story along with playful references to the fact that Victoria Woodhull no longer had two husbands living in the same household. In the _Weekly_ Victoria wrote that she had become the victim of her family's "insatiable vengeance."
As a result of this publicity, the family was ejected from the Twenty-third Street house. The Hoffman House, where they so splendidly began their career, now refused them. Again and again, they were denied lodging. Finally Polly, using the name of her former husband, Ross Burns, secured rooms at the Gilsey House, on the corner of Broadway and Twenty-ninth Street. Letters objecting to their presence soon began to arrive. When the manager asked the family to leave, Victoria thrust her hand into her purse and came up with a fistful of cash. "Here, here," she cried, "we are not paupers—what is wrong with our money?" And he relented.
Victoria appealed to Beecher:
My Dear Sir,
The social fight against me being now waged in this city is becoming rather hotter than I can well endure longer, standing unsupported and alone as I have until now. Within the past two weeks I have been shut out of hotel after hotel and am now, after having obtained a place in one, hunted down by a set of males and females who are determined that I shall not be permitted to live even, if they can prevent it.
Now I want your assistance. I want to be sustained in my position in the Gilsey House, from which I am ordered out and from which I do not wish to go—and all this simply because I am Victoria C. Woodhull, the advocate of social freedom. I have submitted to this persecution just so long as I can endure.... My business, my projects, in fact everything for which I live suffers from it and it must cease. Will you lend me your aid in this?...
Beecher turned the letter over to Moulton. "Will you answer this?" he asked. "Or will you see that she is to understand that I can do nothing! I shall not, at any and all hazards take a single step in that direction, and if it brings trouble—it must come."
Moulton told Victoria that no help from Beecher would be forthcoming, but he did so with regret and apprehension. He and his wife, Emma, had been intrigued by Victoria Woodhull, and Moulton could see the signs of her desperation. With an insight that few others possessed he would later observe, "I always knew what Woodhull really was—a proud, wounded, unlettered creature with some vigor of mind, the more vigorous for her ostracism.... We treated her with the courtesy the case required and by placing her under a higher motive than had generally been imputed to her, that of mercy and sacrifice, we had the scandal suppressed for a year and a half.... We abandoned her only when we found out that there was an adverse influence we could not overreach and that being the want and persecution she was to suffer."
A week after Victoria sent her letter to Beecher, four black lieutenants from the Spencer Grays called for their honorary chairman, Tennie C. Claflin, to take her to inspect a drill of their company. It was just the excuse the manager was waiting for. He told Tennie C., "If you go off with those niggers you need never come back here." Tennie, without answering, hopped into their carriage. When Victoria and Colonel Blood returned to the hotel later that evening they found Zulu Maud and Byron standing on the sidewalk surrounded by their luggage. At midnight they made their way to their newspaper office and hoisted Zulu over the transom. She unlocked the door, and they slept on the floor of the office. Even that did not last long, for the owner of the building raised the rent by $1,000 a year and asked for an entire year in advance. They did not have the money. _Woodhull & Claflin's Weekly_ was forced to suspend publication. With no money, no newspaper, and few supporters, Woodhull's campaign was abandoned.
From the beginning the powerful men behind Victoria Woodhull believed that this campaign was, as Blood put it, "purely educational." Benjamin Butler and Stephen Pearl Andrews delighted in seeing their theories reach a large audience through this beautiful woman, and Colonel Blood wanted to justify his actions. Only Victoria believed that she could actually be president. She believed it irrevocably. Believed, as Joan of Arc, that what her spirit voices told her would come to pass. But with every passing day she became more bitter and desperate, sick with exhaustion, and uncomprehending of why the spirits were leading her down this thorny path. To make matters worse, the summer of 1872 was the hottest that Americans had yet endured. Three hundred people died of sunstroke in New York City in the month of August alone. Persecution and desperation walked hand in hand. Near summer's end, the executive committee of the International Workingmen's Association met in The Hague, and Karl Marx himself denounced Section 12—and Victoria Woodhull personally. He expelled them from the organization, and Vickie's labor support fell away.
Elizabeth Cady Stanton now reconsidered her position. Woodhull no longer had wealth and a constituency. Victoria's blackmail schemes no doubt offended Stanton's sensibilities and perhaps threatened her as well, for as Anthony was to observe, "Human nature is awfully weak and wanting." Lucretia Mott wrote, "Elizabeth... is disposed to be very cautious how she identifies herself in any way with her [Woodhull] now." Henry Blackwell and Lucy Stone, sensing a weakening in Stanton's support for Woodhull, wrote her that she had been mistaken in thinking that Woodhull's party was the road to enfranchisement and that if she allied her interests with theirs, together they might make more progress with Grant in securing the vote for women. Stanton in return wrote Blackwell and Stone placating letters, to which the blunt Anthony commented, "All in the world they want is just what she has given them over and over again this summer, an acknowledgment _from her_ that she has been wrong and they immaculate in their management of affairs."
_Zulu Maud Woodhull—Victoria's loving and loyal daughter_ (Illustration Credit 27.1)
Evidently, Blackwell and Stone made an impression, for an article soon appeared in _Woman's Journal_ that indicated Stanton's withdrawal and that of her group from the support of Woodhull.
The secret of the final rupture... is understood to be an attempt at certain blackmailing operations.... But the friendship must have come to an end, sooner or later, at any rate. There was something not unworthy of sympathy... in the tolerance of Mrs. Woodhull, Tennessee Claflin and that gang... for there was pecuniary aid promised and Miss Anthony and Mrs. Hooker and others felt so acutely the oppression under which they labor in being deprived of suffrage that they cannot help welcoming aid from any source.... They have recovered from the delusion.... The prospect is that most of the prominent ladies known as leaders will work for Grant in one way or another.... Greeley is "a radical gone to seed."
Woodhull immediately wrote Anthony about this article, saying that "the inference... was that either Stanton or Anthony had been the informant." She asked, "Are you willing it should be so recorded?" Anthony wrote Hooker, expressing contempt for Victoria and her supporters.
I have not and shall not reply—because it is merely a trap to get a letter from me to publish in their book and they don't get it—that is all!
So I suppose her long pent up store of slang and slander is now to be emptied into the public maw—If any of those _men have_ ever been on their knees to her it was to little purpose—for her cry is that of the horse leech Give-Give!! If she does gather up every supposed and suspected weakness of every body and give them all in detail it will be a sickening heap.
The one group that still supported Woodhull was the National Association of Spiritualists. _The Word_ , a Spiritualist publication, defended her.
If anything shows the inability of women to vote intelligently it would be the present... attitude of Lucy Stone and Susan B. Anthony. The Republican Party has done nothing worthy of the support of Woman Suffragists. It advocates war, military despotism and the perpetual spoilization of labor. We ask that when they meddle with reforms that they not always take the wrong side as Lucy Stone persistently does.
In September, Victoria Woodhull arrived in Boston for the National Association of Spiritualists convention. She had decided to resign as president of this organization, for she felt so drained and exhausted that she could "carry this load no longer." She mounted the platform determined to tell her supporters of this decision, but suddenly the familiar physical transformation took place: Her skin visibly paled, her eyes became as glazed and bright as azure marbles. "I was seized by one of those overwhelming gusts of inspiration which sometime come upon me." Sparing no one, she told all that she knew of the sexual behavior of Beecher and Mrs. Tilton, of Blackwell and Mrs. Patton, of Livermore and Hanaford, and on and on. Elizabeth Meriwether, the Memphis, Tennessee diarist who had so feared Negro emancipation, was a Spiritualist delegate to the convention. She wrote, "Mrs. Woodhull's speech poured out like a stream of flame.... Editors, teachers, preachers, she spared not.... When she finished off Beecher she came back to Boston and lifted some of its editors high in the air and scorched them with accusations.... Her fiery flame went on until she suddenly stopped and flashed from the platform and out at a side door just as swiftly as she had flashed in."
An immediate repercussion was that Abby Patton, at the behest of her husband, Ludlow, finally severed her relationship with Henry Blackwell. And Julia Ward Howe turned her pen on Victoria Woodhull in the _Woman's Journal_ , calling her "a self-aggrandizing harlot" for destroying countless reputations with her "bogus revelations." In fact, Woodhull's revelations were so scorching that no newspaper dared print the particulars. The _World_ came the closest, stating that Woodhull said that Mary Livermore and Phebe Hanaford were known to support free love practices. The _Boston Journal_ noted only that she had made accusations against people of "impeccable reputation... and had sworn profanely." Woodhull replied, "I swore not profanely. I swore divinely."
The second week in October, Plymouth Church celebrated the twenty-fifth anniversary of Henry Ward Beecher's reign. A beaming Beecher watched as the Plymouth Church Sunday-school children paraded past his house and tossed bouquets of flowers at his feet. Henry Bowen spoke, and with great emotion recounted how he had brought the brilliant preacher from Indianapolis and persuaded him to take the pastorate of Plymouth Church. On the final day of the celebration, Beecher mounted the platform of Plymouth Church as his congregation swarmed about him. At his side stood the Reverend Richard Salter Storrs of the Church of the Pilgrims, who bitterly disapproved of Beecher's permissive doctrine, was jealous of his popularity, and opposed him politically. But Storrs delivered a florid congratulatory speech, after which Beecher embraced him and kissed him firmly on the cheek in front of the three thousand celebrants.
It was then that Woodhull, driven and hounded, announced, "I will make it hotter on earth for Henry Ward Beecher than Hell is below." If she was doomed to perish she would take this hypocrite down with her.
# CHAPTER TWENTY-EIGHT
BURST LIKE A BOMBSHELL
W _oodhull & Claflin's Weekly_ was closed, but Victoria scraped together the money and reactivated the presses. The world must know of the subterfuge and false propriety of those who professed to virtue. Victoria insisted that these hypocrites must be brought down. The spirits commanded it. This was the social revolution for which they had prepared her. Woodhull used a well-known quote that she attributed to Bismarck: "If an omelette has to be made, some eggs have to be broken." And she called this occasion, "Universal Washing Day."
One hundred thousand copies of _Woodhull & Claflin's Weekly_, dated November 2, 1872, were printed, obviously with the expectation that this publication would make enough money to replenish the Claflin family's empty coffers. One of the greatest scandals ever known was, as Victoria noted, about "to burst like a bombshell" on the public. "The Beecher-Tilton Scandal Case," as the exposé was called, appeared on the front page of the _Weekly_ , couched in the form of an interview with Victoria Woodhull by an unnamed reporter. The reporter questioned her views and she answered, as she had on many other occasions, saying that "the marriage institution like slavery and monarchy and many other things which had been good or necessary in their day—are now in a general sense injurious instead of being beneficial to the community." To illustrate this, Woodhull provided the salacious details of Beecher's affair with Mrs. Tilton. She named Elizabeth Cady Stanton, Isabella Beecher Hooker, and Paulina Wright Davis as her sources and published letters from them substantiating the story. She also used this so-called interview to promulgate her free love doctrine.
_Reporter_ —I confess, I cannot understand why you of all persons should have any fault to find with Mr. Beecher assuming everything to be true.
_Mrs. Woodhull_ —I have no fault to find with him in any such sense as you mean, nor as that in which the world will condemn him. I have no doubt that he has done the very best he could do—under all the circumstances—with his demanding physical nature... passional starvation, enforced on such a nature, so richly endowed... is a horrid cruelty. Every great man of Mr. Beecher's type, has had in the past and will ever have, the need for and the right to the loving manifestations of many women.
Woodhull went on to say that free love was a necessity for Beecher, that it was from his strong "amative impulses" that he derived his preaching skills and magnetism. "Mr. Beecher is today, and after all that I have felt called upon to reveal of his life, as good, as pure and as noble a man as he ever was in the past, or as the world has held him to be, and Mrs. Tilton is still a pure, charming, cultured woman. It is then, the public opinion that is wrong and not the individuals." Woodhull emphasized that she did not condemn the practice of free love, for which she had the highest regard. Beecher's sin was not lechery but hypocrisy. She criticized him only "for failing to do what I do, to stand shoulder to shoulder with me and others who are endeavoring to hasten a social regeneration which he believes in."
The interview ended with a barely concealed threat of more revelations. Victoria told the imaginary reporter, "We have five hundred biographies of various persons in high circles of life, many of which persons are the present oracles of society. The facts of these biographies are similar to those presented in this article."
The _Weekly_ was settling scores; Tennie C. too threatened blackmail through a letter that she alleged was written by "the Madam of a first-class house."
From the time that I opened my house, I have kept a sort of diary or record of the men who have visited it and entered in a business way the names and residences and some of the incidents of each visit of all the visitors at my establishment. What occurred to me was this. If you, in the prosecution of your blessed mission as a social reformer, have any need to see more behind the scenes and to understand the real state of New York society better, I will give you access to my two big books, or would even leave them with you in my absence. You will find in them the names of all classes—from doctors of divinity to counter-jumpers and runners for mercantile houses. Make what use of them you please.
Tennie indicated that she had accepted these books and that when the time came she might make use of them. Although several newspapers accused Tennie of writing this letter herself, it is probable that it came from her friend Annie Wood, who kept just such "big books." Madame Wood had a thriving business in prostitution as well as in assignation rooms such as the one she provided for Josie Mansfield and Ned Stokes. Annie was close to both Tennie and Victoria, and she herself admitted to receiving "profits" from Woodhull, Claflin & Co. Her "girls" no doubt also provided the sisters with a great deal of useful information.
The most inflammatory article in the November 2 issue, however, was one that appeared at Tennie's insistence—neither for revenge, nor for blackmail but for the sake of her conscience. Tennie explained that for three years this story had preyed upon her mind and that she must print it as a warning to her "sisters." Victoria agreed. "If this is a social revolution, we should start with a man like him." And so, on page 14 of the _Weekly_ , in an article titled "Beginning of the Battle," the dastardly behavior of one Luther C. Challis was revealed. But what Victoria and Tennie C. had also unwittingly exposed was the world from which they had come, and in so doing they inadvertently stepped into the abyss.
The article began with Tennie's statement that she had lived among prostitutes and knew them well. She felt they were an unfairly reviled and persecuted class, put upon by policemen and courts and customers; persecuted by the very "whoremongers that may marry a pure woman, move in good society and be generally respected." She wrote, "The world shall know the wrongs these women suffer and the men who inflict them."
To illustrate her point she described an evening she and Vickie spent at New York's French Ball in 1869. The annual ball, organized by the Société des Bals d'Artistes, was the most scandalous and anticipated event of the demimonde. At houses of prostitution and in apartments maintained by wealthy men, women assembled risqué costumes. By far the most favored was the skimpy tutu of the ballerina and the revealing bodice of the bayadere and daughter of the regiment. Wigs and masks completed the ensembles and guaranteed anonymity.
The night of the ball, "three thousand of the best men and four thousand of the worst women" arrived at the imposing Academy of Music on the northeast corner of Fourteenth Street and Irving Place. By eleven, almost a thousand men and boys had congregated outside, where the police with their billy sticks beat back a narrow passage for the masqueraders as they alighted from their carriages. Unattended women were greeted by hoots, yells, and whistles. Tennie reported that she and Victoria arrived at midnight "closely dominoed," not having informed Colonel Blood of their intentions. They both wore shepherdess dresses from the court of Louis XV. Tennie C. carried a flower-entwined shepherdess crook. Satin masks concealed their faces.
A thousand gaslights in crystal chandeliers illuminated the seats and stage as they made their way through the crush to a private upstairs box, furnished with red velvet couches, gilt chairs, and heavy red damask curtains trimmed with gold cords. So many dancers crowded the floor that the bodies were compressed into a writhing mass. Leaning over the bronze rail, Tennie C. spotted Luther Challis, a Wall Street broker, and Charles Maxwell, an unusually handsome playboy from a wealthy family, who, having little money of his own, confessed that he "lived on the town." These men were accompanied by "two fresh-faced young schoolgirls perhaps fifteen or sixteen years of age." The party entered Woodhull's box, and Challis ordered wine. Though he and Maxwell drank little, the girls' glasses were constantly refilled. One of the young girls was so besotted that she sank onto the couch. Tennie C., aware of the danger the girl was in, knelt beside her and begged her to stop drinking. With this, Challis turned to Tennie and snapped, "Let them alone!"
By this time, the scene below was one of abandon. Although the women were in costume and masked, the majority of men were not. In the adjoining boxes men stood smoking their cigars and watching the crush of dancers below. Several gestured or shouted to their servants on the floor to pluck a half-naked woman from the crowd and bring her upstairs to their box. The dancing had become wild and orgiastic. According to an account in the _World_ , published four days after the ball,
One of the women is caught up by the crowd and tossed bodily into a proscenium box, where she is dragged by half a dozen brutes in over the sill and furniture in such a manner as to disarrange as much as possible what small vestige of raiment there is on her. The feat awakens general merriment.
Presently the trick is repeated on the other side. A young woman, rather pretty and dressed in long skirts, is thrown up and falls back into the arms of the crowd, who turn her over, envelope her head in her own skirts, and again toss her up temporarily denuded....
A heavier woman than the rest is thrown out of a box and falls heavily on the floor. She is picked up insensible by the police and carried out. There is not a whisper of shame in the crowd, it is now drunken with liquor and its own beastliness. It whirls in mad eddies round and round.
_The infamous French Ball of 1869, attended by "three thousand of the best men and four thousand of the worst women" in New York_ (Illustration Credit 28.2)
Tennie C. saw that Challis was standing behind one of the girls, his hands thrust into the bodice of her dress squeezing her breasts while she leaned against him as if she would fall if he moved away. A moment later, Maxwell announced that they were taking the girls up to "Molly's." Molly de Ford, like Annie Wood, was a client of Woodhull, Claflin & Co. and the madam of a "first-class" house of prostitution. Tennie, overwrought, begged the men not to take these girls to such a place. In reply, Maxwell for the first time removed his hat and with a sweeping gesture thanked her for her hospitality. Then they were gone.
Victoria glanced into the box to her right where a girl lay on the crimson velvet couch, her ballet skirts pulled up over her head while two men mounted her in full view of the public. The son of Mr. Mercier, the manager of the Academy, entered the box and asked them to stop. One of the men pressed a glass of champagne into his hand while the other pulled his hat down over his eyes.
Victoria and Tennie left shortly thereafter. On the dance floor a wild cancan had commenced. The article in the _World_ said that this was "no longer a dance at all, but a series of indecent exposures, a tumultuous orgy in which one man is struck by an unknown assailant, and his cheek laid open with a sharp ring, his white vest and tie splashed with blood.... On the stairs and in the cloakrooms and through the narrow, tortuous passages leading to the stage dressing-rooms were vile tableaus of inflamed women and tipsy men, bandying brutality and obscenity." At a little past two o'clock the band played "Home Sweet Home," the gaslights were extinguished, and another French Ball drew to an end.
The fate of the two innocent schoolgirls troubled Tennie, perhaps because they reminded her of herself at that age. Both she and Victoria believed that if women cared to sell their bodies it was better than the slavery of marriage, but these were innocents who had no idea what they were doing. Within the week, Tennie persuaded her sister to visit the house of Molly de Ford. Here their worst fears were confirmed. The November 2, 1872, issue of _Woodhull & Claflin's Weekly_ included some of the same details that appeared in the _World_ in 1869. But Tennie's first-person account emphasized the plight of the schoolgirls involved. She wrote,
You may be sure I followed those girls up and got the history of their connection with these men. They were seduced by them.... They were taken to a house... of prostitution, then they were robbed of their innocence by each of these scoundrels, Challis and "Smith," taking them to himself. And this scoundrel Challis, to prove that he had _seduced a maiden, carried for days on his finger, exhibiting in triumph, the red trophy of her virginity_. After three days these Lotharios exchanged beds and companions and when weary of this they brought their friends, to the number of one hundred and over, to debauch these young girls—mere children.
On October 26, a week prior to the publication date, issues of _Woodhull & Claflin's Weekly_ were mailed to newspapers in the United States, Canada, and Great Britain. On October 28 a public sale began. Within three hours scores of newsboys hawking the issue had lined up in Broad Street, blocking it for hundreds of feet. By nightfall the paper, which had a face price of 10 cents, was bringing $2.50 a copy. (In the next three days 150,000 new copies were printed and sold, some allegedly for $40.) On the evening of October 28, as the city reeled under this "heap of scandal," Henry Ward Beecher stood in the library of his Columbia Heights home with United States District Attorney Benjamin Tracy, his loyal parishioner and political ally, and Henry Bowen, who had signed the "tripartite agreement" of friendship earlier that year. Beecher took no visible role in what happened next, but at one o'clock in the morning, Tracy's secretary knocked on the door of the Brooklyn home of the commissioner in charge of obscenity and "impure literature" for the Young Men's Christian Association.
_Anthony Comstock—the obsessive crusader against vice_ (Illustration Credit 28.1)
Anthony Comstock, five-feet-ten-inches tall and 210 pounds, his face framed in ginger-colored mutton-chop whiskers, was a fairly new arrival in Brooklyn. Thirteen months before, he had married Margaret Hamilton, a woman ten years his senior who he said reminded him of his mother. The Comstocks then moved from Connecticut to New York City, where he took a job as a dry goods salesman. After boarding in Williamsburg for two months he managed to buy a small house in Brooklyn for $500. But even as he worked as a salesman, sin was Comstock's obsession. For Comstock alcohol was evil, and he energetically harassed two local saloon keepers until one was driven out of business and the other died of a heart attack. But temperance never captured his imagination as much as the war against impure sex. In his adolescence Comstock had been a compulsive masturbator. His strict Calvinist upbringing convinced him that the profane images that accompanied such behavior led irrevocably to moral corruption. Comstock compared erotic feelings to electrical wires connected to the inner dynamite of obscene thoughts. Images of sexuality could cause a deadly explosion, destroying the soul.
Six months after his arrival in Brooklyn, Comstock scrawled a note to Robert R. McBurney, the director of the Young Men's Christian Association, which had been formed at the end of the war to provide the growing population of single young Christian men more wholesome pursuits than the city's gambling parlors, saloons, and brothels. Comstock's note was a desperate plea for help to eliminate the evils to which these men were exposed. The YMCA promptly hired Comstock to ferret out vice and pornography wherever it existed. He left his job as a salesman and began to pursue pornographers with such zeal that, with the aid of the police, as many as seven were arrested in a single day. After six months of employment, he was given $500 by McBurney as a token of appreciation. Comstock wrote in his diary, "God's hand was in this." But he wanted more. He longed to bring his crusade to the attention of the American public and to become known throughout the nation as a brave and famous warrior against sin—a paladin of purity.
Comstock was dressed in his nightshirt and cap when the one o'clock caller dispatched by Benjamin Tracy told him of the publication of "The Beecher-Tilton Scandal Case." He dressed quickly and went to Bowen's office at the _Independent_. At eight o'clock the following morning, Comstock dispatched three of Bowen's clerks to New York City to buy the _Weekly_ , no matter what it cost, and to send it to certain persons by post. On November 2, armed with affidavits and postal receipts, Comstock proceeded to the office of District Attorney Noah Davis, another Beecher parishioner. Although it was a Sunday, Davis was waiting in his office and immediately issued a warrant to shut down _Woodhull & Claflin's Weekly_ and to arrest Woodhull and Claflin for sending obscene material through the mail. The charge was based on a little-known postal law of 1865, which had never before been enforced.
Victoria and Tennie C. were intercepted in a carriage containing five hundred freshly printed copies of the _Weekly_. An overzealous arresting officer plopped himself on Tennie's lap and held her down as the sisters were carted off to the Ludlow Street jail, where they were immediately ushered into a side room. Five officials awaited them, three of whom were Plymouth Church parishioners. During the examination, one of these men blurted out that the sisters were accused not only of obscenity but also of "a gross libel on a reverend gentleman whose character it is well worth the while of the government of the United States to vindicate." Victoria insisted that they would say nothing until they obtained counsel, and therefore the sisters spent Sunday night in jail. On Monday, November 3, they were informed that their bail had been set at $8,000 each. Colonel Blood dipped into the profits from the _Weekly_ 's sales, and they were released before nightfall.
The next morning they were rearrested and informed that the case had been moved to the jurisdiction of United States Commissioner Davenport. In a hearing that day, Davenport said that the charge of obscenity had nothing to do with the Beecher article but was based on a phrase used in the article about Luther Challis, "the red trophy of her virginity." "That came from the book of Deuteronomy," Victoria Woodhull interjected. "The tabloids are full of this, added Tennie. "Why don't you try the _World_ and the _Sun_ , not us?" At the end of the hearing Davenport admitted that "a case of this character was never contemplated" when the obscenity statute was written but added, "I am disposed to, and shall, hold the prisoners." With no further explanation they were returned to their cell.
On Election Day, Susan B. Anthony, accompanied by fourteen Rochester women, including three of her sisters, went to the polls and intimidated the officials into allowing them to vote. "I have been and gone and done it!! Positively _voted_ the Republican ticket...," Anthony noted with pride. Three weeks later a U.S. marshal knocked on the door of her home with an arrest warrant. Anthony knew she was a visible symbol to the women of the nation and demanded to be handcuffed before being taken away.
In a crowded courtroom, in front of the press of the nation, Susan B. Anthony was tried for the crime of voting. The case was brought before an all-male jury and, as a woman, Anthony was refused the opportunity to testify. Late in the day the judge reached into his pocket, drew out a decision he had written before he heard the case, and proceeded to read it to the court. He instructed the jury to find Anthony guilty and insisted that they do so. When the verdict was read, the judge asked, "Has the prisoner anything to say why sentence shall not be pronounced?"
"Yes, Your Honor... in your ordered verdict of guilty you have trampled underfoot every vital principle of our government. My natural rights, my civil rights, my political rights, are all alike ignored...."
"The court must insist—the prisoner has been tried according to established forms of law...," said the judge.
"Yes, Your Honor, but by forms of law all made by men, interpreted by men, administered by men, in favor of men, and against women...."
"The sentence of the court is that you pay a fine of one hundred dollars and the costs of the prosecution."
"... I shall never pay a dollar of your unjust penalty.... And I shall earnestly and persistently continue to urge all women to the practical recognition of the old revolutionary maxim that 'resistance to tyranny is obedience to God,' " declared Anthony.
Though Anthony had struck a blow for women, she violated the conventions of the day. Lucy Stone's group was unsympathetic. Anthony wrote to Isabella Beecher Hooker in exasperation, "Do you know, I have sometimes felt that our Boston friends... try to persuade themselves that my prosecution and verdict and sentence are a disgrace and that they shrink from affiliation with an adjudged criminal. I do hope you can feel out, why it is that you and Mrs. Stanton—the two greatest Woodhull sinners—are more respectable than I. Isn't it too sick."
On Election Day, as Ulysses S. Grant swept into office for a second term with almost a million-vote plurality, Victoria and Tennie were confined to their cell in the Ludlow Street jail. Victoria Woodhull received no votes for the presidency, at least none that were recorded. From her cell, she had written to Anthony,
There is no time, now, to indulge in personal enmity. I have none toward anybody.... I fear they intend to crush out, in your person, the Constitutional Question of Woman's right to suffrage, as they are attempting, in my person, to establish a precedent for the suppression of recalcitrant Journals.... For my individual self, I have a pretty large fight of my own on hand, but... if you can, make use of me. Hoping to hear from you by return mail....
Anthony did not reply.
Horace Greeley, having suffered a humiliating defeat, was broken in health and spirit. The week before the election, Mary Cheney Greeley lay dying at the end of a tortured life. Greeley left her bedside to write, "I have been so bitterly assailed that I hardly know whether I am running for the Presidency or the penitentiary." Mary Greeley died at four o'clock on the morning of October 30, 1872.
The day after the election, Greeley made the announcement that once again he would assume the editorship of the _Tribune_. But in his absence his great love—his newspaper—also had slipped away. While campaigning, the value of his stock had fallen from $10,000 to $6,000 a share. He had only six shares left of his original hundred and had borrowed heavily against them.
On Greeley's first day back, an editorial bitterly denouncing him appeared in his own paper. He wrote a rebuttal, but Whitelaw Reid refused to print it. Greeley was convinced that Reid had betrayed him and that there was a plot to be rid of him. He wrote to a friend, "I dread only the malignity with which I am hounded, and the possibility that it may ruin the _Tribune_. My enemies mean to kill that, if they would kill me instead I would thank them lovingly."
Greeley's instinct was correct. Although Reid was not directly conspiring against him, a syndicate was being formed to oust him from his own newspaper. Greeley wandered in despair about the _Tribune_ offices, until, on November 12, he took to bed at the home of a friend, Alvin Johnson. There in the next few days, he wrote letter after letter disposing of his worldly goods. He wished to pay Elizabeth Cady Stanton's brother-in-law Dr. Edward Bayard the sum of $5,000. He wrote a rambling letter to remind Commodore Vanderbilt of the money he had given to Vanderbilt's epileptic son, Cornelius Jeremiah, and asked that the Commodore, upon Greeley's death, give an equivalent amount of money to his two daughters.
Reid prevailed upon Anna Dickinson to visit Greeley. "You will find him terribly broken," he commented. Greeley received her in a highly emotional state, brushing tears from his eyes. "Whitelaw tells me that since the speech you made on the twenty-fifth of October last, you have had over $14,000 of Lyceum engagements canceled," he said. "Is that true?" When she replied, "Yes," the grateful Greeley said, "Anna Dickinson, I think you are the most generous woman alive.... what you have done for me has cost you dear." Greeley was correct. The $10,000 she had been promised never materialized, and though the newspapers kept printing rumors of her impending engagement to Reid, her private letters indicate that he turned away from her the moment her popularity waned. Reid had persuaded her to support Greeley but joined the rest in deserting her once the election was lost.
The day after Anna's visit, Reid had Greeley committed to Dr. Choate's asylum in Pleasantville and took over as editor of the _Tribune_ , explaining that Greeley was suffering from "nervous prostration." At the end of November, less than a month after the election, Greeley sank into a coma. He'd left instructions for his burial: "No Latin—no embellishments... Whitelaw will distribute locks of my hair." And as a final word, as if he knew all that was to come, he wrote, "Be kind to Tilton. He is foolish—but young."
Even in death, Greeley's wishes were not respected. His funeral was held at the Church of the Divine Paternity at Fifth Avenue and Forty-fourth Street. President Grant and Vice President Schuyler Colfax attended, as did the governors of four states. Whitelaw Reid and Theodore Tilton sat side by side listening to Henry Ward Beecher's effusive ovation. A public that had once excoriated Horace Greeley and heaped abuse on him now saw him as a hero. In death, the doddering fool became a subtle genius; the compromiser was now a conciliator.
Gabrielle and Ida Greeley, having lost both their parents, finally explored the basement of the house in Chappaqua. Here they found the trunks full of china, silver, and linen—the amenities that Mary Cheney Greeley had hidden from her husband during her lifetime. The comforts that Horace Greeley had always lacked were divided between the two daughters who had managed to survive the childhood neglect of their father and the abuse of their mother.
# CHAPTER TWENTY-NINE
I CAN ENDURE NO LONGER
"CAN I HELP YOU?" Isabella wrote this single line on the day that the _Weekly_ appeared. Henry answered,
If you still believe in that woman you can not help me. If you think of her as I do you can perhaps, though I do not need much help. I tread falsehoods into the dirt from whence they spring and go on my way rejoicing. My people are thus far heroic and would give their lives for me.... I trust you give neither countenance nor credence to the abominable coinage that has been put afloat. The specks of truth are mere spangles upon a garment of falsehood.... Thank you for love and truth and silence, but think of the barbarity of dragging a poor, dear child of a woman into this slough.
Isabella was not reassured: she was tortured by the certainty that her brother not only was guilty but that he had, in all probability, incited the legal action against Woodhull. With her husband far away across the sea, she turned to her brothers Thomas and Edward Beecher for guidance. She wrote first to Thomas, a flinty preacher, "At last the blow has fallen," and enclosed Henry's letter with the following observation.
Now, Tom, so far as I can see, it is he who has dragged the dear child into the slough and left her there and who is now sending another woman to prison who is innocent of all crime but a fanaticism for the truth as revealed to her, and I, by my silence, am consenting unto her death.
Tom, can't you go to brother Edward at once... and when you have counseled together as brothers should, counsel me also, and come to me if you can. It looks as if he hoped to buy my silence with my love. At present, of course, I shall keep silent, but truth is dearer than all things else and if he will not speak it in some way I cannot always stand as consenting to a lie. "God help us all."
If you can't come to me, send Edward. I am utterly alone, and my heart aches for that woman even as for my own flesh and blood. I do not understand her, but I know her to be pure and unselfish and absolutely driven by some power foreign to herself to these strange utterances, which are always in behalf of freedom, purity, truth—as she understands it—always to befriend the poor and outcast, and bring low only the proud, the hypocrites in high places.
Thomas Beecher replied,
Dear Belle,
To allow the devil himself to be crushed for speaking the truth is unspeakably cowardly and contemptible. I respect, as at present advised, Mrs. Woodhull while I abhor her philosophy. She only carries out Henry's philosophy against which I recorded my protest some years ago and parted lovingly and achingly from him saying, we cannot work together.... In my judgment Henry is following his slippery doctrines of expediency and in his cry of progress and the nobleness of human nature has sacrificed clear, exact, ideal integrity.... Of the two, Woodhull is my hero, and Henry my coward....
But Thomas lacked his sister's bravery, and he refused to come forward or to involve Edward. He concluded his letter with, "Don't write to me.... You can't help Henry at present.... If Mr. and Mrs. Tilton are brought into court nothing will be revealed. Perjury for good reason is with advanced thinkers no sin."
John Hooker, then visiting in Florence, was unaware that the scandal had been printed in the _Weekly_ and that Victoria and Tennie C. were in jail. He too advised Isabella to distance herself from Mrs. Woodhull.
It looks as if the exposure is near at hand.... Can you not let the report get out after the H. matter becomes public, without being exactly responsible for it, that you have kept up friendship with Mrs. W. in the hope of influencing her not to publish the story, you having learned its truth.... This will give the appearance of self-sacrifice to your affiliation with her and will explain your not coming abroad with me—a fact which has a very unwife-like look. I know that you will otherwise be regarded as holding Mrs. W.'s views, and that we shall be regarded as living in some discord, and probably (by many people) as practicing her principles.
_The members of the Beecher family who became involved in the scandal_ (Illustration Credit 29.1)
Apparently no one was prepared to confirm Woodhull's story. Mrs. Fernando Jones, a prominent member of the Chicago NWSA, gave an interview to the _Mail_ in which she declared that her good friend Elizabeth Cady Stanton found Woodhull's allegations to be false. This was followed by an interview in the _Lewiston_ (Maine) _Telegram_ , reprinted in papers nationwide, in which Stanton was quoted as saying that Woodhull's story was "untrue in every particular." In fact, she had not denied the story, but had merely informed two clergymen from Lewiston that it was not she who had quoted Tilton as saying that Beecher was a "damned lecherous scoundrel." Unaware of this, Anthony, upon reading the Lewiston account, shot off a reproving letter to Stanton. Five days later, Stanton's answer arrived.
Dear Susan,
I had supposed you knew enough of the papers to trust a friend of twenty years' knowledge before _them_. I never made nor authorized the statement made in the Lewiston paper.... I have said many times since the _denouement_ that if my testimony of what I did know would save Victoria from prison, I should feel compelled to give it.
You do not monopolize, dear Susan, all the honor there is among womankind. I shall not run before I am sent, but when the time comes, I shall prove myself as true as you. No, no! I do not propose to shelter a man when a woman's liberty is at stake.
Though Anthony assumed a righteous position, when Isabella asked her to reveal what Mrs. Tilton had told her about the Beecher affair, she declined, writing that to do so would only injure Lib. "I feel the deepest sympathy... for poor, dear, trembling Mrs. Tilton. My heart bleeds for her every hour. I would fain take her in my arms, with her precious comforts—all she has on earth—her children—and hide her away from the wicked gaze of men."
The fact remained that still no one came to Woodhull's defense and thus Beecher was able to fend off the gossip-hungry press. On returning from a Sunday sermon, one reporter followed Beecher up the steps of his house brandishing a copy of the _Weekly_ , and asked why Beecher had made no statement refuting Woodhull's charges. The preacher answered, "When a man is unlucky enough as he passes along on a sidewalk to be drenched by a torrent of dirty water thrown on him from some upper window by some careless or mischievous hand, the best he can do is go home, wipe himself, and say nothing."
"Of course, Mr. Beecher, the whole thing is a fraud from beginning to end," said the reporter.
"Entirely!" replied Beecher.
Anthony wrote Hooker,
_"Entirely._ " Wouldn't you think if God ever did strike any one dead for telling a lie, He would have struck then?... For a cultivated man, at whose feet the whole world of men as well as of women sits in love and reverence, whose moral, intellectual, social resources are without limit—for such a man, so blest, so overflowing with _soul food_ —for him to ask or accept the _body_ of one or a dozen of his reverent and revering devotees, I tell you _he is the sinner—if it be a sin—and who shall say it is not?_ My pen has faltered and staggered; it would not write you for these three days....
Your brother will yet see his way out... and let us hope he will be able to prove himself above the willingness that others shall suffer for weakness and wickedness of his.... If he has no new theories, then he will surely be compelled to admit that he has failed to live or to preach those he has.
On November 23, Victoria Woodhull and Tennie C. Claflin were taken to the Tombs. This squat edifice on Centre Street, built to resemble the Temple of Karnak at Luxor, was the most dreaded of prisons. Open sewage pipes flowed past cells measuring only five by eight feet, some with ceilings so low that prisoners could not stand erect. Sometimes the vapors were so noxious that prisoners passed out. Victoria later said that the first night in the Tombs she had a vision of a fire sweeping toward the prison. She saw people screaming and bodies consumed by fire. She began crying hysterically and threatening to take her own life if she wasn't moved. On the day before Christmas, Victoria and Tennie C. were transferred back to the Ludlow Street jail. That evening a great fire swept Centre Street, but at the last minute a wind turned it away from the Tombs. Nearby, at the Fifth Avenue Laundry, seven scrub girls were reduced to ashes. The wooden roof of Barnum's circus caught fire, and all the wild animals that the circus had acquired over the years burned to death. Victoria speculated that the fire was arson, set by assassins who did not know she had been transferred.
From her prison cell, a despairing Victoria wrote Isabella that she was convinced that a campaign of suppression had been organized against her. Like Christ himself, she was being abandoned. She wrote that Mrs. Stanton knew all the facts and should come forward to deny the _Lewiston Telegram_ story. Anthony too wrote to Gerrit Smith that Stanton should be the one to speak out. "It is she who should annul her Lewiston denial—while _that stands—for me_ to speak _contrary to her_ would place both of us in rather awkward positions before friends and the public."
_The Halls of Justice on Centre Street, known as "The Tombs," was New York's most dreaded prison_. (Illustration Credit 29.2)
Determined to set the record straight, but not wanting to be Henry's "first accuser," Isabella wrote an article under the pseudonym Justitia in the _Hartford Times_ , asserting that as long before as the spring of 1871, Mrs. Stanton "had charged Mr. Beecher... with very much the same offense of which Mrs. Woodhull speaks." In an effort to induce Stanton to come forward, Isabella spoke freely of Stanton's confidences to her, including the fact that Elizabeth had always been in love with her brother-in-law, Edward Bayard. When Stanton learned that her most intimate secret had been revealed, she wrote to Isabella, "What did prompt you to betray all my confidences!... Do you not remember that consideration was in strict confidence. Who can we trust with anything. I begin to think our only safety is in living like oysters, each within a shell of secrecy without human sympathy." And a fortnight later,
I cannot tell you my surprise and sorrow in learning from four different sources how cruelly you are not only betraying my confidences but impeaching my integrity. What I would say I dare not put on paper. I considered that moonlight talk as sacred as if it had been with God himself.
I know the desperate effort that is being made to impeach in every possible way the integrity of Susan and myself, the two unswerving witnesses in the case on which turns the impending revolution. I cannot understand how you can so easily let all you do know of my head, heart, moral purpose and religious earnestness be outweighed by the designing words of an enemy and so cruelly expose the wounds I, in an unguarded moment, unveiled to you as never to mortal eye before. For although several friends know the material fact, to no one but you did I ever reveal the _steadfastness of my affection_ , that instead of seeking solace elsewhere has gone out into my life work.
I have never yet put the least confidence in any woman that it was not betrayed, though I have sacredly guarded everything ever entrusted to my hearing. You have said many things to me that would not bear repeating that have never passed my lips. Be careful what you believe and report to Susan's and my detriment just now. Theodore Tilton, that prince of liars, is moving Heaven and earth to impeach our integrity. You have deeply wronged me already, I pray you go no further.
After the defeat of Greeley, Tilton's problems seemed to engulf him. He desperately wanted to recover his career and his reputation and therefore had as much invested in suppressing the scandal as Beecher did. Because Anthony and Stanton had been Lib's confidantes, Tilton tried to undermine them, writing that they were not above "twisting the truth for their own advantage" and that his wife "deeply regretted her association with them." She had always considered these "public women" dangerous, and he "wished he had listened to her."
Isabella chafed under these attacks and the abandonment of her "Darling Queen." She had worried that her brother might kill himself when the scandal was revealed, but now it had all come out and Henry was "going his merry way" while Victoria Woodhull occupied a jail cell. This injustice preyed upon her mind until, after a sleepless night, she awoke with a solution. Less than a year before, Victoria told Isabella that she would be a "heroine" if she could persuade Henry to admit he was a free lover. Now she knew what she must do: "It seems to me that God has been preparing me for this work and you [Henry] also for years and years," she wrote to her brother. There was "but one honorable way" to deal with the question and that was for Henry to admit he was a practicing free lover and expound his advanced social theories on the subject. Isabella believed that, with her brother at her side, she would lead "the grandest revolution in the world." Victoria Woodhull had foreseen this. Isabella revealed her plan to Henry:
I will write you a sisterly letter, expressing my deep conviction that this whole subject needs the most earnest and chaste discussion... that I have observed for years that your reading and thinking has been profound on this and kindred subjects and now the time has come for you to give the world, through your own paper, the conclusions you have reached and the reasons therefore. If you choose I will then reply to each letter.
I am sure that nearly all the thinking men and women are somewhere near you and will rally to your support if you are bold, frank, and absolutely truthful in stating your convictions.... My own conviction is that the one radical mistake you have made is in supposing that you are so much ahead of your time, and in daring to attempt to lead when you have anything to conceal. Do not, I pray you, deceive yourself with the hope that the love of your church, or any other love, human or divine, can compensate the loss of absolute truthfulness to your own moral convictions.
The unexpected letter was a major threat to a situation that Beecher had felt was finally in hand. The previous April, Isabella had sent him a similar request, but this letter was stronger and more final. After consulting with Frank Moulton, Beecher wrote his sister that a _"calm silence_" was the only course of action he wished to take and added, "I have no philosophy to unfold and no new theory of society."
But Isabella was not to be stopped. The day after Thanksgiving, she spent the afternoon communing with the spirits. That evening she read an interview with the political cartoonist Thomas Nast in the _Hartford Times_. When asked if he didn't think it a great undertaking to attack Mr. Greeley, Nast answered, "The people were fooled with Greeley, as they are fooled with Beecher, and he will tumble further than Greeley yet."
Isabella took it as a sign.
Dear Brother,
I can endure no longer. I must see you and persuade you to write a paper which I will read, going alone to your pulpit, and taking sole charge of the services. I shall leave here on 8 a.m. train Friday morning, and unless you meet me at Forty-second Street station I shall go to Mrs. Phelps' house, opposite Young Men's Christian Association where I shall hope to see you during the day.... I would prefer going to Mrs. Tilton's to anywhere else but I hesitate to ask her to receive me. I feel sure, however, that words from her should go into that paper and with her consent I could write as one commissioned from on high.
Do not fail me, I pray you. Meet me at noon on Friday as you hope to meet your own mother in Heaven. In her name I beseech you, and I will take no denial. Ever yours in love unspeakable.
Belle
Beecher burst into Moulton's study and handed him Isabella's letter. "This is a disaster!" he exclaimed. Moulton thought Beecher should go immediately to Elizabeth Phelps's house, speak kindly to his sister, and exhort her not to take this course. Beecher answered that his sister was "a dribbling old fool" and he would not deal with her. It was decided that Tilton should go instead and, so far as possible, shake Isabella's confidence in the truth of Woodhull's story. Then Edward Beecher arrived and reported that he had just visited Isabella and found her to be "wild and excited." She had threatened to come to Plymouth Church and proclaim from Henry's own pulpit that he was a free lover. Tilton and Beecher took the ferry to New York. Tilton headed for Mrs. Phelps's house while Beecher contacted his sister Harriet Beecher Stowe, who had come to the city for Greeley's memorial service. He told Harriet that Isabella was crazy and persuaded her to occupy the front pew at the Plymouth Church Sunday service in case Isabella entered the church and tried to force herself onto the pulpit.
When Tilton arrived at her home, Mrs. Phelps invited him to remain for dinner with Mrs. Hooker. After dinner, Isabella went upstairs, which gave Tilton the opportunity to question Elizabeth Phelps. She told him, "There is no doubt that Mrs. Hooker will bring Henry to the pillory or down from his pulpit." A moment later, Isabella returned and Tilton drew her into the parlor. Tilton would remember the conversation that followed, for, though he could hardly acknowledge as much, it was the moment he lost all honor.
"What have you to communicate to me?" he asked.
"You know very well, because you know what Henry has been doing," replied Isabella.
Sensing her hesitation, Tilton said, "Madam, confession is good for the soul. Speak plainly. You have nothing to fear from me."
"Very well. I am here to charge Henry Ward Beecher with adultery with Mrs. Theodore Tilton."
"Madam, I anticipated as much. I am here to charge you with adultery at the time you were in Washington with..." And he announced the name of a prominent senator.
Isabella seemed stunned. She stared at Tilton but said nothing. "Where have you heard that Mr. Beecher is guilty?" asked Tilton.
"Don't be absurd. You know Victoria Woodhull and her sister are persecuted, imprisoned for telling the truth."
"If Mrs. Woodhull is the source of your information then I assure you that she is mine also. She has told me that you and the gentleman concerned have been criminally intimate."
As if Tilton's sword had finally penetrated, Isabella sank to the couch. Tilton left to the sound of her sobs. The following morning he reported to Moulton and Beecher in exact detail what had happened. When he'd finished, Beecher laughed, clapped his hands, and cried out, "Bravo!" But Moulton turned with a look of disgust on his face and left the room.
To make sure of the efficacy of Tilton's visit, Beecher himself appeared at Elizabeth Phelps's home the following day. When Mrs. Phelps inadvertently opened the door to the parlor she saw Isabella kneeling in front of Henry Ward Beecher as if imploring him for mercy. Within a fortnight, perhaps for insurance, a story appeared in the _Tribune_ stating that Henry Ward Beecher and two other members of his family were so concerned that Mrs. Hooker had fallen under the Satanic influence of Victoria Woodhull that they had consulted Dr. Harold Butler about her mental condition. Dr. Butler gave his opinion that "Mrs. Hooker was laboring under a monomania superinduced by over-excitement" and that "Mrs. Woodhull had exercised a controlling influence over a too susceptible mind." The doctor then recommended that Isabella be temporarily confined to an asylum for the insane.
Isabella wrote Stanton that her own brother Henry was willing to have the world see her as insane and begged her to defend the truth. At last the floodgates opened. Stanton wrote to Isabella,
Victoria's story is exaggerated, rather higher-colored than I heard it but the main facts correspond with what Susan and I heard. I have not a shadow of a doubt of its truth.... The outrageous persecution of Mrs. Woodhull in our court shows money and power behind.... Your persecutions in another way are as grievous and I am not willing to withhold anything any longer that can help to make things easier for you either on paper or by word of mouth. We are in the midst of a great social battle that will end in the absolute freedom of women and when the victory is gained we shall know that it is worth all that we have suffered. I have been crucified in this matter as much as you.... This is the first thing I have committed to paper on this T [Tilton] and B [Beecher] matter so use it judiciously for your defense.
# CHAPTER THIRTY
WHAT HAVE WE DONE NOW?
AS THE year 1873 began, unemployment escalated in a weakening economy. After seven years of growth following the Civil War, seven years of decline had begun. The corruption and extravagance of William Marcy Tweed's political leadership of New York, which had been allowed to flourish in prosperous times, now came under scrutiny. With Thomas Nast's pen cutting like a scimitar, Tweed soon replaced Victoria Woodhull as the representative of the devil. But Tweed was hard to topple until a disgruntled employee, James O'Brien, came into possession of a copy of the Tweed Ring's books. In exchange for silence he asked the Boss to cancel a loan of $12,000 and give him some $50,000 more. O'Brien had picked a bad morning for blackmail. Tweed ordered his henchmen to toss him out into the street. O'Brien then went to several newspapers trying to sell the books, but, being the recipients of Tweed's patronage, they turned him down. Finally, wanting revenge more than money, he took the books to the incorruptible _New York Times_. Placing them on the desk, he said to the editor, "You have had a hard fight against the Boss," to which the editor replied, "Have still." O'Brien pushed the books forward. "I said you _have had_ it."
The _Times_ 's exposé of the Tweed Ring revealed loot estimated at nearly $200 million. The chief scandal concerned the still incomplete courthouse, which so far had cost the taxpayers more than $17 million. The courthouse thermometers cost $7,500. Tweed's own printing company had received $187,000 for a single order of stationery. His own marble company had received $5 million. John H. Kaiser, the Tweed Ring's plumber, was paid $2 million. The plasterer Andrew Garvey got $3 million, and Tweed's friend James H. Ingersoll was paid $5 million for furniture and carpets. It was figured that Ingersoll could have covered half the island of Manhattan with high-grade Brussels carpeting for the money that had been spent and that Garvey could have applied his plaster six feet thick and used solid gold mesh for reinforcement. On January 2, 1873, Boss Tweed was arrested. After a protracted court battle, Tweed escaped to Spain only to be arrested a year later and shipped back to New York. The rigors of flight and imprisonment had ruined his health. He died in the spring of 1878 in the Ludlow Street jail.
In these hard economic times, the public sought retribution. Rumors circulated that powerful members of Congress, and even the vice president, had accepted at no cost stock and cash from "profits" in the Credit Mobilier, which had drained millions of dollars from the Union Pacific Railroad. A disgruntled congressman supplied newspapers with a list of these men, which he had surreptitiously taken from the office of Massachusetts Congressman Oakes Ames. The suspects denied the allegation. A government committee was formed to investigate the situation, but Republican businessmen exerted their influence to postpone the results of the inquiry until after the presidential election. On February 18, 1873, three months after Grant's reelection, the committee reported its findings to the House of Representatives. The investigation proved that Oakes Ames had kept careful records of payments to all the accused, a fact confirmed by deposits made to their personal accounts. Ames admitted his guilt. James Brooks of New York testified that Oakes Ames had given him free Credit Mobilier stock, but the others denied any involvement. The report stated that the investigating committee could not support the denials made by the congressmen. However, only Brooks and Ames, the two who had come forward with the truth, were found to be functioning from "corrupt motives." The committee recommended that they alone be expelled from Congress.
With the scandal of the Credit Mobilier, people began to look carefully into the financial structure of railroad bonds. By now the Northern Pacific was in deep trouble. With the monies expended to secure Grant's election and other expenses, Jay Cooke found that the Northern Pacific had overdrawn $1.6 million in one year while its previous overdrafts stood at an additional $5.5 million. Its bonds were selling at a heavy discount. Through political manipulation, when all other banks refused, Cooke was able to borrow $500,000 under favorable terms from the Freedmen's Savings and Trust Company, known as the Freedmen's Bank. This bank had been opened in 1865 to encourage the newly freed blacks to invest their money. The bank's advertisements proclaimed that Abraham Lincoln was their sponsor and read, "Cut off your vices—don't smoke—don't drink—don't buy lottery tickets. Put the money you save into the Freedmen's Savings Bank." In its first five years, blacks invested $3,299,201 in branches scattered across the country. Almost all of it came from small depositors—farmers and laborers—saving for a new life. Secretly, the reckless bank board freely squandered bank funds, making unsecured loans to friends, political cronies, and members of the Ku Klux Klan.
_The Freedmen's Savings and Trust Company made loans to members of the Ku Klux Klan_. (Illustration Credit 30.1)
Although Cooke had not paid back a penny of the Freedmen's Bank loan, he tried to raise yet another $300,000 through a bond offering for the Northern Pacific. Suspicion mounted that these bonds, which had been promoted with so much publicity, were, in fact, nearly worthless. As a result, there were so few takers that the offering was withdrawn and pronounced a "resounding failure." Susan B. Anthony wrote a friend, "If the bond plan is a swindle I think it ought to be shown up in the daily papers. It is such a shame for poor, hardworking women and men to be juggleried out of their hard earnings."
ON NEW YEAR'S Day, the portly Anthony Comstock attended church services and resolved "to do something every day for Jesus." He had seen to it that the "indecent" Victoria Woodhull and her sister had been imprisoned for a month, but now they were free on $16,000 bail and awaiting trial. Comstock read that Victoria planned to deliver an address, far and wide, titled "Moral Cowardice and Modern Hypocrisy—The Naked Truth—Thirty Days in the Ludlow Street Jail." He feared this speech would severely criticize him.
Although Comstock was suffering from an ulcerated sore throat, he proceeded by train and sleigh to Greenwich, Connecticut, where he rented a Post Office box under the alias of J. Beardsley and sent a money order for six copies of the November 2 issue of _Woodhull & Claflin's Weekly_. He then went to Norwalk, Connecticut, and entered the same order in the name of a friend who lived there. A few days later, when he received notice from his friend that the papers had been delivered, although he was still ill, he repeated the trip, picked up the twelve newspapers, and secured new warrants for the arrest of Victoria and Tennie C. for sending obscene material across state lines. This time he added the name of Colonel Blood to the indictments.
Victoria planned to inaugurate her speech at Boston's Music Hall, but Harriet Beecher Stowe saw to it that it was canceled and no other hall was made available in that city. Harriet noted, "Those vile women 'jailbirds' had the impudence to undertake to advertise... a lecture.... It appears that lectures cannot be given without a license of the city government which was not to be forthcoming. The impudence of those witches is incredible!" Woodhull managed to get her lecture rescheduled for January 9, at Cooper Institute in New York City, where Mrs. Stowe could not stop her. Now her obstacle was Anthony Comstock, who alerted the police to arrest Victoria before she was able to deliver a speech that would undoubtedly damage him and his purity crusade.
Help now came from an unexpected source. Laura Cuppy Smith, a Woodhull supporter and a famous Spiritualist trance speaker, heard of the impending arrest through a friend on the police force and immediately rushed to warn Woodhull. Mrs. Smith had come to feel the hypocrisy of the present society through the pain she had experienced in her own life and had vowed to support women who wished to take a less traditional path. Widowed at twenty-two, Laura had one daughter, Peggy, to whom she devoted her life. When Peggy was sixteen, she confided to her mother that she was pregnant and that the lover who had promised to marry her had deserted her. "Well-meaning friends" advised an abortion and told Smith, "There is a way. Hide this thing from sight, send her on a journey. Destroy this evidence of youthful folly. All may yet be well." But Laura felt that abortion was a sin against God. She resolved, "No dark secret shall dog my child's footsteps through life, no lie on her lips. This child of my child shall live and has a right to life!"
Together mother and daughter "entered upon our future." Friends turned their backs, doors closed to them. When the grocer refused their money and would sell them no food, "a woman of questionable repute... opened her door to us." When the baby was born Mrs. Smith wrapped her in a blanket and, with her daughter at her side, paraded down the main street of the town, defying all those who had shunned them. "We walked the whole length of the street, running the gauntlet of curious eyes. The worst was over, the world could not wound us much after that. We had 'grasped the nettle,' it would sting us no more."
Smith spirited Victoria Woodhull away and hid her in Taylor's Hotel in Jersey City. Colonel Blood, unaware of the danger, took his usual morning walk down Broad Street. A policeman approached him and asked, "Are you not the publisher of _Woodhull & Claflin's Weekly_?" When Blood answered yes, he was arrested. When the police arrived at their newly rented house neither Victoria nor Tennie C. were to be found. The only occupant was a laundress doing the family wash. After the police left, she lifted her wash-tub and then the large barrel on which it rested. Concealed beneath was Tennie.
That evening marshals were posted at every doorway to Cooper Institute, poised to arrest Woodhull should she appear. Though people were informed there would be no lecture, many pushed inside and waited for something to happen. At eight o'clock, Laura Cuppy Smith appeared onstage and announced that she would read Woodhull's speech. As she was about to begin, an elderly woman engulfed in a gray Quaker's cape and wearing a coal scuttle bonnet tottered down the aisle to a front seat. As Smith began, the Quaker lady limped up the stairway to the platform and walked halfway across the stage. Then with a sudden motion she flung her cape aside and tore off her bonnet. There stood Victoria Woodhull. "Comstock's defeated!" shouted Laura Cuppy Smith, and the audience cheered. The marshals dared not move. "I come into your presence from a cell in the American Bastille... indicted on the preposterous charge of sending obscene literature through the United States mail—a charge which the officers of the government will never dare bring to a trial."
Comstock, she avowed, was Beecher's tool. Officers had seized the _Weekly_ 's type, destroyed its presses, purloined their private papers, and kept them from speaking to anyone. Public opinion had been turned against them. Victoria said that she had read in the _Tribune_ that Comstock had vowed "to run me to the earth even if it took every hour of his life and every dollar of his fortune." If the "zealot Comstock" were allowed to prevail, "soon some woman may be arrested for adultery, for kissing her own baby boy." The case against her, she maintained, was concocted in a conspiracy between Comstock and Plymouth Church.
Woodhull asserted that the leaders of Plymouth Church, many of whom were involved in her prosecution, had said of the charges, "Whether they be true or not, we believe in _you_ Henry Ward Beecher... church and congregation and the public... we accept _you_." Woodhull spoke for an hour and a half. She concluded with, "Sexual freedom—the last right to be claimed for man in the long struggle for universal emancipation, the least understood and the most feared of all the freedoms, but destined to be the most beneficent of any—will burst upon the world." With that she dashed down the steps, ran to the back of the auditorium, threw her arms open in a gesture of crucifixion, and surrendered herself to the waiting officers.
On January 11, Victoria and Colonel Blood were brought forth for a hearing before the grand jury. During the examination Tennie, who had always stood by her sister, entered the room and surrendered herself to a deputy marshal. The sisters and Blood were to be represented by William F. Howe of the firm Howe & Hummel. Howe was a well-known defender of Wall Street confidence men, malcontented mistresses, and radicals. Like Jim Fisk and George Francis Train, he dressed in a flamboyant manner: plaid pants, a purple vest, a blue satin scarf set off by a diamond stickpin. His colleagues considered him effective with a jury but disreputable. Although Mr. Howe insisted that the particulars of the case and the exact charges be revealed, he was overruled. Bail was now raised to $21,000, and when it was forthcoming, the prisoners were released. Eleven days later they were rearrested. "What have we done now?" Victoria asked the arresting officer, who seemed thoroughly confused. "I think this one is for libel," he answered.
By now, cell number eleven in the Ludlow Street jail had become a familiar place. On its floor was a rug that belonged in Victoria's house, and the rough cots were made up with her own linen. Laura Cuppy Smith organized a group of Spiritualists who wrote letters asking for monetary support. Many return envelopes contained small donations, usually from women. That Victoria and Tennie's circumstances were becoming desperate was evinced by a letter Victoria wrote to one Lafayette Beach, stating that she had found an empty envelope from him and asking what had been in it. In a note with many cross-outs she wrote, "We have had so many assisting us in our correspondence it has got somewhat mixed up, but please excuse it on account of our condition."
In February 1873, Comstock, seeking greater enforcement powers to implement his crusade against such "sins" as obscenity, pornography, birth control, and abortion, went to Washington to strengthen the postal law of 1865. In order to keep Victoria and Tennie in jail, he petitioned Congress to add weeklies to the list of publications that fell under this law. Comstock then asked Congress to declare it a crime to advertise contraceptive devices or sexually related materials or to sell them. To give any contraceptive information, even orally, was also to be banned. Furthermore, he urged that the law stipulate that the death of a fetus, or of a woman as a result of an abortion, be punishable by up to twenty years in prison. Benjamin Butler took home the proposed Comstock bill to study it for the House Judiciary Committee and concluded that it violated First Amendment rights.
Undaunted, on February 24 Anthony Comstock appeared before the committee. Setting a small, scuffed, brown leather suitcase on the table, he withdrew from it pornographic engravings and postcards, contraceptives, and devices to implement sexual pleasure—his own private collection. Taking each object, one by one, from the suitcase, he displayed them to the assembled congressmen and implored them in the name of all that was holy to expunge these obscenities. Perhaps his own guilt over his sinful sexual habits and thoughts made him emphasize that impressionable young men must be kept away from images and objects that could "stimulate masturbation." Comstock asked that he be appointed the governments special agent of the Post Office to enforce the law, which would grant him the power to carry a gun and make arrests. Comstock had influential backers among conservative religious members of Congress who believed as he did that abortion was a sin against God. He was also supported by such recent organizations as the Medico-Legal Society, which opposed abortionists and midwives in order to restrict the profession to the male medical establishment.
While waiting anxiously to see if his bill would pass, Comstock attended a presidential reception, but was dismayed by the ladies at the gathering and found them "caricatures of everything but what a modest lady ought to be. They were brazen—dressed extremely so silly—enameled faces and powdered hair—low dresses—hair almost ridiculous and altogether most extremely disgusting to every lover of pure, noble, modest woman.... They disgrace our land." Before dawn on Sunday, March 2, Anthony Comstock walked the deserted streets of Washington and prayed aloud for the passage of his bill. At three o'clock Sunday afternoon he was at his desk at the Young Men's Christian Association when a messenger informed him that the bill had been passed by the House, had moved on to the Senate, and had passed there also. The following morning, Ulysses S. Grant signed the act and Comstock was officially appointed by the postmaster general as the special postal agent for the government. As he dashed toward the White House—oblivious to everything but sin—to thank the president in person, he was surrounded by the sound of music and the beating of drums, by the marching of thousands down Pennsylvania Avenue. He had become caught up in Ulysses S. Grant's second inaugural procession. At twenty-nine, Anthony Comstock had become America's most powerful crusader for purity.
When he returned to New York he attended three hearings in which Victoria, Tennie, and Blood were examined. Then Comstock took the stand. Under Mr. Howes vigorous cross-examination Comstock testified that the Challis article was "disgusting and obscene." However, when Howe read him chapter 22 of Deuteronomy containing the phrase "the token of the damsel's virginity"—which differed only slightly from what had been printed in the _Weekly_ —he claimed it was not obscene. Mr. Howe pressed him, "Even if they say the same thing?" "Yes, even so," replied Anthony Comstock.
When Victoria Woodhull asked to testify in her own behalf, Judge Blatchford denied her petition on the grounds that she was a woman. Under this continued harassment, Victoria began to crack: She read in the _Eagle_ that she was going to be sent to Sing Sing prison, and this rumor so unnerved her that she refused to eat for three days. A reporter from the _Sun_ visited her in her cell and found her pale and listless. She confided to him that the previous evening she had been kneeling on the floor of her cell when she had been visited by Jesus Christ. She described how her cell "was lighted up with spirit-light and the power of Heaven overshadowed us, while a still small voice whispered comfort to our troubled souls.... I went before the throne of grace and asked that Jesus... come and show me the right and He said... 'in the fullness of time all hidden things shall be revealed and you shall be justified where now you stand condemned.' " She told the reporter that she had seen Jesus twice before, but never had she realized so clearly that his martyrdom was to be her own.
Shortly thereafter, Woodhull wrote an open letter to the _Herald_ that was reprinted in newspapers throughout the country.
Cell 11, Ludlow Street Jail
Sick in body, sick in mind, sick at heart, I write these lines to ask if, because I am a woman, I am to have no justice, no fair play, no chance through the press to reach public opinion.... When has it ever been known in this land of so-called religious freedom and civil liberty, that pulpit, press and people tremble before a cowardly public opinion?
Is it not astonishing that all Christian law and civilization seemed to be scared out of their senses at having two poor women locked up in jail? Suppose, Mr. Editor, that some enemies of yours should throw you into a cell for publishing the Challis article... arrest your printers, prosecute your publisher, shut up your business office, close all avenues of press and lecture hall against your honorable defense? Would not every land ring with the outrage?
At this point Benjamin Butler stepped in to help by writing a letter to Woodhull that was published in the _Sun_ and the _World_ , stating that the statute under which the prisoners had initially been arrested had been "wholly misconstrued" and did not apply to the _Weekly_. "The statute was meant to cover, and does cover, sending that class of lithographs, prints, engravings, licentious books and other matters which are published by bad men for the purpose of the corruption of youth, through the United States mail.... If I were your counsel I should advise you to make no further defense but mere matter of law."
Mr. Howe argued Butler's point before Commissioner Davenport, who said that he tended to agree with Butler's premise and that ordinarily he would dismiss the charges, but he felt that the prisoners should remain under indictment until a grand jury made the final determination. Once again, the sisters were returned to their cell. A week later they were released on an aggregate of $60,000 bail, raised by selling the last of their possessions. Mr. Howe pointed out that Boss Tweed, who had been charged with looting the city of nearly $200 million, was released on bail of $51,000.
Anthony Comstock, afraid that the obscenity charge might be dismissed, leaving him with no case, visited Luther Challis and convinced him to sue Woodhull and Claflin for libel. On June 2, this case was called before Judge Noah Davis, a Plymouth Church deacon. Mr. Howe objected that Davis himself had instituted the obscenity proceedings against the defendants when he was district attorney. In the time that it took to shift the matter to another court a remarkable thing happened: Molly de Ford, at whose "house" Challis and Maxwell had debauched the two young women, supplied an affidavit confirming the _Weekly_ account. Molly said that she was prepared to testify in court that every word was true. Molly must have known the great price she might have to pay for this declaration: Her "house" could be closed and her livelihood taken away. But Molly de Ford, the keeper of a brothel, told Victoria that this was a moral issue and that she was willing to take the consequences. Challis dropped the charges. Howe moved swiftly and petitioned the court to bring to trial a perjury case he had lodged against Charles Maxwell, alias "Smith," the man who had accompanied Challis on the evening of the French Ball and had testified that neither of them knew the two girls involved. The judge replied that he "had reasons satisfactory for not trying the case." Howe immediately accused the judge of being in collusion with Anthony Comstock and, by implication, Beecher. In the eyes of the public, if the Challis and Maxwell story could prove true, so could the Beecher-Tilton scandal.
Repeated arrests with no proven charges offended the American sense of fair play. Yet another person now came to Woodhull's defense: the flashy George Francis Train, the millionaire organizer of the Credit Mobilier who had campaigned with Anthony in Kansas and had financed _The Revolution_. Because Woodhull had been charged with obscenity for printing a phrase from the book of Deuteronomy, "red trophy of her virginity," Train published a short document called _The Train Ligue_ , consisting of Old Testament verses dealing with nudity, murder, incest, and adultery. Then, hoping to make a test case, Train dared Comstock to arrest him for printing "disgusting slanders on Lot, Abraham, Solomon and David." Comstock responded, and Train was imprisoned in the Tombs without bail.
When he pleaded guilty to the obscenity charge, adding the words "based on extracts from the Bible," the plea was refused. The judge was none other than Plymouth Church's own Noah Davis, who had been disqualified in the Woodhull case for bringing the original action. Davis directed that the Train case be postponed. Train remained in the squalor of the Tombs for nearly five months. After that he was brought to trial, and Comstock recorded in his diary, "There was present the most disgusting set of Freelovers. The women, thin-faced, cross, sour-looking, each wearing a look of 'Well, I am boss'.... The men, unworthy the name of men, licentious looking, sneakish, mean, contemptible, making a true man blush to be seen near them. This is Free Love."
Judge Noah Davis persuaded Train's counsel to enter a plea of insanity by promising to dispose of the case by instructing the jury to bring in a not guilty verdict. After this was done, Davis, in what the _Sun_ termed "a Plymouth Church betrayal," pronounced that since Train was insane, he must be committed to an asylum in Utica. Anthony Comstock broke into a broad grin, prompting the _Telegram_ to comment, "Mr. Comstock, informer-general to the Young Men's Christian Association, looked quite triumphant at... the prospect of his victim being carried off to a dungeon in a lunatic asylum." Train drove away from the courtroom and boarded the first ship for England. Since he had been judged insane, the federal government impounded his fortune, including land worth approximately $20 million. He returned to America several years later, a pauper and a broken man. Some people said that having been pronounced insane, he became just that. He was known to sit for hours on a park bench feeding the pigeons crumbs of bread, and he would speak only to children.
In the face of this persecution of Woodhull and Claflin and their defenders, public sympathy increased. The Seymour, Indiana, _Times_ wrote, "That these women have been shamefully persecuted admits of no doubt.... Perhaps no class of people are so thoroughly given over to licentiousness as the sleek and well-fed clergy, especially those of our cities." The _Hartford Times_ commented that had the accusations against Beecher appeared in the _New York Times_ there would have been no arrests. Even Woodhull's bitter enemies now defended her. Mary Livermore wrote in the _Woman's Journal_ that both Woodhull and Susan B. Anthony were victims of a campaign to suppress women. And Martha Coffin Wright noted, "Her persecutors will gain nothing by the course they have pursued."
For the first time, Comstock and his obscenity crusade came under criticism. The _Brooklyn Eagle_ stated, "The people of this country are living under a law more narrow and oppressive than any people with a written constitution ever lived under before." Other newspapers agreed. " 'Liberty of the Press,' is ours no longer, when, in the opinion of any single person, the contents of a paper are not exactly moral or high-toned and should therefore be suspended and its publisher imprisoned. The strong hand of the United States Government is felt too often in Commonwealths where the local laws are all sufficient," commented the Easton, Pennsylvania, _Weekly Argus_.
In a sporadic issue of the _Weekly_ an editorial said, "From Maine to California we believe the new order of Protestant Jesuits called the YMCA is dubbed with the well merited title of the American Inquisition." However, Victoria commented that comparing Comstock to Torquemada was no more apt than "contrasting a living skunk with a dead lion." The _Weekly_ also listed many of the books that Comstock sought to burn, works by such diverse authors as "Byron, Cervantes, Swedenborg, Goethe, Dante, Plutarch, Aeschylus, Shakespeare, Hugo, Spencer, Virgil," and, in the Bible, passages concerning "Moses, Ezekiel, Solomon, Isaiah, St. John. What book will be spared?" Victoria stated that portions of the November 2 issue had appeared in no fewer than thirty-five other publications and Comstock had neither attacked them nor banned their now illegal advertising concerning woman's complaints or medical advice. Why, she asked, did Comstock not deal with advertisements in respectable papers offering to buy and sell illegitimate newborn children? She quoted such advertisements, including "A beautiful boy, three months old, for adoption to a wealthy couple." And "Wanted for adoption, a little girl, two to three years of age, light hair and blue eyes preferred." "Here is a game worthy of your steel," she challenged.
_The seal of The New York Society for the Suppression of Vice_
Some members of the YMCA began to feel uneasy about the fanatical zeal of their purity crusader and wished to distance themselves from him. Thus the New York Society for the Suppression of Vice was formed as an offshoot of the YMCA and Comstock installed as its head. The seal of the society depicted on the right side a top-hatted man throwing books into a blazing bonfire, and on the left a marshal with a billy stick in his right hand pushing a man into a dungeon.
The battle against Victoria and her sister was Comstock's first crusade for the Lord, but in that year alone, according to Comstock's records, in his capacity as the government's postal agent he seized "134,000 pounds of obscene books, 194,000 lewd pictures, 60,300 rubber articles, and 5,500 indecent playing cards." He examined and burned each item. With a gun at his side, he arrested abortionists, confiscated radical publications, and destroyed the presses of "impure" publications. In the process, he acquired a long scar from his ear to his ginger mutton-chops when he was stabbed by a desperate book dealer whose stock he had seized and whom he had arrested three times.
Multiple arrests were one means Comstock used to pursue his victims. Victoria and her relatives were repeatedly arrested on a variety of charges brought by members of the Society for the Suppression of Vice and by two Plymouth Church parishioners. One such libel trial lasted ten days. In an effort to expose her to ridicule, Victoria was questioned extensively on her social theories. She was asked such questions as, "What would you advise if a wife deserted her husband to live with another man?" To which she answered, "If her will takes her away from a man, she surely ought to go. I hold that any man or woman, whether married or unmarried, who consorts for anything but love, is a prostitute." Among the witnesses who testified for Mrs. Woodhull was Laura Cuppy Smith, who said that she had stayed by Victoria's side throughout her prison arrests and that Mrs. Woodhull was "a lady, a woman, and a mother." During this testimony Victoria's daughter, twelve-year-old Zulu Maud, sat clasping her mother's hand, and when Mrs. Smith described the closeness between the two, Zulu reached up and kissed her mother's cheek. Later the child told a reporter from the _Sun_ that every evening she sat next to her mother's empty chair and cried. The judge, one William Sutherland, openly favored the prosecution and called Woodhull's testimony "morally indecent." When the time came to instruct the jury, Judge Sutherland said that he felt there was no choice but to return a verdict of guilty. The jurymen retired to a room, where they debated the verdict throughout the night. The following morning at eleven o'clock they filed back into the courtroom and announced their verdict, reached after one hundred ballots: not guilty. Judge Sutherland cried out, "This is the most outrageous verdict ever recorded. It is shameful and infamous.... I am ashamed of the jury."
Victoria, Tennie, and Blood had won at last, but their financial resources were gone and the battle had taken its toll. When Victoria left the courtroom it was raining heavily. She stopped by the offices of the _Star_ and asked them to insert a "card" protesting her unjust treatment in the courts. "They are trying to kill me," she remarked to a reporter. She returned to her boardinghouse and told Colonel Blood she felt ill. She took only a cup of tea for supper and climbed the stairs. Halfway up, Victoria collapsed. Colonel Blood scooped her up and carried her to bed, and Tennie ran for a doctor. When the doctor arrived, he held a looking glass in front of Vickie's mouth, but there was no condensation. He tickled her with a feather, but there was no reaction. Another doctor was summoned. This one pricked her with a pin. Again, no reaction. With her Claflin training to enlist the sympathy of the press, Tennie rushed to the offices of the _Sun_ to inform reporters of what had happened. In the morning, newspaper headlines across the country proclaimed, "Victoria Woodhull—Dead."
# CHAPTER THIRTY-ONE
A MONSTROUS CONSPIRACY
AS TENNIE sat sobbing next to her sisters bed, blood began to ooze from Victoria's mouth and Tennie heard the sound of shallow breathing. For four days Victoria lay in a coma, but on the fifth she spoke about the lawsuits and her concern about the future of the _Weekly_. On the sixth day she was fit enough to receive an unexpected visit from Henry Bowen.
Bowen was ushered into Woodhull's bedroom by an inebriated Utica, who, laughing hysterically, tripped over the doorstep, fell down, and had to be helped to her feet. Victoria was propped up in bed. Bowen told her that it had come to his attention that she had in her possession some letters from Mr. Beecher showing that the preacher's "beliefs on love were freer than his public statements on the subject." He told Victoria that those letters would be valuable to him "for my own vindication."
"I have asked no one to help vindicate me!" replied a contemptuous Victoria. "Eight months ago I began this fight and stood my ground without assistance.... You are all millionaires. You and the press have hounded me and blackguarded me as no woman has ever been hounded before. But I have touched bottom at last. I am on the incoming tide." After a time, however, Victoria did admit that she had several letters from Beecher and intimated that the correspondence "was not one of mere platonic affection." She informed Bowen that she was to be tried for obscenity the following week in a new suit and added cannily, "I will reserve what evidence I have until I see the outcome." The inference was clear: She would trade the incriminating letters for her freedom.
On the day of the trial, before the proceedings began, the judge said he had an important announcement to make to the jury. He declared that the statute on obscene literature did not apply to _Woodhull & Claflin's Weekly_ because a modification to the statute to include the words "weekly" and "paper" had not been added until 1873, while the prosecution of this case had begun in 1872. Therefore, he instructed the jury to bring in a verdict of not guilty. The jury complied.
The following morning, Beecher wrote an open letter to the editor of the _Brooklyn Eagle_.
I have just returned to the city to learn that application has been made to Miss Victoria Woodhull for letters of mine supposed to contain information respecting certain infamous stories against me. I have no objection to have the _Eagle_ state in any way it deems fit that Mrs. Woodhull or any other person or persons who may have letters of mine in their possession have my cordial consent to publish them.... The stories and rumors which have for some time been circulated about me are untrue.
Bowen, having played his part, now wanted a reward, but there was none. He asked for yet another conference with Beecher, who is said to have replied, "There isn't force enough in Brooklyn to draw me into a private interview with Henry C. Bowen." When Bowen began to malign Beecher once again, Samuel Wilkeson lost all patience with him and released to the _Tribune_ and the _New York Daily Graphic_ the "tripartite agreement" in which all three parties had renounced their charges. He wrote, "It is high time that the torrent of slander against Henry Ward Beecher be arrested.... Mr. Bowen has of late repeatedly declared that he had never disavowed his charges against Mr. Beecher, but that he yet insisted on their truth. And now the public can understand the brave silence which the great preacher has kept under this protracted storm."
This was a disastrous miscalculation. People who had grown tired of the old rumors studied them anew. Why would three men sign a document to "repair and reinstate" their relationship no matter what wrong they had done one another? What had caused Beecher to state, "I deeply regret the causes for suspicion and estrangement"? And where were the letters "hereto annexed" that contained the "scandalous charges"?
After reading Beecher's letter in the _Eagle_ and the newspaper account of the "tripartite agreement," Anthony wrote to Isabella, "Mrs. Stanton asked me what I think of the Plymouth pastor's denial. I think it flies straight in the face of assertions and confessions of the Brooklyn male trio [Tilton, Bowen, Beecher] to say nothing of [Woodhull's] allegations. I tell you when God shall take up His old plan of punishing _liars_ there will be a good many people struck dead in Gotham and its suburbs."
Tilton added to the confusion. As the public began to speculate on Beecher's relationship with Mrs. Tilton, Theodore, hoping to put the blame on Beecher but still protect his wife's reputation, changed the accusation against Beecher from adultery to "attempted adultery," writing that Beecher had made "unhandsome advances" to his wife. He then forced the malleable Lib to write, "In July of 1870, prompted by my duty, I informed my husband that H. W. Beecher, my friend and pastor, had solicited me to become _a wife to him, together with all the relations which that term implies_." Theodore delivered this "True Story" to a young journalist, Edward H. G. Clark of the _Troy Daily Press_ , who had become interested in the case. But if Tilton thought that Clark would swallow his story whole, he was mistaken. Clark began an investigation of his own. He baited Paulina Wright Davis, who was then in Paris, by writing her a letter stating that Tilton had said she alone "stood responsible for the story."
In fact Mrs. Davis, who was suffering from tuberculosis, had removed herself from the fray. Her strength was failing. Her husband had taken her to Europe, away from the quarrels of the woman's movement. Because of her illness her family refused "utterly to let me get involved with this subject." Nevertheless, Paulina, who had not seen the November 2, 1872, issue of the _Weekly_ , donned her chinchilla-trimmed blue velvet coat with matching hat, went to her banker's office, and asked him if he was holding any newspapers for her. Instead of answering directly, the stiff-collared banker told her, "The _Weekly_ is unfit for a lady to read."
"Allow me to judge that for myself. If you have my property you cannot legally withhold it." With that, the banker reached into a small cubby and handed her the controversial issue of _Woodhull & Claflin's Weekly_.
Mrs. Davis then answered Edward Clark's letter, stating that she was not the sole source of information on the affair between Lib and the Reverend Beecher. She had heard the story well over a year before from Laura Curtis Bullard, to whom it had been confided by Tilton himself. She described her visit to Mrs. Tilton and Lib's confession and added, "It was even then almost common property." This letter was followed by one to Victoria Woodhull in which Mrs. Davis promised to return to America to defend her friend. Instead, she was sent to a sanitarium in Switzerland, lingered for a year and a half, and then died. Of what had happened to the woman's movement Paulina wrote, "I think our defeat is something like the Communists', the fight has been a brave one for freedom but too soon."
In May 1873, Clark printed a four-page, single-issue newspaper, _The Thunderbolt_ (referring to Beecher's remark that Mrs. Tilton's confession "fell like a thunderbolt on me"). The eight-column headline read, "Beecher-Bowen-Comstock Conspiracy. The Seal Broken at Last. The Account Horrible at Best. Not 'Obscenity,' but God's Truth." Clark was so opposed to Woodhull's radical politics and views on free love that he concluded that she ought to be "hanged" but felt "a monument to her memory [should be] erected at the foot of the gallows." With grudging admiration, Clark wrote that "a monstrous conspiracy" had been mounted against Woodhull to protect Beecher from "deserved shame." He documented how the arrest of Woodhull and Claflin had been engineered by Comstock and Bowen in the offices of the _Independent_ , Bowen being "Mr. Beecher's chief 'supe' and conspirator in combining with that wretched Jesuit of Protestantism, Mr. Anthony J. Comstock, to violate American liberty." Clark declared,
The special friends of Henry Ward Beecher—the skulkers of Plymouth Church and the Young Men's Christian Association—preferred to deflower the laws of their country and the freedom of its people by a gigantic performance of bigotry and chicanery. In the shadow of their false pretenses, the Woodhull slanders, however atrocious, have grown comparatively dim and insignificant.
Clark alleged that because Victoria and Tennie C. were blackmailers and free lovers, Anthony Comstock and members of Plymouth Church found them easy targets and asked, "What is blackmail compared to the great city dailies? How many times was the _World_ blatant with threats at the Tammany Ring and then sopped into silence. Whitelaw Reid has lately elected himself editor and publisher of the _Tribune_ , with half a million dollars behind him. Who owns the dog now that nosed Greeley into the grave?"
Clark asserted that powerful men had banded with Beecher to crush Victoria. He pointed out that Thomas Shearman, known as "Tearful Tommy" for his courtroom histrionics, was now advising Mr. Beecher. Shearman had been involved in many scandals, including the bribery of Tammany judges, and "covered the feculence of his career" by being superintendent of the Plymouth Church Sunday school. Plymouth Church, Clark concluded, was a social, political, and financial institution, and its parishioners could not afford to see their leader fall.
To counteract _The Thunderbolt_ , Tilton now took his True Story to the Reverend Richard Salter Storrs, asking for his advice in this matter. When Tilton told Beecher of this he gasped, "Oh, Theodore, of all the men in world, I wish you had kept clear of Dr. Storrs!" By involving the church in this matter, Tilton had made a dangerous move, for Richard Salter Storrs was a man who deplored Beecher's laxity in eliminating the Articles of Faith, hated his appeal to the emotions, and envied his popularity. Storrs told Tilton that he would consider a church investigation of the matter. Beecher was devastated. He wrote Moulton that, "I have determined to make no more resistance.... I have a strong feeling upon me and it brings great peace with it, that I am spending my _last Sunday_ and preaching my _last sermon_."
_The Reverend Richard Salter Storrs called for a council of churches to look into the scandal_.
_Storrs's Church of the Pilgrims on the corner of Henry and Remsen Streets in Brooklyn_
In his suffering, Beecher turned to Frank Moulton's wife, Emma, who had been his parishioner since she was a child. According to her friends, Emma was an honest and retiring woman. Some thought her shy; in fact she was simply quiet. Emma had not approved when her husband had been drawn into this situation, but she felt that Frank had to carry through as mediator, for there was so much at stake—this scandal would bring disgrace on her close friend Lib and her poor innocent children. Beecher visited Emma Moulton and told her of his pain and regret. He spoke of his love and esteem for Mrs. Tilton, but as he grew more desperate he increasingly blamed Lib for his problems. He asked, "Why did she not tell me of the situation? It is to my great remorse and sorrow that she should ever have confessed to her husband." And why, he asked, did Theodore feel so "hard" toward him, since for over a year "he had condoned his wife's fault"? No longer was this Beecher's sin, but as Mrs. Morse had said of her daughter, it was "her sin." Emma was shocked by this injustice. She recalled that
He walked up and down the room in a very excited manner, with tears streaming down his cheeks and said that he thought it was very hard, after a life of usefulness, that he should be brought to this fearful end.... He sat down in a chair.... I stood behind him and put my hand on his shoulder. "I will always be your friend if you will only go down to the church and confess, because that is the only way out for you."
"You are always to me like a section of the Day of Judgment," he replied.
Then Emma said, "I have never heard you preach since I knew the truth that I haven't felt that I was standing by an open grave. I cannot express to you the anguish and the sorrow it has caused me to know what I have of your life. I believed in you since I was a girl—believed you were the only good man in this world. Now it has destroyed my faith in human nature. Now I don't believe in anybody."
As Beecher feared, Richard Salter Storrs saw his chance to seize power. A small group within Plymouth Church had brought an action to expel Theodore Tilton for his association with Victoria Woodhull although he had voluntarily retired from the church three years earlier. Beecher explained to Tilton that if he would simply appear before this committee, his name would quietly be removed from the rolls. But when Tilton appeared he found himself facing a hostile group that subjected him to a prolonged inquisition and then expelled him for "gross immoral behavior." Using this action as an excuse, Storrs called for a council of churches to review the case, maintaining that Plymouth Church had acted against "the purity of Congregational policy" and should be considered for disfellowship. Beecher was asked to appear before the council to "reaffirm the Articles of Faith and re-establish the administration of Christian and orderly discipline" at Plymouth Church. Beecher refused, maintaining that his church had the right to act independently in expelling Tilton, and the Plymouth Church rules of discipline were quickly amended to exempt him from going before a committee of his fellow ministers. Having at an early age lived through the heresy proceedings against his father, Lyman Beecher, Henry Ward knew how to come down on both sides of an issue. He declared his willingness for a council of churches to convene but said that Plymouth Church would not consent to it. "We can whisk the council down the wind," he told Moulton.
Tilton willingly appeared before the council at the Clinton Avenue Church, expecting the Reverend Storrs to vindicate him, but unexpectedly Storrs turned on Tilton, viciously attacking his character and prodding him to explain himself. Beecher nervously awaited the outcome, writing Moulton,
My Dear Frank:
... I am indignant beyond expression. Storrs' course has been an unspeakable outrage. After his pretended sympathy and friendship for Theodore he has turned against him in the most venomous manner—and it is not sincere. His professions of faith and affection for me are hollow and faithless. They are merely _tactical_. His object is plain. He is determined to _force_ a conflict and to use one of us to destroy the other if possible.... There are one or two reasons, empathetic, for _waiting_ until the end of the Council before taking any action:
1. That the attack on Plymouth Church and the threats against Congregationalism were so violent that the public mind is likely to be absorbed in the ecclesiastical elements and not in the personal.
2. If Plymouth Church is _disfellowshipped_ it will constitute a blow at me and the Church, far severer than at him.
3. That if Council does _not disfellowship_ , then undoubtedly Storrs will go off into Presbyterianism, as he almost, without disguise, _threatened_....
4. At any rate, while the fury rages in Council, it is not wise to make any move.... After the battle is over one can more exactly see what ought to be done....
It was a wise course. In April, the Congregational Church Council disbanded, stating, "It is for Plymouth Church itself to vindicate its pastor." Within Plymouth Church the Beecher political contingent had now established a supremacy that Richard Salter Storrs's accusations of immorality could not erode. That fall, the liberal Republicans of Brooklyn nominated a candidate for Congress in a petition containing the signatures of ninety leading liberals, thirty-five of whom were members of Storrs's Church of the Pilgrims. Beecher and Benjamin Tracy promptly saw to it that a regular Republican candidate was nominated to run against Storrs's candidate, and their man won the election by 76 percent of the vote.
Immediately following the adjournment of the church council, Dr. Leonard Bacon, a powerful Connecticut clergyman who had been a lifelong friend of the Beecher family, delivered a lecture in New Haven before the students of Yale College. He told these young men, "I believe that the infamous women who have started this scandal have no basis for it.... Mr. Beecher would have done better to have let vengeance come on the heads of his slanderers." Bacon went on to say that Beecher's "magnanimity" and "generosity" were commendable and that both he and the council found Theodore Til ton to be "a knave and a dog."
The same week in the _Brooklyn Daily Union_ , Tommy Shearman stated that the council should never have been called. "In regard to the scandal on Mr. Beecher, Tilton was out of his mind, off his balance, and did not act reasonably. He has received pecuniary favors from Mr. Beecher. As for Mrs. Tilton she has occasioned the whole trouble while in a half-crazed condition. She has mediumistic fits and while under the strange power that possessed her often spoke of the most incredible things. Mr. Tilton himself has acknowledged that other things she has told him in her mediumistic spiritual trances are false and impossible. Why should the scandal on Mr. Beecher be the only truth in her crazy words?"
Tilton realized that he and his entire family were to be sacrificed. He went to Beecher and begged him to set the record straight, but although Beecher persuaded Shearman to write a short retraction saying that perhaps he had "confused Mrs. Tilton with Victoria Woodhull," he flatly refused to do any more. After that there was no controlling Tilton. He published an open letter to Leonard Bacon in the _Golden Age_ in which he disavowed all the ameliorative measures he had previously taken and declared, "It was not I _but another man_ who brought dishonor on the Christian name. And yet this other person, a clergyman, permitted his Church to brand me before the Council... as the sinner." Beecher, who had so stoutly maintained that this was an internal Plymouth Church matter that could not be judged by outsiders, was now forced by public pressure to declare a council within his own church. He chose for this council eight members of Plymouth Church who were all close and loyal friends, including Benjamin Tracy, who now controlled Brooklyn politics, and the ruthless Tommy Shearman. The prime organizer of the examining committee was none other than the young man whom Beecher had saved from Fort Lafayette prison, Joseph Howard, the son of his friend and substantial financial backer John Tasker Howard.
Theodore, the Spirits say unto me, "Write and the truth shall make you free."... I told you a year ago that within six months you would fall away from me. "By all that's good, never!" you replied. Nevertheless the fall came. I told you that you were going to lead your friend to his grave. You thought it would be to the Presidential chair. He lies buried—the victim of the ill-starred movement led off by you.
You became a champion of advanced freedom in your support of me and your name was on the lips and treasured in the heart of every radical in the world. You repudiated the course that has won this love, and neither radical nor conservative stands by you.
And now I say, there is a single course of redemption left to you and for your own sake I pray you lead it. Accept the situation. Stand by principle and be not affrighted by public opinion. You have the most glorious opportunity ever vouchsafed to man.... Strike for the glorious and redeemed souls of the near future, and become their hero.
Not surprisingly, Victoria received no answer to her letter. She had risen from her sickbed much weakened. Newspapers across the country debated whether her near-death experience had been genuine or a publicity stunt. In her depleted state the Claflin influence became pervasive. No matter what evil her parents had done, Victoria and her sister Tennie clung to them. The emotionality of their childhood experience had created a unity that no betrayal could sever. Mama Roxy was allowed to move back in with the family. Money was a desperate concern. To supplement her meager lecture income Victoria went to old friends and acquaintances and begged or borrowed or threatened. Issues of the _Weekly_ appeared sporadically as she could fund them. She began selling pictures of herself with a "Spirit Presence" standing behind her holding a crown above her head as if in an act of divine coronation. When the NWSA held its spring convention, Victoria was not invited to attend. At the convention _The Woodhull Memorial_ was praised, but there was no mention of the woman herself. She was "deeply hurt."
The Claflins were still "cats and kits," attacking each other for no reason except their own inner rage at life. First Utica lodged a complaint that Victoria was maltreating her, and then their sister Margaret Ann lodged a complaint against Utica. Victoria responded in a letter to the _Sun_ , "Utica Brooker in a drunken or insane rage, attacked Margaret Ann Miles with a heavy chair... for which Mrs. Miles had her arrested. It was, however, at my special solicitation that Mrs. Miles did not appear against Mrs. Brooker whose complaint is purely malicious and by her own avowal was made to affect the public against me."
Shortly thereafter, Victoria related how she was riding in a horse-car when she clearly heard Utica's voice calling out, "I'm all right now, Vickie." When she arrived at home Colonel Blood informed her that Utica had died. A reporter found Victoria weeping near the coffin of this jealous sister who had conspired to ruin her. "Dead at thirty-one," she said. "Do you wonder that I should feel desperately in earnest to reform the evils of our social life when I remember what I have suffered in my own family? Opposed and misunderstood by my parents and sisters. Compelled to bear an idiot child by a drunken husband. Oh, my God! And the world thinks me only ambitious of notoriety."
Rumors circulated that Utica had died of venereal disease. Victoria insisted on an autopsy and watched it being performed. She followed this with a newspaper interview stating that the results showed that Utica had been addicted to "narcotics and stimulants" and had died of Bright's disease. She informed readers that Utica's "uterus was without disease... as pure as that of a virgin."
With little money left, Victoria, so she said, compensated those who supported her in other ways. A young man named Benjamin Tucker, who later became a well-known editor, reported that he was among Victoria Woodhull's lovers and had accompanied her and Colonel Blood to the Spiritualist camp meeting at Silver Lake, Massachusetts, in the summer of 1873. At the meeting Victoria was exhausted and in despair. When she stepped forward from the black night into the glow of the campfires, someone from the audience asked her to tell them exactly what it meant to live as a free lover. The inquisitor asked her if she was "ready in this connection to uncover individuals from Benjamin Butler on down." Then he asked if she had "prostituted herself—not for love or lust but for the power to carry on this glorious work." A torrent of words poured forth.
I never had sexual intercourse with any man whom I am ashamed to stand side by side before the world with the act. I am not ashamed of any act of my life. At the time it was the best I knew....
And if I want sexual intercourse with one hundred men, I shall have it.... And this sexual intercourse business may as well be discussed now, and discussed until you are so familiar with your sexual organs that a reference to them will no longer make the blush mount to your face any more than a reference to any other part of your body....
Take this as coming from the wisest and best of Spirits... to whom for six years I have yielded a willing and appreciative obedience. I am commanded to declare unto you that in the despised problem of sexuality lies the key that shall serve to open the doors of materiality....
When I came out of prison I came out a beggar. I appealed to the Spiritualists, to the reformers of the country, to send in their money that I might send you my paper. But did you do it? No, you left me to starve in the streets.... I knew my paper had to live or I should assuredly be sent to Sing Sing.... I went to your bankers, presidents of railroads, gamblers, prostitutes, and got the money that has sent you the paper you have been reading and I do not think you are the worse for handling it. The Spirits... have entrusted me with a mission and I have done and shall do everything and anything that is necessary to accomplish it. I used whatever influence I had to get the money and that's my own business and none of yours. And if I devoted my body to my work and my soul to God, that is my business and not yours.
After this revelation she gestured with a sweep of her arm. "This is my lover, but when I cease to love him I will leave him." Observers were unsure whether she had gestured to Colonel Blood or to Benjamin Tucker.
_Chief Ogontz. This medallion of the Sandusky chieftain hung in the hallway of Jay Cooke's mansion_.
ON THE night of September 17, 1873, after a routine visit to Philadelphia, President Grant arrived at Ogontz, the palatial home of Jay Cooke. Ogontz had fifty-two rooms, including a conservatory and a full-scale theater. Its walls were decorated with frescoes. An Italian garden contained fountains and statuary. In the place of honor in the main hall stood a statue of Ogontz, the Indian chief for whom this palace was named. No one remarked that the genocide of the Indians had made possible the fortune that built this edifice.
Dinner was followed by brandy and Cooke's private brand of cigar. The two men sat quietly enjoying the evening's peace. The following morning, while they consumed a breakfast of kippers and eggs, the butler placed in front of Cooke a wire from his partner Harris C. Fahnestock in New York. As soon as President Grant left, Cooke rushed to his Philadelphia office. When he arrived he was told that Fahnestock had closed the New York branch. Just two hours later, the great doors of Philadelphia's Jay Cooke & Co. on Third Street also swung shut, never to open again. Jay Cooke stood behind them weeping. The Northern Pacific had failed.
The panic of 1873 had begun. Like a house of cards, one failure followed another. On September 18 railroad stocks plunged to half their previous value. The following day, the stock market closed and remained so for ten days while the rout continued. By December, five thousand banks, brokerage houses, and businesses had failed.
By the new year it was estimated that one in every seven people in New York City was unemployed. Thousands were on the verge of starvation, in need of clothes and medical attention. A group of the unemployed in New York banded together to appeal to the rich to help restore jobs. They received police permission to parade down East Fourteenth Street to Avenue B and into Tompkins Square, where they would hold a rally. The day before the meeting, however, the police board rescinded its permission. A throng of impoverished people, unaware of this decision, began pouring through the gates of Tompkins Square. No sooner was the square filled with men, women, and children than mounted police, with no warning or provocation, charged in at a gallop, swinging clubs. In the melee that ensued, people fell beneath the blows and were trampled to death by others frantic to escape. Blood ran between the cobblestones of Tompkins Square. The police board issued a statement that the incident was "unfortunate." No official action was taken, and no policemen reprimanded.
All that bitter winter Victoria Woodhull, accompanied by Mama Roxy, traversed the country lecturing night after night, seldom in the same place twice. It was hard going, for now people had little money to spend on lectures. In towns where the halls were large, Victoria reduced her lecture fee to 10 cents a head, of which she received approximately 4 cents after costs. She ate little, slept less. Mama Roxy kept a careful watch, waiting for the moment that she could reclaim her daughter for her own purposes.
Standing on makeshift platforms in icy halls, Victoria presented an hour and a half lecture on free love, censorship, prostitution, social freedom, and sex education. She declared clearly and bluntly that children must be taught about their own bodies and that the false modesty permeating society must be destroyed. She advocated teaching daughters about menstruation. Reflecting on the environment in which she had been spawned, she declared that she would rather see women living outside society, even as prostitutes, than enslaved by the present social conditions.
Every consideration of expediency demands that some one lead the van in a relentless warfare against marriage. Marriage today... gilded over by priestcraft and law... is a license for sexual commerce to be carried on without regard to consent... the most consummate outrage on woman that was ever conceived.... The woman who sells her body promiscuously is no more a prostitute than she is who sells herself in marriage without love.... When I think of the indignities which women suffer in marriage, I cannot conceive how they are restrained from open rebellion.
My most bitter opponents among my own sex are the professional prostitutes who know I am going to break up their business, and the ignorant wives who read little and think less and who are in constant fear of losing their "Paw," over whom they have none except a legal control.... For my part I look beyond the ceremony and the law and observe the facts and if I find people living together in hate and disgust... I say they are prostituting their sexual functions and in the sight of the God of Nature are prostitutes.... Any theory of sexual intercourse for women, so long as they have no control over their maternal functions, is insanity.
_Roxy Claflin, 1874_
The tone of Victoria's speeches now reflected her desperation. Her torments seemed to be bearing down on her with a crushing weight. She seemed close to the edge of some unknown precipice, caught up in visions. The suffering of all women had become her own.
Thousands of poor, weak, unresisting wives are yearly murdered, who stand in spirit-life looking down upon the sickly, half made-up children left behind, imploring humanity for the sake of honor and virtue to look into this matter, to look into it to the very bottom and bring out into the fair daylight all the blackened, sickening deformities that have so long been hidden by the screen of public opinion and a sham morality.... These suffering spirits are with me here. They are with me now.
As the spirits surrounded her, Victoria painted a picture of the hell in which she was living. "Repulsions, discontent and mutual torment haunt the household everywhere. Brothels... crowd the streets... passional starvation, enforced by law and a fractious public opinion... a galling domestic tyranny... sick and weary wives, and even husbands... overwrought, disgusted... in their utter incompetency to meet the legitimate demands of healthy natures. Couple with them ten thousand forms of domestic damnation and everybody crying, 'Peace! Peace!' when there is no peace."
# CHAPTER THIRTY-TWO
HUMAN HYENAS
A WEEK before the church examination was scheduled to begin Benjamin Tracy visited Emma Moulton. He told her that unless she prevented her husband from producing documents damaging to Henry Ward Beecher, he would personally see to it that she would be ruined "socially, financially and in every way." Emma replied that Mr. Tracy was not welcome in her home. At the same time, Sam Wilkeson carried an offer to Theodore Tilton to leave the country with his family. It was too late. Tilton felt he had done no wrong, but was perceived as the villain. He vowed he would clear his name. Even if his judges were loyal to Beecher, he would vindicate himself through the press and public opinion. Thus, the Beecher-Tilton scandal was finally dragged into an open arena.
On July 10, 1874, Tilton appeared before the Plymouth Church Committee and detailed Henry Ward Beecher's adulterous relationship with his wife, supporting it with many of the letters and documents that had been held by Frank Moulton. On the morning after the first session, Mrs. Tilton was visited by Benjamin Tracy, who presented her with a statement prepared by Beecher.
I cannot delay for an hour to defend the reputation of Mrs. Elizabeth R. Tilton, upon whose name, in connection with mine her husband has attempted to pour shame. One less deserving of such disgrace I never knew... she has had my sincere admiration and affection. I cherish for her a pure feeling, such as a gentleman might honorably offer to a Christian woman and which he might receive and reciprocate without moral scruple. I reject with indignation every imputation which reflects upon her honor or my own.
Lib Tilton, by now torn and destroyed, saw in Beecher's statement an expression of love and valor. Evidently it did not occur to her to question why this statement had been seen by no one except herself. Lib felt her husband had betrayed her and their children for the sake of his own selfish pride. The following morning, once and for all, she packed her bags and told Theodore, "I will never take another step by your side. The end has indeed come." Having won Mrs. Tilton's confidence, Beecher was careful not to release his statement. It was found among Benjamin Tracy's papers some years later.
Lib was received by Beecher's close friends, the Edward Ovingtons. The next afternoon Benjamin Tracy visited her and found her so ready to defend her beloved pastor that he rushed to Plymouth Church where the committee was meeting, "whereupon each of the gentlemen hurried to the hat stand, grasped his hat and cane, and repaired to the residence of Mr. Ovington." This hand-picked committee asked no questions that might incriminate Beecher. It was clear from the start that they were looking for an explanation that would justify a complete exoneration. In a five-hour examination before the delighted committee, Lib Tilton did her best to prove Beecher innocent. But in the documents her husband had produced, in her own hand, she had referred to a "dreadful secret" and "the sins" of which she was guilty. This, followed by her confession, seemed almost impossible to explain away. She began a stumbling attempt: The sin, she explained, was that "I felt... I had done him [Theodore] wrong and that I had harmed him in taking any one else in any way, although, on looking it over, I do not think but that I should do it again, because it has been so much to my soul.... I often said 'Theodore, if you had given to me what you give to others, I daresay I should find in you what I find in Mr. Beecher.'... I do not feel any great sin about it now."
Lib went on to say that her initial confession in which she admitted to adultery was dictated to her by Theodore while she was in a weakened mental and physical state and was a lie. The subsequent letter, however, that she had given to Beecher recanting her confession, was accurate. The third letter she gave her husband, in which she once again confessed to adultery, was forced from her at midnight. The "dreadful secret," she explained, was to know that her husband had forced her initial confession. She had told Frank Moulton to burn all the correspondence, but he had betrayed her by giving these documents back to her husband. Lib now referred to Frank Moulton, whom once she had called "a saint" and "a friend indeed," as a man not to be trusted.
In Beecher's interest, anyone who knew the truth of the story must be discredited, but it was hard to explain away her once close association with Susan B. Anthony and Elizabeth Cady Stanton. A fumbling Lib answered a question concerning the allegation that she had told Anthony of her infidelity in the following manner, "I understand that Miss Anthony and another lady have both reported that I made confidantes of them, and I have, full of anguish of soul, many times talked freely to them and on one occasion Susan Anthony stayed all night and I talked with her."
_The trial of the century: A cartoon of the primary players in this drama_ (Illustration Credit 32.1)
This was followed by a sanitized version of the night spent barricaded with Susan in her bedroom. "I told her that Theodore could be so angry that I feared he would be really crazy. He charged me with infidelity with one and another and with Mr. Beecher particularly." In an effort to explain why she had confided in Anthony, Lib lied and said, "I was aroused to tell it by Mrs. Woodhull's being there." (In fact, Lib did not meet Victoria Woodhull until nine months after confiding in Anthony.) Lib told the committee that she was sure that it was Stanton and Anthony who had given Woodhull "a false view of what had transpired... and she printed it." She concluded her testimony by asserting that her husband "alone is responsible for this disruption of my family."
The following day Theodore Tilton came before the committee for a cross-examination that lasted almost three weeks. Under a barrage of questions and accusations he remained firm and asserted that he could prove conclusively that Henry Ward Beecher had engaged in "criminal commerce" with his wife. He was asked by a clever Benjamin Tracy how he could make this accusation and yet maintain that his wife was unblemished. "Have you not frequently asserted the purity of your wife?" asked Tracy, pressing the point.
"I have always had a strange technical use of words.... I have taken pains to say that she was a devoted Christian woman.... I have said that Elizabeth was a tender, delicate, kindly, Christian woman, which I think she is."
"Have not you stated that she was pure?"
"No."
In fact, Tilton skirted close to revealing the free love arrangement that had been practiced in his household. When asked whether he had alienated his wife because he was so free in his pursuit of other ladies, he answered, "If Elizabeth has been troubled concerning my attentions to any lady, take her testimony upon that subject." And later he repeated, "I say if Elizabeth's change of mind was due to the fact that I had loves and affections for other ladies, take her testimony for that fact. I will not deny her."
As to his so-called True Story that had changed the accusation of adultery to "unhandsome advances," Tilton said that he had written it to save his wife's reputation and "to put before the public a plausible answer to the Woodhull accusations that might explain them away." This line of questioning was followed by an extensive examination of Tilton's relationship with Victoria Woodhull. In his answers he tried to distance himself and his wife from her.
"Did Mrs. Tilton express her indignation at Mrs. Woodhull's being at your house the first time she had seen her?"
"I don't know. Oh no, Mrs. Woodhull had taken tea at our house."
"Mrs. Tilton always expressed her indignation at her being there, did she not?"
"Yes, she had a violent feeling against her. She had a woman's instinct that Mrs. Woodhull was not safe.... I wrote a biography about her.... Mrs. Tilton said she thought I would rue the day. She was far wiser than I was."
"Did she not discard Mrs. Woodhull's sentiments and denounce them?"
"Mrs. Woodhull had not then expressed her sentiments.... When I wrote the sketch of Mrs. Woodhull... her ideas were on Spiritualism and woman suffrage."
"Then you never succeeded in convincing your wife that it was necessary to placate Mrs. Woodhull?"
"No, Mrs. Tilton had a strong repugnance to Mrs. Woodhull and to two or three other public women—Mrs. Stanton and Susan Anthony. She would not permit them to come into the house, and some of her letters were very violent against them. She was frequently with them and took part in women's meetings, but then she took a violent antagonism to them after her troubles came on."
In this way, Theodore Tilton equated Victoria Woodhull with Susan B. Anthony and Elizabeth Cady Stanton, as if all three were identical in their views. The committee went even further than Tilton in its efforts to blacken the reputation of these women, implying that all three were free lovers. "Did you ever hear it stated or intimated that you had undue familiarities with those ladies at your house?"
"No, no," replied Tilton.
"I don't mean criminal familiarities, but undue familiarities such as visiting their room or appearing in their room before they were dressed?"
"No, I didn't. I cannot imagine any reason why anybody should."
"Don't you know that your wife's mind has been disturbed in regard to your associations with public women?"
"No sir. I have never associated with public women." Tilton's answer revealed that the term he himself had just used to describe Stanton, Anthony, and Woodhull had a double meaning.
"I don't mean prostitutes. I mean reformers," said his questioner.
"Oh yes, I said before, Elizabeth has been annoyed by my associations with persons out of the realm of religious orthodox ideas. I include Mrs. Stanton and Mrs. Anthony, beyond those persons Lucy Stone was one, she lived in Boston. Elizabeth was a reformer at one time, only now she cannot endure them." Tilton went on to admit that he felt his association with Mrs. Woodhull had been "foolish and wrong" and he "regretted it."
On the last day of Tilton's examination Joseph Howard asserted that Samuel Wilkeson had told him that Tilton had taken $5,000 from Henry Ward Beecher for his silence. Tilton replied, "If Mr. Wilkeson communicates the impression that I ever wanted money from Henry Ward Beecher it is false. I told him I would not take a penny of Mr. Beecher's money if I suffered from hunger or thirst." How then, Howard asked, was it that Mr. Beecher's money had financed the _Golden Age_? "Mr. Moulton never informed me of that!" a shocked Tilton replied. This was followed by another startling question: Had not Mr. Frank Moulton threatened Henry Ward Beecher with a gun, forcing him to give up Mrs. Tilton's retraction of these heinous accusations? Again, Tilton said he knew nothing of this.
Frank Moulton did not wish to appear before the committee or to become involved in what he saw as a public airing of all he had hoped to keep private. Twice he was subpoenaed by the committee, and twice he refused to appear. With the third subpoena he was told by Tracy that his wife, Emma, would be drummed out of Plymouth Church and that he would be dismissed from his position at Woodruff & Robinson if he did not testify. On the night of August 5, Moulton appeared before the committee and said he had prepared a statement that he had read to both Mr. Tilton and Mr. Beecher. Defining himself "as the umpire and peacemaker for the last four years, with a conscientious regard for all the interests involved," he revealed much, but not all, of what had transpired.
He described his visit to Beecher when he had retrieved Mrs. Tilton's letter exonerating her preacher and said that he had been carrying a gun because he had come from an inspection of his company's warehouses and always did so during this procedure. That he should be accused of threatening Beecher was a bold lie that he was sure even Henry Ward Beecher would not confirm. In regard to the charge that Tilton had blackmailed Beecher into giving him the money for the _Golden Age_ , Moulton declared, "Beecher never intimated to me that he thought there was any desire on Tilton's part to blackmail him. I had sole management of the money. Tilton knew nothing of its source."
By his second appearance, it was apparent that every statement Frank Moulton made was corroborated by an appropriate document. Whereas Theodore Tilton had been followed home every night by a horde of reporters and had given interviews on the day's proceedings while reclining on the now infamous red velvet chaise longue in his library, Moulton refused to talk to reporters but sent newspapers exact copies of all the evidence he presented. The impact was considerable. The _Graphic_ stated, "Here are all the original documents that Beecher and his lawyers supposed were destroyed. Mr. Moulton's accompanying explanations are simple, self-consistent, and consistent with the facts that have been brought to light. Whatever their legal value may be, their moral effect is overwhelming." Whitelaw Reid wrote in the _Tribune_ that although Moulton's documentation was "overwhelming... Those who believed in Mr. Beecher's innocence before will believe in it still."
During the examination of Tilton and Moulton, Henry Ward Beecher was secluded at his farm in Peekskill with his lawyer and strategist, Tommy Shearman. On a sweltering morning during the first week in August he came back to Brooklyn for his examination. The committee assembled in the library of Beecher's Columbia Heights home at eight o'clock in the morning. The fact that the committee had come to him signaled that this would be a friendly investigation. Beecher served broiled steak and lemonade for breakfast, after which he settled himself into a comfortable chair and read aloud a statement he had prepared, with the help of Shearman, denying Tilton's charges. He began with an analysis of his former protégé.
Four years ago Theodore Tilton fell from one of the proudest editorial chairs in America... and in a few months thereafter became the associate and representative of Victoria Woodhull and the priest of her strange cause. By his follies he was bankrupt in reputation, in occupation, and in resources.... I can now see that he is and has been from the beginning a selfish and reckless schemer, pursuing a plan of mingled greed and hatred, and weaving about me a network of suspicions, misunderstandings, plots and lies.... It was hard to do anything for such a man. I might as well have tried to fill a sieve with water.
As his testimony began, Beecher recalled in a rose-colored haze of nostalgia his early days with the Tiltons. He pictured himself then as the paterfamilias of a beloved little family. "I was treated as a father or elder brother. Children were born; children died. They learned to love me and to frolic with me as if I were one of themselves. I have for Mrs. Tilton a true and honest regard.... I would as soon as misconceived the confidence of her little girls as the unstudied affection she showed me."
Beecher detailed how over the years his relationship with the Tiltons had deteriorated. The once-upstanding Theodore had turned away from religion, and over the years his infidelities and maltreatment of his wife had become so abominable that finally Beecher, as her pastor, with the counsel of his wife, Eunice, was forced to recommend a separation. He testified that in his opinion Tilton had almost ruined the _Independent_ , "that great religious newspaper," with his radical views that "aroused a storm of indignation among representative Congregationalists."
In a clever combination of praise for the early days followed by accusations of duplicity in the latter days, Beecher also endeavored to undermine Frank Moulton: "In the first few months... he treated me as if he loved me." But then, alleged Beecher, Moulton made him give up Lib Tilton's retraction of her confession. According to the preacher's account, "Moulton called at my house and came up into my bedroom." He said that Tilton had destroyed his wife's first letter and demanded that Beecher give him the letter of retraction he had secured. "He was under great excitement. He made no verbal threats, but he opened his overcoat and with some emphatic remark showed a pistol, which afterwards he took out and laid on the bureau near which he stood. I gave the paper to him and after a few moments' talk he left."
Beecher said that Tilton had indeed tried to blackmail him and provided the following explanation.
I did not at first look upon suggestions that I should contribute to Mr. Tilton's pecuniary wants as savoring of blackmail. This did not occur to me until I had paid perhaps $2,000. Afterwards I contributed at one time $5,000—I mortgaged the house I live in—then I felt very much dissatisfied about it. Finally, a square demand and a threat were made to me by confidential friends that if $5,000 more was not paid, Tilton's charges would be laid before the public. This I saw at once was blackmail in its boldest form, and I never paid a cent of it.
Tommy Shearman had convinced Beecher that the only way to save himself, his church, and his political influence was to renounce Mrs. Tilton. His "angel incarnate" was about to be offered as the final sacrifice. When asked why he had been so generous to the Tiltons, he replied with a clever distortion of the facts.
The family had well nigh been broken up and I had advised it. His wife had long been sick and broken in health and body and I, as I fully believed it, had been the cause of all this wreck by continuing that blind and heedless friendship which had beguiled her heart and had roused her husband into a fury of jealousy, although not caused by any intentional act of mine. Should I coldly defend myself? Should I pour indignation upon this lady? Should I hold her up to contempt as having thrust her affection upon me unsought?
Beecher's testimony revealed that he had used Tilton and Moulton to distance himself from Victoria Woodhull but that they eventually came under her spell.
One wing of the female suffrage party had got hold of the story in a distorted and exaggerated form and wanted it broadcast... and these difficulties were immensely increased by the affiliation of Mr. Tilton with the Woodhull clique.... It was delayed, ostensibly by Mr. Tilton's influence with Mrs. Woodhull, until November 1872.... She became the heroine of Mr. Moulton and Mr. Tilton. She was made welcome by both houses with the toleration, but not the cordial consent, of their wives. I heard the most extravagant eulogies upon her. She was represented as a genius, born and reared among rude influences, but only needed to be surrounded by refined society to show a noble and communing nature.
Of the efforts made to compel him to introduce Woodhull at Steinway Hall, he said that he had refused because "I understood that she was about to avow doctrines which I abhor." Tilton, he said, presided at the meeting "where Mrs. Woodhull gave vent as I understand, for the first time in public, to a full exposition of her free love doctrines."
Using the subject of free love as a springboard, the committee asked about Beecher's involvement with Anna Dickinson, Paulina Wright Davis, Stanton, and Anthony, all of whom knew about his alleged affair with Mrs. Tilton. He impugned them all. Beecher said that Woodhull and the woman's rights advocates who supported her were "the centre of loathsome scandals, organized, classified, and perpetuated with a greedy and unclean appetite for everything that was foul and vile.... It was inexpressibly disgusting to me and I would not associate with these women. Yet Mr. Tilton and Mr. Moulton had some strange theory concerning the management of this particular affair which made it necessary for them to maintain friendly relations with this group of human hyenas."
"Human hyenas!" As the newspapers printed Beecher's testimony, Stanton and Anthony were incredulous. The entire movement was drowning in a wave of recrimination. A _Herald_ editorial was typical of the criticism:
Elizabeth Cady Stanton and the buzzards and creatures in her group are rapacious at this feast. From Mrs. Stanton's lips are the words and fruit of this new philosophy. We spurn it as an offense to human truth and all that is sweet and ennobling in modern life. As it now stands let us hear no more of this "reform" and we should like to hear no more of it for a generation. Mrs. Stanton has in one day undone all the good she ever achieved. Thirty years of agitation in favor of woman suffrage ends in the triumph of a flock of ravens to pick the carcass of Henry Ward Beecher and croak the doctrine of home, marriage, love, motherhood and religion.
Anthony still chose silence. She tried to forget about what was going on in Brooklyn and spent several evenings reading Nathaniel Hawthorne's _The Scarlet Letter_. When followed down the street by a reporter from the _Argus_ , she told him, "During my whole public career I have never answered any personal newspaper allusions to myself, any scandal, charge, gossip or mean thing, not even the charge that I was drunk on the platform in New York last May. And now that I am fifty-five years of age I shall not commence." But an indignant Stanton was ready to tell the public all she knew. In an interview in the _Graphic_ she spoke in vivid detail of the night that Anthony spent with Mrs. Tilton. When asked by the reporter, "You have no doubt in your mind but that Mrs. Tilton made a confession to Susan B. Anthony?" she replied, "Not in the slightest. Susan always speaks the truth." When the reporter said, "And that confession was of criminal intimacy with Henry Ward Beecher?" she answered, "Yes, criminal, as the word is generally understood." She said that Laura Curtis Bullard, Anna Dickinson, and others could confirm this report. When asked if, by corroborating Woodhull's story, was she not destroying the reputation of Mrs. Tilton, she answered, "I don't like to be represented by the press as striking a blow at a woman, but when it comes to the women of the suffrage movement and Mr. Beecher, I prefer to let him kick the beam, though he may take some one woman with him."
In several articles, Stanton outlined in precise detail the monetary and political implications of the scandal and repeated what Samuel Wilkeson had told her about how no matter what Beecher had done he must be supported for the sake of the money and power involved in the control of political patronage, the stock in Plymouth Church, and the investment in the _Christian Union_ and _The Life of Jesus the Christ_.
When asked if she would appear before the committee, she answered, "No, not before _that_ committee.... There is no stronger proof that the committee has a difficult case in sustaining Mr. Beecher than its understood determination to impeach the integrity of every witness against him.... I have no wish to give my testimony. I belong to a family of lawyers and I have great respect for the law."
"Mrs. Stanton, do you believe in the doctrine of free love as it is adopted by Mrs. Woodhull?" asked the reporter.
"No, I believe in law."
Susan B. Anthony was clearly annoyed by Stanton's public commentary. She wrote a friend, "Well, well, aren't they getting deeper and deeper into the bind there in Brooklyn and was there ever such a needless heedless foot splashed into the mud as that of E.C.S. and her boast that she belongs to a family of _lawyers_. Isn't it perfectly killing? Well, I do hope whoever else wants to plunge in they'll go _alone_."
Anthony resented the fact that Stanton had drawn her into the scandal in the _Graphic_ interview, and she wrote and told her so. Stanton felt the entire woman's rights movement was now in mortal danger and answered,
Offended Susan, come right down and pull my ears. I shall not attempt a defense. Of course I admit that I have made an awful blunder in not keeping silent so far as you were concerned on this terrible Beecher-Tilton scandal. The whole odium of this _scandalum magnatum_ has, in some quarters, been rolled on our suffrage movement as unjustly as cunningly, hence I feel obliged just now to make extra efforts to keep our ship off the rocks.... This terrible onslaught on the suffrage movement has made me feel like writing for every daily paper... against this wholesale slaughter of womanhood. When Beecher falls, as he must, he will pull all he can down with him. But we must not let the cause of woman go down in the smash. It is innocent.
Beecher's cross-examination by his parishioners seemed more like an affirmation of all he had previously stated. In no case was he challenged. As witness after witness paraded before the committee, it became increasingly clear that its main purpose was to "impeach" anyone who did not assert Beecher's innocence. Samuel Wilkeson told the committee that Moulton and Tilton had tried to blackmail Beecher and that there was no way he could believe that Tilton did not know the source of the monies he received. Mrs. Tilton persuaded Bessie Turner to return from the West and in four hours before the committee she testified against Stanton, Anthony, and Theodore Tilton. She said that on two occasions, when Mrs. Tilton was away, Mr. Tilton "attempted my ruin." The first was when "Horace Greeley had been staying at the house. I had been sleeping and woke up... he [Mr. Tilton] must have lifted me out of my bed and put me in his. I asked what he was doing that for. He said that he was lonesome. I said that wasn't right and went back to my own room.... The second time he tried to get in bed with me. I got very indignant and as he would not leave my room, I went into another and locked the door after me."
_Bessie (Elizabeth) Turner testifies. Susan B. Anthony called her "half an idiot" for accusing Anthony of sexual improprieties_.
Bessie said that Theodore Tilton "seemed to think a great deal of Mrs. Stanton and Miss Anthony. I saw her sitting on his lap on one occasion when I was coming into the parlor and she jumped up pretty quick."
"Which one?" asked the questioner.
"Susan B. Anthony," Bessie replied.
"What was his conduct with Mrs. Stanton?"
"Well, I never saw him caressing her, but he used to be alone with her a great deal in his study. They used to play chess until two or three o'clock in the morning."
Although Anthony had vowed to say nothing, she could not resist responding to Bessie's testimony and told a _Tribune_ reporter that Bessie Turner was "half an idiot... it is a shame to permit that girl to testify before a committee." And when asked if she sat on Tilton's knee, instead of denying it, she smiled and replied, "All the men have declared that Susan was so sour that she wouldn't get a husband and I thought I would show them I could sit on a young man's knee just like any foolish girl. I did not deny that I sat on Theodore's knee. Why should I not enjoy the opportunity to be womanly and loving, when I have been called an ogre all my life and everybody claims to know that I never had a chance to be married."
Susan's statement about Bessie's testimony constituted a rare breech of self-imposed silence, but her brother Colonel Daniel R. Anthony had no such inhibitions. In the _Leavenworth Times_ he wrote an account, repeated in the _Chicago Tribune_ , in which he said that he heard the scandal story from his sister more than a year earlier. He provided many details, among them: "Tilton accused his wife of adultery with Beecher and she replied with the accusation that he had procured an abortion for a young lady of Brooklyn whom he had seduced."
When asked about her brother's article, Anthony said, "Provided I did tell it to him, which I do not admit, if my brother and Mrs. Stanton have said what has been credited to them, it was a very ungracious thing of them to do. No one is able to repeat what another says... to attempt it is invariably to do injustice." And when pressed for a comment on the case, she responded,
The time is coming when a woman will be answerable only to herself for her own deeds. In an abominable mixture of deceit, either of these men is ready to sacrifice Mrs. Tilton to save himself. The fact is, if a woman gives herself to a man either in marriage or out of marriage, he will trample her into the dirt to serve his own ends. Women sell themselves too cheap. They sacrifice themselves on the spot, and it does not matter whether the man has any brains or not. It is the creation over again. Old Adam said, "The woman tempted me and I did eat." Beecher says, "The woman tempted me and I did _not_ eat." In both cases, she gets the blame.
In a more oblique way she wrote to her brother Daniel,
Whatever comes to those closely united by marriage or by blood, the one lesson from recent developments in Brooklyn is that none of the parties ever should take in an outside person as confidant. If the twain can not themselves restore their oneness, none other can. If parents and children, brothers and sisters, can not adjust their own differences among themselves, it is in vain they look to friends outside....
Whether it be wealth, position, office or the society of one we love, if we have to steal it, though it may be sweet and seemingly real and lasting, the exposure of the illicit means of gaining it is sure to come, and then the thing itself turns to dross. When will the children of men learn this fact, that nothing pays but that which is obtained fairly, openly and honestly?
Isabella Beecher Hooker cast the most doubt on her brother Henry's innocence. When a compilation of the testimony was published under the title _The Romance of Plymouth Church_ , Isabella noted, "It is too bad but they are in sauce up to necks and have got to stay there and keep still or shall be deeper in." And later she scrawled in an unsent letter, "The case has gone out of our hands utterly and out of human hands it seems to me—the verdict will come from on high." The only course left to Henry Ward Beecher's defenders was to utterly destroy Isabella's credibility. Tommy Shearman went before the committee and testified of Isabella: "I know from a private and most reliable source that for a long time she was considered out of her head... insane... deluded. She was laboring under a hallucination... cultivated by her intimacy and strange relationship with Victoria Woodhull. She is well known to be weak minded and to prey upon her brother." He referred to the letter in which she addressed Victoria Woodhull as "My Darling Queen" and added a connotation to it that even her severest critic had not imagined, stating, "I cannot say it outright. The best way I can put it is that she [Isabella] had an _unnatural_ affection for Mrs. Woodhull."
Isabella's own half-brother George Beecher ratified Shearman's testimony in an open letter to the _Eagle:_ "She was devotedly attached to Mrs. Woodhull and has never withdrawn from her. The _strange fascination_ which this woman possessed over her is evinced, among other things, by a letter she wrote commencing as follows, 'My Darling Queen'.... In an interview with our brother Edward, she seemed in a wild excited state of mind and Henry, to soothe and quiet her, refused to deny the stories." In fact, the appellation "Darling Queen" was fairly common parlance among women of the day. Anthony addressed Stanton as "My Darling Empress" and Stephen Pearl Andrews often called his wife, Esther, "Queen of the moral and social world."
In a society where women were frequently isolated from men, relationships among them often became unusually close. Anthony's feelings for other women, particularly Anna Dickinson, were infused with passion. It was not unusual for Anthony to sign a letter to another suffragist, "Your friend and lover." Frequently, she wrote of climbing into bed with one of "her girls" and spending the night. It was in just such a situation that Lib Tilton had confided in her. For women with the moral code of an Anthony or a Hooker, however, a lesbian relationship, with its active sexual implications, was out of the question, and Shearman's barely veiled accusation of such an attachment between Victoria and Isabella, backed by George Beecher, was entirely without foundation. Still, the effect was devastating.
Catharine Beecher joined in the attack, writing in the _Eagle_ that her brother was "an honorable man... while on the other side are... half-crazy women." In a second newspaper article, Catharine maintained that she knew of a woman who, under pressure, confessed that she had been seduced when indeed it was all a fantasy. The most original defense came from Henry's brother, the Reverend William Beecher. "If Henry were in the habit of running after women... why should he choose an old, faded, married woman? There are plenty of young girls that he could have had if he had been so inclined.... It is easier for a preacher than anyone else, except perhaps a doctor, to take advantage of women."
The Hookers, once the social leaders of Nook Farm, were now considered pariahs. Mary Beecher Perkins and Harriet Beecher Stowe also broke off relations with their sister Isabella and barred her from their homes. Friends followed family. Mark Twain's sister-in-law Molly Clemens wrote of him, "Sam says Livy shall not cross Mrs. Hooker's threshold and if he talks to Mrs. H. [Hooker] he will tell her in plain words the reason." Under accusations of sexual perversity, betrayal, and insanity, Isabella could bear up no longer. She fled to Europe.
Although the press had been barred from the committee room, transcripts were leaked by one side or the other to the waiting reporters. The public was obsessed with the case, seemingly sensing that it had to do with issues larger than a simple seduction. It also provided summer's entertainment. At a performance of a play at Niblo's Theater, when a character asked, "What shall I do with these letters?" a member of the audience cried out, "Give them to Moulton."
On August 22, 1874, the church committee concluded its investigation by issuing a short statement that the Reverend Henry Ward Beecher had been exonerated. Two days later, the _Christian Union_ carried excerpts from the committee report, which had concluded that Beecher was the victim of Tilton and Moulton's "vicious and revengeful designs." Mrs. Tilton was found guilty of plaguing the preacher with her "inordinate affection." Beecher himself was reprimanded only for having allowed his "great generosity" to blind him to the perfidy of these people.
Stanton was enraged, and wrote,
What a holocaust of womanhood we have had in the investigation!... What statements and counter-statements they have wrung from her [Mrs. Tilton's] unwilling lips. Then like a withered flower, _the Great Preacher_ casts her aside and tells the world "she thrust her affection upon him unsought," the crowning perfidy in that bill of impeachment that blackens everyone who dared to hear or tell the most astounding scandal of the 19th century!
In common with the rest of the world, members of the National Woman Suffrage Association heard... the scandal as other men and women did and forsooth Mr. Beecher dubs them "human hyenas" and "free lovers," though his own sister, Isabella Beecher Hooker... one of their number... is presented as "insane," "deluded," "weak-minded."
If the secret history of this tragedy is ever brought to light, we shall have such revelations of diplomacy and hypocrisy in high places as to open the eyes of the people to the impossibility of securing justice for anyone when money can be used against him.
To those who take a surface view of the scandal, it is probably prurient, disgusting, nauseating.... This, to my mind, is an evidence, not of a depraved popular taste, but of a vital interest in the social problems that puzzle and perplex the best of us.
# CHAPTER THIRTY-THREE
DANIEL IN THE LION'S DEN
THE PUBLIC reading of the Plymouth Church committee report was merely pro forma, a demonstration of solidarity for Henry Ward Beecher. Emma Moulton felt her husband would be risking his life if he went to the church. "You will be Daniel in the lion's den," she warned him. "But Daniel defeated the lion," he answered. As he began walking toward his carriage, Emma ran after him, burst into tears, grabbed his arm, and implored him, "Oh Frank, if you must go, take me with you. Please Frank." Moulton turned and gently guided Emma back to the front door of 49 Remsen Street. A reporter from the _Argus_ , who had been hiding behind a tree, recorded this scene and overheard Emma say to her husband as she reached the doorstep, "They are not Christians."
Plymouth Church was full. The _Times_ estimated that more than three thousand, four hundred people were in attendance and the street was blocked with hundreds more. One hundred policemen tried to handle the crowd, which had poured through the doors and with high excitement flowed in a circular motion around the platform in the center of the church and scrambled into the balconies above. Some of the most impatient and agile shinnied up onto the window ledges. _Harper's Weekly_ noted, "The Great White Church was like a hive, with the swarming bees hanging in clusters upon the outside." On the left-hand side directly underneath the preaching platform sat members of the press, forty strong, and closest to the door from which Henry Ward Beecher made his dramatic entrances sat two of the clerks from Woodruff & Robinson with an empty seat between them.
John Tasker Howard, his son Joseph, and Sam Wilkeson strode in together, the crowd parting before them like the Red Sea. Joe Howard was in high spirits and, turning to Wilkeson, remarked, "I see we have over three thousand voters here tonight to secure our great city of Brooklyn."
At eight-thirty, half an hour after the meeting was scheduled to commence, Joe Howard, wanting to delay no longer, assumed the platform and began the meeting by calling for a prayer, after which the full report exonerating Henry Ward Beecher was formally entered into the proceedings, and copies were passed from pew to pew.
As the report was circulating, Frank Moulton, who had been escorted by Deacon Talmadge through Beecher's robing room and out a small door under the choir loft, slipped unnoticed into the seat between his two clerks. From the podium, Joe Howard declared, "Mr. Moderator, our report needs no advocacy. It carries conviction with it by the irresistible sweep of its logic.... The committee has taken into consideration everything they should have taken and the report has proven to us the glorious character of the work which has been accomplished... by our guiltless pastor... who has passed through a trying ordeal instigated by reprehensible foes." As Howard paused, a voice called out, "Question. Question."
All eyes turned toward Frank Moulton, and as they recognized him as the questioner, slowly and steadily, a great sound arose in Beecher's church, a steady hiss that permeated the entire room and resounded off the rafters. "Question. Question!" repeated Moulton, his voice almost drowned out by the hissing. Then someone shouted, "Put him out! Put the blackguard out!"
"Quiet. Quiet! No questions," Howard called out, and as the noise subsided he continued to read his speech in praise of Henry Ward Beecher, ending with the words, "When the Reverend Beecher's vacation shall be ended he shall come back to us refreshed in body and mind. Then, in this place so true to him and to us, I trust he shall again appear to ornament the pulpit he has so long adorned and as it never has been adorned before and probably never will again."
With no time for reaction, Professor Rossiter Raymond—a member of the committee, cousin of Henry Ward Beecher, and trained elocutionist—began to speak. In a magisterial voice he read the report of the examining committee, which point by point refuted the evidence against Beecher. During this process Moulton sat quietly, furiously scribbling on a foolscap pad braced upon his knee. Raymond asserted that Theodore Tilton, not Beecher, was the one who had practiced "infidelity" and had exhibited "free love proclivities." Why had Beecher not responded more promptly to the accusations against him? "Mr. Beecher referred the men who had spoken to him of this case to Francis Moulton, who poisoned their minds with infamous lies."
Moulton jumped to his feet. "You sir, are the liar!" he shouted. The hissing began again, uncontrollable, menacing. The evening's moderator, Deacon James Freeland, called out, "This meeting must come to order!" No one listened. The cries of "Put him out!" resumed again and then men began bobbing up from their seats and pushing their way down the aisles toward Moulton. Sensing the danger, one of Moulton's clerks rushed to the back of the church and frantically waved his arms in a signal to the police.
_The Plymouth Church report: Frank Moulton objects to the vindication of Henry Ward Beecher_.
"Order!" yelled Freeland, and turning to Moulton he directed, "This man will sit and hear." Moulton stood his ground, defying them all: "You dare not put me out. You dare not!" Then the Reverend Halliday, who had spoken the opening prayer, said imploringly, "Brethren, brethren, let this man keep his seat." He was answered by cries throughout the church: "No! No! No!" Moulton persevered, "Yes, I will whether you want me to or not."
Men were moving toward Moulton, and one voice louder than the rest cried out, "Kill him! Kill him!" A dozen policemen rushed down the aisles and surrounded the base of the preaching platform. Shaking their billy sticks, they turned to stare down the audience. Samuel Wilkeson, a man adept at handling people, stood, and in a clear, commanding voice instructed, "Return to your seats, gentlemen.... In the name of Plymouth Church, I request that this man shall be allowed to remain if he does not openly disturb the meeting." There was a moment of silence, and then the men retreated.
Once again, Rossiter Raymond resumed. "One of these men charges upon the other the seduction of his wife, and when he repudiates it, Mr. Moulton who stands before you calls that a needless controversy." A voice from the audience yelled, "Hit him again!" and laughter filled the room. The atmosphere was more relaxed as the audience began to enjoy the performance. Raymond proceeded: "A word or two on the subject of Mr. Moulton and blackmail." "Give the blackmailer a shot," a heckler yelled, cupping his hands around his mouth. The merriment continued.
_The attack on Frank Moulton. A mob confronts Moulton to berate him for opposing the report upholding Beecher's innocence_.
We have been informed that a man cannot be blackmailed unless he knows it at the time. I don't think that is reasonable. People can be blackmailed by appealing to their finer feelings, and a man can be induced to give from generosity what threats could not extort from him. The question is purely and simply whether the parties who trafficked in this case against Mr. Beecher took any money. Tilton has confirmed as much. So far as this farce of Tilton not knowing where the money came from, we all know it went from Beecher's pocket to Moulton's pocket to Tilton's own pocket.
Moulton held himself in check, making notes on his pad. Finally, with a surge of mixed metaphors, Professor Raymond reached his conclusion. "The storm is almost over.... I know that in these deep waters Mr. Beecher has been sustained by everlasting arms.... We will stand beside him united by our church against all other churches, doctrines, and creeds. 'The foundation of God standeth sure, having this seal, and the Lord knoweth who are His.' "
A thunderous applause lasting a full four minutes engulfed the congregation, after which the Reverend Halliday called for a vote, asking all who pronounced Henry Ward Beecher innocent of the charges to stand. Thousands stood. Moulton remained seated. As if to emphasize the point, Joe Howard asked all those opposed to the adoption of the report to rise. Frank Moulton rose to his feet and remained standing, looking calmly around the church while on every side the derisive noise and epithets erupted once again. As they reached a crescendo, Howard said, "The resolutions to accept this report will now be voted upon: All in favor say 'Aye.' " "Aye" swept the church. "All opposed?" he asked. The lone voice of Frank Moulton rang out, "Nay."
"I wish to address the audience," Moulton called out in a clear voice, but was shouted down, and almost immediately the audience began singing the Doxology, which halted all discussion. Then the policemen surrounded the still standing figure of Moulton, and forming a human wedge, pushed their way to the small door from which he had entered. Moulton's progress was accompanied by cries of "Lay him out!" "Give it to him!" and many Plymouth Church parishioners raised fists or canes menacingly. At the last moment several men ran toward the line of police, trying to break through, but Moulton disappeared through the door.
On reaching the street, Moulton ran the gauntlet of yet another crowd. A woman dashed in front of him, stopped abruptly, and spat in his face. Another man struck him on the shoulder with his cane. The police, wielding their billy sticks, broke a path to Moulton's carriage. He leaped in, and the vehicle took off at top speed, a police captain from the second precinct perched precariously on the buckboard, an enraged crowd in hot pursuit. After three blocks the pursuers gave up and returned to the street in front of Plymouth Church, where they lingered in a state of high excitement for more than an hour and a half after the closing of the church doors.
When Moulton's carriage pulled up in front of his house, Emma dashed out the door and flung her arms about him. The following morning, as Emma left for the greengrocers, at the foot of her steps on a patch of lawn she saw a crude miniature black wooden coffin on which the name Francis D. P. Moulton had been scrawled in what appeared to be blood.
About the same time that Emma Moulton discovered this grisly relic on her lawn, Theodore Tilton presented himself at the Brooklyn City Court to file suit for "alienation of affection" against Henry Ward Beecher, asking $100,000 in damages.
ON THE day that the Plymouth Church committee report was read Victoria Woodhull delivered a lecture, "Tried as by Fire," in Des Moines, Iowa. She was still touring the country, trying to support the Claflin family, but the renegade glamour that attached itself to her had vanished in these depressed times. As she grew weaker the jackals attacked: Joseph Treat, a former associate and contributor to the _Weekly_ , published a pamphlet purporting to support the philosophy of free love but deploring the methods used by Mrs. Woodhull, which he alleged had corrupted the movement. The _Treat Pamphlet_ was a compendium of salacious gossip. Treat wrote that in the presence of Colonel Blood he had heard Victoria say that "it was right for a woman to prostitute herself when she could make $50 by it," and that she did so. Both sisters were "prostitutes" and had used this means "not only to carry on their paper but... to make money and enjoy life." Treat wrote that he had split with them on this issue because "prostitution can _never_ be made the basis of reform.... Prostitutes should be brought out of their condition instead of being upheld as examples for the world to follow." He claimed that when Buck Claflin arrived in New York, he took Tennie C. to the brothels of Greene Street "and bade her get her living there." That was how both sisters had become so familiar with the life led in these places.
Treat wrote that Victoria and Tennie C. had sent their sisters Utica and Polly on assignations and that within Treat's hearing Colonel Blood had instructed Tennie to have assignations with several men and told her, "We must have money and you must get it for me." Tennie, Treat added, was addicted to gambling and had lost thousands of dollars "in a policy shop in the rear of Clute's brokerage office, 206 Broadway." She "was in the habit of telling young girls that she would find men who would love and support them." Psyche Club meetings at Woodhull's house were used to arrange assignations, and it was announced at every meeting that upstairs rooms were furnished and both ladies and gentlemen could stay the night. Treat's pamphlet sold more than forty thousand copies at 10 cents each.
Now when Victoria appeared on platforms, people came more to gape than to listen. But still she spoke, night after night, telling of her deepest feelings.
I am charged with seeking notoriety, but who among you would accept any notoriety and pay a tithe of its cost to me? Driven from my former beautiful home, reduced from affluence to want, my business broken up and destroyed, dragged from one jail to another... for telling the truth.
I have been smeared all over with the most opprobrious epithets, and the vilest names, am stigmatized as a _bawd_ and a _blackmailer_. Now, until you are ready to accept my notoriety, with its conditions—to suffer what I have suffered and am yet to suffer—do not dare to impugn my motives. As to your approval or dissent, your applause or your curses, they have not a feather's weight with me. I am set apart for a high and sacred duty, and I shall perform it without fear or favor.
But she could not. She seemed to have lost her force. On returning home one night she stood in front of an oval mirror contemplating her image; then she took a shears, raised it, and with one swift, deliberate motion lopped off part of her long brown hair. Portraits from that time show a haunted, anxious face surrounded by ragged wisps of hair. She bore little resemblance to the charismatic, beautiful woman she had been. In a new lecture, "The Scarecrows of Sexual Freedom," she railed against enforced sexual intercourse, unwanted children, venereal disease. On a particularly cold night she stood before a Chicago audience and stared out at the crowd. "Can anyone tell me where I am?" she asked. Mama Roxy came onstage and led her daughter off.
MORE THAN six years after "A Day Memorable" had appeared in Mrs. Tilton's diary, on January 11, 1875, the civil trial of _Tilton v. Beecher_ began. Ice in the East River prevented the ferries from operating, and the New York lawyers were unable to come to the Brooklyn courthouse. By the time the trial ended on July 2, the temperature in the courtroom had touched 98 degrees and one lawyer and two jurors had fainted. For 112 days the testimony was front-page news. Reporters were allowed in the courtroom, and every gesture, every glance of the principals was keenly observed. Tilton was said to sweat in cold weather. Beecher appeared "the center of beatific calm." The trial became an American obsession, a supershow, far transcending the question of innocence or guilt. Every day the courtroom was full; mobs flooded the street and jammed the corridors. One day an estimated three thousand people were turned away. Food vendors prowled the halls selling pretzels, sandwiches, and drinks. The nearby restaurants and saloons enjoyed an unexpected boom business. Tickets to the trial were sold on the black market at $10 apiece, and special seats were reserved for prominent members of the press, the famous, and the notorious. In the third week a lottery was instituted and the eight winners were allowed a seat in the courtroom.
Beecher's chief lawyer was none other than William Evarts, the Plymouth Church parishioner who had successfully defended President Andrew Johnson in the impeachment proceedings. He had never forgiven Theodore Tilton for being the first to suggest Johnson's impeachment and was glad to volunteer his services against him. Sentiment was clearly on Henry Ward Beecher's side. When he swept into the courtroom, flowers fell at his feet; his dropped handkerchief was seized upon so eagerly that the crowd tore it to shreds. When the time came for his testimony, he ascended the stand as if it were his pulpit. Every word he said was infused with high drama. In florid, romantic terms, he described his early relationship with the Tilton family. And sometimes, as he did in his sermons, he skated close to the edge of destruction. Looking over at his wife, Eunice, who followed his every word, once again he explained that if there was fault here it was Mrs. Tilton's, for she had developed a passion for him and had created this problem by not warning him that Theodore Tilton suspected him of being the object of her unwanted affections.
"Did you kiss Mrs. Tilton?" the prosecution's lawyer asked.
"Sometimes I did, and sometimes I did not."
"What prevented you upon the occasions when you did not?"
"It may be that the children were there then."
Beecher went on to relate the scene in which he said that as Mrs. Tilton sat on his knee he kissed her. Then he kissed Mr. Tilton, and then "I believe they kissed each other." George Templeton Strong, a diarist of the time, wrote,
_February 20_... Tilton v. Beecher. The evidence begins to pinch Monsignor Beecher very hard. He is probably ruined by his utterly fatuous confidences and confessions. But Plymouth Church is a nest of "psychological phenomena," _vulgo vocato_ lunatics, and its chief Brahmin is as moonstruck as his devotees. Verily they are a peculiar people. They all call each other by their first names and perpetually kiss one another. The Rev. Beecher seduces Mrs. Tilton and then kisses her husband, and he seems to acquiesce in the osculation.... They all seem, on their own shewing, to have been behaving like bedlamites and to have been afflicted with both moral and mental insanity.
_The Brooklyn City Court, January 11, 1875, during the opening session of the_ Tilton v. Beecher _civil trial_ (Illustration Credit 33.1)
_Henry Ward Beecher testifies. Public sympathy was on his side_.
To prove Tilton's bad character, Evarts subpoenaed three former servants of "the notorious Mrs. Woodhull." All three testified that Mr. Tilton had been seen in Mrs. Woodhull's bedroom in the most intimate postures and that her house was without doubt a house of assignation. To rebut this testimony Tilton's lawyers called to the stand Stephen Pearl Andrews.
Andrews, whose philosophy of free love had brought on much of this trouble, clung to his theories with the fervor of a pedant. He said that Woodhull had run a salon not unlike that of Madame Roland during the French Revolution, and he reeled off a list of politicians and business leaders who had visited at her Murray Hill mansion. Next, Victoria Woodhull herself was called to testify. On May 12, 1875, Woodhull entered the Brooklyn city courtroom. She wore black, and her face was concealed by a veil. As soon as she was seated, Eunice Beecher stood, turned on her heel, and left the courtroom. When asked to produce all her correspondence with Theodore Tilton, she reached into a red Moroccan case, brought out a slim packet, and said sweetly, "I have reason to believe that some of my private letters are in the hands of the defense as well as the prosecution." Benjamin Tracy whispered in William Evarts's ear, after which he came forward and made a motion to dismiss Mrs. Woodhull without further testimony. The motion, unopposed, was granted. After Victoria Woodhull left the courtroom, Theodore Tilton once again stated that he deeply regretted his relationship with her. She had scarcely arrived home when a reporter brought her news of this. She commented, "I believe Mr. Tilton would make quite a man if he should live to grow up."
The public keenly anticipated new revelations from Francis De Pau Moulton. He was one of the last witnesses to testify but when he did there was no holding back. He released every document and every letter that had been entrusted to him. In the course of eleven days, Moulton tried to refute with documentation the accusations against Tilton and himself, and to establish Beecher's guilt. Most damning were Beecher's suicide threats, his writing repeatedly of having sinned, as well as the clandestine correspondence with Mrs. Tilton in which he declared his abiding love for her and told her exactly when she was "permitted" to write, which depended on when his wife was away. But most penetrating were Moulton's own insights into the individuals involved in this matter. Said Evarts, "You are criticized, Mr. Moulton, for bringing Mrs. Woodhull into your family. How do you reply to this?"
If I could tolerate Mr. Beecher in my family after what I knew of him, why not Mrs. Woodhull?... The scandal she published was wrung out of her by taunt and scorn and general persecution on Beecher's account. Besides, there was at her elbow old Stephen Pearl Andrews pushing her on to prove society falsely constructed.... She had reason to see from the class of people who called at her house that from the clergy to the press she was the object of both fear and desire.
"Did Mr. Beecher also fear his sister Mrs. Hooker, Mr. Moulton?"
He did. She was not a crazy woman but a bolder Beecher than he with equal appetite for the world.... Mr. Beecher's sister—the amiable, intelligent, enthusiastic, and clearheaded Mrs. Hooker, now happily, for her peace, abroad—who became the recipient of the knowledge of Beecher's guilt, has been placarded as insane because she advised him to make a clear and full confession in the interest of truth and justice, to rescue a woman from jail who Mrs. Hooker believed was incarcerated for having told simply the truth.
"Do you think the Beecher case, Mr. Moulton, will be of any harm to society?"
No, it has only brought down an impostor who could not be saved from the weakness of himself. We respected her [Mrs. Tilton] even after her fall because we had studied Beecher out and knew him to have a fine mind, a powerful animal nature, and between the two he got his power. He never could have preached the sermons he has, addressing the weakness of our flesh, but for the animality which drew him into libertinism and was followed by self-reproach. The fact is he has been sifted out of the little principle that he possessed, by the flattery of mankind. Everybody took care of him, paid his bills, wanted his society, and encouraged his selfishness. He has had bursts of emotion and tenderness, but they are not reliable, and he was too mean to lose his fame.
Day after day the prematurely aged Eunice Beecher sat in the courtroom listening to the evidence that characterized her as "a virago... cold... complaining... inhospitable." She heard how her husband had fled the "uncongenial atmosphere" of her home to find warmth and attention at Mrs. Tilton's. And the press, observing her wizened countenance—pursed mouth, wrinkled face, gray hair covered by a white lace mantle—ratified this opprobrium. Still she assumed the stand and testified that her husband was the most noble of men, a victim of his own "kindness and sympathy." All the Tiltons had taken advantage of her husband's goodwill: "It has been hard to convince the dear, guiltless, simple-hearted man that such baseness and treachery could exist... how weakly he has trusted... how fearfully he has been muted or blackmailed." When Eunice finished her testimony, her husband rose from his seat, walked to the witness stand, and escorted his wife out of the courtroom. She had become, after all, the perfect preacher's wife.
After Eunice Beecher testified, Mrs. Tilton stood and appealed to be heard by the court. Since neither side wished to hear what she had to say, her appeal was denied, but she released to the press the enigmatic statement "I would like to tell the _whole_ sad story _truthfully_ —to acknowledge the frequent falsehoods wrung from me by compulsion."
By the end of the trial millions of words had been written about this matter. Tilton's side had been represented by a dozen witnesses, a meager showing. Beecher's case had been presented by ninety-five witnesses. The summations took twenty-five days; the jury was sequestered for eight. As the summer heat increased the jurors were assigned another room on the cooler side of the courthouse. Reporters climbed the lampposts trying to see what was going on inside. For lack of real news, they reported on the most mundane of actions: which juror was asleep, what the jurors were eating. At eleven-thirty on the sweltering morning of July 2, 1875, the jurors filed into the courtroom. After fifty-two ballots, there was still no agreement. On the first ballot the vote had been 8 to 4 against Tilton; on the last, 9 to 3. The judge declared a mistrial. The circus was over.
The economic depression had brought sterner times, and the gospel of self-indulgence and sentimentality was losing its power. The _Times_ noted,
There is only one good result which can possibly follow from this exposure and trial. It may lead people in Brooklyn and elsewhere to distrust the Gospel of Love and to allow no priests or ministers to come between husband and wife, or to interfere with family ties or sully family honor. Lastly it may induce them to return to the older and safer moorings which alone can prevent society from drifting into chaos. If this should be the fruit of the trial, a scandal which poisoned the air for six long months will not have been dragged to the light in vain.
_Henry Ward Beecher and Eunice Beecher in the courtroom. The press nicknamed the worn Mrs. Beecher "The Griffon._ "
_Lib Tilton appeals to be heard. No one wanted to hear her explanation_.
# CHAPTER THIRTY-FOUR
A METEOR'S DASH
ALTHOUGH THE case ended in a mistrial, the Beecher forces clearly had triumphed. At their preacher's suggestion a Plymouth Church committee, nicknamed "the scandal bureau," was appointed specifically to quash further rumors. Most of the blame for the scandal fell on Henry Bowen. The _Eagle_ pointed out that he was the Iago who had created the friction between Beecher and Tilton. Several members of Plymouth Church brought suit against Henry Bowen to recover the monies owed them, which had been forfeited through the bankruptcy of Jay Cooke & Co. Testimony given in the Brooklyn City Court showed that Bowen had used the _Independent_ to promote Northern Pacific Railroad bonds while fully aware that they were nearly worthless, and that Tilton's opposition to this swindle was a factor in his dismissal. According to the _Eagle_ , testimony also established that Bowen "was the link by which Wall Street was connected to the religious world." After several months the case was dismissed.
In December 1875, a Plymouth Church committee brought a formal action to dismiss both Emma Moulton and Henry Bowen from their rolls. Emma was dropped without a hearing, but Henry Bowen insisted on appearing before the committee. In his statement he branded Henry Ward Beecher "an adulterer, a perjurer, and a hypocrite." He vowed that he had done no wrong and would not willingly leave the church that he had founded. Bowen pointed out that this committee, like the first committee to investigate Mr. Beecher, was composed of partisan men, including Beecher's brother, his assistant pastor, his publisher, and his attorney. "This is no impartial court," he declared with some understatement. He concluded by saying, "I have been held up as a slanderer and a liar until my character has been blackened.... You have forced me into this trying and unwelcome position.... I now leave my case with you well understanding that you will continue to asperse my character and that I shall suffer.... So be it." Bowen was expelled by a unanimous vote.
In January 1876, the Plymouth Church Council organized a victory party of sorts in the form of a meeting to give its preacher a vote of support. Once again Plymouth Church was full. When the council commended Beecher for the way he had handled himself during the case, he stood upon the preaching platform, silent tears rolling down his cheeks. Then in a voice cracking with emotion he told his flock, "I have not been pursued as a lion is pursued. I have not been pursued even as wolves and foxes. I have been pursued as if I were a maggot in a rotten corpse."
Carried away, he provided as close an explanation for his behavior as one would ever hear. "When you shall find a heart to rebuke the twining morning glory or any other plant that holds on to that which is next to it... you may rebuke me for loving where I should not love. It is not my choice. It is my necessity.... And I have loved on the right and on the left, here and there, and it is my joy that today I am not ashamed of it, I am glad
_The final victory meeting at Plymouth Church. Henry Ward Beecher's powers were undiminished_. (Illustration Credit 34.2)
_Isabella Beecher Hooker. In her old age her spirit communication sustained her_. (Illustration Credit 34.1)
of it." No one seemed affected by the truth of his confession, only by the emotion with which he presented it. In response, men and women alike wept. Then the mercurial Beecher lashed out at the assemblage: "I do not care for you or anybody else.... I do care and I don't—just as I happen to feel.... I am tired of you.... I am tired of the world."
Although Henry Ward Beecher had emerged unscathed, other lives had been destroyed. In her hotel room in Paris, during her self-imposed exile, Isabella Beecher Hooker woke one morning from a deep sleep and mechanically began repeating to herself, "Thanks be unto God, the Father of our Lord Jesus Christ, who giveth us the victory." She repeated this litany over and over until the rhythmic flow of the chant induced "a deep and solemn peace" that settled into her very soul. The thought occurred to her that though she had been reviled, somewhere there was love, for "Love begets love—love is God." Then miraculously, her mother, dead forty years, appeared before her. Isabella had "drawn the veil" between herself and her mother, but now and henceforth forever she would be nurtured by mother love. She wrote, "From that time she has never left me, and such communion as we have had only the angels know, and He who is above all angels, Jesus of Nazareth our brother, friend and Savior."
In the past, sometimes her visions had seemed "but a delusion and a mockery," but now "the light of the eternities, past and future, shines into my soul and all is peace and a great patience of waiting takes the place of all unrest." For as Victoria Woodhull had predicted, past, present, and future were as one.
As Moulton's documents had been revealed to the public, the feelings of Nook Farm residents toward Isabella had softened. John Hooker wrote a letter to Susan Howard in which he spoke of his wife's sorrow and how cruelly she had been dealt with. Finally, Susie Howard mustered the courage to do what Isabella had asked her to do six years previously, to think for herself and come to her own conclusions. Despite the support her husband and son, John Tasker and Joseph Howard, had given Beecher and the attacks they had made on Isabella, she wrote a letter stating that, in her opinion, Isabella was innocent of everything but telling the truth. She wrote that all of Brooklyn knew of Lib Tilton's attachment to her preacher. This letter along with John Hooker's was shown to Mark Twain. It convinced him and other residents of Nook Farm that Beecher had indeed been guilty. He told his wife, Livy, that he would no longer forbid her to see Isabella.
Harriet Beecher Stowe stubbornly continued to believe in her brother's innocence, but she justified Isabella's behavior in a letter she wrote to their sister Mary Perkins. She felt that "the Woodhull" had gained "an ascendancy" over "Belle" by showing her the apology Henry had written. (In fact, Stanton had showed Isabella this letter.) She wrote that Woodhull, Stanton, and Anthony had prevailed upon Henry that if he came out with free love views there would be "an immediate rush to his standard of all the emancipated." Perhaps confusing Woodhull and Anthony, Stowe blamed the latter.
In the letter which Belle sent to Henry when she urged this course was a postscript by S. B. Anthony saying in so many words, that the _needs_ of a nature so large as his were evident and that the _only_ sin he had committed was living in a false marriage and they hoped he would come out and confess that and show from his own experience from Sunday to Sunday and from day to day the evil of the present system of marriage. I don't think they would have got Belle over, for she made strong resistance at first, if they had not produced a conviction on her mind that the only way to save her brother was to attack the laws of marriage.
When at last the scandal subsided, Isabella returned to Nook Farm, where she found solace and acceptance in the spirit world. She spent many hours in a spirit room she created from one of her upstairs bedrooms, communing with those who had "passed over." Frequently, Isabella sank into a trance, and the spirits spoke through her. On Sundays, she held meetings with Spiritualists, convinced that she was forming "the beginnings of the Church on Nook Farm which I have seen so distinctly in visions over and over again." On sunny days Isabella would climb out onto the tin roof below her spirit room window, open her silk parasol, and sit in the sun, waiting for the popping sounds that she interpreted as spirit messages.
Once the trial was over, Benjamin Tracy lodged libel suits against both Tilton and Moulton. The suits finally were dropped, but Tilton emerged bankrupt and the _Golden Age_ was closed for lack of funds. In the winter of 1877, he moved to Paris, never to return. Mrs. Tilton was left with little money and four children to raise. She was rapidly going blind. Lucy Stone wrote of the case, "Of all the persons in the sad scene which has been passing before our eyes, Mrs. Tilton seems to me to be more wronged and injured than any other... carrying to the graves of her children the dead hopes of all her life. Driven to the wall in utter despair, endured for five miserable years, Elizabeth Tilton said what her tormentors required her to say." The only friends left to Mrs. Tilton were a small group of Spiritualists who called themselves The Christian Friends. They supported and sustained her emotionally and convinced her that she would never be at peace until she was no longer the instrument of others but became her own woman and lived in the light of truth. At their behest, she wrote a public letter to the _Times_.
A few weeks ago, after long months of mental anguish, I told... a few friends whom I bitterly deceived, that the charge brought by my husband, of adultery between myself and the Reverend Henry Ward Beecher, was true and that the lie I had lived so well the last four years had become intolerable to me. That statement I now solemnly re-affirm and leave the truth to God, to whom I also commit myself, my children, and all who must suffer.
With so many accusations and retractions and the interest of the public moving on to other events, this final confession engendered little interest. When Beecher was asked by a reporter to comment on the letter, he said that Mrs. Tilton was "an unbalanced clairvoyant" and asserted that Lib used to "grovel in the dust and roll in the gutter, even kissing the feet of those to whom she felt under obligation."
MAMA ROXY had brought Victoria back to New York, where they lived in a furnished flat in a brownstone on Eighteenth Street. Here they seemed to revert to their former living conditions. Mama or Tennie took the large pitcher from the washstand and walked to Third Avenue to have it filled with beer. Victoria was so weak that she remained in bed until it was nearly time to lecture, and then she roused and dressed herself. There was little call for her services.
One evening Tennie entered her sister's bedroom to find her kneeling on the floor, her palms pressed together in a gesture of supplication. "Oh, Tennie," she said. "I am afraid of dying. I don't know what will become of me." As she grew increasingly upset she confided to her sister that she no longer had the support that had sustained her all her life. Her beloved Demosthenes and Bonaparte and Josephine had deserted her. Her spirit voices were stilled. Like Joan of Arc, she had been abandoned. No longer was she on fire with the word of what the world could become. No longer could she be the savior of women. It was over.
Tennie, thinking that her sister's ailment was physical, took her to see a physician who diagnosed Victoria as suffering from "a female ailment amenable to surgery but aggravated by neglect." It was then that Mama Roxy moved to reclaim her daughter. In Victoria's weakened state her mother convinced her that the reason her beloved spirits had deserted her was that Colonel Blood was poisoning her in some undetectable way. Roxy also said Blood was possessed by the devil, whose presence blocked Victoria's spirit guides. Finally, Victoria agreed to go with her mother to see a group of monks who were said to be skilled in ridding one of evil spirits according to Catholic Church rituals. In the winter of 1875, Victoria Woodhull was "exorcised" by these monks. Roxy now was able to persuade her daughter to file for divorce from James H. Blood. Even though Victoria had no papers to prove she had remarried Blood, she nonetheless filed suit on charges of adultery. Colonel Blood never met the witness Victoria procured to testify that he had taken her to a house of prostitution, but he did not contest the suit. Later his only comment was, "The grandest woman in the world went back on me."
At the time that Victoria Woodhull was deserted by her spirits, Spiritualism itself was coming under attack. Margaretta Fox had converted to Catholicism, and although for some time professed not to believe in Spiritualism she had recently returned to the séance table to earn a living. Her sister Kate performed in London at séances, where she was often so hysterical and inebriated that she could make no contact with the spirit world.
The simple faith that had once brought comfort to women in their intolerable lives, and the promise of joy beyond, had become a highly profitable form of entertainment. Laura Cuppy Smith and her trance speaking had been the main event of the 1874 Spiritualist camp meeting at Silver Lake Grove, Plympton, Massachusetts, but the following year the meeting had become a carnival featuring a carousel, a singing midget, a Mrs. Suydam, who held fire in her hands, and a so-called Spiritualist who painted flowers while blindfolded.
The Foxes' eldest sister, Leah, was the first to produce a "full-form materialization" in which a spirit figure walked among those gathered at the séance table. Now spirits could not only speak but be seen. And soon, in the light of day, Spiritualists were seen as frauds. A document was discovered that instructed bogus mediums on how to produce the desired diaphanous and luminous effects when impersonating a ghost. A common recipe was also circulated: "One jar Balmain's Luminous Paint, half a pint of Demar varnish, one pint odourless benzine, and fifty drops lavender oil." Everywhere tricks began to be revealed—fraudulent ectoplasmic materializations, false trance messages, séances where instruments were used to produce eerie music and spirit messages. As Spiritualism weakened, the church condemned trance speakers, many of whom had replaced preachers as conduits for words from on high, while the male medical establishment attacked faith healers. In 1876, a medical report stated that the majority of women's diseases were caused by Spiritualism. Perhaps inevitably, adherents to Spiritualism now began to attack each other with accusations of occultism and personal aggrandizement.
The 1876 meeting of the loosely organized Association of Spiritualists was attended by only a few hundred people, and at the first session Victoria Woodhull resigned as president. Feeling that she had to defend herself, she told an _Argus_ reporter,
It has been supposed in some quarters that because I resigned the presidency... I had become weak in the faith.... I wish at this time to publicly assert... though every professional medium in the country should be exposed, as most of them, apparently at least, are going to be, from their earliest childhood all of my sisters were mediums and were controlled, were subject to trances, to having visions, and to seeing spirits. This was before the commonly acknowledged advent of modern Spiritualism through the Fox girls at Rochester and was widely known in the country round about where we resided in Ohio.
Spiritualism had resisted organization, for it assumed that all human beings contained truth and power in their own souls. By the end of the decade other movements, Theosophy and Christian Science, drew the support of many former Spiritualists. On October 21, 1888, Margaretta Fox delivered what the press termed "a death blow" to Spiritualism by confessing that the rappings of the spirits had been produced by an abnormality of her big toe which sounded like "a muffled hammer." She reproduced the sounds to an astounded and disillusioned audience. Then Kate Fox confessed that at the Corinthian Hall demonstration in Rochester forty years earlier, she had arranged for a Dutch servant girl to rap from the cellar whenever she heard the sisters' voices calling upon the spirits. Margaret and Kate then wrote a book detailing their fraud. However, soon after, Margaret claimed that financial pressure had caused her to renounce the spirits, but she could not live with this lie. At her death in 1893 she was once again a practicing medium.
With the departure of her own sustaining spirits, Victoria became a pale replica of her earlier self. Stephen Pearl Andrews too was no longer Victoria's guide. Isabella Beecher Hooker had written her concerning Andrews, begging her "to shake off a serpent who is simply bringing you evil," and Victoria severed all connection with him. On her next tour she began her speeches by reading verses of Scripture. She devised a new lecture called "Breaking the Seals," which was, unbelievably for the woman she had been, a defense of the institution of marriage. In her desperation to secure an audience, she began to twist the meaning of free love into something mundane, respectable, and self-evident. In Kokomo, Indiana, she announced, "Free Love is the free love of God to the world. I have been traduced, vilified, and imprisoned for maintaining that right." In Boston she said, "God is love, and love is God. Who dares to tell me tonight that the love of God is not free.... Free Love is not what I asked for nor what I pleaded for. What I asked for was educated love."
Woodhull now lectured on purity, the sanctity of the body, and on "stirpiculture," a kind of eugenics advocating a scientific way of breeding to create superior human beings. Her lectures became murky, convoluted, pseudo-scientific. _Woodhull & Claflin's Weekly_, which appeared infrequently, also seemed to have lost its vigor and purpose. June 10, 1876, was the last issue, and in it Victoria Woodhull severed all connection with free love, stating that she had always advocated "the sanctity of marriage... I believe in the institution as a divine provision.... I do not believe in the loose system of divorces now so much in vogue."
As Victoria Woodhull vanished from the woman's rights platform she left behind her the wreckage of the movement. Women, had they stood together, might have found enfranchisement within their grasp. But the moral hypocrisy of domestic society that Woodhull clearly illuminated through her views on free love, marriage, and divorce had completely split an already divided movement. As Elizabeth Cady Stanton had predicted, the "scandalum magnatum" had been "rolled on our suffrage movement," and the chance of enfranchisement had "gone down in the smash." But Stanton, with a clearer view than most, realized as well that the work Victoria Woodhull had done on behalf of women would have a lasting effect no matter how she recanted her views or how quickly she became a mere footnote to women's history. In an interview in 1876, Stanton indignantly told a reporter who dismissed Woodhull's work,
Victoria Woodhull has done a work for woman that none of us could have done. She has faced and dared men to call her the names that make women shudder, while she chucked principle, like medicine, down their throats. She has risked and realized the sort of ignominy that would have paralyzed any of us who have longer been called strong-minded.
Leaping into the brambles that were too high for us to see over them, she broke a path into their close and thorny interstices with a steadfast faith that glorious principle would triumph at last over conspicuous ignominy, although her life might be sacrificed. And when, with a meteor's dash, she sank into a dismal swamp, we could not lift her out of the mire or buoy her through the deadly waters. She will be as famous as she had been infamous, made so by benighted or cowardly men and women.... In the annals of emancipation, the name... Victoria Woodhull will have its own high place as a deliverer.
The new world Victoria Woodhull had envisioned had not come about. The dream that a powerful nation would protect the rights of all its people was at best premature. Radical Reconstruction had failed. In the 1874 congressional election the Democrats had won in a landslide, and would remain in power for more than a decade. In Southern states, White Leagues were formed, openly dedicated to restoring white supremacy. The violence began again. During the 1876 presidential election campaign, blacks were prevented from voting by parades of white supremacists carrying rifles. Unlike the hooded Klan members, these men paraded in broad daylight. In Coahoma County, Mississippi, six black and three white men were killed, and yet President Grant failed to intervene.
A dubious Frederick Douglass, seeing the lack of interest in a real Reconstruction, asked a gathering of political leaders, "Do you mean to make good to us the promises in your Constitution?" The answer was soon to become obvious. The idealistic Douglass had been a pawn in the greedy profligacy of the postwar era. In March 1874, because of his prestige among his own people, he was invited to become president of the Freedmen's Bank, in which reposed the hopes of poor, hardworking black people who had literally invested their pennies to save for a better life. Douglass was so delighted with his job that he invested $10,000 of his own money in the bank. He was in office only three weeks, however, when by accident he came across a ledger that showed that the week he'd arrived the majority of the trustees of the bank had withdrawn their own money. Douglass investigated and found that the bank was insolvent. The trustees knew it and even the Senate Committee on Finance knew it. When the bank collapsed, Douglass was deluged with letters from people who had trusted and revered him. The best he could arrange was for them to receive 18 cents for every dollar they had invested. The Freedmen's Bank "has been the black man's cow but the white man's milk," noted a despondent Douglass.
Totally discouraged, Douglass began to see what others had already seen, the greed and corruption of the new America. "The moral atmosphere is more than tainted, it is rotten," he wrote. "Avarice, duplicity, falsehood, corruption, servility, fawning and trickery of all kinds, confront us at every turn." Of the shining hope of freedom for his people, Douglass wrote, "You say you have emancipated us. You have and I thank you for it.... But when you turned us loose, you gave us no acres. You turned us loose to the sky, to the storm, to the whirlwind, and, worst of all, you turned us loose to the wrath of our infuriated masters."
In 1877 all troops were removed from the South, leaving the blacks to fend for themselves. The federal government continued to make remarkable concessions to the railroad interests and land-hungry citizens. On the Great Plains, farmers and cattlemen were quickly disposing of the Indians.
_Frederick Douglass never gave up the fight for equality. "Agitate, agitate, agitate, " he instructed_. (Illustration Credit 34.3)
Although Indian warfare would not end until the final massacre of the Sioux at Wounded Knee in 1890, their world had vanished.
AT THE age of eighty-three, in January 1877, Commodore Cornelius Vanderbilt lay dying. The previous year, the Commodore had supervised the laying of rails and the building of bridges that cut the running time of the New York Central trains between New York and Chicago from fifty hours to twenty-four. His obsession with speed had become the American obsession. Reporters were stationed across the street from 10 Washington Place awaiting the death of this titan whose fortune was estimated at $110 million, $5 million more than the United States had in its treasury. In his failing state he begged his wife, Frank, to call in Spiritualist healers to relieve his pain, and as they came he said, "Why have I been deprived of this so long? It is doing me so much good." He experienced visions of life after death and said he had no fear of "passing over" to where his mother awaited him.
At ten o'clock on the morning of January 4, thirty relatives gathered in the Commodore's bedroom. He requested the singing of a hymn, after which Frank read the Lord's Prayer. "That was a good prayer," the Commodore whispered. His eyes were like glass. One physician whispered to another that he was now sightless. The Commodore heard him, raised his hand, and with his own fingers pressed his eyelids shut. Ten minutes later he died.
Vanderbilt was survived by ten children; to his eight daughters he left modest bequests, the largest being a trust fund of $500,000. He left Cornelius Jeremiah, his epileptic son, only the income on a trust fund of $200,000. His entire residuary estate was left to his son William Henry. Cornelius Jeremiah and one of his sisters brought suit to invalidate the Commodore's will on the grounds that his long-held belief in the spirits and his rampant sexuality were signs of mental incompetency. Shortly thereafter, there was a knock at the door of the modest boardinghouse where Victoria and Tennie now lived. Their caller was William Henry Vanderbilt.
Laura Cuppy Smith, who was visiting at the time, later said that Mrs. Woodhull informed William Henry Vanderbilt that he had nothing to fear. Based on the correspondence Victoria possessed, there could be no doubt that his father had been brilliant and not unsound of mind, and she and Tennie were prepared to say so in open court. She would testify that the conversations with his dead mother had been beneficial to him, as had the advice she had given him while in her trances. Tennie interjected that the attentions paid to her were remarkable considering the Commodore's age, and was not virility considered a sign of health? Victoria also politely informed William Henry that she knew the details of how he had made the necessary arrangements to confine his mother, Sophia, to Dr. McDonald's insane asylum so that his father could pursue other women. She added that the Commodore had always appreciated his help in convincing physicians of Sophia's derangement.
A few weeks after William Henry's visit, Victoria was directed by the court to produce any correspondence she had with Commodore Vanderbilt; several of his servants had testified that from 1868 to 1872 they had carried numerous notes to her. However, Victoria replied by affidavit that she had none in her possession. During 1877, "The Great Vanderbilt Will Trial" made headlines across the nation, as all manner of Spiritualists—healers, clairvoyants, trance speakers, mediums—were subpoenaed. The testimony of the notorious sisters was eagerly awaited. But when they were summoned, Victoria and Tennessee were on a ship halfway to Southampton. Awaiting them in England was a cozy house at 8 Gilston Road, West Brompton, in the name of Mrs. Victoria Woodhull. Also awaiting them was a small fortune in Lloyd's Bank in London, enough for two gentlewomen to start a new life. Six months later, when she was comfortably settled, Victoria wrote William Henry Vanderbilt a short note requesting the return of "my box of letters concerning your father."
# CHAPTER THIRTY-FIVE
THE LAST ENEMY
AS THE battle for Radical Reconstruction had been lost, so had the battle for woman's rights. Many of the women who had fought bravely found themselves disillusioned and disenfranchised. With the passing of time, the truth of what had gone wrong was gradually revealed, though few cared to see. In 1877, as Victoria and Tennessee sailed for England, Anna Dickinson's career was collapsing; her luster had tarnished. The previous year, she had written Whitelaw Reid saying that she was gravely ill, but was acting the part of Anne Boleyn, and a favorable mention in the _Tribune_ would be of great service. He read the letter, wrote on the envelope "unanswered" and filed it away. That same year Whitelaw Reid chose a bride, the daughter of Darius Ogden Mills, a vastly wealthy banker who had made his money in California after the gold rush. Anna wrote, "I suspect Whitelaw is _actooally_ engaged _this_ time. I infer so from what I hear—also that the lady is rich and _humly_."
Benjamin Butler remained fascinated by Anna, although with her waning fame, he saw her as a possible mistress but no longer as a wife. Soon after his wife died in 1876, Butler tried to persuade Anna to go with him to Atlantic City. She put him off and then canceled. He wrote, "I give it up.... I will have no more." But they stayed on friendly terms, and he continued to send Anna money until 1887, when he wrote her, enclosing a clipping from the _Graphic_ stating that he had proposed marriage to her. An embarrassed Butler wrote that the article "pained" his daughter with the thought that he was trying "to fill her mother's place." He continued, "My lame and trembling hand bids me stop but I go on.... Will you therefore in the words that your own good sense will teach... write me a frank denial that I have tried to get another wife so far as you know. I must wish this could come from your pen."
Anna, undoubtedly realizing that this excuse was fabricated and that he thought her too low to marry, replied that Butler had made an "astounding and dastardly request." She reminded him that he had indeed proposed marriage to her and she would not disgrace herself by lying about the matter. After this, their correspondence became acrimonious. Butler wrote that he would send Anna no more money unless she wrote the letter denying his proposal. She answered,
I want to see or hear from you _at once_. If I do not, _you_ will see every letter you have written to me in print within the next month. This is neither blackmail nor the letter of a mad woman. You know what I was, I am. _You_ know what _you_ have written about many matters.... You may remember a letter written while your wife was dying. It will make good food for the public's palate.... I have endured all I am going to suffer. Enough _is_ enough. Do you understand? I need some money. I need it immediately.
There is no record that Butler replied to this letter, and she never saw him again. At this time Anna suffered severe headaches. Her sister, Susan, said that she needed chloroform in order to sleep. On a frosty February day in 1891, Anna carried her press clippings into her bedroom and spread them across the floor until every inch of the rug was covered. Day after day she sat cross-legged, reading them obsessively. She allowed no one to enter, and Susan was obliged to leave Anna's meals on a tray outside the door. One evening Anna failed to return the tray. The next night was the same. On the third night, Susan entered the room to retrieve the trays, and Anna grasped her by the throat. For three days thereafter Anna ate nothing at all. Her sister then committed her to the state hospital for the insane in Danville, Pennsylvania.
Two months later Anna managed to write William Howe, Woodhull's former lawyer, of the intolerable conditions to which she was being subjected. He took her case and later arranged a press conference in which she professed her sanity and attacked the male establishment for suppressing her views and driving her to poverty. In a subsequent sanity hearing Anna recalled how Benjamin Butler had courted her and how Whitelaw Reid had persuaded her to speak for Greeley in 1872 but after the election had abandoned her. She said that Reid subsequently boasted that she had been his mistress and told many people that his shoes could be found outside her hotel room door at night. The truth, she said, was that he had proposed marriage to her, but she had rejected him because he was an epileptic who had inherited the disease from "the tainted blood of his father and the epileptic blood of his mother." He was forced to take powders "to make a man" of himself. The jury rendered a split verdict of 8 to 4 in favor of Anna's sanity.
In November 1895, Susan B. Anthony received a letter from someone long estranged.
As I opened the envelope I said, "Why this looks like Anna Dickinson's writing & turning to the last page sure enough there it was, Anna E. Dickinson, the same dear old name
I'm awfully glad to know you still live and that I have a chance to tell you that my _motherly love_ —my elderly sister's love has never abated for my _first Anna_ —I have had several lovely _Anna_ girls— _"nieces_ " they call themselves now a day—since my _first Anna_ but none of them ever has or ever can fill the niche in my heart that you did, my dear.
Anna lived to see the Lyceum lecture circuit replaced by movies and radio. At times this frail, white-haired old woman would visit the candy store in Goshen in the foothills of the Catskill Mountains and there would tell schoolchildren how she had known Lincoln and how she and a man named Theodore Tilton had been instrumental in securing for black men the right to vote. Anna died six days before her ninetieth birthday, on October 22, 1932, two weeks before Franklin D. Roosevelt was elected president.
IN 1890, after an eighteen-year separation, the warring women's groups were reunited as the National American Woman Suffrage Association, but their agenda had become conservative—"toothless," a newspaper called it. No longer did they deal with issues such as divorce and the exploitation of working-class women. Their ranks were filled, almost exclusively, with white, middle-class members. At their first convention Lucy Stone commented that her morning speech was "cheapened" by the knowledge that the people listening to her "did not care so much for us as if we had votes." By the fall of 1893, Stone was confined to her bed. On the evening of Wednesday, October 18, she signaled her daughter Alice Blackwell and whispered in her ear, "Make the world better." Then she died.
Elizabeth Cady Stanton never ceased fighting for the rights of women. Over the years she came to realize that the vote was only one element in securing woman's rights: "I feel that suffrage is but the vestibule of woman's emancipation!" A major effort of her later years was _The Woman's Bible_ , wherein she reprinted passages from the Bible with her own commentary, emphasizing woman's strength and illustrating how this was a historical, not a sacred, document. _The Woman's Bible_ created a controversy within the woman's movement, but it was a ripple compared to the tidal wave that Victoria Woodhull had caused and all that had followed.
Elizabeth Cady Stanton stayed on cordial terms with her brother-in-law Sam Wilkeson. In 1880, he brought suit against Henry Ward Beecher to recover the $10,000 that he'd advanced from his personal funds toward the publication of _The Life of Jesus the Christ_. The case was dismissed, but Wilkeson told Stanton that on the witness stand he'd said once again that "The Beecher-Tilton trial knocked 'The Life of Christ' higher than a kite" and it was "worth ten thousand dollars to see the expression on the faces of judge and jury."
Stanton was to see Victoria Woodhull one more time. In December 1882, she visited London but seldom left her hotel, for one of the most terrible fogs in British history had engulfed the city. Even during the day, she observed, "the gas was lighted in all the houses and streets, carriage lamps were burning, and other conveyances had flaming flambeaux." Early one evening "a heavily veiled lady" came to call, whereupon she "threw off her concealment and there stood Victoria Woodhull." Stanton wrote, "She has passed through great suffering. May the good angels watch and guard her. I will not condemn."
In February 1902, Stanton, then eighty-six, realized that the vote for women probably would not be achieved in her lifetime. She wrote in her diary, "Logically, our enfranchisement ought to have occurred in... Reconstruction days. And that was what I argued in the sixties. Our movement is belated and like all things too long postponed now gets on everybody's nerves." She died eight months later. The ever-organized Mrs. Stanton had instructed her daughter Margaret on the details of her funeral. "I should like to be in my ordinary dress, no crepe or black, no fripperies or fandangos of any sort, and some common sense women to conduct the services."
At the head of her coffin was placed the mahogany McClintock Spirit Table, recalling the time when the woman's movement had begun. The table, now the property of the Smithsonian Institution in Washington, is identified only as the table on which the original Woman's Declaration of Rights and Sentiments was written in 1848. There is no reference to the help of the spirits.
_Susan B. Anthony and Elizabeth Cady Stanton, c. 1898. The fight for equality would last throughout their lives and beyond_. (Illustration Credit 35.1)
ONE BY one the old friends and enemies slipped away. On February 20, 1895, seventy-eight-year-old Frederick Douglass had accompanied the seventy-six-year-old Susan B. Anthony to a Washington woman suffrage meeting. That evening Douglass was relating the day's events to his wife when his heart failed. At the time of his death much that he had gained for his people had been lost. In his native Talbot County the only memorial to Douglass was a segregated schoolhouse.
Like Stanton, Susan B. Anthony would not live to see women vote, but when she was asked if they eventually would, she assured her listeners, "Failure is impossible." On March 13,1906, Anthony followed her friend to the grave at the age of eighty-seven. In accordance with the Quaker custom, her house bore no symbols of mourning. In 1920, seventy-two years after the fight began, women at last were granted the vote. The amendment, the nineteenth to the Constitution, was named for Susan B. Anthony.
A few months after Anthony's death, Isabella Beecher Hooker, then eighty-six, died of a massive cerebral hemorrhage. She too had continued her work for woman's rights, serving as president of the Connecticut Woman Suffrage Association until 1905. Her reliance on spirit guidance, however, made many in the movement question her credibility. In later years she was to write, "I have been often ignored and never treated with the old time courtesy and appreciation of the value of my devoted service."
As a result of the Beecher-Tilton scandal, Isabella had become alienated from her daughter, Mary, who felt that her mother had endangered her father's health and well-being by supporting Victoria Woodhull. But after Mary died in 1884, Isabella erected a shrine to her, and they became reconciled. Isabella felt her daughter's love and approval and spoke with her spirit almost daily. Isabella cut herself off completely from the brothers who had conspired to destroy her. After a time, Harriet grudgingly reinstated her relationship with Isabella, but Catharine Beecher and Mary Perkins never forgave her for supporting Victoria Woodhull over their brother. However, once they had "passed over," Isabella said, these two sisters returned to her in friendship and forgave her. When John Hooker chastised his wife, Catharine's spirit took over her hand and in Catharine's own handwriting instructed John to be a more sympathetic husband. As the years passed, Isabella and John estimated that they had communicated with some 450 spirits, including Byron, Dickens, Beethoven, Haydn, and Harriet Beecher Stowe's son Fred, who had disappeared and when contacted, Isabella declared, was suffering from delirium tremens.
In the last decade of Henry Ward Beecher's life he enjoyed peace and prosperity as the acclaimed shepherd of his flock. In 1887, as Beecher lay dying, Isabella ventured to Brooklyn to see him, only to be turned away from his Columbia Heights home by Eunice. As he lay in his coffin in Plymouth Church, a veiled Isabella joined the end of a long line and filed past, thinking that she might see a sign of apology on the face of the brother she so loved. There was none. However, when she returned to her spirit room at Nook Farm, Henry's spirit appeared, and with tears streaming down his face he knelt before her, as she had been forced to kneel before him, and begged forgiveness.
IN 1883 Theodore Tilton rented a room on the Ile St.-Louis in Paris and remained there for the next twenty-four years. He wrote poetry, a novel, infrequent articles for French journals, nothing at all for American consumption. Often he spent afternoons sipping Pernod and playing chess at the Café de la Régence. When he died on May 25, 1907, his daughter Florence (Florry) issued a statement that no one in the family would attend the funeral and that they had no plans to bring his body back to America. Theodore Tilton's accomplishments vanished: Several histories confuse him with the presidential candidate Samuel J. Tilden; others relate that Abraham Lincoln came to Brooklyn to ask the advice of Henry Ward Beecher, not Tilton, and that Beecher's exile to England during the Civil War was not attributable to his adulterous relationship with Mrs. Bowen, but to a secret mission, assigned him by the president, to gain support for the North.
Lib [Elizabeth] Richards Tilton preceded her husband to the grave by ten years. From the time Lib issued her last letter admitting to adultery with the Reverend Henry Ward Beecher, she wrote no more letters. She spent the last decade of her life blind and a recluse, sustained only by the Spiritualist group who believed in the final salvation of the soul through truth, atonement, and the sacrifice of Christ. Lib's Spiritualist belief in a glorious world beyond, according to these friends, brought her peace in her final days.
ANTHONY COMSTOCK, whose battle to silence Victoria and Tennessee launched his vindictive career, continued his purity crusade for almost another half a century. In February 1878, a stocky, shabbily dressed man presented himself at the basement office of Madame Restell's mansion at 1 East Fifty-second Street and asked for an abortion powder and later for a contraceptive device for a woman who could support no more children. Madame Restell asked the gentleman to return the following week. He did, accompanied by two friends. When she gave him the contraceptive device he revealed himself as Anthony Comstock, special agent of the Post Office and the Society for the Suppression of Vice, and his friends as _World_ and _Tribune_ reporters. He arrested her, and she was imprisoned in a cell in the Tombs.
In her youth Ann Lohman, alias Madame Restell, had served a year on Blackwell's Island, and the bitter memory remained. She could not face another prison sentence. The night she was released on bail she was found in her marble bathroom with the frescoed ceiling, lying in the crimson water of her bathtub. She had so thoroughly slit her throat with an eight-inch ebony-handled carving knife that she severed the carotid artery and both jugular veins. When Comstock was told of her suicide he wrote in his diary, "A bloody ending to a bloody life."
_A rendering of Fifth Avenue, four years after the death of the abortionist Madame Restell_
As the Victorian era faded, Comstock became an object of ridicule. He was lampooned for his action against the Art Students League's use of nude models and for seeking to ban the Paul Chabas painting of a nude young bather, _September Morning_ , which had received the medal of honor in the Paris Salon. In 1905, he sued George Bernard Shaw to prevent an American production of _Mrs. Warren's Profession_ , and called him "this Irish smut dealer." Shaw in return coined the word "Comstockery" to connote buffoonery in censorship. A cartoon of the day represented Comstock as saying to a judge, "Your Honor, this woman gave birth to a naked child." Comstock's last campaign was against Margaret Sanger, the renowned birth control advocate. He sent her husband, William, to jail for thirty days for distributing her pamphlet "Family Limitation."
At seventy-four Comstock proudly recorded in his diary that he could fill a train sixty-one carriages long, sixty seats per carriage, with the sex-obsessed, immoral people he had brought to justice and that he had destroyed more than 160 tons of obscene material, including such "pornographic" books as _Fanny Hill_ and the _Decameron_. Two years later, on September 21, 1915, Anthony Comstock died. The Comstock laws to keep America "pure" are still on the books and were lately revitalized by Congress in an attempt to police the Internet, but the Supreme Court, in a unanimous decision, found (as had Benjamin Butler one hundred twenty-four years earlier) that they were unconstitutional.
WHEN THE eminent British banker John Biddulph Martin first saw Victoria Woodhull, she was standing on the stage of St. James Hall in London delivering a quasi-religious, quasi-scientific lecture, "The Garden of Eden" (which she held to be the body itself). The moment he laid eyes on her, Martin determined to marry this extraordinarily vital woman, as if her force would invigorate his own pale life. It took six years to convince members of his family that Victoria was respectable. Antecedents—it had come down to that. Victoria invented them with no regard for truth. In a printed pamphlet she traced her family back to King Robert III of Scotland and King James of England. Her daughter, Zulu, was now Zula, and Roxy was now Anna. Her father was "an eminent barrister." Since the United States was far away, she found it easy to claim both Alexander Hamilton and George Washington as ancestors.
In an effort to escape her past, for several years she changed the spelling of her name, calling herself "Woodhall." And in an effort to impress British society with her leadership of the woman's movement, in 1879 she decided, although living in London, to run once again for president of the United States. Her campaign this time consisted mostly of publishing a supplement to the September issue of _The American Traveler_ in London that showed her handsome face over the caption "Victoria C. Woodhall. Candidate for the Presidency of the United States." In the spring of 1892 she made a third mock run for the presidency, but plagued with scandalous revelations about her past, she never attempted another campaign.
By the time Victoria Woodhull married John Biddulph Martin in 1883, she had distanced herself from the woman she had been and abandoned everything that had once made her unique. Henry James (though he later denied it) was said to have written _The Siege of London_ about Woodhull's efforts to launder her past in order to marry an affluent member of British society. His self-aggrandizing heroine, Nancy Headway, cleverly insinuates herself into a rich upper-class British family by convincing them that she is respectable. Nancy Headway says of herself, "I'm burying my past. You can't be delicate when you're trying to save your life.... I have done things I don't understand myself... but I've completely changed and I want to change everything."
In the pursuit of social acceptance Victoria started a conservative publication, _The Humanitarian_ , and wrote, "During no part of my life did I favor Free Love even tacitly.... I regarded it with loathing when once I got a slight idea of its character and the deep infamy to which it led." She avowed that the _Weekly_ had printed its most radical articles while she was "a thousand miles away." Woodhull denounced her former mentor, Stephen Pearl Andrews, declaring that this "high priest of debauchery—actually had the audacity and unblushing effrontery to affix Mrs. Woodhull's signature to his filthy effusions." She asserted that in 1856 Andrews had formed a club, subsequently closed by the police, where he taught free love principles to Elizabeth Phelps and Elizabeth Cady Stanton. Another article began, "Stephen Pearl Andrews! I impeach thee before the judgment bar... archblasphemer!"
Colonel James Blood too was sacrificed; she alleged in _The Humanitarian_ that he had written articles without her knowledge and added that she was "near being the victim of slow poison" administered by him. After their divorce, for a time the impoverished Colonel James Blood had ventured out only at night because his clothes were so shabby he was ashamed to be seen in daylight. Eventually, he secured a clerkship on Governor's Island. After that he ran a bakery shop, a refreshment booth at Coney Island, a carnival show with exhibitions of mesmerism and clairvoyance. It was there that an African seer told him of a gold mine in West Africa. Blood sailed to Accra to find it. One hundred miles into the jungle he died of an unknown fever. The death date given was December 29, 1885, although by the time his body was recovered no one could tell exactly when he had died.
In the Martins' desperate attempt to establish Victoria's credibility, in 1893 they sued the trustees of the British Museum, in the first libel action brought against this institution, for housing two books on the Beecher-Tilton scandal. In a protracted trial, Victoria took the stand and with great hauteur eluded the lawyer's questions by garrulous vagueness. She said that she could not remember if she had ever performed on the stage, and she answered a question about free love by saying, "I never knew that love was anything but free."
The one thing she would not disavow, however, were her spirits. She said that Demosthenes and Bonaparte and Josephine had once been her advisers and announced dramatically, "There is an apparition appearing to me right now!" But she did come close to admitting how her spirits had deserted her. "I have been abandoned by those whom I would never abandon," she said plaintively. After all the melodramatics, the case was decided in favor of the British Museum, but the offending books were withdrawn.
In her quest for respectability, Victoria Woodhull Martin distanced herself from Isabella Beecher Hooker. Isabella wrote Victoria asking her to consult with her deceased mother, Harriet Porter Beecher, who she said possessed great wisdom. Victoria wrote back that she had indeed spoken with Isabella's dead mother and that she had told her that Victoria and Isabella should no longer work together in this world but surely would meet again on the "other side."
Still society shunned her. When they married, John Martin's friends stopped visiting 17 Hyde Park Gate, but the Martins' isolation only drew them closer. Slights became a way of life. Once at an Athletic Club dinner in London to which John Biddulph Martin, as president, was permitted to invite his wife and her sister, the wife of a member, a Mrs. Taylor, declared that these women were not "proper persons to be associated with." When Vickie and Tennie arrived they found themselves alone with Martin in the vast banquet room.
Unlike her sister, Tennessee cared little about what people thought of her and simply by retaining her lighthearted American enthusiasm she fared better than Victoria. In London in 1884 she met the vastly wealthy widower Francis Cook and told him that the spirit of his dead wife said he should marry her. And he did. At Doughty House, his mansion on the Thames, and at his marble castle at Montserrat, in Portugal, Tennessee entertained lavishly and invited everybody from the local gentry to the blacksmith. The British found Lady Cook, the Viscountess of Montserrat, witty and amusing. She lived in quiet security and died in 1923.
Mama Roxy came to live with Tennessee, and Buck moved into Victoria's home at 17 Hyde Park Gate, where he died at age eighty-nine with his daughter at his side. Victoria was convinced that an anonymous letter to the London _Times_ alleging that his death was the result of foul play was from her sister Polly Sparr. The Martins offered a reward if the identity of the writer was revealed. It never was.
_Tennessee Claflin as Lady Cook, the Viscountess Montserrat_
_Victoria Woodhull as Mrs. John Biddulph Martin, a proper English matron_ (Illustration Credit 35.1)
Mrs. Martin was fifty-nine in 1897 when John Martin died, leaving her both his London residence and his country estate, along with a comfortable fortune of 171,779 pounds sterling ($831,410 in American dollars at that time). The dowager Martin spent more and more time at Bredon's Norton, becoming the virtual ruler of this small feudal kingdom beside the river Avon. At the Manor House, which one approached through an arch over which was chiseled the date 1585, she held court in a room arranged for her comfort. Sturdy oak chairs stood beside dark-green velvet sofas with Brussels lace antimacassars and several tables were covered with a profusion of trinkets: bottles, inlaid boxes, and photographs of the Martins with the few important personages they had managed to cultivate. One table was covered with pamphlets containing the history of the estate and Victoria's own bogus genealogy. Another was littered with literature on palmistry and astrology. A niche at the far end of the room was cordoned off with heavy green damask portieres behind which was a shrine to the goddess Nike. Mrs. Martin occasionally confided to visitors that in another life she herself had been Nike.
Victoria was a benevolent despot. She chose the local vicar, paid his salary, and provided a house for him. She established a flower show at Bredon, endowed a local elementary school, repaired rutted roads, and modernized the cottages of the tenant farmers, who for generations had lived without electricity or running water. A villager said that she was generous to the poor in a way that only someone who had been poor herself could be.
All her life Victoria Woodhull Martin had been intoxicated by ideas. Spiritualists, clairvoyants, fortune-tellers, trance speakers, astrologers, palmists, and assorted local gentry, including the Earl of Coventry, were eager to be invited for lunch and to stay for her "international country salon." Finally, Victoria achieved social acceptance of a sort: King Edward VII, when he was still Prince of Wales, visited Bredon's Norton, and on this occasion the dining room was decorated with Victoria's native American flag and the Union Jack intertwined.
After her eightieth year, when her heart began to fail, Mrs. Martin frequently retreated to Brighton for the sea air. No longer able to ride horseback, and her sex life behind her, she was still enamored of speed and insisted that she be driven at a breakneck pace about the countryside in her Aston-Martin. No chauffeur had nerves steady enough to stay long in her employ. Victoria offered a substantial reward to the first man or woman to fly the Atlantic. In May 1927 Charles A. Lindbergh did just that, but by that time all Victoria's energies were concentrated on outwitting death.
To avoid germs she refused to shake hands or be kissed. Visitors were permitted to come no closer than six feet. She ordered her drinking water boiled. Long ago, when she still lived in the United States, she had told Theodore Tilton that she "would rather die than live—such was her infinite estimate of the other world over this." Her fervent belief in the afterlife had assuaged both grief and loss and inspired others. Then the spirits had been her "constant companions." Then she had prophesied, quoting Saint Paul, that, "The last enemy that shall be destroyed is death." Then she had not been afraid. Now she was.
Facing death, she seemed to regret her decision to negate all that she had been. One morning she arose from her bed and in her shaky hand wrote in large letters, "Sitting here today in this north room... dreary, smoky, foggy... I am thinking with all the bitterness of my woman's nature how my life has been warped and twisted out of shape in this environment until, as I catch a glimpse of my haggard face in the mirror opposite, I wonder whether I shall be able to pen the history of my turbulent existence." She never could.
Night after night she sat erect in a straight-backed wooden chair and defied death. At 2 a.m. on the morning of June 10, 1927, death visited her nonetheless. Victoria Woodhull left instructions that she be cremated and her ashes cast into the sea. She willed her fortune to Zulu, who never married, having devoted her life to her mother. Victoria requested that after her daughter's death whatever monies remained be donated to the Society for Psychical Research.
Sixty years earlier in Chicago, Victoria Woodhull had nursed a woman who received "extreme unction from her priest" and was "given up by the physicians." For ten days this gifted spiritual healer had stood over her patient day and night neither sleeping nor eating. At the end of that time the woman recovered and Victoria found that her own body, instead of being weary or exhausted, "was more fresh and bright" than it had ever been before. Her skin wore "an unearthly look of transparency." In that moment Victoria was struck by "the idea that the time will come when the living human body, instead of ending in death by disease and dissolution in the grave, will be gradually refined away until it is entirely sloughed off and the soul only remains."
Perhaps that is true of Victoria Woodhull. She was a woman before her time in a world that was not ready to receive her. Her "soul," however, remains with us. A month before her death, she scrawled what might have been her epitaph:
_Victoria Woodhull, 1874_
The deeper I delve for a sure footing, the higher I reach for light, the more convinced am I that only here and there do we find an instrument capable of responding to the hungry heart's desire for Truth....
On the retina of our brain the outline of Truth is revealed to those attuned to the music of the spheres.... Therefore, I feel well assured that whatever be the misrepresentations to which I may be subject, the events must be committed to time, who relentlessly unravels all distortions and rights all wrongs.... Whoever I am, whatever I have done, belongs to the spirits.
# Abbreviations
_BTS_ | Beecher-Tilton Correspondence and trial and the correspondence of Lib (Elizabeth) Tilton, Theodore Tilton, Francis De Pau Moulton, Mrs. Morse, and Tucker-Sachs.
---|---
_CT_ | _The Official Report of the Trial of Henry Ward Beecher with Notes and References_ , Austin Abbot; _The Great Brooklyn Romance: All the Documents on the Famous Beecher-Tilton Case_ , Orson S. Fowler; and _The Great Scandal: History of the Beecher-Tilton Case_
_ECS & SBA_ | The Papers of Elizabeth Cady Stanton and Susan B. Anthony
_EYM_ | _Eighty Years and More: Reminiscences 1815–1897_ , Elizabeth Cady Stanton
_GAT_ | "Victoria C. Woodhull: A Biographical Sketch: Mr. Tilton's Account of Mrs. Woodhull," Theodore Tilton
_HWB_ | Letters of Henry Ward Beecher
_HWS_ | _History of Woman Suffrage_ , Elizabeth Cady Stanton, Susan B. Anthony, and Matilda Joslyn Gage, eds.
_IBH_ | Isabella Beecher Hooker Project, letters and papers
_MCW_ | Martha Coffin Wright Correspondence
_PCH_ | Transcript of the Plymouth Church Hearing of Henry Ward Beecher
_TS_ | _The Terrible Siren: Victoria Woodhull (1838–1927)_ , Emanie Sachs
_VCWM_ | Victoria Claflin Woodhull (Martin) Papers
_WCW_ | _Woodhull & Claflin's Weekly_
# Notes
## Prologue A Page of History
"bright, glorious, young and strong spirit": Susan B. Anthony to Victoria Woodhull, February 4, 1871, _The Papers of Elizabeth Cady Stanton and Susan B. Anthony_ , The Stanton and Anthony Project, ed. Ann D. Gordon and Patricia G. Holland (New Brunswick, NJ: Rutgers University) 110 reels, microfilm edition, Reel 15:387. Footnote citations taken from material originally at the University of Massachusetts supplied by Kathleen A. McDonough may differ from later _ECS & SBA_, the guide to the microfilm, Ann D. Gordon and Patricia G. Holland, eds., and Kathleen A. McDonough and Gail K. Malmgreen, assoc. eds., _The Papers of Elizabeth Cady Stanton and Susan B. Anthony, Guide and Index_ , Vol. 1 (New Brunswick, NJ: Rutgers University. 110 Reels, microfilm edition, 1997). Whenever possible I have given both citations. Abbreviated as _ECS & SBA_.
"In the annals of emancipation": Elizabeth Cady Stanton interview on the subject of Victoria Woodhull, Tenafly, New Jersey (December 1875), printed in the Newark _Sunday Call_ (January 2, 1876); _ECS & SBA_, Reel 18:510.
"I believe you were raised": Paulina Wright Davis to Victoria Woodhull, Providence, Rhode Island (February 4, 1871), Victoria Woodhull Papers, reprinted in _Woodhull & Claflin's Weekly_ (May 13, 1871).
"Heaven sent": Isabella Beecher Hooker to Anna Savery, Hartford, Connecticut (November 12, 1871), Isabella Beecher Hooker Collection, Project #46 C3–12, the Harriet Beecher Stowe Center Library (formerly the Stowe-Day Foundation), Hartford, CT. "If my political campaign": _Chicago Mail_ (May 6, 1872); _New York World_ (April 16, 1892); Victoria Claflin Woodhull (Martin) Papers, Southern Illinois University Special Collections, Morris Library, Carbondale.
"The statement that Mrs. Biddulph Martin": Lucy Stone statement issued May 11, 1892; reprinted in _Chicago Mail_ (May 13, 1892).
"Knowing your love of _truth_ ": _ECS & SBA_, Reel 30.
"Mrs. Hooker, I am glad" and ff.: Interview between John Martin and Isabella Beecher Hooker at the Sherman House (May 1, 1892), typed May 13, 1892; _VCWM_ , Reel 2: 37–40. "Her dress was peculiar" and ff.: Ibid.
"Tennie and Her Vickie" and ff.: _Chicago Mail_ (May 8, 1892), p. 1.
"It was so long ago": _Chicago Mail_ (May 14, 1892).
"the look of pain and anguish": _VCWM;_ Victoria Woodhull Autobiographical Sketch, London, 1895, p. 171.
## Chapter 1 Born Again in the Lamb's Blood
"a young Greek God": Paxton Hibben, _Henry Ward Beecher: An American Portrait_ (New York: George Doran, 1927), p. 31; reprint, Press of the Readers Club, New York, 1942).
"We are connected": Ibid., p. 32.
"We will love and watch": Ibid., p. 39.
"demonstrating phrenology": Webster's dictionary defines phrenology as a study based on the now outdated theory that mental faculties and dispositions can be judged by observing the shape of the skull as a whole and different parts of its surface. The _Oxford English Dictionary_ defines it as the branch of inquiry that deals with the shape and size of the cranium as supposed indicators of character and mental faculties.
"obliged to make the bed": Milton Rugoff, _The Beechers: An American Family in the Nineteenth Century_ (New York: Harper & Row), pp. 260, 613. Taken from interview with Mrs. Henry Ward Beecher, _Ladies' Home Journal_ (December 1891).
"a respectable woman": _History of Woman Suffrage_ , 2 volumes, ed. Elizabeth Cady Stanton, Susan B. Anthony, and Matilda Joslyn Gage (Salem, NH: Ayer Publishers, 1985), Vol. 1, p. 335. Reprint from an original copy in the State Historical Society of the Wisconsin Library. Originally published by Fowler & Wells, 753 Broadway, 1882; also _Weld-Grimke Papers and Diaries_ , Willard L. Clements Library, University of Michigan.
"lashed them with cowskin": William S. McFeely, _Frederick Douglass_ (New York: W. W Norton, 1991), p. 17.
"disrupt domestic harmony": "The Silent Feminist Revolution: Women and the Law (in New York State from Blackstone to the Beginnings of the American Woman's Rights Movement)," Margareth Rabkin, Ph.D. dissertation, State University of New York, Buffalo, 1975.
"the husband and wife are one": Sir William Blackstone, _The Great Commentaries on the Laws of England_ , 1765–69.
"The royal blood of Germany": Emanie Sachs, _The Terrible Siren: Victoria Woodhull (1838–1929)_ (New York: Harper & Bros., 1928; reprint, New York, Arno Press, 1978), p. 317; _VCWM_ , Victoria Woodhull Autobiographical Sketch, 1895, London, p. 173.
"Nowhere": _Chicago Daily Mail_ (May 9, 1892); also _TS_ , p. 7.
"a real Hummel": _Chicago Daily Mail_ (May 9, 1892).
"the holy mudder in Israel": _TS_ , p. 12; also Johanna Johnston, _Mrs. Satan: The Incredible Saga of Victoria C. Woodhull_ (New York: G. P. Putnam's Sons, 1967; London: Macmillan 1967), p. 17. A medium is a person who acts as a channel of communication between human beings and spirits. It is a passive ability. A psychic is sensitive to nonphysical forces but is in no way controlled by spirits. He or she receives information not available through the five senses, and thus uses what is known as the sixth sense. The words "psychic" and "medium" are not interchangeable.
"Religious love is": William Hepworth Dixon, _Spiritual Wives_ (London: Hurst and Blacken, 1868), Vol. 2, p. 177.
"into a state of frenzy": Ibid., p. 28.
"After his departure": Ibid., p. 30.
"to praise the Lord": Ibid., p. 32.
"smote him on the face": Ibid.
"fiery hail": Ibid.
"Glory": Testimony of Frank Warner at civil trial in Chicago, _John Biddulph Martin v. John Dunlop_ , excerpted in the _Chicago Mail_ (May 9, 1892), p. 1.
"I am born again": Ibid.
## Chapter 2 A Child Without a Childhood
"Squalid vice": Stanley Weintraub, _Victoria: An Intimate Biography_ (New York: Truman Talley Books, E. P. Dutton, 1987), p. 117. Excerpted from the journal of Queen Victoria (December 30, 1838).
"tormented and harried her children": Theodore Tilton, "Victoria C. Woodhull: A Biographical Sketch: Mr. Tilton's Account of Mrs. Woodhull," _Golden Age_ (New York: 1874), tract 3, p. 6.
"She was worked like a slave" and ff.: _GAT_ , p. 7. Of Buck Claflin's brutality, Tilton wrote:
In a barrel of rain-water he kept a number of braided green withes made of willow or walnut twigs, and with these stinging weapons, never with an ordinary whip, he would cut the quivering flesh of the children till their tears and blood melted him into mercy. Sometimes he took a handsaw or a stick of firewood as the instrument of his savagery. Coming home after the children were in bed, on learning of some offence which they had committed, he has been known to waken them out of sleep, and to whip them till morning.
"He named this process animal magnetism": Mesmer wrote, "There exists a mutual influence between celestial bodies, the earth, and animated bodies.... This influence exhibits, particularly in the human body, properties analogous to those of a magnet. One can distinguish diverse and opposing poles which can be changed, diminished, and reinforced.... The property in an animal body which renders it susceptible to this influence... I have decided to call animal magnetism." Robert C. Fuller, _Mesmerism and the American Cure of Souls_ (Philadelphia: University of Pennsylvania Press, 1982), pp. 4, 5. "magnetized": Ibid., p. 10.
"The Marquis de Puységur": Full name is Amand Marie Jacques de Puységur.
"other eyes": _VCWM_ , Reel 2, 1890, articles on the "Invisible World."
"the power of the soul": Fuller, _Mesmerism_ , p. 11.
"sanctified": _GAT_ , p. 5; also _VCWM_ , Reel 2, undated.
"skills demonstrated by mesmerized subjects": Modern psychiatry (with the exception of the theories of Carl Jung) has repudiated or ignored this evidence of supernormal powers, providing explanations of suggestibility and the control exercised by the mesmerist. "The sexual implications were obvious": A Boston mesmerist confessed that he had "telepathically" induced several young ladies to thrust their affections upon him. Ralph Waldo Emerson cringed at the idea that someone "should attempt to put me asleep by the concentration of his will without my leave" (Fuller, _Mesmerism_ , p. 34), and Nathaniel Hawthorne warned his fiancée, who wanted to use mesmerism to cure her headaches, "My spirit is moved to talk to thee today about these magnetic miracles, and to beseech thee to take no part in them.... The sacredness of an individual is violated by it" (Ibid., p. 35).
"He succeeded in almost throwing me": Rugoff, _The Beechers_ , p. 267.
"it would require": _Samuel F. B. Morse, His Letters and Journals_ , Volumes 1 and 2, ed. Edward Lind Morse (Boston: Houghton Mifflin and New York: Riverside Press, 1914), p. 72.
"in Heaven": _The Banner of Light_ (Boston, November 20, 1870).
"Catharine Beecher, the eldest child": Isabella often spells her sister's name "Catherine," but others spell it "Catharine," as she herself does in her books and writing.
"To American Mothers": Catharine E. Beecher, _A Treatise on Domestic Economy, For The Use of Young Ladies At Home, And At School_ (Boston: Marsh, Capen, Lyon, and Webb, 1841).
"The physical and domestic education": Ibid., p. 38.
"her first spiritual vision": The details and circumstances were to change in Victoria Woodhull's written accounts of this incident—"I was four," "five," "a neighbor died," "a woman died," "Rachel Scribner, my nurse, died"—but the vision remained the same.
"felt herself gliding": _GAT_ , p. 11; _VCWM_ , Reel 3, p. 11.
"I saw the spirits descending": _GAT_ , p. 11; _Boston Post_ (October 20, 1876); _VCWM_ , Reel 3.
"a child without a childhood": _GAT_ , p. 4.
"a household drudge": Ibid., p. 7.
"She has been lifted": Ibid., p. 10.
"I often performed": Chapter in the unfinished autobiography of Victoria Woodhull, _VCWM_ , Reel 1.
"ride around in their own carriages": Testimony of Roxanna Claflin in _Roxanna Claflin v. Col. James Blood_ , May 11–May 21, 1871. Also, quoted in many newspapers, ex. _The World, Herald_ , New York Public Library Scrapbook III.
"A red silk handkerchief": Chapter in the unfinished autobiography of Victoria Woodhull, _VCWM_ , Reel 1.
"I saw not a booted foot": Ibid.
"pale white hands": _Boston Post_ (October 20, 1876), reprinted in _The Banner of Light_ (November 20, 1876).
"So shall sinners fall in Hell": Ibid.
"I am the Word" and ff.: Ibid.
"bleeding feet and haggard face": _GAT_ , p. 6.
"There is a red glow" and ff.: _TS_ , p. 14.
## Chapter 3 The Spiritual Telegraph
"Follow me" and ff: _TS_ , p. 19.
"Now, do as I do" and ff.: Ibid.
"I did not think": Ruth Brandon, _The Spiritualists: The Passion for the Occult in the Nineteenth and Twentieth Centuries_ (New York: Knopf, 1983), pp. 2, 3. E. W. Capron and H. D. Barron, _Spiritualism_ , pp. 15, 16.
"I should love much": Frederick Douglass to Amy Post (April 11, 1848), Amy and Isaac Post Family Papers, University of Rochester Library, Department of Rare Books and Special Collections; Ann Braude, _Radical Spirits: Spiritualism and Women's Rights in Nineteenth-Century America_ (Boston: Beacon, 1989), p. 11.
"moved by the spirit": Ida Husted Harper, _Life and Work of Susan B. Anthony_ (Indianapolis: Bowen-Merrill, 1898), Vol. 1, p. 40. Taken from letters of Daniel B. Anthony. "six-penny farm": Ibid., p. 48.
"Of what use is preaching": Katharine Anthony, _Susan B. Anthony: Her Personal History and Her Era_ (Garden City, NY: Doubleday, 1954), p. 92. _ECS & SBA_, taken from letter, Daniel B. Anthony to Dear S. (Susan B. Anthony, 1849).
"the deep throbbing of": Braude, p. 14. Much of this account of spiritual phenomena is taken from Emma Hardinge, _Modern American Spiritualism: A Twenty Years' Record of the Communication Between Earth and the World of the Spirits_ , 1869, and from Albert Cronise, _Beginnings of Modern Spiritualism in and Near Rochester_ (Rochester, NY: Rochester Historical Society, 1925).
"automatic writing": There are numerous accounts of automatic writing in the letters and diaries of Spiritualists, including Isaac and Amy Post, Isabella Beecher Hooker, Paulina Wright Davis. Isaac Post, _Voices from the Spirit World, Being Communications from Many Spirits, by the hand of Isaac Post, Medium_ (Rochester, NY: C. H. McDonell, 1852), p. 258. Andrew Jackson Davis also explains this phenomenon in _The Principles of Nature, Her Divine Revelations and a Voice to Mankind_. George Lawton, "The Drama of Life After Death: A Study of the Spiritualist Religion," _Dictionary of American Biography_ (New York: Henry Holt and Company, 1932), p. 500.
"the God principles at work": Braude, p. 119.
"an unbroken chain" and ff.: Ibid., p. 13. Taken from Waterloo Friends, _Proceedings_ , pp. 9, 20.
"Spiritual Telegraph": Hal D. Sears, _The Sex Radicals: Free Love in High Victorian America_ (Lawrence, KS: Regent Press of Kansas, 1977), p. 13; Braude, p. 5.
"passed over": The common term used by Spiritualists to refer to those who had died and were now, they believed, residents of heaven or, as they called it, Summerland.
"Instead of the social existence" and ff.: Sears, p. 10.
"the Spirit World is derived" and ff.: Andrew Jackson Davis, _The Philosophy of Spiritual Intercourse_ (New York: Fowler & Wells, 1851), p. 413.
"Charles Fourier": Born François Marie Charles Fourier but commonly called Charles Fourier.
"a common system of movement": Fuller, p. 5; Robert Darnton, _Mesmerism and the End of the Enlightenment in France_ (Cambridge, MA: Harvard University Press, 1968), p. 143.
"incessant labor": Sears, p. 16. A complete account of this incident appears in John M. Spear, _Boston New Era_ (June 29, 1854).
"living principle": Sears, p. 18.
"a crisis": Ibid.
"at precisely the time designated": Ibid.
"lofty electrical position": Ibid., p. 19.
"tore out the heart": Ibid.
"an awful foreboding": Letters of Louise Chandler Moulton, Library of Congress, Manuscript Division.
"in the depth of the night": John S. Hart, _The Female Prose Writers of America_ , 1857, p. 533.
"rather than mourn": A typical spirit message read, "Gentle mother, your little Seraph boy is not _dead_ but _liveth_. In his uncontaminated love, find comfort for the ills of life." See Braude, p. 40.
"Not in Church": Ibid., p. 24; Andrew Jackson Davis, _Philosophy of Spiritual Intercourse_ , p. 96.
"are not gone": Braude, p. 55.
"In the funeral oration": Warren Felt Evans, _The New Age and Its Messenger_ (Boston, 1864), p. 25.
## Chapter 4 My Long-Accumulating Discontent
"poured out... the torrent of": Elisabeth Griffith, _In Her Own Right: The Life of Elizabeth Cady Stanton_ (New York: Oxford University Press, 1984), p. 51; Alma Lutz, _Created Equal: A Biography of Elizabeth Cady Stanton, 1815–1902_ (New York: John Day, 1940), P. 48.
"convention to discuss": _ECS & SBA_, 6:708. Call to Woman's Rights Convention at Seneca Falls, NY, published in the _Seneca County Courier_ , July 14, 1848. Authors: Elizabeth Cady Stanton, Lucretia Coffin Mott, Martha Coffin Pelham Wright, Mary Ann Wilson McClintock, and Jane C. Hunt.
"felt as helpless and": Griffith, p. 52; Lutz, _Created Equal_ , p. 45; Mari Jo Buhle and Paul Buhle, eds., _The Concise History of Woman Suffrage: Selections from the Classic Work of Stanton, Anthony, Gage and Harper_ (Urbana: University of Illinois Press, 1978), p. 92.
"We hold these truths": Elizabeth Cady Stanton, _Eighty Years and More: Reminiscences 1815–1897_ , new introduction by Gail Parker (New York: Schocken Books, 1971), p. 149. Reprinted from the T. Fisher Unwin edition of 1898; Buhle and Buhle, p. 94.
"the first organized protest": _EYM_ , p. 149.
"mental hunger": _Elizabeth Cady Stanton, as Revealed in Her Letters, Diary and Reminiscences_ , 2 volumes, Theodore Stanton and Harriot Stanton Blatch, editors (New York: Arno & The New York Times, 1969), Vol. 1, p. 144. Originally published by Harper & Bros., NY, 1922.
"spiritlike": Christian Frederich Samuel Hahnemann, _Organon of Medicine_ , translated by J. Kunzl et al. (London: Vic Gollancz, 1983), p. 269; Stanton and Blatch, p. 114.
"got up, bathed myself": Elizabeth Cady Stanton to Elizabeth Smith Miller, Seneca Falls, New York (February 10, 11, 13, 1851); _ECS & SBA_, 7:036–038; Stanton and Blatch, Vol. 2, pp. 26, 27.
"We know what not": Lutz, _Created Equal_ , p. 160, taken from _The Revolution_ , January 22, 1868.
"As an amusement": Ibid., p. 58, taken from the article "Women," January 1, 1850.
"silent chamber of death": _EYM_ , p. 20.
"As he took no notice of me": Ibid.
"I resolved... to be at the head": Ibid., p. 21.
"prefers boys": Ibid.
"But I suffer not": Timothy 2:12; Lutz, _Created Equal_ , p. 5.
"I soon noticed that": _EYM_ , p. 48.
"I left him": Ibid., p. 49.
"To think that all in me": Griffith, p. 9.
"streak of fire": Constance Mayfield Rourke, _Trumpets of Jubilee: Henry Ward Beecher, Harriet Beecher Stowe, Lyman Beecher, Horace Greeley, P. T. Barnum_ (New York: Harcourt, Brace, 1927), p. 46.
"I can see him now": _EYM_ , p. 42.
"a monster of iniquity": Ibid., p. 41; Griffith, p. 21.
"puzzling and harrowing": Griffith, p. 21.
"I cannot understand what": _EYM_ , p. 42.
"Repent and believe" and ff.: Ibid., p. 43; Griffith, p. 21.
"the nature of the delusion": _EYM_ , p. 47.
"the physical conditions": Ibid., p. 48.
"slowly through a beautiful grove": Ibid., p. 60.
"dreaded the influence of Mr. Bayard": Griffith, p. 31.
"in an unguarded moment" and ff.: Elizabeth Cady Stanton to Isabella Beecher Hooker, Tenafly, New Jersey (January 21, 1873); _ECS & SBA_, 16:968–973.
"RESOLVED—That it is the duty": "From the First Convention Ever Called to Discuss the Civil and Political Rights of Women," Seneca Falls, New York (July 19, 20, 1848), Woman's Rights Convention, _ECS & SBA_, 6:719.
"Oh, Lizzie! If thou": Lutz, _Created Equal_ , p. 46.
"He stood there like an African Prince": Ibid., p. 39.
"The ballot": Ibid., p. 47.
"Our doctrine is that": July 28, 1848, _The North Star;_ Philip S. Foner, _The Life and Writings of Frederick Douglass_ , 5 volumes (New York: International Publishers, 1955), Vol. 1: Early Years, p. 321.
"foolish conduct": Griffith, p. 59.
"make the puddings and": _ECS & SBA_, microfilm edition, Reel 8.
"Oh dear, dear! If the spirits": Susan B. Anthony to Elizabeth Cady Stanton (June 5, 1856); _ECS & SBA_, Reel 8:443; _Elizabeth Cady Stanton, Her Letters, Diary and Reminiscences_ , pp. 65–66. This is an example of where Anthony's exact words were altered. Here it reads "right thing" in place of "rights," which appears in the original letter of June 5.
"Gerrit Smith's daughter Elizabeth": Elizabeth Smith came for a visit. She was called Libby by her parents, but Lizzie by many of her friends. Elizabeth Smith married Charles Dudley Miller on October 18, 1843, at the age of twenty-one. Libby Smith would introduce "Bloomers." In the winter of 1850–1851 Elizabeth Smith Miller visited Elizabeth Cady Stanton and appeared on the streets of Seneca Falls wearing this costume. _Notable American Women 1607–1950: A Biographical Dictionary_ , 3 volumes, ed. Edward T. James (Cambridge, MA: Belknap Press of Harvard University Press, 1971), Vol. 1, p. 180.
"As to the rapping heard": George Willets to Isaac Post, October 23, 1848; Amy and Isaac Post Family Papers.
## Chapter 5 You Ugly Creature
"But where is the best fruit?" and ff.: _TS_ , p. 16. Excerpted from _The Spiritualist_ (November 4, 1870).
"The Wonderful Child": Sachs, p. 30.
"in my own home": _GAT_ , p. 10.
"a woman before her time": Ibid., p. 14.
"All that I am": Martha Coffin Wright to Susan B. Anthony (April 6, 1871) _MCW;_ from _ECS & SBA_, Reel 15:537–540.
"a majestic guardian": Rourke, p. 201.
"she would rise to": _GAT_ , p. 11.
"I would talk to them": Ibid.; also in _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876).
"gracious guests": _GAT_ , p. 8; also in _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876).
"My little puss": _GAT_ , p. 13.
"not unwilling to be rid": Ibid., p. 14.
"My marriage was an escape": Ibid.
" 'New Age' Philosophy": Madeleine B. Stern, _The Pantarch: A Biography of Stephen Pearl Andrews_ , p. 5. Stephen Pearl Andrews wrote, "I see now a new age beginning to appear." "more troublesome than a crown": Sears, pp. 3, 4. Stern, excerpted from Josiah Warren, _Practical Applications of the Elementary Principles of "True Civilization,_ " p. 21.
"the antithesis of enslaved love": Stephen Pearl Andrews, "Fragments," _New York Daily Tribune_ (November 8, 1858), p. 3. Repeated in _Love, Marriage and Divorce and The Sovereignty of the Individual_ , (A Discussion by) Henry James, Horace Greeley, and Stephen Pearl Andrews (New York: Stringer &Townsend, 1853).
"All sexual relations": Debate with Stephen Pearl Andrews, _New York Daily Tribune_ (November 10, 1853).
"adultery": _New York Tribune_ (November 10, 1858), repeated in _Love, Marriage and Divorce_ , 1853.
"convinced beyond the shadow": Braude, p. 16. Excerpted from Hardinge, 1869, p. 71.
"Take your hands": William Harlan Hale, _Horace Greeley: Voice of the People_ (New York: Harper & Bros., 1950), p. 123.
" 'dream boy' Pickie were soon communicating": Henry Luther Stoddard, _Horace Greeley: Printer, Editor, Crusader_ (New York: G. P. Putnam's Sons, 1946), pp. 108, 119. Greeley recorded a typical spirit conversation with his dead son.
Horace Greeley: "Pickie, tell the story of that last day that you were well—how you came up to my room where I was writing and kept trying to pull my pen from me—how about that?"
Answer by rapping: "You put me out" (as was the fact...)
Mary Greeley: "I do not see why you should bring up that disagreeable, melancholy reminiscence."
Pickie: "No, mother it is _not_ melancholy. I _disturbed_ father."
Horace Greeley descriptions of "spirit rapping" seance and other rappings, New York Public Library, Manuscript Division.
"I am very lonely": Catherine Fox to John Fox (October 26, 1850), Catherine Fox to Amy Post (June 19, 1850), Catherine Fox to Amy Post (October, November 1850), Amy and Isaac Post Family Papers. Excerpted from Braude, p. 16.
"crazy for learning": Hale, p. 32. This description is attributed to Margaret Fuller.
"My wife is in bad health": Ibid., p. 34; documents in Horace Greeley Papers, New York Public Library; Horace Greeley Letters to Margaret Fuller, Harvard University Library.
"because her fire would not burn": Ibid.
"Mary is terribly ill": Ibid.
"Mrs. Greeley is in her usual bad health": Ibid.
"Mrs. Greeley's life is": Ibid., p. 120.
"Castle Doleful": Stoddard, p. 121.
"bad magnetism": Horace Greeley to Margaret Fuller (January 27, 1848), ibid., p. 114; Hale, p. 120.
"Castle Rackrent": Hale, p. 112.
"Catharine Beecher's _Treatise on Domestic Economy_ ": Chapter "On Washing," pp. 308–319; Chapter "On Whitening, Cleansing And Dyeing," pp. 327–336.
"a woolen ironing-blanket": Ibid., p. 324.
"inability to accomplish anything": Hale, p. 183.
"dirty food": Stoddard, p. 110.
"Your sore, sick father" and ff.: Horace Greeley, Papers; also in Hale, p. 116; Glyndon G. Van Deusen, _Horace Greeley, Nineteenth-Century Crusader_ (Philadelphia: University of Pennsylvania Press, 1953), p. 153.
"while Mother often said": Hale, p. 119.
"Pickie regarded her": Ibid.
"I never saw a creature": Horace Greeley to Margaret Fuller (July 29, 1847), Harvard University Library; Stoddard, p. 111.
"A week after the funeral": Ibid.
"I had a pretty little sister": Pickie (dictated letter to his father, Horace Greeley) to be sent to Margaret Fuller, New York (August 14, 1849), Harvard University Library; Stoddard, p. 122.
"When I came home": Horace Greeley to Margaret Fuller (July 29, 1847), Harvard University Library; Hale, p. 119; Stoddard, p. 111.
"He is governed and restricted": Horace Greeley to Margaret Fuller (June 27, 1848), Harvard University Library; Stoddard, p. 116.
"the piano was sweetly played": Catherine Fox to Amy Post (November 1850), Amy and Isaac Post Family Papers; also Braude, p. 17.
"ring bells and move tables": Catherine Fox to Amy Post (October 30, 1850), Amy and Isaac Post Family Papers; also Braude, p. 17.
"How I hate her!": Catherine Fox to John Fox (October 26, 1850); Braude, p. 16.
## Chapter 6 True Wife
"walking unsteadily across": _VCWM_ , Reel 1. Reprinted in _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876).
"Did you marry that child": _GAT_ , p. 14.
"mostly with his cups": Ibid.
"half-drunken husband": Ibid., p. 15.
"icicles clinging to her bed post": Ibid.
"the iron door": Ibid., p. 13.
"wrestling with God": Ibid., p. 15.
"true wife": Ibid., p. 16.
"the harlot to pack her trunk": Ibid.
"Go West, young men, and grow up with the country!": Rourke, p. 241, excerpted from _New York Daily Tribune_.
"Tied down the safety valve": Edwin P. Hoyt, _The Vanderbilts and Their Fortunes_ (New York: Doubleday and Co., 1962), p. 119.
"cigar girl": _GAT_ , p. 17.
"have somebody who can rough it": Ibid.
"It is no use": Ibid.
"I am meant for": _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876).
"Victoria, come home": _TS_ , p. 25.
"Come home!": Ibid.
"thrown into such vivid": Ibid.
"send the spirits": Ibid.
"A WONDERFUL CHILD!": Ibid., p. 30.
"Miss Tennessee's Magnetic Life Elixir": Ibid., pp. 31–32.
"It was a hard life": Testimony of Tennessee Claflin in _Roxanna Claflin v. Colonel James H. Blood_ (May 15, 1871), Essex Market Police Court, New York City.
"Blue Book": Several such books are available in the Manuscript Division of the New York Public Library and in the collection of the American Psychical Society of New York. "through the power of her mind": A phrase used in such books as _Mind Cure in New England: From the Civil War to World War I_ , by Gail Thain Parker (Hanover NH: University Press of New England, 1973) and Braude's _Radical Spirits_.
"Now we will sit up": Margaret Sanborn, _Mark Twain: The Bachelor Years: A Biography_ (New York: Doubleday, 1990), pp. 380–381 ff.
"I've done considerable": Mark Twain, _Huckleberry Finn_ (Cleveland, OH: World Publishing Co., reprint 1947), p. 169.
"thousands of women": Victoria Woodhull, _Tried as By Fire_ , reprinted in _The Victoria Woodhull Reader_ , pp. 6, 7, 8 ff.
"twisting and braiding": We have only Victoria's word that these phenomena took place. However, a new, twelve-year study at Princeton University headed by Professor Robert Jahn finds conclusively that such things occur. Tests on hundreds provided evidence of psychokinesis and the ability to transmit messages through the power of thought.
"No. I will not permit": _GAT_ , p. 21; also _TS_ , p. 34.
"servant girls": _Indianapolis Herald_ , undated, preprinted in _Chicago Mail_ (May 8, 1892), courtesy Chicago History Works, Oak Park, Illinois.
"disorderly conduct": Ibid.
"Zulu Maud": One cannot know if Victoria named her daughter because at birth the baby resembled a member of that African tribe or if Victoria just liked the name. (There were so many unusual names in her family.) In all Woodhull's early writings and in documents, the child is referred to by her birth name, Zulu. However, after Victoria moved to England in 1877, for a time Zulu's name was changed to Zula.
"staggering up the steps": _GAT_ , p. 26.
"Why should I any longer live": Ibid., p. 21.
## Chapter 7 Willfully Did Kill
"women in the fields": Katharine Anthony, p. 159.
"I cannot meet my Maker": Henry Bowen's statement, Plymouth Church Committee, December 1875, Beecher-Tilton trial, Scrapbook IV, Beecher-Tilton Correspondence, Lib Tilton Correspondence, Theodore Tilton Correspondence, Francis De Pau Moulton legal and personal Correspondence, Mrs. Morse Correspondence, newspaper articles, clippings, pamphlets, church and civil trial-related material, etc., from the offices of the _Independent_. Tucker-Sachs Correspondence, Scrapbooks I—VIII, New York Public Library, Manuscript Collection, New York.
"a failure": Hibben, p. 167.
"soul-food": A term used by Lib (Elizabeth) Tilton to Theodore Tilton (hereafter abbreviated as ET to TT), January 4, 1870, _BTS_ II.
"I sometimes feel": Henry Bowen to Theodore Tilton, June 16, 1863. Reprinted in the _Brooklyn Eagle_ , January 21, 1875, _BTS_ IV.
"cowardice and treason": Giraud Chester, _Embattled Maiden. The Life of Anna Dickinson_ (New York: G. P. Putnam's Sons, 1951), p. 29.
"At Antietam in Sharpsburg": The battle of Antietam took place September 17, 1862. James M. McPherson, _Battle Cry of Freedom: The Civil War Era_ (New York: Oxford University Press), 1988, p. 541.
"a human being cannot": Excerpt from the _London Spectator_ , as cited in Kenneth M. Stampp, _The Era of Reconstruction, 1865–1877_ (New York: Knopf, 1978, Vintage, 1965 reprint edition), p. 44.
"the most brutal mob": Elizabeth Cady Stanton to Mrs. Gerrit Smith, Johnstown, New York (July 20, 1863); _ECS & SBA_, Reel 10: 543–544; Stanton and Blatch, Vol. 2, p. 94. The house was at 75 W. 45th Street.
"Here's one of those": Ibid.
"Let's go in, fellows!": Ibid., p. 544; Stanton and Blatch, pp. 94–95.
"servants and the children": Ibid., pp. 543–544; Stanton and Blatch, p. 95.
"Colored Orphan Asylum": Ibid.; Stanton and Blatch, p. 94.
"three years of age": Mattie Griffith to Mary Estlin (July 27, 1863); Eric Foner, _Reconstruction: America's Unfinished Revolution, 1863–1877_ (New York: Harper & Row, 1988), p. 33, footnote 58; excerpted from Estlin Papers, Dr. Williams' Library, London.
"weak" and ff.: _EYM_ , p. 42; Henry B. Stanton, _Random Recollections_ (Johnstown, NY: Blunck & Leaning, Printers, 1885), pp. 22, 23.
"For years before": _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876); also in _Banner of Light, Original Essays, VCWM_ , dated Boston, Saturday, November 20 (year obscured).
"When Lincoln was on his way": Ibid.
"American King of Cancers": Johnston, p. 32. Buck Claflin also claimed he could cure "all kinds of chronic diseases, fever sores, bone diseases, scrofula, piles, sore eyes in the worst stages, heart and liver complaints, female weaknesses, constipation, inflammatory rheumatism, asthma, neuralgia, sick headache, dropsy in the chest, and fits in various forms."
"cult of love": _Ottawa Free Trader_ (April 4, 1863), courtesy of Chicago History Works, "hobbled in on crutches": _The Romance of Plymouth Church, Plymouth Church and Its Pastor, or Henry Ward Beecher and His Accusers_ , compiled by J.E.P. Doyle (Hartford, CT: Park Publishing, 1874), p. 440.
"There are only three cures": _VCWM_ , Reel 3.
"Mrs. Rebecca Howe, recovering": _Ottawa Free Trader_ (June 1, 1864).
"Miss Claflin is": Rebecca Howe letter printed in the _Ottawa Republican_ (June 4, 1864); Sachs, p. 35.
"On the first day of November": Indictment issued June 1864, Ottawa, Illinois, La Salle County Courthouse, courtesy of Chicago History Works.
## Chapter 8 God Bless This Trinity
"I could never be": ET to TT (April 1, 1864).
"My friends, fellow citizens": _Brooklyn Eagle_ (April 20, 1865), p. 5.
"brother soldiers": Ibid.; also in N. D. Hillis, _Lectures and Orations of Henry Ward Beecher_ (New York: AMS Press, 1913, reprint 1970).
"Christ can save you": Leon Oliver, _The Great Sensation_ (Chicago: Beverly Co., 1973), p. 210.
"What is orthodoxy?": Doyle, p. 413.
"I cannot think, only feel": ET to TT (February 7, 1869).
"Cool, fragrantly airy": _Brooklyn Eagle_ (November 6, 1858), p. 8. Excerpted from pamphlet _Old Brooklyn Heights, The Brooklyn Savings Bank_ , 1827–1927.
"most ideal woman": Theodore Tilton to Lib (Elizabeth) Tilton (hereafter abbreviated as TT to ET (January 14, 1867), _BTS_.
"fulfill his every need": Statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 28, 1874) transcript, New York Public Library, New York Historical Association.
"a home": ET to TT (January 16, 1865), _BTS_.
"kind of ecstasy": ET to TT (December 28, 1866), _BTS_.
"Beecher's Bibles": Hibben, pp. 134, 148.
"to be sold by" and ff.: Hibben, p. 136.
"dazzled" and ff.: Statement of Theodore Tilton, Plymouth Church Hearing of Henry Ward Beecher (August 1, 1874).
"His mind was opening": Charles E Marshall, _The True History of the Brooklyn Scandal_ (Philadelphia: National Publishing Co., 1874), p. 256.
"At the birth of": Ibid., pp. 189–190; also in statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 23, 1874).
"I had a very severe and prolonged sickness": Ibid. In a statement, Lib (Elizabeth) Tilton calls the doctor "Doctor Putnam."
"beautiful Mattie": Statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 28, 1974).
"cold" and ff.: _Theodore Tilton v. Henry Ward Beecher_ , civil trial, court testimony of Francis D. P. Moulton (February 11, 1875), New York Public Library Scrapbook VI; Marshall, p. 448.
"scolded and chided": Statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher Quly 31, 1874); Marshall, p. 189.
"He spent a great deal": Lib (Elizabeth) Tilton Testimony, _PCH_ , New York Public Library Scrapbook V; Altina L. Waller, _Reverend Beecher and Mrs. Tilton: Sex and Class in Victorian America_ (Amherst: University of Massachusetts Press, 1982), p. 48; Marshall, p. 190. "manias and frenzies": Statement of Theodore Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 21, 1874); Marshall, p. 129; _Brooklyn Argus_ (July 23, 1874), p. 1, New York Public Library Scrapbook V.
"make a name for himself": Statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 31, 1874); New York Public Library Scrapbook V; Marshall, p. 193.
"And into whatsoever house": Marshall, p. 182.
"luxurious carelessness; the wallpaper was": _Brooklyn Eagle_ (August 1, 1874).
"Alone I can do": Marshall, p. 191; statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 23, 1874).
"small woman, without presence": _Theodore Tilton v. Henry Ward Beecher_ , civil trial, court testimony of Francis D. P. Moulton (February 11, 18, 1875); New York Public Library Scrapbook VI.
"he had married her": Ibid.
"Here is Theodore": Ibid.
"going back to my own house": Hibben, p. 179.
"hell on earth": Robert Shaplen, _Free Love and Heavenly Sinners: The Story of the Great Henry Ward Beecher Scandal_ (New York: Alfred A. Knopf, 1954), p. 38. Originally published in _The New Yorker_ (June 5 and 12, 1954).
"the vainest, the most vapid": Hibben, p. 179; also Shaplen, p. 37.
"childlike in appearance": Marshall, p. 257. Statement of Henry Ward Beecher, Plymouth Church Hearing of Henry Ward Beecher, New York Public Library Scrapbook V. "O Theodore, God might": Hibben, p. 178.
"I was almost in despair" and ff.: Statement of Henry Ward Beecher, Plymouth Church Hearing of Henry Ward Beecher, New York Public Library Scrapbook V; Hibben, p. 182.
"People have said that I": Ibid.
"tell whether a speech": Marshall, p. 159.
"I never felt a bit": Ibid., p. 197; statement of Lib (Elizabeth) Tilton, Plymouth Church Hearing of Henry Ward Beecher (July 28, 1874), _BTS_ , Scrapbook V.
"I took my first walk": ET to TT (January 2, 1865), _BTS_.
"I am glad Mr. Beecher called": TT to ET (January 13, 1865), Cleveland, Ohio, _BTS_.
"a blessed Trinity": ET to TT (January 25, 1867), _BTS_.
"Tomorrow the whole nation": _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876).
## Chapter 9 An Accident of Fate
"I had thought that states rights": Thursday, February 16, 1871, Lincoln Hall, Washington, before presenting the minority report on the Woodhull memorial. Buhle and Buhle, Vol. 1, p. 293.
"Reconstruction": Eric Foner, p. 5.
"soft peace": Stampp, p. 50.
"to unify the nation": Several historians posit that Benjamin Butler sought this Union Party nomination himself and therein lay the root of his antagonism toward Andrew Johnson.
"yet to be ratified": Indeed, Mississippi did not ratify this amendment until March 1995, 130 years later.
"temporary arrangements for the Freed People": Stampp, p. 46.
"Ile à Vache": (December 31, 1862); James M. McPherson, _The Negro's Civil War: How American Negroes Felt and Acted during the War for the Union_ (Urbana and Chicago: University of Illinois Press, originally Pantheon Books, 1965), p. 96.
"The real trouble is": Stampp, p. 102.
"murderous and wasteful war": Ibid., p. 101.
"wretched treatment of black people": _Independent_ (September 14, 1865, September 23, 1865), specifically attributed to Theodore Tilton in a sermon by Henry Ward Beecher, October 22, 1865, and in Theodore Tilton's letter to Lib Tilton, October 25, 1865.
"while the defeated South": _Independent_ (September 23, 1865), p. 2.
"And may I not be": Henry Ward Beecher sermon (October 8, 1865), reprinted in the _Independent_ (October 2, 1865).
"A Great Political Sermon": _New York Times_ (October 24, 1865); also in the _New York Tribune_ (October 26, 1865). Beecher wrote a letter to the _Tribune_ stating that the sentence quoted from his sermon, "The laws and interests of the government and of ourselves will prove to no avail if they are hostile and unpleasant to the _white_ people of the south," did not contain the word "hostile" and that he was "inaccurately reported." Therefore, I have removed "hostile" from the quote, but the sense is essentially the same.
"There is no such kindness" and ff.: _Independent_ , byline Theodore Tilton (October 27, 1865).
"remote": Ibid.
"On the contrary" and ff.: Ibid.
"I have not seen Mr. Beecher": TT to ET (October 25, 1865), _BTS_.
"My friend, from my boyhood": Theodore Tilton to Henry Ward Beecher, Brooklyn (November 30, 1865).
"To demonstrate that there were": Theodore Tilton statement, _CT_.
"Woman's Influence in Politics": _Independent_ (February 16, 1860).
"Well, old girl": Harper, Vol. 1, p. 234.
"re-examination of the Declaration": _Independent_ (February 23, 1865).
"When Andrew Johnson began": _ECS & SBA_, Elizabeth Cady Stanton to Martha Coffin Wright (January 6, 1866); also cited in Stanton and Blatch, Vol. 2, p. 112.
"both classes of disenfranchised citizens?": Elizabeth Cady Stanton to Wendell Phillips, in the _National Antislavery Standard_ (December 26, 1865).
"I have about made up my mind": _HWS_ , Vol. 2, p. 168; also cited in Stanton and Blatch, Vol. 2, p. 106.
"While I could continue arguing": Wendell Phillips to Elizabeth Cady Stanton (February 18, 1866), _HWS_ , Vol. 2, p. 152.
"We have fairly boosted the negro": Elizabeth Cady Stanton to Martha Coffin Wright (December 20, 1865), _ECS & SBA_, 1865–66; also cited in Stanton and Blatch, Vol. 2, p. 108. Gloria Steinem points out that until this time black and white women were allied in their fight for woman suffrage. Stanton's demand for enfranchisement based on the superior social position and education of white women adversely affected black women and their rights. Although limitations of space prohibit my dealing with this important issue, Steinem has written extensively on this subject.
"It mattered very little": Lib (Elizabeth) Tilton's statement to the Plymouth Church Committee, _PCH_ , 1874; Marshall, p. 192.
"sick-bed" and ff.: Ibid., p. 197.
"I would give $500": Ibid., p. 198.
"You seldom meet a woman": Elizabeth Cady Stanton to Sara Jane Lippincott (May 30, 1873), _ECS & SBA_, 1873.
"a changed woman": ET to TT (February 7, 1869), _BTS_.
"dead to rights": Deposition of James Kerr reprinted in _Chicago Mail_ (May 14, 1892), p. 2, courtesy of Chicago History Works.
"Not me. I'm just": Ibid.
"the most notorious woman": Ibid.
"Negro squatters were everywhere" and ff.: Elizabeth Avery Meriwether, _Recollections of 92 Years, 1824–1916_ (Nashville: The Tennessee Historical Commission, 1958), p. 167.
"Blood saw Missouri": Missouri was ripped apart over the question of slavery. In Missouri there were 1, 162 battles and skirmishes; only Virginia and Tennessee had more.
"on the spot": _GAT_ , p. 24.
"wonderful cures": Handbill of Victoria Woodhull, _VCWM_ , Box 2; also in _GAT_ , p. 19.
"a debt of $3,700": _New York Times_ (May 11, 1871) stated that the debt was "in excess of $5,000."
"Father, at times a Mephistopheles": _GAT_ , p. 23.
"should earn all the money": Ibid., p. 7.
"Tennie has had ten men": Joseph Treat, M.D., _Beecher, Tilton, Woodhull, The Creation of Society: All Four of Them Exposed, and if Possible Reformed and Forgiven, in Dr. Treat's Celebrated Letter to Victoria C. Woodhull_ (New York: Published by the author, 1874), p. 9.
"My God, have I got": Ibid.
"I was almost lost": Tennessee Claflin testimony at the trial _Roxanna Claflin v. Colonel James Harvey Blood_ , Essex Market Police Court (May 15, 1871); _BTS_ , Scrapbook II; _World_ (May 17, 1871).
## Chapter 10 Draw Its Fangs
"This species of legislation": Eric Foner, p. 245.
"the rising young star of America": _Argus_ (April 18, 1866); _BTS_ , Scrapbook IV; _Independent_ (February 29, 1866). Tilton's suggestion was endorsed by William Lloyd Garrison in editorials in the _Independent_ the following month. In aligning himself with the Radical Republicans, Tilton wrote, "I shall endeavor to make and keep the _Independent_ a member of Congress during the winter." Theodore Tilton to Thaddeus Stevens, Papers of Thaddeus Stevens, Library of Congress.
"I have argued constantly": Elizabeth Cady Stanton to Susan B. Anthony (August 11, 1865), _EYM_ , Vol. 2, p. 105; _ECS & SBA_, 1865.
"even if it meant": Stampp, p. 116.
"to take control of the city": _Mass Violence in America: Memphis Riots and Massacres_ , Facsimile Edition, House Report No. 101, U.S. 39th Congress, First Session (New York: Arno Press, 1969), p. 20.
"spontaneous": Ibid.
"One good result of the Memphis Riot": Meriwether, p. 91.
"They are the natural terriers": _Independent_ (May 4, 1865).
"Mrs. President, this convention is" and ff.: _HWS_ , Vol. 2, p. 155.
"According to the programme": Ibid., p. 154.
"You may... ask me" and ff.: Ibid., pp. 161, 162.
"There is neither Jew nor Greek" and ff.: Ibid.
"In the midst of" and ff.: Ibid., p. 167.
"I remember that": Ibid.
"praying for the enfranchisement": Harper, p. 323.
"I would rather cut off": Katharine Anthony, p. 193.
"What does ail Susan?" and ff.: Ibid.
"Oh, Susan" and ff.: Ibid.
"the most passionate affection": Ibid., p. 176.
"almost every day": Ibid., p. 177.
"her vim, her energy": Sanborn, p. 318.
"the possession of her soul": Katharine Anthony, p. 199.
"I must buy butter": Elizabeth Cady Stanton to Elizabeth Smith Miller (October 22, 1866); _ECS & SBA_, 1866; Stanton and Blatch, p. 115; Theodore Tilton, _Elizabeth Cady Stanton: Eminent Woman of the Age_ (Hartford, CT: S. M. Batts & Co., 1869), p. 354; Griffith, p. 126.
"I only _scold now_ ": Susan B. Anthony to Antoinette Brown Blackwell (September 4, 1858), Griffith, p. 96.
"the people": Stampp, p. 114.
"much excited": Eric Foner, p. 276.
"the people of the South": Ibid.
"I say Thaddeus Stevens": Stampp, p. 133. Hans L. Trefousse, _Andrew Johnson: A Biography_ (New York: W. W. Norton, 1989), p. 244. The Trefousse quote differs: "I say Thaddeus Stevens of Pennsylvania; I say Charles Sumner of Massachusetts; I say Wendell Phillips of Massachusetts."
"vulgar, vindictive and": Stampp, p. 114.
"The people have been witness": _Independent_ (September 13, 1866); Trefousse, p. 266.
"If they [the Negroes] have": Henry Ward Beecher, _Patriotic Addresses_ , p. 741; Rugoff, _The Beechers_ , p. 395. The most detailed account of the so-called "Cleveland letter" affair appears in _Beecher to Special Committee of the Soldiers'and Sailors' Convention_ (August 30, 1866), reprinted in _Theodore Tilton v. Henry Ward Beecher_ (McDivitt Campbell and Company, 1875), pp. 476, 477, New York Public Library, Tucker-Sachs Collection.
"unqualified endorsement": _New York Times_ (September 3, 1866).
"He has done more harm": _Independent_ (September 6, 1866).
"During the first three": _Independent_ (October 25, 1866); Hibben, p. 174.
"the worst candidate possible": Ibid.
"The spectacle at the": Ibid.
"full pardon": Louis M. Starr, _Bohemian Brigade: Civil War Newsmen in Action_ (New York: Knopf, 1954), p. 316.
"the weeping bosom of his family": Hibben, p. 168.
## Chapter 11 A Dangerous Man
"with which to save": Philip S. Foner, _The Life and Writings of Frederick Douglass_ , Vol. 4, pp. 21–22.
"in a manner which indicated" and ff.: Ibid., pp. 25, 241.
"The President no more expected... those damned sons of bitches" and ff.: Philip Ripley, private secretary to President Johnson, to Manton Marble (February 8, 1866), Marble Papers, Library of Congress; Trefousse, _Andrew Johnson_ , p. 242.
"but not under such conditions": Buhle and Buhle, Vol. 2, p. 397.
"Gentlemen, with all respect" and ff.: _Life and Times of Frederick Douglass, Written by Himself_ , ed. Rayford W. Logan (New York: Collier Books, 1962), pp. 397–399; ibid., Vol. pp. 330, 331.
"Philadelphia Full of Miscegenists" and ff.: Stampp, p. 116; Frederick Douglass, _Life and Times of Frederick Douglass_ (New York: Citadel Press, 1983), p. 386. Facsimile Edition.
"ashamed and afraid": Ibid., p. 397; Buhle and Buhle, Vol. 2, p. 331.
"I was the ugly": Ibid.
"He came to me in": Douglass, ibid., p. 398; Buhle and Buhle, ibid.
"A good many people here": Thaddeus Stevens to Dr. Kelley (May 5, 1870), Papers of Thaddeus Stevens, Stampp, p. 26.
"Miss Dickinson, Mr. Tilton and": Douglass, _Life and Times_ , p. 405.
"By law December 18, 1866": In the South many former slaves knew nothing of the Thirteenth Amendment (submitted to Congress February 1, 1865, and ratified December 18, 1865), and their former masters kept them ignorant for a year, until the eve of December 18, 1866.
"I never thought that Abraham": TT to ET, Springfield, Illinois (December 20, 1866), _BTS_.
"The more I think of": TT to ET, Akron, Ohio (December 2, 1866), _BTS_.
" 'vile women' ": The concept of a "vile woman" used here and in the following chapters is explained in a letter, Isabella Beecher Hooker to Mary Livermore (March 15, 1871), _IBH_. Such a woman, commonly a prostitute or one of loose sexual behavior, was considered both seductive and dangerous. Examples of the day were contained in frequent sermons citing how Samson was shorn of his hair and destroyed by the "vile woman" Delilah, and in the accepted wisdom of the day that prostitutes who came to a bad end deserved their fate.
"Men and women who": TT to ET, on the cars, northern Indiana (December 7, 1866), _BTS_.
"rein and curb": Henry Ward Beecher, _Norwood: or Village Life in New England_ (New York: J. B. Ford, 1868), p. 307.
"If love was proper": Second statement of Frank Moulton, civil trial, 1875, _BTS, CT_.
"I believe that I have": TT to ET (December 31, 1866), _Chicago Tribune_ , New York Public Library Scrapbook IV.
"totally and thoroughly" and ff.: TT to ET, Altoona, Pennsylvania (November 26, 1866), _BTS_.
"falling into sin" and ff.: Lib (Elizabeth) Tilton statement to Plymouth Church Committee, 1875; also partially quoted in Marshall, p. 189.
"I live in profound wonder": ET to TT (January 11, 1867), _BTS_.
"No temptation or fascination": Ibid. (February 3, 1868).
"I am afraid!": Ibid. (February 20, 1867).
"I think any man": TT to ET (December 12, 1866), _BTS_.
"her sensuality" and ff.: Lib (Elizabeth) Tilton statement to Plymouth Church Committee, 1874.
"When Theodore said" and ff.: Ibid.
"I tried to give him a full accounting" and ff.: Ibid.
"Now that the _other_ man": TT to ET (December 27, 1866), _BTS_.
"My espoused saint" and ff.: Ibid. (December 6, 1866).
"During the early part": ET to TT (January 14, 1867), _BTS_.
"My Own True Mate": ET to TT (December 28, 1866); _BTS_. In Rugoff, this letter appears in slightly different form. I chose the version from the Frank Moulton exhibit submitted to the courts in 1874.
"I like Mr. Beecher": TT to ET (December 30, 1866), Akron, Ohio, _BTS_. (Misdated in New York Public Library Scrapbook VI as December 2, 1866.)
## Chapter 12 Written in Fire
"This is a critical" and ff.: Elizabeth Cady Stanton to Martha Coffin Wright (June 27, 1867), Martha Coffin Wright correspondence, Sophia Smith Collection, Smith College. Abbreviated as _MCW;_ Stanton and Blatch, p. 116; Lutz, pp. 140, 141; _HWS_ , Vol. 2, p. 284.
"The ballot and the bullet": Another version of this appears in Harper, where it is stated as: "Certainly, Mr. Greeley, just as you fought in the late war—at the point of a goose quill." However, in the original letter Elizabeth Cady Stanton sent to Martha Coffin Wright (June 27, 1867) in Albany, New York, she writes, "Yes, we are ready to fight, sir, just as you did in the late war, by sending our substitutes." I used Stanton's own version.
"Mr. Chairman, I hold": Lutz, _Created Equal_ , p. 141.
"Your committee does not": Ibid., p. 142.
"Prepare for a storm" and ff.: Elizabeth Cady Stanton to Emily Howland, New York (September 1, 1867), _ECS & SBA;_ Stanton and Blatch, Vol. 2, pp. 116–118. The version that appears in Vol. 2 is abridged, and the text altered. I have used a longer and more detailed version.
"I saw the reporters" and ff.: Stanton and Blatch, Vol. 2, p. 117. In Lutz, p. 142, the conversation is as follows:
"You are so tenacious about your own name. Why did you not inscribe my wife's maiden, Mary Cheney Greeley, on the petition?"
"Because I wanted all the world to know that it was the wife of Horace Greeley who protested against her husband's report."
"Well, I understand the animus": _HWS_ , Vol. 2, p. 287.
"I can not sign": Ibid., p. 317; also quoted in _The Elizabeth Cady Stanton-Susan B. Anthony Reader: Correspondence, Writings, Speeches_ , ed. Ellen Carol DuBois (Boston: Northeastern University Press), 1981, p. 119.
"Would Horace Greeley": _ECS & SBA_ (May 10, 1867), Reel 12:181.
"I am not alone": Elizabeth Cady Stanton to Susan B. Anthony, New York (August 22, 1864), _ECS & SBA;_ Stanton and Blatch, Vol. 2, p. 100.
"Man has been molding": Address of Elizabeth Cady Stanton, AERA Convention (May 15, 1869), _ECS & SBA_.
"I wish you were a boy!": _EYM_ , p. 23.
"This is the country": Elizabeth Cady Stanton to Henry Stanton from Kansas. In Lutz, _Created Equal_ , p. 150. The letter reads "contracted" Eastern existence, but on microfilm I read the word as "constricted" and have used it accordingly.
"low-down nigger men" and ff.: Lutz, _Created Equal_ , p. 151; Alice Stone Blackwell, _Lucy Stone, Pioneer of Woman's Rights_ (Boston: Little, Brown, 1930), p. 212.
"a charlatan" and ff.: Griffith, p. 130.
"mortified and astonished beyond measure": Ibid.; Katharine Anthony, p. 217.
"Not one leading politician": Susan B. Anthony to Anna Dickinson, _ECS & SBA_ Reel 12.
"All the old friends": _ECS & SBA_ (January 1, 1868), Reel 12; Harper, p. 295.
"Not lack of brains" and ff: Lutz, _Created Equal_ , p. 153.
"Mr. Train... has some" and ff.: Elizabeth Cady Stanton to Martha Coffin Wright (January 8, 1868), _MCW;_ repeated in Stanton and Blatch, Vol. 2, pp. 119–120; _The Revolution_ (January 29, 1868); Griffith, p. 130.
"The old religious teachings": TT to ET (February 12, 1867), _BTS_.
"I have no ambition": TT to ET (December 2, 1866), _BTS_.
"I was struck with a little": TT to ET, Alton, Illinois (December 23, 1866), _BTS_.
"The _Banner of Light_ , optimistically estimated": _Banner of Light_ (January 18, 1868), New York Public Library, Manuscript Collection.
"How long have you" and ff.: Testimony of Colonel James H. Blood, _Roxanna Claflin v. James Blood_ , Essex Market Police Court (May 15, 1871).
"By inspiration": _GAT_ , p. 18.
"temporary sojourn": Ibid., p. 12.
"Your work is about to begin" and ff.: _Victoria Woodhull Prophesy, Boston Post_ (October 20, 1876), fragmentary autobiography, Woodhull Reel 1, _VCWM_.
"Who are you?" and ff: Ibid.
## Chapter 13 To Equal Account
"(now called Park Avenue)": Park Avenue was the name given in 1860 to Fourth Avenue between Thirty-fourth and Thirty-sixth Streets, but by 1868 the avenue was referred to as Park Avenue all the way up to Forty-second Street.
"Office Female Physician" and ff.: Clifford Browder, _The Wickedest Woman in New York: Madame Restell, The Abortionist_ (Hamden, CT: Archon Books, 1988), pp. 117, 118.
"The Revolution": The following year Stanton and Anthony came under such criticism from the Sorosis Society and other groups housed in the Women's Bureau that they moved _The Revolution_ to offices on Park Place.
"fit for an Empress's crown": _The Revolution_ (December 31, 1868).
"Oh, you've come about" and ff.: _VCWM_ , Box 1; Johnston, p. 42; from Woodhull's account, _Boston Post_ (May 18, 1872).
"What do I care": Barbara Goldsmith, _Little Gloria... Happy at Last_ (New York: Knopf, 1980), p. 77, from Allen Churchill, _The Splendor Seekers_ (New York: Grosset & Dunlap, 1974), P. 82.
"Female physicians": Allan Keller, _Scandalous Lady: The Life and Times of Madame Restell, New York's Most Notorious Abortionist_ (New York: Atheneum Encore Editions, 1981), p. 11; Browder, 1988, p. 73; James C. Mohr, _Abortion in America: The Origins and Evolution of National Policy, 1800–1900_ (New York: Oxford, 1978), p. 197, illustrates these personal advertisements.
"quackery and charlatanry" and ff.: G. J. Barker-Benfield, _The Horrors of the Half-Known Life: Male Attitudes Toward Women and Sexuality in 19th-Century America_ (New York: Harper & Row), 1976, p. 28; Mohr, pp. 92–96.
"heroic medicine": James Harvey Young, _The Medical Messiahs: A Social History of Health Quackery in Twentieth-Century America_ (Princeton, NJ: Princeton University Press, 1967), p. 18, provides many sources for this and a specific description. Barker-Benfield, pp. 61–69, provides a concise explanation, titled "Midwives to Gynecologists," on how medical doctors tried to drive women from the practice of female medicine and childbirth.
"rest cure": This cure, identified with Dr. S. Weir Mitchell, is explained in Barker-Benfield, p. 130.
"recycled and taught" and ff.: Ibid., taken from the Reverend John Todd, "A Sermon Before the American Board of Commissioners for Foreign Missions," in Pittsburgh, Pennsylvania, on October 5, 1869 (Boston: T. R. Marvin & Son, 1869), pp. 259–267, 277–279.
"Allied to the sexual organs": Barker-Benfield, p. 277, taken from "Our Children" by Augustus Kinsley Gardner, p. 20.
"over-exciting the sexual appetites": Barker-Benfield, p. 124.
"in close proximity to": Ibid., pp. 132, 270. A full explanation of the unnatural and contaminated nature of women's sexuality appears on pp. 229–307.
"as pretty as the dimple": Ibid., p. 132.
"According to the talk" and ff.: _The Diary of George Templeton Strong: Post-War Years 1835–1875_ , ed. Allan Nevins and Milton Halsey Thomas (New York: MacMillan, 1952), Vol. 2, p. 42.
"The fashionable _demi-monde":_ George Ellington, _The Women of New York, or the Underworld of the Great City_ (New York: New York Book Company, 1869), p. 243. Reprinted by Arno Press, New York, 1971.
"There are many popular fallacies": "Tried as by Fire, or The True and The False, Socially. An Oration Delivered by Victoria C. Woodhull in All the Principal Cities and Towns of the Country During an Engagement of One Hundred and Fifty Consecutive Nights To Audiences Together Numbering A Quarter of a Million of People" in _The Victoria Woodhull Reader_ , ed. Madeleine B. Stern (Weston, MA: M & S Press, 1974), Part 5: Sociology, p. 31.
"$3.00 to $10.00 a week": _WCW_ (February 11, 1871).
"We are told that prostitution" and ff.: "Tried as by Fire; or, The True and The False, Socially." This speech, delivered on 150 consecutive nights, was first presented at the National Spiritualist Association convention in Boston September 1872. Stern, Part 5: Sociology, pp. 22, 23.
_A Gentleman's Guide_ is available at the New York Historical Society Library (by appointment).
## Chapter 14 Consult the Spirits
"The old man is bound": The Vanderbilt Will Case Clippings, November 1877, New York Public Library, Room 315.
"I would never cross": Wayne Andrews, _The Vanderbilt Legend_ (New York: Harcourt, Brace and Co., 1941), p. 8; Edwin P. Hoyt, _The Vanderbilts and Their Fortunes_ , p. 177.
"lived nearly seventy-four years": _New York Tribune_ (August 24, 1868).
"When I have to give up smoking": Wayne Andrews, p. 143.
"old boy" and ff.: Hoyt, p. 187.
"a thousand" and ff.: Treat, p. 9.
"Phebe Hand Van der Bilt": Although Victoria Woodhull was in touch with the commodore's mother, she had died fourteen years earlier, on January 22, 1854.
"After all, they're not Vanderbilts": Wayne Andrews, p. 30.
"Who the devil": Van Deusen, p. 423.
"If this printing press": W. A. Swanberg, _Jim Fisk: The Career of an Improbable Rascal_ (New York: Charles Scribner's Sons, 1959), p. 41.
"going astray": _New York Herald_ (November 25, 1871); Swanberg, p. 256; Ellington, pp. 37, 38.
"I want more": _New York Herald_ (January 9, 1872), p. 1, in an account of the death of Jim Fisk.
"she obtained not only Fisk's money": _VCWM_ , Box 3; _Argus_ interview with Victoria Woodhull (December 18, 1872).
"It's bound to go up" and ff.: Commodore Vanderbilt to Susan A. King. Commodore Vanderbilt's remark, "Do as I do. Consult the spirits," was the seed from which this book grew. In the sections dealing with Commodore Vanderbilt I used material from my own _Little Gloria... Happy at Last_ , which in turn was based on many sources, including biographies of Commodore Vanderbilt by Wayne Andrews, A. Croffut, and Edwin P. Hoyt and the court testimony of the so-called "Great Vanderbilt Will Contest," 1877.
"great luxury" and ff.: TT to ET, Newport, Rhode Island (August 23, 1866), _BTS_.
"I was not a happy woman": _Notable American Women 1607–1950_ , Vol. 1, p. 444.
"Winslow's Soothing Syrup" is called "Dr. Winslow's Soothing Syrup" in Griffith, p. 133.
"Spiritual Wife": a full explanation of this philosophy appears in Dixon's _Spiritual Wives_.
"a great friendship": ET to TT (February 24, 1868), _BTS_.
"It is like the breath": Henry Ward Beecher, _Norwood_ , p. 40.
"It would seem": Ibid., p. 171.
"My endeavor was entirely": Lib (Elizabeth) Tilton to Henry Ward Beecher; Doyle, p. 524; New York Public Library Scrapbook VII.
"I am hungry to see your children" and ff.: ET to TT (January 13, 1867), _BTS;_ statement of Lib (Elizabeth) Tilton, _PCH_.
"You will be amused": ET to TT (March 13, 1868), _BTS_.
"a terrible wrestling" and ff.: TT to ET, Empire House, Akron, Ohio (January 10, 1870), _BTS_.
"a paroxysmal kiss": statement of Henry Ward Beecher, 1875, _CT_.
"I regard my": TT to ET, Crawfordsville, Indiana (February 9, 1868), _BTS_.
"Theodore, do you _know_ ": ET to TT (February 8, 1868), _BTS_.
"Innocence or guilt": TT to ET, on the cars from Independence to Cedar Rapids, Iowa (April 4, 1868), _BTS_.
"My brain has run wild": TT to ET, At Your Desk (April 6, 1868), _BTS_.
"Your last note reminded me": TT to ET, Washington (March 26, 1868), _BTS_.
"I have less faith": Ibid. (March 21, 1868).
"The country is going to the devil": This abbreviated account of the impeachment trial of President Andrew Johnson was drawn from many sources, including articles in the _Independent_ , March 23, April 4 and 15, May 2 and 10, 1868; Eric Foner, pp. 333–337; Trefousse, _Andrew Johnson_ , pp. 311–334; The Papers of Benjamin F. Butler, Library of Congress, Manuscript Division.
"Filled with sunshine": Harper, p. 200.
"Do you believe that" and ff.: Cross-examination testimony of Theodore Tilton civil trial, 1875, _CT_.
"A Day Memorable": October 10, 1868, diary entry (entered into evidence _CT_ , 1875) quoted in many newspapers.
"The man who has been wallowing": Henry Ward Beecher's sermon, December 20, 1868; Rugoff, _The Beechers_ , p. 476.
## Chapter 15 We Are Ready. We Are Prepared.
"Twenty-year-old Hester Vaughn": The case involving Hester Vaughn began on Friday, December 25, 1868. My account is drawn from articles in _The Revolution_ over five days: January 21, 1869, January 28, 1869, February 18, 1869, May 12, 1869, November 12, 1869. All are from the Sophia Smith collection, Smith College. See also Kathleen Barry, _Susan B. Anthony: A Biography of a Singular Feminist_ (New York: New York University Press, 1988), pp. 216–217; Lutz, _Created Equal_ , pp. 162–163; Ellen Carol DuBois, _Feminism and Suffrage: The Emergence of an Independent Women's Movement in America 1848–1869_ (Ithaca, NY: Cornell University Press, 1978), pp. 145–147, 179; _New York Tribune_ and _New York Herald_.
In all contemporary accounts, including that of Anna Dickinson herself, the name of the doctor involved was Dr. Susan A. Smith. However, in Harper and in several other later accounts the doctor is referred to as Dr. Lozier. In the _ECS & SBA_ microfilm index, the spelling of Vaughn is Vaughan. However, in all other accounts, including court records, the name is spelled as I have spelled it.
"the Queen of the Lyceum": Chester, p. 86.
"so I suppose thee'll": Ibid., p. 113.
"splendor on": Louis M. Starr, _Bohemian Brigade: Civil War Newsmen in Action_ (New York: Knopf, 1954), p. 140.
"impending marriage to": Whitelaw Reid to Susan Dickinson (January 21, 1870), Papers of Anna E. Dickinson, Library of Congress, Microfilm Edition, 14:72.
"her love," "her darling," and "her chick a dee dee": these as well as other expressions of affection occur in such letters as Susan B. Anthony to Anna Dickinson, _ECS & SBA_, Reel 12 (July 12, 1867; February 15, 1868).
"My soul goes out": Susan B. Anthony to Anna Dickinson (January 15, 1868), _ECS & SBA_.
"I get two dollars" and ff.: Alma Lutz, _Susan B. Anthony: Rebel, Crusader, Humanitarian_ (Boston: Beacon Press, 1959); Barry, pp. 211, 212; account in _The Revolution_ by Elizabeth Cady Stanton (January 8, 1868), pp. 1–12, differs slightly from Barry's source. Stanton wrote,
These are the prices paid to the lower classes who do the world's real work: for heavy cloth pantaloons, lined, finished and pressed shopwork, eighteen to twenty cents a pair; for linen coats with three pockets and six buttonholes, a dollar a dozen, eight cents each; for shirts, best quality, a dollar fifty a dozen; for shirts, second quality, retailing at two dollars each, a dollar twenty-five a dozen; for shirts, third quality, seventy-five cents a dozen; for fancy flannel shirts, lapel or breast, turned over collar, cuffs, gussets, buttonholes, six cents each. For jumpers (blue overshirts) ending at waist in a band with long sleeves, five cents a dozen.
These "Women are starving today": _The Revolution_ , April 22, 1869.
These "The male element is": _The Revolution_ (June 3, 1869); _HWS_ , pp. 334–335. Stanton believed that women could not rely on the support of men. She wrote, "We must not trust any of you. All these men... have pushed us aside for years.... We are right in our present position. We demand in the reconstruction, suffrage for all the citizens of the Republic."
"We are now in the midst": Eleanor Flexner, _Century of Struggle: The Woman's Rights Movement in the United States_ (Cambridge, MA: Belknap Press of Harvard University Press, 1959), p. 150; Lucy Stone to Olympia Brown (January 6, 1868).
"Omit my name": _The Revolution_ (May 1, 1869; May 8, 1869).
"I have written to": Mary Livermore to Susan B. Anthony (April 4, 1869); Harper, Vol. 1, p. 321; _HWS_ , Vol. 2, p. 396.
"generous, sweet atmosphere will prevail": _HWS_ , Vol. 2, p. 380. The rest of this account of the meeting is on pp. 380–392, 397–401. Elizabeth Cady Stanton's speech is on pp. 348–356; Frederick Douglass's speech appears on pp. 382, 383; Lucy Stone's speech appears on pp. 384–389. These scenes also appear in Harper, Vol. 1, pp. 323–325 and Lutz, _Created Equal_ , pp. 162–165. Stephen Foster was a farmer and is not to be confused with the famous composer of the same name, who died in 1864.
"The Republican party to-day congratulates": Stanton was primarily directing her protest against _male_ domination rather than berating the Negro. Her sympathy for Negro women had been evident when she wrote, "The Negro's skin and the woman's sex are both [used as] _prima facie_ evidence that they were intended to be in subjection to the white Saxon man." The battle to enfranchise black males before giving the vote to women hurt black women who had been included in the woman suffrage movement. It was at this point that Elizabeth Cady Stanton began arguing for the enfranchisement of "women of wealth and education," i.e., white women.
"This howl comes from": _HWS_ , Vol 2, p. 390.
"I am unwilling": Ibid., p. 389.
"the Hutchinson family": For a short biography of this abolitionist family see _HWS_ , Vol. 2, p. 398. Roseville, New Jersey, referred to as the residence of Abby Hutchinson Patton and her husband, is now part of Newark, New Jersey.
"grand and secret love" and ff.: Andrea Moore Kerr, _Lucy Stone: Speaking Out for Equality_ (New Brunswick, NJ: Rutgers University Press, 1992), p. 144.
"I can say nothing" and ff.: Elizabeth Blackwell to Emily Blackwell (August 20, 1869), ibid.
"Mrs. P." and ff.: Emily Blackwell to Elizabeth Blackwell (September 14, 1869), ibid.
"butterfly streak": George Washington Blackwell to Elizabeth Blackwell (February 18, 1870), ibid., p. 149.
"I feel crushed": Lucy Stone to Antoinette Brown Blackwell (October 31, 1869), ibid., p. 145.
"The American Equal Rights Association": Harper, Vol. 1, p. 328; Barry, p. 150; Andrea Moore Kerr, p. 141.
"Having been drilled" and ff.: DuBois, _Feminism and Suffrage_ , p. 191.
## Chapter 16 Soup for Three
"I have not authorized": This account of the attempt to corner the gold market is drawn from House Report no. 31, 41st Congress, Second Session, 1870, The Investigation into the Causes of the Gold Panic, Library of Congress. See also Swanberg, pp. 134–142; Maury Klein, _The Life and Legend of Jay Gould_ (Baltimore: Johns Hopkins University Press, 1986), pp. 99–116. Victoria Woodhull's appearance and involvement is detailed in the _New York Sun_ (October 1, 1869) and _TS_ , pp. 49, 50.
"a ladies' drawing room": _New York Herald_ (February 13, 1870).
"Simply to Thy Cross I Cling": _TS_ , p. 49.
"Madame de Ford... profit" and ff.: _WCW_ (February 3, 1873).
"All gentlemen will state their business": _Ibid._ , p. 52.
"If I'd a married her" and ff.: Goldsmith, _Little Gloria_ , p. 79, from Wayne Andrews, p. 145.
"Just wait until we" and ff.: _The Revolution_ (March 10, 1870).
"If you wear that": _VWCM, New York Sun_ (October 8, 1870).
"We assumed a gentleman" and ff.: _VWCM_ , Box 3; also, _TS_ , p. 63.
## Chapter 17 A Hard Place
"with their sorrows": _ECS & SBA_ (June 27, 1870); Stanton and Blatch, p. 127.
"Innocents Abroad": Published in July 1869.
"measure up" and ff.: Isabella Beecher Hooker to John Hooker (June 27, 1852), _IBH_. The sources for this section on Isabella Beecher Hooker and other vital material were initially found in Jeanne Boydston, Mary Kelly, and Anne Margolis, _The Limits of Sisterhood: The Beecher Sisters on Women's Rights and Woman's Sphere_ (Chapel Hill: University of North Carolina Press, 1988), pp. 80–111, which led me to the Harriet Beecher Stowe Center (formerly the Stowe-Day Foundation), in Hartford, Connecticut, and to the Isabella Beecher Hooker Project, containing heretofore unpublished letters and diaries of Isabella Beecher Hooker. Isabella Beecher Hooker is referred to as Belle, or as Bell, by family members; usually Belle is used.
"I do regret that": Isabella Beecher Hooker to John Hooker (August 30, 1859), _IBH_.
"I should enjoy reading": Isabella Beecher Hooker to John Hooker (February 21, 1847), _IBH_.
"At sixteen and a half": Isabella Beecher Hooker to Rachel Burton (January 25, 1859), _IBH_.
"Oh my _soul_ " and ff.: Isabella Beecher Hooker to John Hooker (January 24, 1860), _IBH_.
"gently and persuasively": Journal entry, Isabella Beecher Hooker (March 17, 1852), _IBH_.
"She was frightened" and ff.: Ibid.
"If absolute power": Isabella Beecher Hooker to John Hooker (January 11, 1849), _IBH_.
"I would give an individual": Mrs. John Hooker, _Shall Women Vote? A Matrimonial Dialogue_ (February 18, 1860), unpublished, _IBH_.
"A Mother's Letters to a Daughter": _Putnam's Magazine_ , November–December 1868, _IBH_.
"Mrs. Stanton... is a noble woman": Isabella Beecher Hooker to Caroline M. Severance (August 27, 1869), _IBH;_ Lutz, _Created Equal_ , p. 182; Harper, p. 332.
"The True Story of Lady Byron's Life": _Atlantic Monthly_ (October 1869); Rugoff, _The Beechers_ , p. 526.
"revolting, obscene garbage": Rugoff, _The Beechers_ , p. 526; _Independent_ (August 26, 1869).
"worthy of a place" and ff.: Elizabeth Cady Stanton to Isabella Beecher Hooker (September 23, 1869), _ECS & SBA_, Reel 13.
"I did my best": (October 29, 1869), _ECS & SBA_, Reel 14; Lutz, _Created Equal_ , pp. 182–183; Harper, Vol. 1, pp. 332, 325.
"injuriously and foolishly keeps" and ff.: Emily Blackwell to Elizabeth Blackwell (October 11, 1869), Andrea Moore Kerr, p. 145.
"underhanded": Boydston et al., p. 215.
"I _hope_ that you": Lucy Stone to Elizabeth Cady Stanton (October 19, 1869), Andrea Moore Kerr, pp. 145–146.
"so dreadful an incubus": Lucy Stone to Antoinette Brown Blackwell (October 31, 1869), Andrea Moore Kerr, p. 146.
"hypocrites" and ff.: _The Revolution_ (October 28, 1869); Andrea Moore Kerr, p. 146; Susan B. Anthony quoted in _New York Tribune_ (November 25, 1869).
"So help me, Heaven!": _ECS & SBA_, Reel 14; Lutz, _Created Equal_ , p. 184; Lutz, _Susan B. Anthony_ , p. 172; Katharine Anthony, p. 238.
"rashness for courage": _New York Tribune_ (November 25, 1869), _ECS & SBA_, Reel 14; Griffith, p. 139.
"At 5:30 in the afternoon" and ff.: This account of McFarland's assassination of Richardson is drawn from many sources. George Cooper, _Lost Love: A True Story of Passion, Murder, and Justice in Old New York_ (New York: Pantheon Books, 1994) is a study in depth, but I have favored the court records of Daniel McFarland's trial as well as contemporaneous accounts appearing in the _New York Times_ , the _New York Herald, The Revolution_ , the _Argus_ , the _Brooklyn Eagle_ , etc., and most notably the _New York Times_ of November 26, 27, 1869, December 6, 13, 1869, and May 7 (triple sheet), 11, 12, 13, 14, 15, 1870. This case was emblematic of the wildly divergent views of the marriage and divorce laws of the day and called into question rigid Victorian mores. Starr's _Bohemian Brigade_ provided material on Richardson's career and the background. Harper, Vol. 1, pp. 351–355, covers the Apollo Hall protest.
"He would lock himself": Trial affidavit of Abby Sage McFarland.
"Lucia P. Calhoun": Calhoun was self-supporting and had a keen awareness that other members of her sex needed to earn money in a respectable way. At the end of the war, she began writing political articles for the _New York Tribune_ along with notes on the glamorous activities of society. She was one of Hester Vaughn's most eloquent defenders.
"I am in need of a surgeon" and ff.: _New York Tribune, New York Sun_ , and _New York Times_ (November 26, 27, 28, 29, 30, 1869).
"Do you think I will live?" and ff.: _New York Times_ (November 27, 1869), p. 2.
"Henry Ward Beecher marry him": _New York Times_ (May 7, 1870), p. 2. When Beecher spoke at Richardson's funeral on December 22, criticism had already begun to mount. Beecher defended Richardson by comparing him to "a lion who in his strength and fastness is able to defend himself, but no sooner has the cruel arrow of the huntsman laid him low than he is set upon by every wild thing, every fly, every crawling worm."
"Consider, married men of New York!": _New York Tribune_ (November 26, 1869). That the tide turned against Beecher was evident in the mounting criticism. Reverend J. M. Pullman of the Sixth Universalist Church on Thirty-fifth Street expressed the morality of the day when he wrote,
Richardson the meddler was a social outlaw. If he got hurt in going where he should not, people, although they would not uphold the man who fired the shot, would agree that a man who cared for his home and believed in its purity should be allowed the natural justice of a revenging blow. It was that instinct that justified it. ( _New York Times_ , December 6, 1869, p. 8)
"I took every": _New York Times, New York Herald_ (December 11, 1869); Marshall, pp. 561, 562.
"Once in a while" and ff.: Chester, p. 118.
"Marriage today is": Lutz, _Created Equal_ , p. 190.
"Loose women" and ff.: _Woman's Journal_ , 1870; Andrea Moore Kerr, pp. 156, 157.
"It is not good": Lucy Stone to Emily Blackwell (April 13, 1870), Andrea Moore Kerr, p. 151.
"has come to look upon" and ff.: Isabella Beecher Hooker to Susan Howard (January 2, 1870), _IBH_.
"Susie, you are in": Ibid.
"From what little I have": John Hooker to Isabella Beecher Hooker (January 5, 1870), _IBH;_ Boydston et al., p. 293.
## Chapter 18 The Evangel
"wondrous dome a flood" and ff.: _New York Sun_ (April 7, 1870).
"Modern Palace Beautiful": _New York Star_ (April 15, 1870).
"Oh Doc, you poor fellow!": _Chicago Mail_ (May 4, 1892), p. 1, col. 4; _VCWM_ , Box 3, Victoria Woodhull.
"it costs us over": _New York Herald_ (August 10, 1870); Johnston, p. 58.
"Your work is about to begin": _Boston Post_ (October 20, 1876); see also _Banner of Light_ , Boston (November 20, 1876).
"New Age" and ff.: Stern, _The Pantarch_ , p. 5.
"a grand domestic revolution": Ibid., p. 82.
"Eclectic Medical College": Ibid., p. 99.
"the bereaved wife": Carol Farley Kessler, _Elizabeth Stuart Phelps_ (Boston: Twayne Publishers, 1982), p. 31.
"heresy": Ibid., p. 33.
"The name of the gentleman was" and ff.: Chester, pp. 116, 117; Whitelaw Reid to Anna Dickinson (March 12, 1870), Papers of Anna E. Dickinson, 14:73.
"at night": _Washington, D.C., Daily Morning Chronicle_ , interview with Victoria Woodhull (October 24, 1873), _VCWM_ , Boston.
"an opportunity to feast" and ff.: Beecher-Tilton correspondence, newspaper articles, clippings, pamphlets, etc. from the offices of the _Independent_. Tucker-Sachs correspondence, Scrapbooks I-VIII, New York Public Library, Manuscript Division, Scrapbook V.
"edited in one world": _GAT_ , 1871, p. 27.
"the ruler of the whole world": Isabella Beecher Hooker to Susan B. Anthony (March 11, 14, 1871), _ECS & SBA_, Reel 15; Boydston et al., p. 206.
"I am the evangel": The quote that appeared in _WCW_ (November 2, 1872), p. 12, read, "I am the evangel—I am a savior if you would but see it. But I, too, come not to bring peace but a sword." This appears in slightly different form in Stern's _Victoria Woodhull Reader_ , p. 13.
"beg at the pockets": Isabella Beecher Hooker interview with John B. Martin, Sherman House (May 1, 1892), _VCWM_ , Reel 2.
"The Reverend May has adopted" and ff.: Buhle and Buhle, Vol. 2, p. 422.
"Like a shark" and ff.: Elizabeth Cady Stanton to Martha Coffin Wright, Tenafly, New Jersey (March 21, 1871), _ECS & SBA_, Reel 15; Stanton and Blatch, Vol. 2., p. 131.
"the husbands right" and ff.: Lutz, _Created Equal_ , pp. 190, 191.
"I hate the whole doctrine": Lutz, _Susan B. Anthony_ , p. 142.
"No, no," and ff.: _ECS & SBA_, Reel 15; Katharine Anthony, p. 239; Lutz, _Created Equal_ , p. 175.
"I will form a new": Stone's account of reconciliation, National American Woman Suffrage Association Collection, Library of Congress; Andrea Moore Kerr, p. 151. "grand times" and ff.: Elizabeth Cady Stanton to Martha Coffin Wright, _ECS & SBA_, Reel 15; Griffith, p. 145; Andrea Moore Kerr, p. 153.
"unconstitutional": Andrea Moore Kerr, p. 272.
"popular man": Buhle and Buhle, Vol. 2, p. 427.
"If Anna Dickinson will": _ECS & SBA_, Reel 14; Harper, Vol. 1, p. 361.
"was like signing my own death-warrant": (May 22, 1870), _ECS & SBA_, Reel 14; Harper, Vol. 1, p. 362.
"I feel a great calm": _ECS & SBA_, Reel 14; Anthony, pp. 244, 245; Harper, Vol. 1, p. 362.
"The Revolution is": Theodore Tilton to Anna Dickinson (June 1870), Anna Dickinson Papers, Letter no. 38.
"female demagogue" and ff.: William Lloyd Garrison to Theodore Tilton (April 5, 1870), Garrison Family Papers, Boston Public Library; Griffith, p. 119.
"stood firmly": _ECS & SBA_, Reel 23, quoted in Griffith, p. 142.
## Chapter 19 Your Child Is Not My Child
"One, two, three—jump,": Hibben, p. 200; _New York Tribune_ (July 6, 1870), New York Public Library Scrapbook V.
"Beecher's boys.": Ibid. (December 21, 1869).
"desire" and ff.: Waller, p. 71.
"Lying in defense": Ibid., p. 81.
"second rate people": _Independent_ (March 25, 1869).
"If we do not educate": _Independent_ (December 2, 1869).
"There is no end" and ff.: Samuel Wilkeson to Jay Cooke (July 11, 1869; July 26, 1869); Ellis Paxson Oberholtzer, _Jay Cooke: Financier of the Civil War_ (Philadelphia: George W. Jacobs, 1907), Vol. 2, pp. 120, 121.
"influencing the public": Hibben, p. 198.
"My friend, the time has come" and ff.: Marshall, p. 480.
"on this our glorious anniversary" and ff.: _New York Times_ and _New York Herald_ (July 8, 1870).
"I have come to tell you" and ff.: _PCH_. The alleged confession of Lib (Elizabeth) Tilton is drawn from Theodore Tilton's court testimony and his statement at the Plymouth Church hearing. It is repeated in Hibben, Marshall, and Doyle.
"affinity... to save her health": Lib (Elizabeth) Tilton's statement, _PCH;_ Waller, p. 122; Marshall, p. 194.
"playfully reproached her" and ff.: Lib (Elizabeth) Tilton's statement, _PCH_.
"Henry Ward Beecher" and ff.: This account is drawn from _WCW_ (November 2, 1872) and from _Victoria Woodhull's Complete and Detailed Version of the Beecher-Tilton Affair_ , a pamphlet published by J. Bradley Adams, Washington, D.C., in Stern, _Victoria WoodhullReader_, Section A, Sexual Relations, pp. 1–22. Elizabeth Cady Stanton corroborated this, saying that the story was "substantially true" but that she had not used the phrase "damned lecherous scoundrel."
"Mrs. Tilton told essentially the same": Elizabeth Cady Stanton to Isabella Beecher Hooker (November 3, 1873), _ECS & SBA_, Reel 16.
"a scoundrel" and ff.: This account is drawn from Theodore Tilton's civil court trial testimony ( _CT_ ) and is repeated in Marshall, pp. 527–531, and in various newspapers of the day.
"Oh Theodore, Theodore! What shall I": ET to TT, written from Marietta, Ohio, to Brooklyn, New York (November 1870); _CT:_ entered into evidence.
"So the harlot has returned!" and ff.: _CT;_ Marshall, p. 536.
"Marriage without love": _Independent_ (December 1, 1870).
"Women know their own wants": Blackwell, pp. 220, 221.
"the comfort of a paramour" and ff.: The following account is drawn from Tilton and Beecher's testimony in the civil trial, _CT_. It also appears in numerous newspapers and in Hibben, Marshall, and Doyle.
"all this torture" and ff.: Statement of Lib (Elizabeth) Tilton, _PCH_.
## Chapter 20 The Yawning Edge of Hell
"What is the meaning" and ff.: Testimony of Theodore Tilton, _CT_ , also Theodore Tilton to Frank Moulton (February 7, 1871); Marshall, p. 260.
"If anybody in King's County" and ff.: _Independent_ (September 26, 1870); Edwin W. Terry, _Theodore Tilton as Social Reformer, Radical Republican, Newspaper Editor, 1863–1872_ (Ph.D. dissertation, St. John's University, 1971; University Microfilms International Dissertation Information Service, Ann Arbor, Michigan), p. 219.
"I understand what Mr. Tilton" and ff.: _BTS_ , Scrapbook VI; _Brooklyn Eagle_ (October 22, 1870).
"I will not mince words" and ff.: _Brooklyn Daily Union_ (December 22, 1870).
"Then I surmise" and ff.: Testimony of Theodore Tilton (December 20, 1870), _CT_.
"In the nine years": Junius Henri Browne, _The Great Metropolis: A Mirror of New York_ (Hartford, CT: American Publishing Co., 1870), p. 313.
"control federal patronage" and ff.: Waller, pp. 80, 86, 87; Harold Coffin Syrett, Ph.D., _The City of Brooklyn, 1865–1989: A Political History_ (New York: Columbia University Press, 1944), p. 61.
"miscarriage" and ff.: Elizabeth Tilton to Laura Curtis Bullard (January 13, 1871), Doyle, p. 527.
"You infernal villain!": Letters of Mrs. Morse entered into evidence, _CT_. All the sources for the following events have been drawn largely from the testimony of Theodore Tilton, Henry Ward Beecher, Henry Bowen, and Frank Moulton during the civil trial found in Austin Abbot, _The Official Report of the Trial of Henry Ward Beecher With Notes and References_ (New York: George W. Smith & Co., Law Booksellers and Publishers, 1875, 1876), 2 volumes, and in _BTS_ , the New York Public Library Scrapbooks I—VIII, compiled in the offices of the _Independent_. They are repeated in Marshall, Doyle, Hibben, and Shaplen. All letters quoted were entered into evidence in the course of the trial.
"Dear Mr. Bowen, The understanding": December 21, 1870, made public when printed in the _New York Argus_ , June 29, 1875, and later in transcripts of court testimony.
"Theodore, I am in a dream": Beecher was later to deny this wording but admitted that the news of Lib's confession "fell like a thunderbolt on me." See _CT_ and Shaplen, p. 92.
"My Dear Friend Moulton": Full letter not entered in court trial but printed in its entirety in the _New York Times_ (July 30, 1874).
## Chapter 21 The Woodhull Memorial
"I shall go to": Isabella Beecher Hooker to Susan B. Anthony (1870), _ECS & SBA_, Reel 14; Harper, p. 357. (Paxton Hibben incorrectly cites this as Susan B. Anthony to Isabella Beecher Hooker, p. 232.)
"Let us exalt Mrs. Hooker": Elizabeth Cady Stanton to Susan B. Anthony (December 1870), _ECS & SBA_, Reel 15.
"You know when I drop": Ibid., Reel 14.
"looked disgusted": Susan B. Anthony to Martha Coffin Wright (December 1870), _ESC & SBA_, Reel 14.
"I don't know": Isabella Beecher Hooker to Elizabeth Cady Stanton, _IBH;_ Harper, p. 372.
"Mrs. Hooker's attitude": Susan B. Anthony to Elizabeth Cady Stanton (January 2, 1871), _ECS & SBA_, Reel 15; _MCW_.
As she had vowed, Elizabeth Cady Stanton did not attend the convention, although in _Frank Leslie's Illustrated_ newspaper she is depicted as sitting directly behind Victoria Woodhull as Victoria presents the Woodhull Memorial. On May 24, 1873, this illustration was reprinted in _WCW_ , still without correction.
As "Miss Anthony, Congress has" and ff.: Isabella Beecher Hooker to Anna Savery, Hartford, Connecticut (November 12, 1871), _IBH_. Senator Pomeroy of Kansas was against linking Negro and woman suffrage, but since the Fifteenth Amendment had been added to the Constitution, he was now willing to help women.
"beautiful enough to win": Washington, D.C., _Daily Morning Chronicle_ (December 19, 1870), _IBH_.
"It would ill become" and ff.: The writing of Victoria Woodhull (November, 1872), _WCW_. See also Stern, _Victoria Woodhull Reader_ , Sociology, Part 2, p. 10.
"fire and freedom": Isabella Beecher Hooker to Anna Savery (November 21, 1871), _IBH;_ Stern, _Victoria Woodhull Reader_ , p. 112.
"All persons born": This and the quotations following it are drawn from the Memorial of Victoria C. Woodhull to the joint houses of Congress.
"The Heavenly Father": Isabella Beecher Hooker to Anna Savery, Hartford, Connecticut (November 12, 1871), _IBH_.
"a great vision": Victoria Woodhull to Isabella Beecher Hooker (August 8, 1871), _IBH; TS_ , pp. 93, 94, no; Marshall, pp. 333, 334.
"dead Congress": Isabella Beecher Hooker to Anna Savery (November 18, 1871), _IBH_.
"The Memorial of Victoria C. Woodhull": Isabella Beecher Hooker noted that franking privileges were "an immense advantage—we could not work without it.... We cannot begin to supply individual demand for tracts. These women are all so hungry." Isabella Beecher Hooker to Susan B. Anthony (March 11, 14, 187), _IBH_ , 44 D7, E14.
"In declaring that women": Elizabeth Cady Stanton to Victoria Woodhull (June 21, 1871), _ECS & SBA_, Reel 15.
"Bravo! My Dear Woodhull!": Susan B. Anthony to Victoria Woodhull (February 4, 1871), _ECS & SBA_, Reel 15.
"petitioner's prayer": Stern, _Victoria Woodhull Reader_ , Vol. 2, p. 1.
"His [God's] own instrument": Isabella Beecher Hooker to Elizabeth Cady Stanton (May 12, 1872), _IBH_ , 47 D9–E1; _ECS & SBA_, Reel 16.
"Burn this as soon as sent": Isabella Beecher Hooker to Victoria Woodhull, undated (written in 1872), _IBH_ , 53 B10–13, also _VCWM_ , dated 1872.
"her conviction": Isabella Beecher Hooker to Susan B. Anthony (March 11, 14, 1871), _IBH;_ Boydston et al., p. 206.
"I have this moment read": Isabella Beecher Hooker to Victoria Woodhull, _IBH_ (undated; February 16 given as approximate date); _VCWM_.
"did not dare": Isabella Beecher Hooker to Victoria Woodhull (February 4, 1889), _IBH_ , 63 B10–13.
"As I am about to speak": Victoria Woodhull to Isabella Beecher Hooker (November 22, 1872), _IBH;_ Victoria Claflin Woodhull Correspondence, Butler Library, Columbia University, New York.
"It flowed out of my inner" and ff.: Isabella Beecher Hooker to John Hooker (February 4, 1889), _IBH_.
"a triumph": Elizabeth C. Stanton to Susan B. Anthony (January 31, 1871), _ECS & SBA_, Reel 15.
"Her hands are unclean": Susan B. Anthony to Laura DeForce Gordon (February 9, 1871), _ECS & SBA_, Reel 15.
"If I had time or space": Ibid.
"It is a great": Isabella Beecher Hooker to Elizabeth Cady Stanton (May 12, 1872), _IBH_.
"Dear Friend Susan": Isabella Beecher Hooker to Susan B. Anthony (May 11, 14, 1871), _IBH_ , 44 D–7 E14.
"No one can be": Martha Coffin Wright to Susan B. Anthony (April 6, 1871), _MCW_.
"When we begin to search" and ff.: Susan B. Anthony to Isabella Beecher Hooker (March 21, 1871), _IBH_.
## Chapter 22 Silence, Time and Patience
"There came scarcely" and ff.: Paulina Wright Davis's account of her visit to the Tilton house as told to Victoria Woodhull, appearing in _Victoria C. Woodhull's Complete and Detailed Version of the Beecher-Tilton Affair_ , Victoria C. Woodhull pamphlet (Washington, D.C.: J. Bradley Adams); _WCW_ (November 2, 1872), pp. 9–13.
"I know of twelve persons": Letter of Mrs. "Judge" Morse to Henry Ward Beecher (January 27, 1871), entered in evidence, _CT_ , reprinted in the _Daily Argus_ (March 13, 1875); _BTS_ , Scrapbook VII. The other sources for the quotes herein are Marshall, pp. 328–330, Hibben, pp. 228, 251, 269, 273.
"We have Plymouth Church": Isabella Beecher Hooker to Henry Ward Beecher, _IBH;_ Marshall, p. 330; Hibben, p. 273.
"that the destruction of": Elizabeth Cady Stanton to Isabella Beecher Hooker, Tenafly, New Jersey (November 3, 1873), _IBH;_ Hibben, p. 228.
"I have been the centre": Henry Ward Beecher to Frank Moulton (February 5,1872), _CT;_ Marshall, pp. 359–360.
"The church property is": _CT;_ Hibben, p. 269.
"the lie" and ff.: Elizabeth Cady Stanton to Isabella Beecher Hooker (November 3, 1873), _IBH; ECS & SBA_, Reel 17.
"You are in trouble": Sam Wilkeson to Theodore Tilton (January 11, 1871); _Brooklyn Eagle_ (May 20, 1875); _BTS_ , Scrapbook VII; Hibben, p. 370.
"take back": Susan B. Anthony to Anna Dickinson (September 19, 1870), _ECS & SBA_, Reel 14.
"Can you help" and ff.: Mrs. "Judge" Morse to Henry Ward Beecher (January 27, 1871), _CT_ , and _Daily Argus_ (March 13, 1875); _BTS_ , Scrapbook VII; Doyle, p. 320.
"When I saw you last": Henry Ward Beecher to Elizabeth Tilton (February 7, 1871), _CT;_ Doyle, p. 524.
"Dear Friend and Sister": Lib (Elizabeth) Tilton to Laura Curtis Bullard (January 13, 1871), _CT; Doyle_ , p. 527.
"Does you heart bound": Lib (Elizabeth) Tilton to Henry Ward Beecher (March 6, 1871), _CT; Daily Argus_ (March 13, 1875); _BTS_ , Scrapbook VII.
"as a talisman" and ff.: _GAT_ , p. 20.
"My Dear, dear friend": Victoria Woodhull to Isabella Beecher Hooker (August 8, 1871), _IBH;_ Marshall, p. 333.
"I could not ask": Rugoff, _The Beechers_ , p. 46.
"No salvation!": Rourke, p. 50.
"As things are now" and ff.: Catharine Beecher's address on female suffrage delivered at the Boston Music Hall, 1870, _IBH;_ Boydston et al., p. 232.
"Many intelligent and benevolent": Catharine Beecher, Appeal to American Women, _American Woman's Home_ , pp. 463–468; Boydston et al., p. 257.
"just as much in love": Isabella Beecher Hooker to Catharine Beecher, _IBH_ , Document 112.
"son, Henry, drowned": Isabella Beecher Hooker to Catharine Beecher, _IBH_ , Document 112; Rugoff, p. 343.
"Why will you not speak" and ff.: Rourke, p. 131; Rugoff, _The Beechers_ , p. 343.
"a reaction from the intense": Rugoff, _The Beechers_ , p. 349.
"Mrs. Woodhull is": Isabella Beecher Hooker to Susan Howard (April 1871), _IBH_.
"Disastrous influences, the teachings": "Woman Suffrage and Woman's Profession," Catharine Beecher, p. 34; Boydston et al., p. 251.
"must come from a misunderstanding": _WCW_ (November 2, 1872), reprinted April 1873. This account is solely Victoria Woodhull's. Catharine Beecher never gave the details of this interview, but thereafter referred to Victoria Woodhull variously as a "witch," "harlot," "free lover," and, later, "jailbird."
"Sister Catharine returned": (February 6, 1871), _IBH_ , 115 E4–9; Harper, p. 379.
"bring her back to God": Harriet Beecher Stowe (and Mary Perkins) to John Hooker (March 2, 1871), _IBH_.
"Do you believe I" and ff.: Isabella Beecher Hooker to Harriet Beecher Stowe (March 16, 1871), _IBH;_ also in _MCW_.
"Mrs. Woodhull's antecedents" and ff.: _ECS & SBA_, Reel 15; _New York Tribune_ (January 28, 1872), Susan B. Anthony at Lincoln Hall Convention, January 1872.
"She and her five sisters": Martha Coffin Wright to (her daughter) Elizabeth Wright Osborne (March 16, 1871), _MCW_.
"Under all the curses": Victoria Woodhull to Isabella Beecher Hooker, _IBH_ , 44 D–7 E14. In a letter to Susan B. Anthony (March 11, 14, 1871), Isabella Beecher Hooker encloses a copy of this letter from Victoria Woodhull. Isabella answers Victoria Woodhull's letter by writing, "In my observation all human souls are more or less lonely who have only human forces of sympathy and consolation" ( _IBH_ , 48 C4–10).
## Chapter 23 The Worst Gang
"that damn scoundrel Blood": This and quotes that follow are found in the court testimony of _Roxanna Claflin v. Colonel James Blood_ , Essex Market Police Court (May 10, 15, 16, 1871), and accounts in the _New York Herald, New York Sun, New York World, New York Argus_ , and _New York Times_ (May 11–21, 1871), _BTS_ , Scrapbook VII.
"They h'aint no friends of mine": Commodore Cornelius Vanderbilt, _New York Tribune_ (May 21, 1871); _BTS_ , Scrapbook VII.
"great trouble allowing Mrs. Woodhull": Susan B. Anthony to Martha Coffin Wright (March 31, 1871), _ECS & SBA_, Reel 15.
"to give her respectability": Johnston, p. 102.
"a small splinter of": _TS_ , p. 77; Hibben, p. 233; excerpted from the _Philadelphia Press_.
"Why do I war": Apollo Hall speech (May 11, 12, 1871), _VCWM_ , Box 2, reprinted in _WCW_ (May 27, 1871), p. 9; _TS_ , p. 87; Johnston, p. 103.
"Ridiculous string of resolutions": Martha Coffin Wright to Elizabeth Cady Stanton (May 27, 1871), _MCW_.
"Professor Purlo's hypotheses": Susan B. Anthony to Isabella Beecher Hooker (May 31, 1871), _IBH; ECS & SBA_, Reel 15.
"No subject discussed": _BTS, Independent_ (May 28, 1871), Scrapbook VI; _TS_ , p. 93.
"one who has two husbands": _BTS, New York Tribune_ (May 20, 1871), Scrapbook VI; Boydston et al., p. 164.
"Mr. Greeley's home": _VCWM, WCW_ (Mzy 27, 1871).
"I would like to have": Sara Burger Stearns to Mrs. Griffing (June 20, 1871), _ECS & SBA_, Reel 15.
"I do not understand" and ff.: Isabella Beecher Hooker to Sara Burger Stearns of Minnesota, May 23, 1871 _IBH;_ Boydston et al., p. 306.
"with a meaning of her own" and ff.: Isabella Beecher Hooker to Anna Savery (November 12, 18, 1871), _IBH_. Isabella explains her view of free love in a letter to Victoria Woodhull: "Lest you should misunderstand my position on the question of Social Freedom let me state it briefly here. Human law should not attempt to regulate marriage—this is a sacrament of Souls owing allegiance to God and their own consciences only. But the ideal marriage is between two only": Isabella Beecher Hooker to Victoria Woodhull (July 28, 1872), _IBH_ , 48 C49.
"kept women": Susan B. Anthony to Isabella Beecher Hooker (April 22, 1871), _IBH_. See Isabella Beecher Hooker to Paulina Wright Davis (May 29, 1871). Also see Paulina Wright Davis to Victoria Woodhull, Providence, Rhode Island (May 1871).
"I will take by the hand": Oliver, p. 43; Susan B. Anthony to the _New York Tribune_ (January 14, 1872).
"In regard to the": Harper, Vol. 1, p. 379; Lutz, _Created Equal_ , p. 211; _TS_ , p. 79.
"Mrs. Stanton whispered": Doyle, p. 57.
"were belching forth": Account of Victoria Woodhull as told to Allen Putnam, _Banner of Light_ (November 20, 1875).
"communists... given the opportunity": _The Telegram_ (October 28, 1871).
"a menace" and ff.: _New York World_ (November 2, 1871).
"Come on, come on": _New York Tribune, Brooklyn Eagle_ (May 28, 1871), _VCWM_ , B3.
"a woman of the town": _Brooklyn Eagle_ (May 28, 1871); _VCWM_ , B3.
"Vickie, Colonel and Tennie included": _Brooklyn Eagle_ (May 28, 1871); _TS_ , p. 110.
"to lead the movement" and ff.: Blackwell, p. 223.
"What effort more than Herculean": Susan B. Anthony to Martha Coffin Wright (June 14, 1871), _ECS & SBA_, Reel 15.
"The division is so senseless": Martha Coffin Wright to William Lloyd Garrison II, her daughter Ellen's husband (May 15, 1871), _MCW_. Theodore Tilton concurred. See Terry pp. 160, 201.
"Without pretending to a perfect knowledge": National American Woman Suffrage Association Collection, Library of Congress, Washington, D.C.; _WCW_ (June 8, 1871).
"This fresh howl": Susan B. Anthony to Martha Coffin Pelham Wright (May 20, 1871), _MCW_.
"Because I am a woman": _New York World_ and _New York Times_ (May 22, 1871); _TS_ , pp. 96–97.
## Chapter 24 This Girl Is a Tramp
"Who do you mean" and ff.: _WCW_ (November 2, 1872), pp. 9–13; _Victoria C. Woodhull's Complete and Detailed Version of the Beecher-Tilton Affair_ , pamphlet; Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 2, p. 12.
"some influence might be brought": Statement of Frank Moulton, _PCH_ , abridged in Marshall, p. 562.
"Don't take any steps now" and ff.: The quotations are taken from various newspaper interviews with Victoria Woodhull, the November 2, 1872, issue of _WCW_ , pp. 9–13, and Woodhull's speeches throughout the course of the scandal—November 2, 1872–June 30, 1875.
"Be kind to and sympathize": This quotation is repeated in the Woodhull speech, Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 2, p. 30.
"unexpected modesty": Statement of Emma Moulton (May 23, 1875), _CT_ , Vol. 2, p. 118.
"I want you to take a look": Ibid.
"a broad-minded, self-possessed woman": Ibid.
"life on the theory of human imperfection": Ibid.
"Such another family-circle": _GAT_ , pp. 6–7.
"I must now let out a secret": _GAT_ , pp. 8–9.
"If apples are wormy": _TS_ , p. 120.
"Can anything but infatuation": Marianna Pelham Mott to Martha Coffin Wright, (September 24, 1871), _MCW_.
"A popular objection against Free Love": "Tried as by Fire," a speech by Victoria Woodhull, appears in Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5, p. 35.
"Mr. Tilton was my devoted lover": Doyle, p. 315; _Chicago Times_ (June 4, 1874).
"Civilization is festering": _WCW_ (June 3, 1871), received on May 22, 1871; Hibben, p. 272.
"My Dear Victoria, I have arranged": Theodore Tilton to Victoria Woodhull (June 5, 1871), _CT_ , entered into evidence, exhibit UV 130.
"Marriage is the grave of love" and ff.: _WCW_ (November 2, 1872), pp. 9–13; _Victoria C. Woodhull's Complete and Detailed Version of the Beecher-Tilton Affair_ , Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 2, p. 16.
"Why then do you not preach" and ff.: Ibid.
"held society to be upside down" and ff.: Frank Moulton's final statement, _CT_ , reprinted from Abbot, Vol. 2, p. 343.
"a most indignant and rebuking letter": Isabella Beecher Hooker in a letter to Henry Ward Beecher (February 14, 1872) explains Woodhull's assertion of October 1871. Isabella, who frequently tried to influence Victoria Woodhull, had suggested that the Steinway Hall convention be restricted to women only. Woodhull, however, seemed impervious to Hooker's suggestions and wrote her that she had "carefully read" her letter and "I think it an _error_ to exclude men from the hall. In doing that you pattern after the old custom. There should be nothing exclusive about it." Victoria Woodhull to Isabella Beecher Hooker (October 19, 1871), _IBH_ , 142 D10–13.
"As to what Mrs. Woodhull means": Isabella Beecher Hooker to Anna Savery (November 12, 1871), _IBH_ , 46 C3–12.
"A Lady of Connecticut": Sometimes even on the same page the spelling of Catharine changes to Catherine. Josephine Blatti of the Harriet Beecher Stowe Center informs me that the great preponderance of correspondence favors the _a_. I too have found this to be true. In this particular passage, for the reader's convenience I have used _a_ even if _e_ appears in the original letter.
"Every drop of Beecher blood": Reverend Phebe Hanaford to Isabella Beecher Hooker (September 13, 1871), _IBH_. In Boydston et al., p. 295, this letter is dated August 9, 1871.
"That private note of mine": Isabella Beecher Hooker to Anna Savery, Hartford, Connecticut (November 12, 1871), _IBH_ , 46 C3–12.
"Mrs. Woodhull received me cordially": Whitelaw Reid to Charles W. Warner (October 13, 1871), _IBH_.
"I shall break all engagements": Whitelaw Reid to Anna Dickinson (March 24, 1871), Anna E. Dickinson Papers.
"I must and do always": Whitelaw Reid to Charles Dudley Warner (October 13, 1871), _IBH, VCWM_. "Demi-rep" was a common euphemism of the day for prostitute. Clearly Whitelaw Reid used this term as a snide reference to Victoria Woodhull.
"a snake and should be given": Johnston, p. 126. This quote is slightly altered in Rourke, p. 145. It reads, "The Woodhull is a snake and we should hit her with a shovel."
"My Wife and I": The complete title is _My Wife & I or Harry Henderson's History_. This work by Harriet Beecher Stowe was serialized in the _Christian Union_ , 1870, 1871 (New York: J. B. Ford & Co., 1871; Boston: Houghton Mifflin, 1896).
"Mrs. Cerulean": Cerulean is defined in the _Oxford English Dictionary_ as "sky-blue, the color of the cloudless sky, deep blue, azure."
"Dear Sir: For reasons in which" and ff.: Victoria Woodhull to Henry Ward Beecher (November 19, 1871), Marshall, p. 358. This encounter is detailed in _WCW_ (November 2, 1872, pp. 9–13). Also in speeches of Victoria Woodhull, "Tried As By Fire," Stern, _Victoria Woodhull Reader_.
"Mr. Beecher, if I am compelled" and ff.: Testimony given by Stephen Pearl Andrews, Frank Moulton, and Theodore Tilton, letter entered into evidence March 3, 1875, _CT:_ Abbot, Vol. 2, p. 60.
## Chapter 25 Yes! I Am a Free Lover!
"Victoria C. Woodhull": _TS_ , p. 129; Johnston, p. 127. The Steinway speech, "The Principles of Social Freedom" (November 20, 1871) was reprinted in _WCW_ (August 16, 1873), pp. 2–7, 11–15.
"I hope by God": _New York Tribune_ (November 22, 1871). In _TS_ , p. 129, the quote differs: "I hope, by gosh I haven't come here for nothing in all this rain."
"Your paper has never misrepresented": _New York Tribune_ (November 22, 1871); _TS_ , p. 129.
"There isn't one brave man" and ff.: _TS_ , pp. 130–131; Marshall, p. 814; Frank Moulton statement reprinted in Abbot, Vol. 2, p. 346.
"The basis of society" and ff: All quotations that follow are from the speech "The Principles of Social Freedom" given by Victoria Woodhull November 20, 1871, reprinted in _WCW_ (August 16, 1873), pp. 2–7, 11–15. Also excerpted in abridged form in Stern's _Victoria Woodhull Reader_ , Part A, Sexual Relations, 1, pp. 17–26. I have also used accounts from the _New York Tribune, New York Sun, New York Times_. The various transcripts of this speech contain small but not substantive variations. I have, whenever possible, used the version that Woodhull herself printed. In this case, I used the pamphlet _A Speech on The Principles of Social Freedom_ (New York: Woodhull, Claflin & Co., 1871). The subtitle reads: "To an audience of 3,000 people, to which hundreds found it impossible to obtain admission; which, in consideration of the dreuching [ _sic_ ] rain of the evening, is the best evidence of what subject lies nearest the hearts of the people."
"Free Lover Lectures on Free Love": _Weekly Argus_ (November 22, 1871).
"Died of Free Love... The Woman Suffrage Movement": _Lancaster Gazette_ (November 25, 1871).
"It was not the printed speech": _New York Herald_ (November 22, 1871).
"The free love panic" and ff.: Martha Coffin Wright to Elizabeth Cady Stanton (March 22, 1872), _MCW_.
"The Impending Revolution": This speech, delivered February 1, 1872, can be found in its entirety in Stern, _Victoria Woodhull Reader_ , Section A, Part 3, and in the pamphlet "The Impending Revolution" (New York: Woodhull, Claflin & Co., 1872).
"there would be a riot": _Victoria Woodhull Prophesy_ , "Honor to the Martyrs," _Boston Post_ (October 20, 1876).
"Some people carp": Elizabeth Cady Stanton, _WCW_ (December 18, 1871); _GAT_ (December 9, 1871); Harper, Vol. 1, p. 379.
"It is only four years" and ff.: Swanberg, p. 250.
"I feel just as I used to" and ff.: Ibid., p. 275.
## Chapter 26 A Heavy Load
"Dear Mrs. Woodhull, Will you ask Demosthenes": Elizabeth Cady Stanton to Victoria Woodhull, Tenafly, New Jersey (December 29, 1871), _ECS & SBA_, Reel 15, pp. 852–857. "Felicita": Also known as Felicité.
"I do not mean": Henry Ward Beecher to Frank Moulton, Brooklyn, New York (January 2, 1872), _CT_.
"I am ashamed": Frank Moulton to Henry Ward Beecher (February 5, 1872), _CT;_ Marshall, p. 359.
"This whole affair [is]": Sam Wilkeson to Frank Moulton (April 2, 1872), _CT;_ Marshall, p. 370.
"It seems a change": Henry Ward Beecher to Frank Moulton (February 5, 1872), _CT;_ Hibben, p. 238; Marshall, p. 361. During the trial this letter became known as "The Ragged Edge" letter and the object of intense press scrutiny.
"The whole case has terminated": Theodore Tilton to Anna Dickinson (January 4, 1872), Anna E. Dickinson Papers, Reel 26.
"closing act": Sam Wilkeson to Frank Moulton (April 2, 1872); Marshall, p. 370.
" 'Grantism' became a synonym": Mark Twain commented in a manner that is easily applicable to today: "We will not hire a school teacher who does not know the alphabet... but when you come to our civil service we serenely fill great numbers of our minor public offices with ignoramuses. We put the vast business of a Custom House in the hands of a flathead who does not know a bill of lading from a transit of Venus.... Under our consular system we send creatures all over the world who speak no language but their own."
"You will be responsible": _VCWM; Boston Post_ (October 20, 1876); _Banner of Light_ (November 20, 1875).
"None but the liberals": (July 26, 1872), Harper, Vol. 1, p. 419.
"Baltimore Warned—Susan B. Anthony": _ECS & SBA_, scrapbooks; _New York World_ (June 12, 1872), p. 1.
"the honest demands": Griffith, p. 153.
"Baltimore will not recognize": Susan B. Anthony to Martha Coffin Wright (June 13, 1872), _MCW_.
"Both parties were utterly corrupt": Martha Coffin Wright to her daughter Ellen Garrison (March 15, 1871), _MCW_.
"We have no element": Susan B. Anthony to Elizabeth Cady Stanton and Isabella Beecher Hooker, Leavenworth, Kansas (March 13, 1872), _IBH; ECS & SBA_, Reel 16; Lutz, _Created Equal_ p. 217; Harper, Vol. 1, p. 413.
"Oh, if I were—" and ff.: Isabella Beecher Hooker to Olympia Brown (February 14, 1872), _IBH_ , 47 B2–9.
"Mrs. Stanton told me precisely": Isabella Beecher Hooker to Henry Ward Beecher, Hartford, Connecticut (November 1, 1872), _IBH;_ Doyle, pp. 506–507.
"The only reply I made": Hartford, Connecticut (November 1, 1872), _IBH;_ Marshall, p. 330.
"My Dear Belle... I do not intend": Henry Ward Beecher to Isabella Beecher Hooker (April 25, 1872), _IBH_. Letter transcribed by Henry E. Burton, son-in-law of Mrs. Hooker, and a copy then sent to John Hooker on October 31, 1872.
"My mind flew back": Isabella Beecher Hooker to Henry Ward Beecher (April 25, 1872), _IBH;_ Marshall, p. 334.
"We believe the time has come": Harper, Vol. 1, p. 413; Lutz, _Created Equal_ , p. 218.
"I do not believe": Susan B. Anthony to Lillie Devereux Blake (May 14, 1872), _ECS & SBA_, Reel 16.
"millionairess" and ff.: This story appeared in _WCW_ (March 18, 1871), pp. 5–6.
"the sexual liaisons and free love": Accounts of this blackmail scheme (which in one case included a death threat if the recipient of the letter did not pay up) are in the National American Woman Suffrage Association Collection, Reel 58; _TS_ , pp. 164–165; Susan B. Anthony to Isabella Beecher Hooker (June 19, 24, 1872), _IBH_. Woodhull defended her use of blackmail, writing that for women, blackmail was "the only method of righting themselves" and a way to "avenge the oppressions of [their] sex," _WCW_ (April 6, 1872).
"Phebe Hanaford": Mrs. Hanaford was living alone in New Haven while her husband, the Reverend Joseph Hanaford, remained in Reading, Connecticut, where he vowed to stay.
"practice what I preach": _WCW_ (November 2, 1872), p. 13; _TS_ , p. 95; Oliver, p. 300.
"Called on Mrs. Phelps": Diary entry of Susan B. Anthony (Monday, May 6, 1872), _ECS & SBA_, Reel 16.
"I concluded to shut the mouths": _WCW_ (April 2, 1873), by Victoria Woodhull; Oliver, p. 299.
"My Dear Victoria, Driven to bay": Paulina Wright Davis to Victoria Woodhull, Oliver, p. 202.
"Narrow minded and domineering": Harper, Vol. 1, p. 414.
"claimed right to possess the meeting": Diary entry of Susan B. Anthony (Tuesday, May 7, 1872), _ECS & SBA_
"There was never such a foolish _muddle_ ": Diary entry of Susan B. Anthony (Wednesday, May 8, 1872).
"Friday: National Convention—again small audience": Diary entry of Susan B. Anthony (Friday, May 10, 1872).
"Saturday: I was never so hurt": Diary entry of Susan B. Anthony (Saturday, May 11, 1872).
"I never before came so near": Diary entry of Susan B. Anthony (Friday, May 10, 1872), _ECS & SBA_.
"Woodhull paper comes freighted": Diary entry of Susan B. Anthony (Wednesday, May 22, 1872), _ECS & SBA_.
"At the Saturday Apollo Hall meeting": In her first speech to the convention, Victoria declared that her party would ensure that "the millions now paid into the pockets of wealth will remain in the pockets of industrial people. Instead of there being the _very rich_ few and the _very poor_ many, all will be rich enough to have all the comfort that wealth and enjoyment demand." She then advocated violence if no other method would secure an eight-hour workday and a fair wage. This speech may be found in Stern, _Victoria Woodhull Reader_ , Political Theory, Part II, B, 9, pp. 11–14.
"Apollo Hall was a success": Isabella Beecher Hooker to Elizabeth Cady Stanton (May 12, 1872), _IBH_ , 47 D9–E1. In this same letter Isabella writes, "I do wish that her Suffrage friends who think the cause was lost through the advocacy of Victoria and our advocacy of her would show us where the money and brains and unceasing energy of that paper would come from if she had not been moved to present the _Memorial_ and follow it up with the prodigious outlays of the last year and a half. I verily believe she has sunk a hundred thousand dollars in Woman Suffrage beside enduring loneliness of soul immeasurable—let us never forget this. Let us still look over her faults if she has them, remembering that she is human like ourselves."
"overpowered by the sublimity of this hour": Isabella Beecher Hooker to Susan B. Anthony (May 6, 1872), _IBH_.
"Yes, I am overpowered": Susan B. Anthony to Isabella Beecher Hooker (May 19, 1872), _IBH_.
"Mrs. Woodhull, The nominee": _Broadway's_ article appeared in _WCW_ (June 1, 1872), p. 10.
"What a ridiculous letter": Susan B. Anthony to Elizabeth Cady Stanton (June 1, 1872), _ECS & SBA_, Reel 14; Barry, p. 237.
"resorting to blackmail intentionally": _WCW_ (November 2, 1872), p. 2.
"Surprise found her home": Diary entry of Susan B. Anthony (June 7, 1872), Ibid.
"fabricated terrible charges against T. C.": (Monday, June 10, 1872), ibid.
"We do not believe the charge": Martha Coffin Wright to Susan B. Anthony, Auburn, New York (May 20, 1872), _MCW_
"That letter of Dr. A. Orvis": Susan B. Anthony to Isabella Beecher Hooker (June 19, 1872), Rochester, New York, _IBH_.
"I shall welcome her work": Susan B. Anthony to Isabella Beecher Hooker (June 24, 1872), _IBH_.
"The charges... as made": _WCW_ (December 28, 1872).
"marplot" and ff.: Diary entry of Susan B. Anthony (May 29, 1872), _ECS & SBA_.
"By the way I have been hearing": Whitelaw Reid to Anna Dickinson (August 6, 1872), Anna E. Dickinson Papers, 14:112; Chester, p. 130.
"What you write us is shocking": John Hooker to Isabella Beecher Hooker (October 31, 1872), _IBH_.
## Chapter 27 Mrs. Satan
"Whoever is set up to be President": Boydston et al., p. 284, excerpted from Stowe, _My Wife & I_.
"We must now all pull": Susan B. Anthony to Lillie Devereux Blake (May 17, 1872), _ECS & SBA_.
"The Grant folks": Hale, p. 345.
"terriers to watch such rats": _Independent_ (May 4, 1865).
"We believe that the anti-slavery": _Golden Age_ (June 15, 1872).
"a floundering old buffoon": Hale, p. 343.
"Republican Party and all women": Susan B. Anthony to Martha Coffin Wright (September 30, 1872), _ECS & SBA_, Reel 16:422.
"We hope they got rid": Lutz, _Created Equal_ , pp. 220–221.
"Woodhull & Claflin tribe" and ff.: Leslie Wheeler, _Loving Warriors: Selected Letters of Lucy Stone and Henry B. Blackwell, 1853–1893_ (New York: Dial Press, 1981), p. 236.
"And now, dear Miss Anthony": Henry Blackwell to Susan B. Anthony (September 7, 1872, copied by Martha Coffin Wright), _MCW_.
"Under suspicious circumstances": Isabella Beecher Hooker to Victoria Woodhull (April 10, 1872), _IBH_ , 47 C12–D1. Isabella Beecher Hooker wrote a letter of condolence to Victoria in which she stated:
I cannot always see truth just as you do, but I see that you love the truth and follow after it with all your soul—and that is just what I am trying to do. I feel that we are sisters—children of the same Father and bound to the same home—ever the kingdom of righteousness which is from on high.
"insatiable vengeance": _TS_ , p. 154.
"My Dear Sir, The social fight": Victoria Woodhull to Henry Ward Beecher (June 3, 1872, sent by hand), _VCWM_ , Box 3; _CT_ , Vol. 2, Exhibit UU.
"Will you answer this?": Henry Ward Beecher to Frank Moulton (June 3, 1872); reprinted in the _New York Tribune_ (November 8, 1872).
"I always knew what Woodhull": Frank Moulton's final statement, _CT_ , Vol. 2, p. 239.
"only Victoria believed": Woodhull later described her feeling of hopelessness during this period. "The press suddenly divided between the other two great parties, refused all notice of our reformatory movement, and a series of pecuniary disasters... forced us into a desperate struggle for mere existence" _WCW_ (November 2, 1872).
"Human nature is awfully weak and wanting": Susan B. Anthony to Isabella Beecher Hooker, Rochester, New York (August 7, 1872), _ECS & SBA_, Reel 16; Barry, p. 248.
"All in the world": Susan B. Anthony to Martha Coffin Wright (September 12, 1872), _MCW_.
"The secret of the final rupture": _Woman's Journal_ , Boston, Chicago, and St. Louis (August 3, 1872). (Sent to subscribers July 27.)
"The inference... was that" and ff.: Victoria Woodhull to Susan B. Anthony (August 2, 1872), _VCWM_.
"I have not and shall not reply": Susan B. Anthony to Isabella Beecher Hooker, Rochester, New York (August 7, 1872), _IBH_ , 47 D10–C2; Barry, p. 248.
"If anything shows the inability of women": _The Word_ (September, 1872), New York Public Library, Manuscript Division.
"I was seized": _TS_ , p. 154; _WCW_ (December 28, 1872).
"Mrs. Woodhull's speech poured out": Elizabeth Avery Meriwether, _Memphis Appeal_ (November 17, 1872); The Tennessee Historical Commission, Nashville, Tennessee, publishers of _Recollection of 92 Years; 1824–1916, The Diaries of Elizabeth A. Meriwether; TS_ , p. 170.
"a self-aggrandizing": _Woman's Journal_ (December 1872).
"I swore not profanely": _WCW_ (December 28, 1872).
"The second week in October": Stoddard, p. 314.
"I will make it" and ff.: _WCW_ (November 2, 1872).
## Chapter 28 Burst Like a Bombshell
"If an omelette has to be made": The Beecher-Tilton scandal and the story of Luther C. Challis and all the details in this account are drawn from _WCW_ (November 2, 1872) and subsequent issues where these events are amplified and reiterated. The behavior of the people concerned is drawn from numerous contemporary newspaper accounts (see bibliography for a complete list of newspapers consulted).
"From the time that I opened my house": (alias) Mary Bowles to Tennie C. Claflin (December 16, 1871, and repeated August 19, 1873), _WCW_.
"three thousand of the best men": _New York World_ (December 24, 1869); James D. McCabe Jr., _Lights and Shadows of New York Life; or, The Sights and Sensations of the Great City_ (Philadelphia: National Publishing Co., 1872; New York: Farrar, Straus and Giroux, 1970), p. 604; Stern, _Victoria Woodhull Reader_ , "Tried As By Fire or The True and The False Socially," Part A, Sexual Relations, 5.
"closely dominoed": The following quotations and events are drawn from reports in the _New York World_ (December 24, 1872), _WCW_ (November 2, 1872), and a subsequent interview in the _Argus_ (November 19, 1872) as well as from testimony in the obscenity trials of Victoria Woodhull and Tennessee Claflin.
"Molly de Ford": In _TS_ and McCabe, p. 604, the name is spelled Mollie, but in "Tried as by Fire, or The True and The False, Socially," Victoria Woodhull's speech, 1874, it is spelled Molly. See speech in Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5.
"On October 26": Although _WCW_ is dated November 2, 1872, it was issued October 26, 1872.
"God's hand was in this": Heywood Broun and Margaret Leech, _Anthony Comstock: Roundsman of the Lord_ (New York: Albert & Charles Boni, 1927), p. 91.
"a gross libel": Ibid., p. 103; "Tried as by Fire or, The True and The False, Socially," Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5.
"the red trophy of her virginity": The quotation is from Deuteronomy, Chapter XXII. Hibben, p. 285.
"The tabloids are full": Johnston, p. 169.
"a case of this character" and ff.: _WCW_ (November 2, 1872); _New York World_ (December 24, 1869); Johnston, p. 169; _TS_ , p. 209; McCabe, p. 604. The prostitution quotations are from "Tried as by Fire," _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5, p. 31.
"I have been and gone and done it!!": This was perhaps Susan B. Anthony's most publicized moment and is recounted in Harper, Barry, Lutz, and other books and articles.
"Do you know, I have sometimes felt": Susan B. Anthony to Isabella Beecher Hooker (October 13, 1873), _IBH_.
"There is no time": Victoria Woodhull to Susan B. Anthony, New York (January 2, 1873), _ECS & SBA_.
"I have been so bitterly assailed": Stoddard, pp. 315–316.
"I dread only the malignity": Ibid., p. 319.
"You will find him" and ff.: Chester, pp. 141–142.
"No Latin—no embellishments": Hale, p. 352.
"Be kind to Tilton": Ibid.
## Chapter 29 I Can Endure No Longer
"If you still believe": Henry Ward Beecher to Isabella Beecher Hooker (October 28, 1872), _IBH_ , original and copy by Henry E. Burton.
"At last the blow has fallen" and ff.: Isabella Beecher Hooker to Thomas Beecher (November 3, 1872), _IBH_.
"Dear Belle, To allow the devil": Thomas Beecher to Isabella Beecher Hooker (November 5, 1872), _IBH_.
"Don't write to me": Ibid.
"damned lecherous scoundrel": _Hartford Times_ (November 25, 1872); the _Patriot_ of Chariton, Iowa; Doyle, pp. 93–94. Elizabeth Cady Stanton to Isabella Beecher Hooker (November 19, 1872), _IBH_ ,. 11:16, confirms the above. Stanton writes, "Only think of that item going the rounds of the papers that I said in Maine that I was convinced that Mrs. Woodhull's statements were false. I said no such thing, but... I should be ashamed to use such abominable language as she put in my mouth."
"Dear Susan, I had supposed you knew": Elizabeth Cady Stanton to Susan B. Anthony, _ECS & SBA_, Reel 16; Lutz, _Created Equal_ p. 222.
"When a man is unlucky" and ff.: _New York Sun_ (November 8, 1872), New York Public Library Scrapbook VII.
" 'Entirely.' Wouldn't you think if God": Susan B. Anthony to Isabella Beecher Hooker, Rochester, New York (November 16, 1872), _ECS & SBA_. Reel 16; "Mrs. Tilton's Story," _New York Tribune_ (September 19, 1874), New York Public Library Scrapbook VII.
"It is she who should annul": Susan B. Anthony to Gerrit Smith (December 25, 1872), _ECS & SBA_. Reel 16.
"first accuser": Isabella Beecher Hooker to Mary Porter Chamberlin (November 4, 1874), _IBH_. 51 A2–9. Isabella's full explanation reads,
I have always thought and do still think that the evidence before her, _she was fully justified in her action_. I told my brother the same and added as long as women were _slain by the mere breath of suspicion against their chastity_ , I would never help to conceal the wrong doings of a man, though he were my own brother or son. I would not become his first accuser but when he was accused by trustworthy testimony I would not defend him and the higher his station and influence the less would I try to shield him.
I "has charged Mr. Beecher": _IBH_. "Justitia" (November 25, 1872), _Hartford Times;_ Doyle, pp. 92–97.
I "What did prompt you to betray": Elizabeth Cady Stanton to Isabella Beecher Hooker, Willard Hotel Washington (November 19, 1872), _IBH_.
"I cannot tell you my surprise": Elizabeth Cady Stanton to Isabella Beecher Hooker (January 21, 1873), _IBH_. "Steadfastness of my affection"—emphasis added.
"twisting the truth for their own advantage": Theodore Tilton's "True Story" as given to Edward H. G. Clark. Reprinted in the _Thunderbolt_ , Albany and Troy, New York, May 1873. Tilton made a similar statement in _CT_ , Vol. 2, p. 339.
"It seems to me that God": Isabella Beecher Hooker to Henry Ward Beecher (November 1, 1872), _IBH;_ Marshall, p. 331; Doyle, p. 587.
"but one honorable way": Ibid.
"I will write you a sisterly letter": Ibid.
"calm silence": Henry Ward Beecher to Isabella B. Hooker (November 9, 1872), _IBH_. original and copy by Henry E. Burton.
"Dear Brother, I can endure no longer": Isabella Beecher Hooker to Henry Ward Beecher (November 27, 1872), _IBH_.
"This is a disaster!" and ff.: _CT_ , Vol. 2, pp. 745–748. This and quotes that follow concerning this incident are excerpted from the testimony of Frank Moulton.
"Victoria's story": Elizabeth Cady Stanton to Isabella Beecher Hooker (November 3, 1873), _IBH_.
## Chapter 30 What Have We Done Now
"You have had a hard fight" and ff.: Denis Tilden Lynch, _"Boss" Tweed: The Story of a Grim Generation_ (New York: Boni and Liveright, 1927), p. 362.
"corrupt motives": This account is based on contemporary newspaper accounts of the Credit Mobilier Scandal and on information in Matthew Josephson, _The Robber Barons: The Great American Capitalists_ (New York: Harcourt, Brace, 1934), pp. 164, 355.
"Cut off your vices": W.E.B. Du Bois, _Black Reconstruction in America 1860–1880: An Essay Toward a History of the Part Which Black Folk Played in the Attempt to Reconstruct Democracy in America_ (New York: Russell & Russell, 1966), pp. 599–601.
"If the bond plan is a swindle": Susan B. Anthony to Lillie Devereux Blake (May 20, 1872), _ECS & SBA_, Reel 16.
"to do something every day for Jesus": See Broun and Leech, p. 116, and also Chapter 8, pp. 108–127, by Leech, for a full account of Comstock's repeated arrests of Victoria Woodhull and Tennessee Celeste Claflin.
"Those vile women jailbirds": Harriet Beecher Stowe to Hattie and Eliza Stowe (December 19, 1872), Harriet Beecher Stowe Center; Boydston et al., p. 288.
"Well-meaning friends" and ff.: Laura Cuppy Smith, "How One Woman Entered the Ranks of Social Reform: or, a Mother's Story," _WCW_ (March 1, 1873), pp. 3, 4, 5.
"Are you not the publisher": _New York Sun_ (January 10, 1872), New York Public Library Scrapbook IV.
"Comstock's defeated!": Laura Cuppy Smith quoted William Cullen Bryant, "The terrible syren has defeated you and charmed your cohorts and battalions to silence and inaction" _New York Evening Post_ (January 9, 1873). At this time the tabloid newspapers were putting out extra sections to cover Woodhull's speech as well as Tweed's trial and the appeal of Edward Ned Stokes for Fisk's murder.
"I come into your presence" and ff.": _WCW_ (January 25, 1873), pp. 3–8, 14, 15; Stern, _Victoria Woodhull Reader_ , Part A, Sexuality, 3, "The Naked Truth; or the Situation Reviewed," pages unnumbered.
"to run her to the earth" and ff.: Excerpted from "The Naked Truth; or the Situation Reviewed" (January 9, 1873), Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 3, pages unnumbered; _WCW_ (January 25, 1873), pp. 3–7, 13–15.
"What have we done now?" and ff.: _TS_ , p. 200.
"We have had so many": Victoria Woodhull to Lafayette Beech, Ludlow Street Jail, Cell 1, New York (December 2, 1872), collection of the author.
"caricatures of everything": Excerpted from Comstock's diary, February 25, 1873, as quoted in Broun and Leech, pp. 134–141. I have searched in vain for years for the original Comstock diaries. No institution or relative claims to know what has become of them. Although in 1927 Broun and Leech were permitted access to these diaries, and facsimile pages appear in their book, subsequent books and articles simply quote Broun and Leech but add no new information. The archives of the Young Men's Christian Association was the last place the diaries were said to exist. It is the opinion of several researchers and scholars that Anthony Comstock's diary entries concerning his compulsive masturbation were considered inappropriate in a purity crusader and were therefore destroyed by persons unknown.
"Disgusting and obscene" and ff.: This and the quotations that follow are drawn from the testimony of Anthony Comstock in the obscenity trials of Victoria Woodhull, Tennie C. Claflin, and Colonel James Blood, which took place January–March 1873, from the New York Public Library Scrapbook IV, as well as from contemporary newspaper accounts. An excellent summary of these proceedings appears in Broun and Leech, pp. 93–107, 114–127.
"was lighted up with spirit-light": _TS_ , p. 234.
"Sick in body, sick in mind": _VCWM_ , 3, Victoria Woodhull to the _New York Herald, TS_ , p. 288.
"wholly misconstrued" and ff.: _New York Sun_ and _New York World_ (February 3, 1873); _TS_ , p. 201.
"had reasons satisfactory": Ibid., p. 209.
"disgusting slanders on Lot": This account has been drawn from the _New York Times_ , March 29, p. 2, April 1, p. 1, April 3, p. 2, April 4, p. 4, April 14, p. 4, April 19, p. 2, April 22, p. 4, April 23, p. 3, April, 24, p. 3, April 29, p. 3, April 30, p. 3, May 3, p. 3, May 7, p. 2, May 13, p. 3, May 17, p. 2, May 20, p. 2, May 21, p. 2, May 23, p. 2, May 30, p. 4, May 31, p. 4.; _WCW_ April 12, 1873; "The Naked Truth; or the Situation Reviewed" in Stern, _Victoria Woodhull Reader_ , Sociology, Part 3, pp. 26–27; _The Train Ligue;_ Broun and Leech, Chapter 8; D.R.M. Bennett, _Anthony Comstock: His Career of Cruelty and Crime_ (New York: Da Capo Press, 1971), pp. 1022–1023; George Francis Train _My Life in Many States and in Foreign Lands_ (New York: D. Appleton and Company, 1902).
"There was present": Diary of Anthony Comstock (April 15, 1873), Broun and Leech, p. 112.
"a Plymouth Church betrayal": _VCWM, New York Sun_ , May 23, 1873.
"Mr. Comstock, informer-general": _VCWM, New York Telegram_ (May 20, 1873).
"That these women": _Indiana Times_ (May 9, 1873).
"Her persecutors will gain": Martha Coffin Wright to her daughter Ellen Garrison (March 12, 1873), _MCW_.
"The people of this country": _Brooklyn Eagle_ (November 7, 1873).
" 'Liberty of the Press' ": _Weekly Argus_ of Easton, Pennsylvania, as excerpted in Broun and Leech, p. 124.
"contrasting a living skunk" and ff.: _WCW_ (February 8, 1873).
"A beautiful boy" and ff.: _WCW_ (February 15, 1873).
"What would you advise" and ff.: _VCWM_ , New York Public Library, Scrapbook VI, trial testimony (January 16, 1874); _Brooklyn Eagle_ (January 20, 1874).
"a lady, a woman, a mother": _VCWM_ , New York Public Library Scrapbook VI, _Banner of Light_ (November 1, 1873).
"This is the most outrageous verdict": _TS_ , p. 227, trial testimony (March 12, 1874).
"Victoria Woodhull—Dead": _TS_ , p. 210. This headline appeared in the _New York Sun, New York Daily Graphic, Cleveland Leader_ , and various other publications.
## Chapter 31 A Monstrous Conspiracy
"for my own vindication" and ff.: Johnston, pp. 194–195.
"was not one of mere platonic affection": _Brooklyn Eagle_ , New York Public Library Scrapbook V (June 22, 1873).
"I will reserve": _TS_ , p. 214.
"I have just returned": Henry Ward Beecher to the editor of the _Brooklyn Eagle_ (June 30, 1873), New York Public Library Scrapbook V.
"There isn't force enough": Doyle, p. 361.
"It is high time" and ff.: The tripartite agreement was signed by Tilton, Bowen, and Beecher on April 2, 1872. Wilkeson released it to the press on May 29, 1873, along with his statement concerning Bowen's treachery. Doyle, p. 362.
"Mrs. Stanton asked me": Susan B. Anthony to Isabella Beecher Hooker Rochester, New York (July 14, 1873), _ECS & SBA_, Reel 17:242.
"unhandsome advances": Theodore Tilton's "True Story" published in the _Thunderbolt_ , Edward H. G. Clark, New York, Albany, and Troy: n.p., May 1873, Library of Congress; Marshall, p. 355.
"stood responsible for the story" and ff.: _IBH_. This account is derived from the letters of Paulina Wright Davis to Victoria Woodhull and Isabella Beecher Hooker, 1873, and from Elizabeth Cady Stanton's pseudonymous article under the nom de plume "Enquirer" in the _New York Daily Graphic_ (July 24, 1874); Doyle, pp. 353–354.
"It was even then almost common property": "True Story," Clark, the _Thunderbolt_.
"I think our defeat": Paulina Wright Davis to Elizabeth Cady Stanton (July 20, 1872), _ECS & SBA_, Reel 16.
"ought to be hanged" and ff.: "True Story," Clark, the _Thunderbolt_.
"True Story": Johnston, p. 186.
"Oh, Theodore, of all the men": Ibid., p. 187.
"I have determined": Henry Ward Beecher to Frank Moulton (June 1, 1873); Marshall, p. 365.
"Why did she not tell me" and ff: Statement of Emma Moulton, _CT;_ Rugoff, pp. 497–498.
"He walked up and down": Rugoff, _The Beechers_ , pp. 497–498.
"We can whisk the council" and ff.: Henry Ward Beecher to Frank Moulton. These proceedings are thoroughly documented in Doyle, pp. 114–142 (Chapters V and VI). "My Dear Frank:—I am indignant": Henry Ward Beecher to Frank Moulton (March 25, 1874); Marshall, pp. 367–368.
"I believe that the infamous women" and ff.: Dr. Bacon's speech and Tilton's reply are to be found in Marshall, pp. 40–42, 43–63.
"In regard to the scandal": _Brooklyn Union_ (March 28, 1874), Thomas Shearman interview. Mr. Shearman was a founding partner of a law firm still in existence, Shearman and Stirling. Marshall, p. 48.
"confused Mrs. Tilton with Victoria Woodhull": Thomas Shearman to the _Brooklyn Union_ (April 2, 1874); Marshall, p. 49.
"It was not I": Theodore Tilton to Reverend Leonard Bacon, D.D., Brooklyn (April 3, 1874).
"Theodore, the Spirits say unto me": Marshall, p. 60; Oliver, p. 197.
"deeply hurt": _TS_ , p. 207.
"Mrs. Brooker in a drunken or insane rage": _TS_ , p. 216.
"Dead at 31," and ff.: _WCW_ (July 23, 1873); _TS_ , p. 217; "Tried as by Fire," Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5, pp. 27, 28.
"ready in this connection" and ff.: The account of Benjamin Tucker, _TS_ , p. 254.
"I never had sexual intercourse": _TS_ , p. 254.
"This is my lover": _TS_ , pp. 222–223.
"would hold a rally": This account of the Tompkins Square Rally of January 13, 1874, is taken from the _New York Times_ , January 16, 18, 24, 1874.
"Every consideration of expediency" and ff.: "Tried as by Fire," Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5, pp. 6–8.
"Repulsions, discontent and mutual torment": Ibid., 5, pp. 9, 22; _TS_ , p. 197.
## Chapter 32 Human Hyenas
"socially, financially and in every way": _CT_ , Vol. 2, p. 528.
"I cannot delay": Henry Ward Beecher's statement to the _Brooklyn Daily Argus_ , published July 22, 1874.
"I will never take": Marshall, p. 188; Shaplen, p. 187; _The Great Scandal: History of the Beecher-Tilton Case_. All the documents and letters (New York: The American News Company, 1874), TB transcript.
"dreadful secret" and ff.: _PCH_. All quotes and exhibits are contained in Marshall, the statement of Lib (Elizabeth) Tilton, pp. 181–188, Tilton's cross-examination, pp. 130–177, 189–214.
"criminal commerce" and ff: Theodore Tilton's statement and cross-examination before the Plymouth Church Committee, _PCH_. All quotes and exhibits are contained in Marshall, Theodore Tilton's "Sworn Statement," pp. 112–130, Theodore Tilton's cross-examination, pp. 130–177, 189–214.
"as the umpire and peacemaker" and ff.: Frank Moulton's testimony to the Plymouth Church Committee, _CT_ , Vol. 1, pp. 307–390, with exhibits, Moulton's formal first statement, pp. 109–112, Moulton's second appearance and exhibits, pp. 227–240.
"Four years ago" and ff.: _PCH_. All in Marshall: the first statement of the Reverend Henry Ward Beecher, pp. 177–181, "Mr. Beecher's Defense," pp. 251–286, "Mr. Beecher's Cross Examination," pp. 286–307, "Final Attack," pp. 566, 603–609.
"Human hyenas!": _PCH;_ Beecher "final attack," Marshall, p. 273.
"Elizabeth Cady Stanton and": The Friday _New York Herald_ (July 31, 1874), supplement, New York Public Library Scrapbook VII.
"During my whole public career": _New York Argus_ , correspondent, Rochester, New York (July 3, 1874), also in the fourth edition of the _New York Argus_ , (August 2, 1874).
"You have no doubt" and ff.: Elizabeth Cady Stanton, _New York Daily Graphic_ (August 6, 1874).
"Well, well, aren't they": Susan B. Anthony to Olympia Brown, Rochester, New York (August 6, 1874), _ECS & SBA_, Reel 18.
"Offended Susan, come right down": Elizabeth Cady Stanton to Susan B. Anthony, Tenafly, New Jersey (July 30, 1874), _ECS & SBA_.
"attempted my ruin" and ff.: _PCH_. Testimony of the Tilton's ward, Bessie Turner, Marshall, pp. 390–400; Anthony on lap, p. 393.
"half an idiot" and ff.: Statement made by Bessie Turner, Marshall, p. 393. Susan B. Anthony's answer and explanation, "All men have declared," appeared in the _Rochester Union_ (August 24, 1874), p. 3.
"Tilton accused his wife": _Chicago Tribune_ (October 13, 1874).
"Provided I did tell it": _ECS & SBA_, Reel 18.
"Whatever comes to those": Susan B. Anthony to Daniel R. Anthony (August 22, 1874), _ECS & SBA_, Reel 18; Katharine Anthony, p. 317; Harper, Vol. 1, p. 459.
"It is too bad": Isabella Beecher Hooker's comment on Doyle's _The Romance of Plymouth Church_ , fragment of letter, _IBH_. 100 C11, 1874.
"The case has gone out": Fragment of letter, _IBH_. 100 C12, 1874.
"I know from a private" and ff.: _New York Daily Graphic_ (August 21, 1874); Marshall, p. 387.
"She was devotedly attached": George H. Beecher's letter, Marshall, p. 43 (emphasis added).
"an honorable man": _Brooklyn Eagle_ (September 25, 1874), New York Public Library Scrapbook VIII.
"If Henry were in the habit": Interview with Reverend William H. Beecher, _Chicago Post and Mail_ Quly 24, 1874), New York Public Library Scrapbook VIII.
"Sam says Livy shall not cross": Molly Clemens to her mother-in-law, Jane Clemens (November 26, 1872), Kenneth R. Andrews, _Nook Farm: Mark Twain's Hartford Circle_ (Cambridge, MA: Harvard University Press, 1950), p. 39.
"What a holocaust of womanhood": New York Public Library Scrapbook VIII, _Chicago Tribune_ (September 18, 1874), Lutz, _Created Equal_. pp. 226–227.
## Chapter 33 Daniel in the Lion's Den
"You will be Daniel" and ff.: This account of Friday evening, August 28, 1874, and all quotations contained therein are drawn from Marshall, "The Report of the Investigating Committee," pp. 405–433, "Scene at Plymouth Church," pp. 433–446, as well as from newspapers such as the _New York Argus_ (August 30, 1874) and _New York Times_ (August 30, September 4, 1874), and New York Public Library Scrapbook VIII.
"Tried as by Fire": "Tried as by Fire, or The True and The False, Socially," 1874, in Stern, _Victoria Woodhull Reader_ , Part A, Sexual Relations, 5, pp. 1–44.
"published a pamphlet" and ff.: The Treat pamphlet, as it was commonly called, contains the quotations and accusations that follow. The proper name of the document is _Beecher, Tilton, Woodhull, The Creation of Society: All Four of Them Exposed, and if Possible Reformed and Forgiven, in Dr. Treat's Celebrated Letter to Victoria C. Woodhull_ , by Joseph Treat, M.D. Treat's pamphlet, though never substantiated, has, I believe, framed the common perception that Victoria Woodhull and Tennessee C. Claflin were nothing more than self-aggrandizing prostitutes—clearly a misconception.
"I am charged with seeking": Doyle, p. 431.
"Can anyone tell me where I am?": _Chicago Mail_ (January 2, 1875), New York Public Library Scrapbook VII.
"A Day Memorable" and ff.: The _Tilton v. Beecher_ civil trial for alienation of affection was covered extensively in newspapers and later in books and compendiums of trial testimony. A full account of the testimony and exhibits can be found in Doyle, Marshall, and _CT_.
Literally millions of words were written on this trial, which touched a nerve in the American psyche of the day. So extensive was the coverage that one might compare it to that of the O. J. Simpson trial. There were, or course, no television cameras, but sketches of all the subjects and proceedings appeared in the newspapers daily. I counted twelve New York City newspapers on one day alone with virtually the same extensive coverage of this trial, which was then called "The Trial of the Century."
"I believe they kissed each other" and ff.: George Templeton Strong was also a well-known New York lawyer and the comptroller of Trinity Church. See _The Diary of George Templeton Strong: 1835–1875_ , 4 volumes, edited by Allan Nevins and Milton Halsey Thomas (New York: The Macmillan Co., 1952), Vol. 4, p. 552. At another point, after a quarrel, Henry impetuously kissed Tilton on the lips, and later testimony indicated that Lib (Elizabeth) Tilton entered her bedroom and found Henry sitting on Theodore's knee. See Rugoff, _The Beechers_ , p. 485.
"I believe Mr. Tilton": _TS_ , p. 233.
"kindness and sympathy" and ff.: The testimony of Eunice Beecher, _CT_ , Vol. 2, pp. 632–640. See "The Beecher Trial: A Review of the Evidence," _New York Times_ (July 3, 1875), fifth edition.
"There is only one good result": "The Beecher Trial: A Review of the Evidence," _New York Times_ (July 3, 1875), fifth edition.
## Chapter 34 A Meteor's Dash
"was the link": New York Public Library Scrapbook VIII, _Brooklyn Eagle_ (October 13, 1875).
"an adulterer, a perjurer" and ff.: Shaplen, p. 263.
"I have not been pursued" and ff.: Rugoff, _The Beechers_ , p. 500.
"I do not care for you": Ibid.
"Thanks be unto God" and ff.: Isabella Beecher Hooker diary (Thursday, May 11, 1876), Hartford, Connecticut, _IBH_. 1001 D4–9, 75.
"a deep and solemn peace": Ibid.
"Love begets love—love is God.": Isabella Beecher Hooker diary (Monday, September 18, 1876), _IBH_. 1002 D7–14, 75.
"drawn the veil" and ff.: Isabella Beecher Hooker diary, Hartford, Connecticut (August 8, 1876), _IBH_. pp. 84–88, the Connecticut Historical Society.
"From that time": Ibid.
"but a delusion and a mockery": Isabella Beecher Booker to Mr. Fishbough, Hartford, Connecticut (July 15, 1876), _IBH_. 1001 B8–75.
"the light of the eternities": Ibid.
"Susie Howard": Susan Howard wrote this letter February 5, 1875. Her letter and Mark Twain's reaction are detailed in Kenneth R. Andrews, _Nook Farm_ , p. 40.
"an ascendancy" and ff.: Harriet Beecher Stowe to Mary Beecher Perkins (May 2, 1875), _IBH_
"In the letter which Belle sent": Harriet Beecher Stowe to Mary Beecher Perkins (February 28, 1875), _IBH_.
"the beginnings of the Church": Rugoff, _The Beechers_ , p. 584.
"Of all the persons": Lucy Stone in the _Woman's Journal_ (August 29, 1874), Blackwell, p. 250.
"A few weeks ago": Lib (Elizabeth) Tilton to her lawyer, _New York Times_ (April 13, 1878); Shaplen, p. 266.
"an unbalanced clairvoyant": Ibid., p. 267.
"a female ailment" and ff.: _TS_ , p. 267.
"The grandest woman": (October 6, 1876), _VCWM_ , Reel 1, Nos. 33–34.
"One jar Balmain's Luminous Paint": Ruth Brandon, _The Spiritualists: The Passion for the Occult in the Nineteenth and Twentieth Centuries_ (New York: Knopf, 1983), pp. 100–101. "It has been supposed": _New York Argus_ (October 20, 1876).
"death blow": Howard Kerr, _Mediums, and Spirit-Rappers, and Roaring Radicals: Spiritualism in American Literature, 1850–1900_ (Chicago: University of Illinois Press, 1972), p. 119.
"a muffled hammer": Brandon, p. 228.
"to shake off a serpent" and ff.: Isabella Beecher Hooker to Victoria Woodhull, _IBH_. circa 1876, uncatalogued.
"God is love and love is God": General correspondence, _VCWM_ , Reel 1, Box 3, Folder 3, p. 1.
"scandalum magnatum" and ff.: Years later _The Nation_ strongly corroborated this by observing that the woman's rights movement "received a severe blow from the figure cut by many of its leading supporters in the Tilton-Beecher scandal and by the adhesion to Victoria Woodhull.... From this it can hardly be said ever to have recovered. It came to have in the eyes of the general public the air of being the first stage... of "free love." If it were not for the greater respectability of the Associations in Massachusetts... the movement would have kept itself alive here with great difficulty." _The Nation_ (February 19, 1885), taken from Andrea Moore Kerr.
"Victoria Woodhull has done": Stanton's interview on Victoria Woodhull, Tenafly, New Jersey. Printed in the _Newark Sunday Call_ (January 2, 1876), _ECS & SBA_, Reel 18.
"Do you mean to make good": Eric Foner, p. 567.
"The moral atmosphere is": Frederick Douglass to Gerrit Smith (September 24, 1874), Gerrit Smith Papers, Syracuse University; Philip S. Foner, _The Life and Writings of Frederick Douglass_ , 534.
"You say you have emancipated us": Frederick Douglass, 1876. Dorothy Sterling, _The Troubles They Seen: Black People Tell the Story of Reconstruction_ (Garden City, NY: Doubleday & Company, 1976). (This title later changed to _Nobody Knows the Trouble I Seen: The Story of Reconstruction in the Words of African Americans_.) A version that differs somewhat appears in Philip S. Foner, _The Life and Writings of Frederick Douglass_ , p. 96.
"Why have I been deprived": Wayne Andrews, p. 176.
"That was a good prayer": Barbara Goldsmith, _Little Gloria... Happy at Last_ , p. 80.
"my box of letters": _VCWM_ , Reel 2, 79–80, Box 2.
## Chapter 35 The Last Enemy
"unanswered": Chester, p. 217.
"I suspect Whitelaw is _actooally_ engaged": Chester, p. 216.
"I give it up": Benjamin Butler to Anna Dickinson, Atlantic City (July 22, 1878), Anna E. Dickinson Papers, 7:767.
"pained" and ff.: Benjamin Butler to Anna Dickinson, Boston (June 14, 1888), ibid., 7:805.
"astounding and dastardly request": 7:813. Butler's letter is undated but Anna gives a specific date when she writes to Butler, "your last letter to me—that of June 14, 1888..."
"I want to see or hear": Anna Dickinson to Benjamin Butler (April 13, 1982), ibid. (A notation on the letter indicates that it is misdated and is actually from c. 1888.) 7:809.
"the tainted blood": Chester, pp. 267, 268.
"to make a man": Ibid., p. 268.
"As I opened the envelope": Susan B. Anthony to Anna Dickinson (November 5, 1895), Anna E. Dickinson Papers, 6:289.
"cheapened" and ff.: Andrea Moore Kerr, p. 236.
"not care so much": Ibid.
"Alice, make the world better!": Blackwell, p. 282.
"I feel that suffrage": Elizabeth Cady Stanton, Tenafly, New Jersey, in an interview on the subject of Victoria Woodhull (December 13, 1875), printed in the _Newark Sunday Call_ (January 2, 1876); _ECS & SBA_, Reel 18:510.
"worth $10,000 to see the expression": Tenafly, New Jersey (December 1, 1880), Stanton and Blatch, Vol. 2, p. 180.
"the gas was lighted" and ff.: London (December 11, 1882), Vol. 2, p. 200.
"Logically, our enfranchisement": (February 20, 1902) Vol. 2, p. 363.
"I should like to be": Griffith, p. 218.
"Failure is impossible": Lynn Sherr, _Failure Is Impossible: Susan B. Anthony in Her Own Words_ (New York: Times Books, 1995), p. 324.
"I have often been ignored": Isabella Beecher Hooker to Clara B. Colby (February 16, 1903), _IBH_. If Isabella's belief in the spirits called her credibility into question, she herself never doubted the validity of the spirits. In 1888, Isabella wrote her daughter Alice Hooker Day, "[N] early all the women I meet are more or less interested in some form of spiritual healing. It is simply amazing how the domain of the spirit is extending itself and what is before us in the near future no one can prognosticate." Isabella Beecher Hooker to Alice Hooker Day, 1888, _IBH_. 60 A5.
"passed over" and ff.: Isabella Beecher Hooker diary (Monday, September 18, 1876), _IBH_. 102 D7–14, p. 45.
"A bloody ending": Browder, p. 188.
"this Irish smut dealer": Broun and Leech, p. 230.
"Your Honor, this woman": _The Masses_ (September 15, 1915), reproduced in Broun and Leech, p. 254.
"an eminent barrister": Victoria Woodhull autobiographical sketch, 1895, London, _VCWM_ , Reel 3, no. 173; _New York World_ (February 17, 1897).
"During no part of my life" and ff.: _VCWM_ , Reel 3, no. 136; Johnston, p. 264.
"high priest of debauchery": Ibid., p. 267.
"Stephen Pearl Andrews!": Ibid.
"near being the victim": _The Humanitarian_ (February 12, 1892).
"I never knew that love": _TS_ , p. 368.
"There is an apparition" and fT.: Ibid., p. 369.
"other side": Isabella Beecher Hooker to Victoria Woodhull (September 14, 1906), _IBH_. So convinced was Isabella Beecher Hooker that her mother was guiding her that on September 14, 1906, she wrote Victoria Woodhull asking her to work with Isabella and with her deceased mother, Harriet Porter. Victoria Woodhull Martin, by then a conventional English matron, undoubtedly trying to maintain her reputation, wrote back that she had visited the spirit of Isabella's mother, who told her that they might work together in the heavenly realm but should avoid each other on earth. _VCWM_ , Reel 2, Box 2.
"international country salon": _VCWM_ , Reel 3; _New York World_ (February 17, 1897).
"would rather die than live": _GAT_ , p. 14.
"constant companions": Ibid., p. 8.
"the last enemy": Ibid., p. 33; _VCWM_ , Reel 2, no. 85–88, taken from Victoria Woodhull Martin letter from London to the _Boston Globe_ , circa 1888.
"Sitting here today": Victoria Woodhull fragmentary autobiography, 1895, London, _VCWM_ , Reel 1, no. 1, 12.
"extreme unction from her priest" and ff.: _GAT_ , p. 32, also in Victoria Woodhull Martin letter from London to the _Boston Globe_ , circa 1888; _VCWM_ , Boston Public Library Manuscript Collection.
"the idea that the time": Ibid., p. 33.
"The deeper I delve": _VCWM_ , Reel 3, no. 45–48 and a duplicate no. 49–52, written by Victoria Woodhull Martin (July 1895).
# _Bibliography_
## _Books_
Abbot, Austin. _The Official Report of the Trial of Henry Ward Beecher with Notes and References_. 2 Volumes. New York: George W. Smith & Co., Law Booksellers and Publishers, 1875, 1876. Abbreviated as _CT_.
Abbott, Lyman, D.D. _Henry Ward Beecher, A Sketch of His Career: With Analyses of His Power as a Preacher, Lecturer, Orator, and Journalist, and Incidents and Reminiscences of His Life_. New York: Funk & Wagnalls, 1883.
Abbott, Lyman, D.D. _Henry Ward Beecher_. Boston: Riverside Press, 1903. Reprint Houghton Mifflin.
Abbott, Lyman, D.D. _Reminiscences_. Boston: Houghton Mifflin, 1915.
Abels, Jules. _Man On Fire: John Brown and the Cause of Liberty_. New York: Macmillan, 1971.
Allen, Frederick Lewis. _The Big Change_. New York: Harper & Brothers, 1952.
Alter, Robert, and Frank Kermode, eds. _The Literary Guide to the Bible_. Cambridge, MA: Belknap Press of Harvard University Press, 1987.
Andrews, Kenneth R. _Nook Farm: Mark Twain's Hartford Circle_. Cambridge, MA: Harvard University Press, 1950.
Andrews, Wayne. _The Vanderbilt Legend_. New York: Harcourt, Brace, 1941.
Anthony, Katharine. _Susan B. Anthony: Her Personal History and Her Era_. Garden City, NY: Doubleday, 1954.
Aries, Philippe. _Centuries of Childhood: A Social History of Family Life_. New York: Vantage Books, 1962.
Asbury, Herbert. _The Gangs of New York_. New York: Alfred A. Knopf, 1928.
Asbury, Herbert. _The Barbary Coast_. New York: Alfred A. Knopf, 1933.
_A Sense of History: The Best Writings From the Pages of American Heritage_. Introduction by Byron Dobell. New York: American Heritage Press, 1985.
Asimov, Isaac. _The Golden Door: The United States From 1865 to 1918_. Boston: Houghton Mifflin, 1977.
Atherton, Lewis. _Main Street on the Middle Border_. Bloomington: Indiana University Press, 1954.
Auerbach, Nina. _Woman and the Demon: The Life of a Victorian Myth_. Cambridge, MA: Harvard University Press, 1982.
Badeau, Adam. _Grant in Peace from Appomattox to Mount McGregor: A Personal Memoir_. Hartford, CT: Books for Libraries Press, 1887.
Barker-Benfield, G. J. _The Horrors of the Half-Known Life: Male Attitudes Toward Women and Sexuality in 19th Century America_. New York: Harper & Row, 1976.
Barkun, Michael. _Crucible of the Millennium: The Burned-Over District of New York in the 1840s_. Syracuse, NY: Syracuse University Press, 1986.
Barry, Kathleen. _Susan B. Anthony: A Biography of a Singular Feminist_. New York: New York University Press, 1988.
Beebe, Lucius. _The Big Spenders_. Garden City, NY: Doubleday, 1966.
Beecher, Catharine E. _A Treatise on Domestic Economy, For The Use of Young Ladies At Home, And At School_. Boston: Marsh, Capen, Lyon, and Webb, 1841.
Beecher, Catharine E., and Harriet Beecher Stowe. _The American Woman's Home Or, Principles of Domestic Science; Being A Guide To The Formation And Maintenance of Economical, Healthful, Beautiful And Christian Homes_. 4th ed. Hartford, CT: Stowe-Day Foundation, 1987.
Beecher, Henry Ward. _Norwood: or Village Life in New England_. New York: J. B. Ford, 1868.
Beecher, Henry Ward. _The Life of Jesus Christ_. 2 volumes. New York: J. B. Ford, 1871.
Beecher, Henry Ward. _The Original Plymouth Pulpit: Sermons of Henry Ward Beecher In Plymouth Church, Brooklyn_. [1868–1873]. From stenographic reports by T. J. Ellinwood. 10 volumes. New York: Fords, Howard & Hulbert, 1893.
Beecher, Henry Ward. _Autobiographical Reminiscences_. New York: F. A. Stokes, 1898.
Beecher, Henry Ward. Plymouth Church. New York Public Library Compilations. Printed sermons taken from 4 volumes.
Beecher, Lyman, D. D. _Autobiography, Correspondence, Etc. of Lyman Beecher, D. D_. Edited by Charles Beecher. Vol. 1, New York: Harper & Brothers, Franklin Square, 1864. Vol. 2, New York: Harper & Brothers, Franklin Square, 1865.
_Beecher-Tilton Investigation: The Scandal of the Age_. Philadelphia: Barclay & Co., 1874.
Beecher, Wm. C., and Rev. Samuel Scoville, assisted by Mrs. Henry Ward Beecher. _A Biography of Rev. Henry Ward Beecher_. New York: Charles L. Webster, 1888.
Benet, Stephen Vincent. _John Brown's Body_. New York: Holt, Rinehart and Winston, 1928.
Bennett, D. R. M. _Anthony Comstock: His Career of Cruelty and Crime_. New York: Da Capo Press, 1971.
Bentley, George R. _A History of the Freedmen's Bureau_. Philadelphia: University of Pennsylvania Press, 1955.
Berg, Barbara. _The Remembered Gate: Origins of American Feminism 1800–1860_. New York: Oxford University Press, 1978.
Bernstein, Iver. _The New York City Draft Riots: Their Significance for American Society and Politics in the Age of the Civil War_. New York: Oxford University Press, 1990.
Bernstein, Samuel. _The First International in America_. New York: Augustus M. Kelley, 1962.
Bigelow, John. _Retrospections of an Active Life_. Vol. 5. New York: Doubleday, Page, 1913.
Blackstone, Sir William. _The Great Commentaries on the Laws of England 1865–69_.
Blackwell, Alice Stone. _Lucy Stone, Pioneer of Woman's Rights_. Boston: Little, Brown, 1930.
Blanch, Lesley. _The Wilder Stories of Love_. New York: Simon and Schuster, 1954.
Blatch, Harriot Stanton, and Alma Lutz. _Challenging Years: The Memoirs of Harriot Stanton Blatch_. New York: G. P. Putnam's Sons, 1940.
Blight, David W. _Frederick Douglass' Civil War: Keeping Faith in Jubilee_. Baton Rouge: Louisiana State University Press, 1989.
Bloch, George. J., trans. _Mesmerism: A Translation of the Original Medical and Scientific Writings of F. A. Mesmer, M. D_. Los Angeles: William Kaufmann, 1980.
Blum, Stella, ed. _Ackermann's Costume Plates: Women's Fashions in England, 1818–1828_. New York: Dover Publications, 1978.
Boller, Paul F., Jr. _Presidential Campaigns_. New York: Oxford University Press, 1984.
Bontemps, Arna. _Free At Last: The Life of Frederick Douglass_. New York: Dodd, Mead, 1971.
Boydston, Jeanne, Mary Kelly, and Anne Margolis. _The Limits of Sisterhood: The Beecher Sisters on Woman's Rights and Woman's Sphere_. Chapel Hill: University of North Carolina Press, 1988.
Boyer, Paul S. _Purity in Print: The Vice-Society Movement and Book Censorship in America_. New York: Charles Scribner's Sons, 1968.
Brandon, Ruth. _The Spiritualists: The Passion for the Occult in the Nineteenth and Twentieth Centuries_. New York: Knopf, 1983.
Braude, Ann. _Radical Spirits: Spiritualism and Women's Rights in Nineteenth-Century America_. Boston: Beacon, 1989.
Bredvold, Louis I., ed. _Lord Byron: Don Juan and Other Satirical Poems_. New York: Odyssey Press, 1935.
Britten, William. _Art Magic; or mundane, sub-mundane and super-mundane spiritualism..._ (1876) 1898 reprint.
Brockett, Linus P. _Men of Our Day_. Philadelphia: Ziegler & McCurdy, 1868.
Brockway, Beman. _Fifty Years in Journalism. Embracing Recollections and Personal Experiences with an Autobiography_. Watertown, NY: Daily Times Printing and Publishing House, 1891.
Brough, James. _The Vixens: A Biography of Victoria and Tennessee Claflin_. New York: Simon and Schuster, 1980.
Broun, Heywood, and Margaret Leech. _Anthony Comstock: Roundsman of the Lord_. New York: Albert & Charles Boni, 1927.
Browder, Clifford. _The Wickedest Woman in New York: Madame Restell, the Abortionist_. Hamden, CT: Archon Books, 1988.
Brown, Vinson. _Great Upon the Mountain_. New York: Macmillan, 1977.
Browne, Junius Henri. _The Great Metropolis: A Mirror of New York_. Hartford, CT: American Publishing Co., 1870.
Bruce, Robert V. _1877: Year of Violence_. Indianapolis: Bobbs-Merrill, 1959.
Buhle, Mari Jo. _Women and American Socialism, 1870–1920_. Urbana, Chicago, London: University of Illinois Press, 1981.
Buhle, Mari Jo, and Paul Buhle, eds. _The Concise History of Woman Suffrage: Selections from the Classic Work of Stanton, Anthony, Gage and Harper_. Urbana: University of Illinois Press, 1978.
Butler, Benjamin F. _Autobiography and Personal Reminiscences of Major-General Benjamin F. Butler; Butler's Book_. Boston, 1892.
Cadwallader, M. E. _Hydesville in History_. Chicago: Progressive Thinker Publishing House, 1917, 1922.
Callow, Alexander B., Jr. _The Tweed Ring_. New York: Oxford University Press, 1966.
Campbell, Joseph. _The Hero with a Thousand Faces_. Princeton, NJ: Princeton University Press, 1949.
Capers, Gerald M. _Occupied City: New Orleans Under the Federals 1862–1865_. Lexington, KY, 1865.
Carman, Harry J. _Social and Economic History of the United States_. Vol. 2, _The Rise of Industrialism 1820–1875_. New York: D. C. Heath, 1934.
Carter, Hodding. _The Angry Scar: The Story of Reconstruction_. Garden City, NY: Doubleday, 1959.
Carter, Paul. _The Spiritual Crisis of the Gilded Age_. DeKalb: Northern Illinois University Press, 1971.
Caskey, Marie. _Chariot of Fire: Religion and the Beecher Family_. New Haven, CT, and London: Yale University Press, 1978.
Catton, Bruce. _A Stillness at Appomattox_. New York: Washington Square Books, 1958.
Caverno, C. _The Brooklyn Council_. Chicago: Brewster & Bro., 1874.
Chester, Giraud. _Embattled Maiden. The Life of Anna Dickinson_. New York: G. P. Putnam's Sons, 1951.
Churchill, Allen. _The Splendor Seekers_. New York: Grosset & Dunlap, 1974.
Claflin, Tennessee, Lady Cook. _Essays on Social Topics_. London: Roxburghe Press, c. 1895.
Clark, Clifford E., Jr. _Henry Ward Beecher: Spokesman for a Middle-Class America_. Urbana: University of Illinois Press, 1978.
Cleveland, Cecilia. _The Story of a Summer, or Journal Leaves from Chappaqua_. New York: G. W. Carleton, 1874.
Clews, Henry. _Fifty Years in Wall Street_. New York: Irving Publishing, 1908.
Cohen, Daniel. _Curses, Hexes and Spells_. Philadelphia and New York: J. B. Lippencott, 1974.
Conant, Mrs. J. H. _Flashes of Light from the Past Spirit Land_. Boston: Colby & Rich.
Cooper, George. _Lost Love: A True Story of Passion, Murder, and Justice in Old New York_. New York: Pantheon Books, 1994.
Cox, La Wanda. _Lincoln and Black Freedom_. Chicago: University of Illinois Press, 1985.
Crabtree, Adam. _Animal Magnetism, Early Hypnotism, and Psychical Research 1766–1925_. White Plains, NY: Kraus International Publications, 1988.
Crabtree, Adam. _From Mesmer to Freud: Magnetic Sleep and the Roots of Psychological Healing_. New Haven, CT: Yale University Press, 1994.
Crichton, Michael. _Travels_. New York: Knopf, 1988.
Crimmins, John D. _Irish-American Historical Miscellany_. New York: The author, 1905.
Croffut, A. _The Vanderbilts and the Story of Their Fortune_. New York: Belford, Clarke, 1886.
Cronise, Adelbert. _The Beginnings of Modern Spiritualism in and Near Rochester_. Rochester, NY: Rochester Historical Society, 1925.
Cross, Barbara M., ed. _The Autobiography of Lyman Beecher_. 2 volumes. Cambridge, MA: Belknap Press of Harvard University Press, 1961.
Cross, Whitney R. _The Burned-over District: The Social and Intellectual History of Enthusiastic Religion in Western New York, 1800–1850_. New York: Harper & Row, 1950.
Danciger, Elizabeth. _Homeopathy: From Alchemy to Medicine_. Rochester, VT: Healing Arts Press, 1987, 1988.
Darewin, G. S. _Synopsis of the Lives of Victoria C. Woodhull (now Mrs. John Biddulph Martin) and Tennessee Claflin (now Lady Cook), The First Two Lady Bankers and Reformers of America_. London: 1891.
Darnton, Robert. _Mesmerism and the End of the "Enlightenment in France._ " Cambridge, MA: Harvard University Press, 1968.
Davenport, Reuben Briggs. _The Death-Blow to Spiritualism: Being the True Story of the Fox Sisters. As Revealed by Authority of Margaret Fox Kane and Catherine Fox Jencken_. New York: G. W. Dillingham, 1897.
David, Deirdre. _Intellectual Women and Victorian Patriarchy: Harriet Martineau, Elizabeth Barrett Browning, George Eliot_. Ithaca, NY: Cornell University Press, 1987.
Davis, Andrew Jackson. _The Philosophy of Spiritual Intercourse_. New York: Fowler & Wells, 1851.
Davis, Andrew Jackson. _The Principles of Nature, Her Divine Revelations and a Voice to Mankind_. (Trance lectures)
Davis, Burke. _Sherman's March_. New York: Random House, 1980.
DeLeon, David. _The American as Anarchist: Reflections on Indigenous Radicalism_. Baltimore: Johns Hopkins University Press, 1978.
DeMorgan, Sophia. _From Matter to Spirit_. (1863).
Depew, Chauncey M., L. L. D., United States senator from the state of New York, ed-in-chief; Nathan Haskell Dole, Caroline Ticknor, Thomas Charles Quinn, assoc. eds. _The Library of Oratory: Ancient and Modern with Critical Studies of the World's Great Orators by Eminent Essayists_. Vol. 15. New York: E. R. DuMont, 1896.
Dickerson, K. _The Philosophy of Mesmerism..._ 1842.
Dijkstra, Bram. _Idols of Perversity_. New York: Oxford University Press, 1986.
Dingwall, Eric John, ed. _Abnormal Hypnotic Phenomena: A Survey of Nineteenth-Century Cases_. Vol. 4. United States of America, by Allan Angoff; Great Britain, by Eric J. Dingwall. London: J. & A. Churchill, 1967–68.
Ditzion, Stanley. _Marriage, Morals and Sex in America_. New York: Bookman Associates, 1953.
Dixon, William Hepworth. _Spiritual Wives_. 2 volumes. London: Hurst and Blackett, 1868.
Doctorow, E. L. _Ragtime_. New York: Random House, 1975.
Douglas, Ann. _The Feminization of American Culture_. New York: Knopf, 1977.
Douglass, Frederick. _Life and Times, Written by Himself_. Edited by Rayford W. Logan. New York: Collier Books, 1962. Facsimile edition, New York: Citadel Press, 1983.
Douglass, Frederick. _Narrative of The Life of Frederick Douglass, An American Slave_. Edited by Houston A. Baker Jr. New York: Penguin Group, 1986.
Douglass, Frederick. _My Bondage and My Freedom_. Edited by William L. Andrews. Urbana & Chicago: University of Illinois Press, 1987.
Doyle, J. E. P., compiler. _The Romance of Plymouth Church. Plymouth Church and Its Pastor, or Henry Ward Beecher and His Accusers_. Hartford, CT: Park Publishing Co., 1874.
Druesedow, Jean L. _In Style: Celebrating Fifty Years of the Costume Institute From Queen to Empress Victorian Dress 1837–1877_. New York: Metropolitan Museum of Art.
DuBois, Ellen Carol. _Feminism and Suffrage: The Emergence of an Independent Women's Movement in America 1848–1869_. Ithaca, NY: Cornell University Press, 1978.
DuBois, Ellen Carol. ed. _The Elizabeth Cady Stanton-Susan B. Anthony Reader: Correspondence, Writings, Speeches_. Revised edition. Boston: Northeastern University Press, 1981.
Dursey, Leslie, and Janice Devine. _Fare Thee Well_. New York: Crown, 1964.
Edmiston, Susan, and Linda D. Cirino. _Literary New York: A History and Guide_. Boston: Houghton Mifflin, 1976.
Edmonds, I. G. _The Girls Who Talked to Ghosts: The Story of Katie and Margaretta Fox_. New York: Holt, Rinehart and Winston, 1979.
Edwards, Stewart, ed. _The Communards of Paris, 1871_. Ithaca, NY: Cornell University Press, 1973.
Eliade, Mircea. _The Sacred and the Profane: The Nature of Religion_. New York: Harcourt Brace Jovanovich, 1959.
Ellington, George. _The Women of New York, or the Underworld of the Great City_. New York: New York Book Company, 1869. Reprint, New York: Arno Press, 1971.
Ellis, Dr. John B. _Free Love and Its Votaries_. New York, 1870.
Evans, Warren Felt. _The New Age and Its Messenger_. Boston, 1864.
Everdell, William R., and Malcolm Mackay. _Rowboats to Rapid Transit: A History of Brooklyn Heights_. New York: Brooklyn Heights Association, 1973.
Ewing, Elizabeth. _Underwear: A History_. New York: Theatre Arts Books, 1972.
Farren, David. _Living with Magic_. New York: Simon and Schuster, 1974.
Feaver, William. _Masters of Caricature_. New York: Knopf, 1981.
Firestone, Shulamith. _The Dialectic of Sex_. New York: William Morrow, 1970.
Flexner, Eleanor. _Century of Struggle: The Woman's Rights Movement in the United States_. Cambridge, MA: Belknap Press of Harvard University Press, 1959.
Foner, Eric. _Reconstruction: America's Unfinished Revolution, 1863–1877_. New York: Harper & Row, 1988.
Foner, Eric, and Olivia Mahoney. _A House Divided: America in the Age of Lincoln_. Chicago: Chicago Historical Society in association with W. W. Norton & Company, 1990.
Foner, Philip S. _The Life and Writings of Frederick Douglass_. Volume 4: _Reconstruction and After_. New York: International Publishers, 1955.
Foner, Philip S. _Frederick Douglass_. New York: Citadel Press and International Publishers, 1964.
Fowler, Orson S. _Phrenology Proved, Illustrated and Applied_. Philadelphia: Fowler and Brevoort, 1836.
Folwer, Orson S. _The Great Brooklyn Romance: All the Documents in the Famous Beecher-Tilton Case_. Unabridged. New York: J. H. Paxon, 1874. Abbreviated as _CT_.
Franklin, Allan. _The Trail of the Tiger: 1789–1928_. New York: Allan Franklin, 1928.
Fuller, Robert C. _Mesmerism and the American Cure of Souls_. Philadelphia: University of Pennsylvania Press, 1982.
Fuller, Robert H. _Jubilee Jim: The Life of Col. James Fisk, Jr_. New York: Macmillan, 1928.
Garrison, Wendell Phillips, and Francis Jackson Garrison. _William Lloyd Garrison, 1805–1879; the Story of His Life as Told by His Children_. Vol. 4. Boston: Houghton Mifflin, 1889.
Gauld, Alan. _A History of Hypnotism_. Cambridge, UK: Cambridge University Press, 1992.
Gay, Peter. _The Bourgeois Experience, Victoria to Freud_. Vol. 1, _Education of the Senses_. New York: Oxford University Press, 1984.
Gay, Peter. _The Bourgeois Experience, Victoria to Freud_. Vol. 2, _The Tender Passion_. New York: Oxford University Press, 1986.
Geller, Uri, and Guy Lyon Playfair. _The Geller Effect_. Grafton Books, 1988.
Goethe, Johann Wolfgang von. _Elective Affinities_. Translated and with an introduction by Victoria Woodhull. Boston: D. W. Niles, 1872. See also the edition translated and with an introduction by R. J. Hollingdale, New York: Penguin, 1983. Originally published in 1809 as _Die Wahlverwandtschaften_.
Goldsmith, Barbara. _Little Gloria... Happy at Last_. New York: Knopf, 1980.
Goldsmith, Margaret. _Franz Anton Mesmer: A History of Mesmerism_. Garden City, NY: Doubleday, Doran, 1934.
Goldstone, Harmon H., and Martha Dalrymple. _History Preserved: A Guide to New York City Landmarks and Historic Districts_. New York: Simon and Schuster, 1974.
Gordon, John Steele. _The Scarlet Woman of Wall Street: Jay Gould, Jim Fisk, Cornelius Vanderbilt, the Erie Railway Wars, and the Birth of Wall Street_. New York: Weidenfeld & Nicolson, 1988.
Gordon, Ann D. and Patricia G. Holland, eds. _The Papers of Elizabeth Cady Stanton and Susan B. Anthony, Guide and Index_ , Vol. 1. New Brunswick, NJ: Rutgers University, 1997.
Gordon, Linda. _Woman's Body, Woman's Right: A Social History of Birth Control in America_. New York: Grossman, 1976.
Grafton, John. _New York in the Nineteenth Century: 317 Engravings from Harper's Weekly and Other Contemporary Sources_. 2nd ed. New York: Dover, 1980.
Gray, Eden. _A Complete Guide to the Tarot_. New York: Bantam Books, 1972.
_The Great Scandal: History of the Beecher-Tilton Case_. All the documents and letters. New York: American News Company, 1874. Abbreviated as _CT_.
Green, Harvey. _The Light of the Home: An Intimate View of the Lives of Women in Victorian America_. New York: Pantheon Books, 1983.
Gregory, William. _Animal Magnetism: or, Mesmerism and its Phenomena_. 5th ed. London: Nichols & Co., 1909.
Griffith, Elisabeth. _In Her Own Right: The Life of Elizabeth Cady Stanton_. New York: Oxford University Press, 1984.
Gruber, Dorothy Whitney. _The Chappaqua Life of Horace Greeley: An Illustrated History of His Farm, His Family and His Houses_. New Castle, NY: Chappaqua Historical Society, 1974.
Hacker, Louis M., and Benjamin B. Kendrick. _The United States Since 1865_. New York: F. S. Crofts & Co., 1933.
Hahnemann, Christian Frederick Samuel. _Organon of Medicine_. Translated by J. Kunzl et al. London: Vic Gollancz, 1983.
Haight, Gordon. _George Eliot: A Biography_. Oxford University Press, 1968.
Hale, William Harlan. _Horace Greeley: Voice of the People_. New York: Harper & Bros., 1950.
Handford, Thomas W. _Beecher: Christian Philosopher, Pulpit Orator, Patriot and Philanthropist_. New York: Belford, Clark, 1887.
Harding, Vincent. _There Is a River: The Black Struggle for Freedom in America_. New York: Vintage Books, 1983.
Hardinge, Emma. _Modern American Spiritualism: A Twenty Years' Record of the Communication Between Earth and the World of the Spirits_. 1869.
Harlow, Alvin F. _Old Waybills: The Romance of the Express Companies_. New York: D. Appleton-Century, 1934.
Harnsberger, John Lewis. _Jay Cooke and Minnesota: The Formative Years of the Northern Pacific Railroad 1868–1873_. New York: Arno Press, 1981.
Harper, Ida Husted. _The Life and Work of Susan B. Anthony_. Volumes 1 and 2. Indianapolis, IN: Bowen-Merrill, 1898.
Hart, John S. _The Female Prose Writers of America_ , 1857.
Haskell, Molly. _From Reverence to Rape_. Canada: Holt, Reinhart and Winston of Canada Ltd., 1973, 1974.
Hawthorne, Nathaniel. _The Scarlet Letter_. 2nd ed. New York and Boston: Thomas Y. Crowell, 1850.
Hay, John. _Lincoln and the Civil War in the Diaries and Letters of John Hay_. Edited by Tyler Dennett. New York: Dodd, Mead, 1939.
Hays, Elinor Rice. _Morning Star: A Biography of Lucy Stone, 1818–1893_. New York: Harcourt, Brace & World, 1961.
Heininger, Mary Lynn Stevens, Karin Calvert, Barbara Finkelstein, Kathy Vandell, Anne Scott MacLeod, and Harvey Green. _A Century of Childhood 1820–1920_. Rochester, NY: Margaret Woodbury Strong Museum, 1984.
Henretta, James. _The Evolution of American Society 1700–1815_. Lexington, MA, 1973.
Hibben, Paxton. _Henry Ward Beecher: An American Portrait_. New York: George H. Doran, 1927. Reprint, Press of the Readers Club, New York, 1942.
Hibbert, Albert. _Smith Wigglesworth: The Secret of His Power_. Tulsa, OK: Harrison House, 1982.
Hillis, N.D. _Lectures and Orations of Henry Ward Beecher_. New York: AMS Press, 1913. Reprint, 1970.
Himmelfarb, Gertrude. _Marriage and Morals Among the Victorians_. New York: Knopf, 1986.
Hobson, Barbara Miel. _Uneasy Virtue: The Politics of Prostitution and the American Reform Tradition_. Chicago: University of Chicago Press, 1990.
Holbrook, Stewart H. _The Age of the Moguls_. Garden City, NY: Doubleday, 1953.
Holbrook, Stewart H. _The Story of American Railroads_. New York: Crown, 1947.
Hole, Judith, and Ellen Levine. _Rebirth of Feminism_. New York: Quadrangle Books, 1971.
_Holy Bible, The: Containing the Old and New Testaments and Commonly Known as the King James Version_. New York: American Bible Society, 1816.
Holzman, Robert S. _Stormy Ben Butler_. New York: Macmillan, 1954.
Howard, Joseph, Jr. _Life of Henry Ward Beecher, The Eminent Pulpit and Platform Orator_. Philadelphia: Hubbard Brothers, 1887.
Howard, Robert West. _The Great Iron Trail: The Story of the First Trans-Continental Railroad_. New York: G. P. Putnam's Sons, 1962.
Hoyt, Edwin P. _The Vanderbilts and Their Fortunes_. Garden City, NY: Doubleday, 1962.
Huggins, Nathan Irvin. _Slave and Citizen: The Life of Frederick Douglass_. Edited by Oscar Handlin. Boston: Little, Brown, 1980.
Hungerford, Edward. _Men of Erie_. New York: Random House, 1946.
_Instructive and Amusing: An Essay on Toys, Games and Education in New England_. Volume 123, No. 2. Essex Institute, April 1987.
Jahn, Robert G., and Brenda J. Dunne. _Margins of Reality: The Role of Consciousness in the Physical World_. New York: Harcourt Brace Jovanovich, 1987.
James, Henry. _Siege of London_. London: Macmillan, 1883.
James, Henry. _The Great Short Novels of Henry James: The Siege of London_. New York: Dial Press, 1944; New York: Penguin, 1983. ( _The Siege of London_ first appeared in _Cornbill Magazine_ , January–February 1883).
James, Henry, Horace Greeley, and Stephen Pearl Andrews (A Discussion by). _Love, Marriage and Divorce, and the Sovereignty of the Individual_. New York: Stringer & Townsend, 1853.
Johnston, Johanna. _Mrs. Satan: The Incredible Saga of Victoria C. Woodhull_. New York: G. P. Putnam's Sons, 1967; London: Macmillan, 1967.
Jones, Willoughby. _The Life of James Fisk, Jr_. Philadelphia, Chicago: Union Publishing Co., 1872.
Jordan, Robert Paul. _The Civil War_. New York: National Geographic Society, 1969.
Josephson, Matthew. _The Robber Barons: The Great American Capitalists_. New York: Harcourt, Brace and Company, 1934.
Keller, Allan. _Scandalous Lady: The Life and Times of Madame Restell, New York's Most Notorious Abortionist_. New York: Atheneum, 1981.
Keller, Morton. _The Art & Politics of Thomas Nast_. New York: Oxford University Press, 1968.
Kenner, Hugh. _Magic and Spells_. Bennington, VT: Bennington Chapbooks in Literature, 1987.
Kerr, Andrea Moore. _Lucy Stone: Speaking Out for Equality_. New Brunswick, NJ: Rutgers University Press, 1992.
Kerr, Howard. _Mediums, and Spirit-Rappers, and Roaring Radicals: Spiritualism in American Literature, 1850–1900_. Chicago: University of Illinois Press, 1972.
Kerr, Howard, and Charles L. Crow, eds. _The Occult in America: New Historical Perspectives_. Chicago: University of Illinois Press, 1983.
Kessler, Carol Farley. _Elizabeth Stuart Phelps_. Boston: Twayne Publishers, 1982.
Klein, Maury. _The Life and Legend of Jay Gould_. Baltimore: Johns Hopkins University Press, 1986.
Kneeland, George Jackson. _Commercialized Prostitution in New York City_. Montclair, NJ: Patterson Smith, 1969.
Koch, Freda Postle. _Civil War Heroines: The Coggeshall Ladies_. Columbus, OH: PoKo Press, 1992.
Koffit, Avodah, M.D. _The Sexual Self_. J. B. Lippencott, 1977.
Kouwenhoven, John A. _The Columbia Historical Portrait of New York (1857–1900)_. New York: Harper & Row, 1972.
Kugler, Israel. _From Ladies to Women. The Organized Struggle for Woman's Rights in the Reconstruction Era_. Contributions in Women's Studies, Number 77. New York, Westport, CT; London: Greenwood Press, 1987.
Lancaster, Clay. _Old Brooklyn Heights: New York's First Suburb_. 2nd ed. New York: Dover, 1979.
Lane, Wheaton J. _Commodore Vanderbilt: An Epic of the Steam Age_. New York: Knopf, 1942.
Larson, Henrietta. _Jay Cooke: Private Banker_. New York: Greenwood Press, 1968.
Latham, Jean. _Victoriana: A Guide for Collectors_. New York: Stein & Day, 1971.
Latimer, Margaret, ed. _Brooklyn Almanac: Illustrations/Facts/Figures/People/Buildings/Books_. New York: Brooklyn Rediscovery, Brooklyn Educational & Cultural Alliance, 1984.
Lawton, George. _The Drama of Life After Death: A Study of the Spiritualist Religion_. New York: Henry Holt, 1932.
Leach, William. _True Love and Perfect Union: The Feminist Reform of Sex and Society_. New York: Basic Books, 1980.
Levine, George, and William Madden. _The Art of Victorian Prose_. New York: Oxford University Press, 1968.
Lindsay, Gordon. _The Chaos of the Psychics_. Dallas: Christ for the Nations, 1978.
Lindsay, Gordon. _Astrology, Reincarnation, Flying Saucer Men, Edgar Cayce, Psychics_. Sorcery In America Series, Volume 2. Dallas: Christ for the Nations, 1982.
Lindsay, Gordon. _Spiritualism, Telepathy, ESP, Ouija Board, Hypnotism_. Sorcery In America Series, Volume 1. Dallas: Christ for the Nations, 1985.
Lindsay, Gordon. _Jeane Dixon—Prophetess or Psychic Medium_. Dallas: Christ for the Nations, 1987.
Longford, Elizabeth. _Queen Victoria: Born to Succeed_. New York: Harper & Row, 1964.
Longford, Elizabeth. _Byron_. London: Cox & Wyman, 1976.
Lundberg, Ferdinand. _America's 60 Families_. New York: Vanguard Press, 1937.
Lutz, Alma. _Created Equal: A Biography of Elizabeth Cady Stanton 1815–1902_. New York: John Day, 1940.
Lutz, Alma. _Susan B. Anthony: Rebel, Crusader, Humanitarian_. Boston: Beacon, 1959.
Lynch, Denis Tilden. _"Boss" Tweed: The Story of a Grim Generation_. New York: Boni and Liveright, 1927.
Mabee, Carleton. _The American Leonardo. A Life of Samuel F.B. Morse_. New York: Knopf, 1943.
Maddow, Ben. _A Sunday Between Wars: The Course of American Life 1865–1917_. New York: W. W. Norton, 1979.
Maher, James T. _The Twilight of Splendor_. New York: Little, Brown, 1976.
Mandelbaum, Seymour J. _Boss Tweed's New York_. New York: John Wiley & Sons, 1965.
Marberry, M. M. _Vicky: A Biography of Victoria C. Woodhull_. New York: Funk & Wagnalls, 1967.
Markun, Leo. _Mrs. Grundy: A History of Four Centuries of Morals Intended to Illuminate Present Problems in Great Britain and the United States_. New York: D. Appleton, 1930.
_Marriage: For & Against_. Introduction by Harold H. Hart. New York: Hart Publishing Co., 1972.
Marshall, Charles F. _The True History of the Brooklyn Scandal_. Philadelphia: National Publishing Company, 1874.
Martin, Edward Winslow. _The Secrets of the Great City: A Work Descriptive of the Virtues and the Vices, the Mysteries, Miseries and Crimes of New York City_. Philadelphia: Jones Brothers & Co., 1868.
Martin, Victoria (Claflin) Woodhull. _A Fragmentary Record of Public Work Done in America. 1871–1877_. London: Self-published, 1877.
Martin, Victoria (Claflin) Woodhull. _Stirpiculture: or, The Scientific Propagation of the Human Race_. London: Self-published, 1888.
Martin, Waldo E., Jr. _The Mind of Frederick Douglass_. Chapel Hill: University of North Carolina Press, 1984.
_Mass Violence in America: Memphis Riots and Massacres_. New York: Arno Press, 1969. Facsimile edition, House Report No. 101, U.S. 39th Congress, First Session.
McCabe, James D., Jr. _Lights and Shadows of New York Life; or, The Sights and Sensations of the Great City_. Philadelphia: National Publishing Co., 1872. Facsimile edition, New York: Farrar, Straus and Giroux, 1970.
McCaffrey, Lawrence. _The Irish Diaspora in America_. Bloomington: Indiana University Press, 1976.
McCaughey, Elizabeth P. _Government By Choice: Inventing the United States Constitution_. New York: New York Historical Society and Basic Books, 1987.
McClinton, Katherine Morrison. _Antiques of American Childhood_. New York: Bramhall House.
McFeely, William S. _Grant: A Biography_. New York: W. W. Norton, 1982.
McFeely, William S. _Frederick Douglass_. New York: W. W. Norton, 1991.
McLoughlin, William G. _The Meaning of Henry Ward Beecher: An Essay in the Shifting Values of Mid-Victorian America_. New York: Knopf, 1970.
McPherson, James M. _The Negro's Civil War: How American Negroes Felt and Acted during the War for the Union_. Urbana and Chicago: University of Illinois Press, originally Pantheon Books, 1965.
McPherson, James M. _Battle Cry of Freedom: The Civil War Era_. New York: Oxford University Press, 1988.
Meade, Marion. _Free Woman: The Life and Times of Victoria Woodhull_. New York: Knopf, 1976.
Ment, David. _The Shaping of a City: A Brief History of Brooklyn_. New York: Brooklyn Rediscovery, Brooklyn Educational & Cultural Alliance, 1979.
Meriwether, Elizabeth Avery. _Recollections of 92 Years, 1824–1916_. Nashville: Tennessee Historical Commission, 1958.
Milbauer, Barbara. _The Law Giveth_. New York: McGraw-Hill, 1984.
Mill, John Stuart. _The Subjection of Women_. Introduction by Wendell Robert Carr. Cambridge, MA: MIT Press, 1970.
Minnigerode, Meade. _Certain Rich Men_. Freeport, NY: Books for Libraries Press, 1927. Reprint, 1970.
Mohr, James C. _Abortion in America: The Origins and Evolution of National Policy, 1800–1900_. New York: Oxford University Press, 1978.
Moore, R. Laurence. _In Search of White Crows: Spiritualism, Parapsychology, and American Culture_. New York: Oxford University Press, 1977.
Morris, Lloyd. _Incredible New York: High Life and Low Life of the Last Hundred Years_. New York: Random House, 1951.
Morse, Edward Lind, ed. _Samuel F.B. Morse: His Letters and Journals_. Volumes 1 and 2. Boston: Houghton Mifflin; New York: The Riverside Press, 1914.
Mumford, Lewis. _Sticks and Stones_. New York: Dover, 1955.
Nash, Howard P., Jr. _Stormy Petrel: The Life and Times of General Benjamin F. Butler, 1818–1893_. Rutherford, NJ: Fairleigh Dickinson University Press, 1969.
Nast, Thomas. _Thomas Nast, Cartoons and Illustrations_. New York: Dover, 1974.
Nevins, Allan. _A History of American Life in Twelve Volumes_. Edited by Arthur M. Schlesinger and Dixon Ryan Fox. New York: Macmillan, 1927.
Nevins, Allan, and Milton Halsey Thomas, eds. _The Diary of George Templeton Strong: Post-War Years 1865–1875_. New York: Macmillan, 1952.
Nevins, Deborah, ed. _Grand Central Terminal: A City Within the City_. New York: Municipal Art Society of New York, 1982.
Newman, Richard. _Words Like Freedom: Afro-American Books and Manuscripts in the Henry W. and Albert A. Berg Collection of English and American Literature_. New York Public Library, 1989.
Northrup, Flora. _Record of a Century, 1834–1934_. New York: American Female Guardian Society, 1934.
_Notable American Women 1607–1950: A Biographical Dictionary_. 3 volumes. Edited by Edward T. James. Prepared under the auspices of Radcliffe College. Cambridge, MA: Belknap Press of Harvard University Press, 1971.
Noyes, Alexander Dana. _Forty Years of American Finance: A Short Financial History of the Government and People of the United States Since the Civil War, 1865–1907_ (the second and extended edition of "Thirty Years of American Finance"). New York: G. P. Putnam's Sons, the Knickerbocker Press, 1898, 1909.
Oates, Stephen B. _To Purge This Land With Blood: A Biography of John Brown_. 2nd ed. Amherst: University of Massachusetts Press, 1984.
Oberholtzer, Ellis Paxson, Ph.D. _Jay Cooke: Financier of the Civil War_. 2 volumes. Philadelphia: George W. Jacobs, 1907.
_Old Brooklyn Heights: 1827–1927_. To Commemorate the One Hundreth Anniversary of the Founding of The Brooklyn Savings Bank, 1827–1927.
Oliver, Leon. _The Great Sensation_. Chicago: Beverly Co., 1973.
Pagels, Elaine. _Adam, Eve and the Serpent_. New York: Random House, 1988.
Papachristou, Judith. _Women Together: A History in Documents of the Women's Movement in the United States_. New York: Knopf, 1976.
Parker, Gail Thain. _Mind Cure in New England: From the Civil War to World War I_. Hanover, NH: University Press of New England, 1973.
Patterson, Jerry E. _The City of New York_. New York: Harry N. Abrams, 1978.
Peterson, Merrill D. _Lincoln in American Memory_. New York: Oxford University Press, 1994.
Phelps, Elizabeth Stuart (Ward). _The Story of Avis_. Boston: James R. Osgood & Co., 1877.
Phelps, Elizabeth Stuart (Ward). _Our Famous Women: An Authorized Record of Their Lives and Deeds_. 1883. Reprint, Freeport, NY: Books for Libraries Press, 1975.
Pike, E. Royston. _Golden Times (Human Documents of the Victorian Age)_. New York: Frederick A. Praeger, 1967.
Pivar, David J. _Purity Crusade: Sexual Morality and Social Control, 1868–1900_. Westport, CT: Greenwood Press, 1973.
Post, Isaac. _Voices from the Spirit World, being Communications from Many Spirits, by the Hand of Isaac Post, Medium_. Rochester, NY: C. H. McDonell, 1852.
Potter, George W. _To the Golden Door_. Boston: Little, Brown, 1960.
Potts, E. Daniel, and Annette Potts, eds. _A Yankee Merchant in Goldrush Australia: The Letters of George Francis Train 1853–55_. Melbourne: William Heinemann, 1970.
Preston, Dickson J. _Young Frederick Douglass: The Maryland Years_. Baltimore: Johns Hopkins University Press, 1980.
Pullen, C. H. _The Fight at Dame Europa's School_. Illustrated by Thomas Nast. New York: F. B. Felt, 1871.
Pullen, C. H. _Miss Columbia's Public School or Will it Blow Over?_ Illustrated by Thomas Nast. New York: F. B. Felt, 1871.
Putnam, Allen. _Mrs. J. H. Conant: The World's Medium of the Nineteenth Century_. Boston: William W. & Co., 1873.
Quarles, Benjamin. _The Negro in the Civil War_. Boston: Little, Brown, 1953.
Rabinowitz, Richard. _The Spiritual Self in Everyday Life: The Transformation of Personal Religious Experience in Nineteenth-Century New England_. Boston: Northeastern University Press, 1989.
Rable, George C. _Civil Wars: Women and the Crisis of Southern Nationalism_. Chicago: University of Illinois Press, 1989.
Raleigh, Dr. A. S. _Metaphysical Healing_. Volume 1. San Francisco: Hermetic Publishing Co., 1916.
Raleigh, Dr. A. S. _Scientifica Hermetica_. San Francisco: Hermetic Publishing Co., 1916.
Randolph, P. B. _The New Mola!: The Secret of Mediumship! A Handbook of White Magic, Magnetism and Clairvoyance_. 1873.
Reade, Charles. _Griffith Gaunt or Jealousy_. London: Chapman and Hall, 1866.
Reed, Henry Hope, and Sophia Duckworth. _Central Park: A History and Guide_. New York: Clarkson N. Potter, 1972.
Reynolds, David S. _Beneath the American Renaissance_. New York: Knopf, 1988.
Reynolds, Moira Davison. _Nine American Women of the Nineteenth Century: Leaders into the Twentieth_. Jefferson, NC: McFarland, 1988.
Richmond, Rev. J. F. _New York and Its Institutions_. New York: E. B. Treat, 1872.
Rose, Phyllis. _Parallel Lives: Five Victorian Marriages_. New York: Knopf, 1984.
Rourke, Constance Mayfield. _Trumpets of Jubilee: Henry Ward Beecher, Harriet Beecher Stowe, Lyman Beecher, Horace Greeley, P.T. Barnum_. New York: Harcourt, Brace, 1927.
Rugoff, Milton. _Prudery & Passion: Sexuality in Victorian America_. New York: Putnam, 1971.
Rugoff, Milton. _The Beechers: An American Family in the Nineteenth Century_. New York: Harper & Row, 1981.
Rugoff, Milton. _America's Gilded Age: Intimate Portraits from an Era of Extravagance and Change, 1850–1890_. New York: Henry Holt, 1989.
Russell, Sharman Apt. _Frederick Douglass_. New York: Chelsea House Publishers, 1988.
Russet, Cynthia Eagol. _Sexual Science: The Victorian Construction of Womanhood_. Cambridge, MA: Harvard University Press, 1989.
Ryan, Mary P. _Women in Public: Between Banners and Ballots, 1825–1880_. Baltimore: Johns Hopkins University Press, 1990.
Sachs, Emanie. _The Terrible Siren: Victoria Woodhull (1838–1927)_. New York: Harper & Brothers, 1928. Reprint, New York, Arno Press, 1978. Abbreviated as _TS_.
Sanborn, Margaret. _Mark Twain: The Bachelor Years: A Biography_. New York: Doubleday, 1990.
Schickel, Richard. _Intimate Strangers: The Culture of Celebrity_. Garden City, NY: Doubleday, 1985.
Schulze, Suzanne. _Horace Greeley: A Bio-Bibliography_. New York: Greenwood Press, 1992.
Sears, Hal D. _The Sex Radicals: Free Love in High Victorian America_. Lawrence, KS: Regent Press of Kansas, 1977.
Seely, Lida. _Mrs. Seelys Cook Book_. New York, London: Macmillan, 1902.
Seitz, Don Carlos. _Uncommon Americans: Pencil Portraits of Men and Women Who Have Broken the Rules_. Indianapolis: Bobbs-Merrill, 1925.
Seitz, Don Carlos. _Horace Greeley: Founder of the New York Tribune_. Indianapolis: Bobbs-Merrill, 1926.
Seitz, Don Carlos. _The Dreadful Decade: Detailing Some Phases in the History of the United States from Reconstruction to Resumption, 1869–1879_. Indianapolis: Bobbs-Merrill, 1926.
Shaplen, Robert. _Free Love and Heavenly Sinners: The Story of the Great Henry Ward Beecher Scandal_. New York: Knopf, 1954. Originally published in _The New Yorker_ , June 5 and 12, 1954.
Sherr, Lynn. _Failure Is Impossible: Susan B. Anthony in Her Own Words_. New York: Times Books, 1995.
Showalter, Elaine. _The Female Malady: Women, Madness, and English Culture 1830–1980_. New York: Pantheon Books, 1985.
Silver, Nathan. _Lost in New York_. New York: Schocken Books, 1971.
Simon, John Y., ed. _The Personal Memoirs of Julia Dent Grant_. New York: G. P. Putnam's Sons, 1975.
Sinclair, Andrew. _The Better Half: The Emancipation of the American Woman_. New York: Harper & Row, 1965.
Sklar, Kathryn Kish. _Catharine Beecher: A Study in American Domesticity_. New Haven & London: Yale University Press, 1973.
Sloane, Florence Adele. _Maverick in Mauve_. Garden City, NY: Doubleday, 1983.
Smith, Gene. _Lee & Grant_. New York: McGraw-Hill, 1984.
Smith, Matthew Hale. _Sunshine and Shadow in New York_. Hartford, CT: J. B. Burr, 1868.
Smith, Matthew Hale. _Twenty Years among the Bulls and Bears of Wall Street_. Hartford, CT: J. B. Burr, 1870.
Smith, Page. _The Rise of Industrial America_ , Volume 6. New York: McGraw-Hill, 1984.
Smith, Thomas E. V. _The City of New York: In the Year of Washington's Inauguration 1789_. New York: Trow's Publishing and Bookbinding Co., 1889.
Smith-Rosenberg, Carroll. _Disorderly Conduct: Visions of Gender in Victorian America_. New York: Knopf, 1985.
Squire, Geoffrey. _Dress and Society 1560–1570_. New York: Viking, 1974.
Stampp, Kenneth M. _The Era of Reconstruction, 1865–1877_. New York: Knopf, 1965, Vintage Books, 1978 Reprint Edition.
Stansell, Christine. _City of Women: Sex and Class in New York 1789–1860_. New York: Knopf, 1986.
Stanton, Elizabeth Cady. _Eighty Years and More: Reminiscences 1815–1897_. New York: Schocken Books, 1971. Reprinted from the T. Fisher Unwin edition of 1898. Abbreviated as _EYM_.
Stanton, Elizabeth Cady, Susan B. Anthony, and Matilda Joslyn Gage, eds. _History of Woman Suffrage_. Volumes 1 and 2 of 6 volumes. Salem, NH: Ayer Publishers. Reprint edition 1985 from an original copy in State Historical Society of the Wisconsin Library. Originally published by Fowler & Wells, New York, 1882. Abbreviated as _HWS_.
Stanton, Henry B. _Random Recollections_. Johnstown, NY: Blunck & Leaning, Printers, 1885.
Stanton, Theodore, and Harriot Stanton Blatch, eds. _Elizabeth Cady Stanton, As Revealed in Her Letters, Diary and Reminiscences_. 2 volumes. New York: Arno and the _New York Times_ , 1969. Originally published by Harper & Bros., New York, 1922.
Starr, Louis M. _Bohemian Brigade: Civil War Newsmen in Action_. New York: Knopf, 1954.
Stein, Dorothy. _Ada: A Life and a Legacy_. Cambridge, MA: MIT Press, 1985.
Sterling, Dorothy, ed. _The Troubles They Seen: Black People Tell the Story of Reconstruction_. Renamed _Nobody Knows The Trouble I Seen: The Story of Reconstruction in the Words of African Americans_. Garden City, NY: Doubleday, 1976.
Stern, Madeleine B. _We the Women: Career First of Nineteenth-Century America_. New York: Schulte Publishing Company, 1963.
Stern, Madeleine B. _The Pantarch: A Biography of Stephen Pearl Andrews_. Austin: University of Texas Press, 1968.
Stern, Madeleine B. _Heads and Headlines: The Phrenological Fowlers_. Norman: University of Oklahoma Press, 1971.
Stern, Madeleine B. ed. _The Victoria Woodhull Reader_. Weston, MA: M & S Press, 1974.
Stiles, Henry R. _A History of the City of Brooklyn: Including the Old Town and Village of Brooklyn, the Town of Bushwick, and the Village and City of Williamsburgh_. 3 volumes. Brooklyn: Published by subscription. Volume 1, 1867; Volume 2, 1869; Volume 3, 1870.
Stiles, Henry R. _The Civil, Political, Professional and Ecclesiastical History and Commercial and Industrial Record of the County of Kings and the City of Brooklyn, New York, 1683–1884_. 2 volumes. New York: W. W. Munsell & Co., 1884.
Stoddard, Henry Luther. _Horace Greeley: Printer, Editor, Crusader_. New York: G. P. Putnam's Sons, 1946.
Stoehr, Taylor. _Free Love in America: A Documentary History_. New York: AMS Press, 1979.
Storrs, Richard S. _The Church of the Pilgrims, Brooklyn, New York: Its Character and Work, with the Changes around It, during Forty Years of Pastoral Service_. New York: A. S. Barnes, 1886.
Stowe, Harriet Beecher. _Lady Byron Vindicated_. Boston: Fields Osgood & Co., 1870.
Stowe, Harriet Beecher. _My Wife & I or Harry Henderson's History_. New York: J. B. Ford, 1871. Boston: Houghton Mifflin, 1896. Serialized in the _Christian Union_ , 1870, 1871.
Strasser, Susan. _Never Done: A History of American Housework_. New York: Pantheon Books, 1982.
Strong, George Templeton. _The Diary of George Templeton Strong: 1835–1875_. 4 volumes. Edited by Allan Nevins and Milton Halsey Thomas. New York: Macmillan, 1952.
Sutherland, Daniel E. _Americans and Their Servants_. Baton Rouge: Louisiana State University Press, 1981.
Swanberg, W. A. _Jim Fisk: The Career of an Improbable Rascal_. New York: Charles Scribner's Sons, 1959.
Syrett, Harold Coffin, Ph.D. _The City of Brooklyn, 1865–1989: A Political History_. New York: Columbia University Press, 1944.
Talese, Gay. _Thy Neighbor's Wife_. New York: Dell, 1981.
Taylor, W. G. Langworthy. _Katie Fox: Epochmaking Medium and the Making of the Fox-Taylor Record_. New York: G. P. Putnam's Sons, 1933.
Teltscher, Harry O. _Handwriting, A Revelation of Self: A Source Book of Psychographology_. Alexandria, VA: Red Dragon Press, Hawthorn Books, 1971.
_Theodore Tilton v. Henry Ward Beecher: Action for Criminal Conduct Tried in the City Court of Brooklyn, Chief Justice Joseph Neilson, Presiding. Verbatim Report by the Official Stenographer_. 3 volumes. New York: McDivitt, Campbell & Co., Law Publishers, 1875.
Thomas, George M. _Revivalism and Cultural Change: Christianity, Nation Building, and the Market in the Nineteenth-Century United States_. Chicago: University of Chicago Press, 1989.
Thurston, Carol. _The Romance Revolution_. Chicago: University of Chicago Press, 1987.
Tilton, Theodore. _The True Church_. Philadelphia: J. P. Lippincott, 1867.
Tilton, Theodore. _Sanctorum_. New York: Sheldon, 1870.
Tilton, Theodore. _Tempest-Tossed: A Romance_. New York: Sheldon, 1874.
Tozer, Jane, and Sarah Levitt. _Fabric of Society: A Century of People and Their Clothes 1770–1870_. Wales: Laura Ashley Ltd., 1983.
Tracy, Benjamin F. _The Case of Henry Ward Beecher: Opening Address of Counsel for the Defendant_. New York: G. W. Smith, 1875.
Trager, James, ed. _The People's Chronology: A Year-by-Year Record of Human Events from Prehistory to the Present_. New York: Holt, Rinehart and Winston, 1979.
Train, George Francis. _The Great Epigram Campaign of Kansas: Championship of Women_. Leavenworth, KS: Prescott & Hume, 1867.
Train, George Francis. _My Life in Many States and in Foreign Lands_. New York: D. Appleton, 1902.
Train, George Francis. _Young America in Wall Street_. New York: Greenwood Press, 1968. Originally published in 1857, Derby and Jackson.
Train, George Francis. _Yankee Merchant in Gold Rush Australia_. Melbourne: William Heinemann, 1970.
Trefousse, Hans L. _Ben Butler: The South Called Him BEAST!_ New York: Twayne Publishers, 1957.
Trefousse, Hans L. _Andrew Johnson: A Biography_. New York: W. W. Norton, 1989.
Trumbull, Charles Gallaudet. _Anthony Comstock, Fighter: Some Impressions of a Lifetime of Adventure in Conflict with the Powers of Evil_. 2nd ed. New York: Fleming H. Revell, 1913.
Truth, Sojourner. _Narrative of Sojourner Truth_. Edited by Olive Gilbert. Battle Creek, MI, 1878.
Tuchman, Barbara W. _The Proud Tower: A Portrait of the World Before the War: 1890–1914_. New York: Macmillan, 1966.
Twain, Mark. _Huckleberry Finn_. Cleveland: World Publishing Co. Reprint, 1947.
_Uncontradicted Testimony in the Beecher Cases, Compiled From the Official Records_. New York: D. Appleton, 1876.
Underhill, Lois Beachy. _The Woman Who Ran for President: The Many Lives of Victoria Woodhull_. Bridgehampton, NY: Bridge Works Publishing Co., 1995.
Vandal, Gilles. _The New Orleans Riot of 1866: Anatomy of a Tragedy_. Lafayette: Center for Louisiana Studies, University of Southwestern Louisiana, 1983.
Van Deusen, Glyndon G. _Horace Greeley, Nineteenth Century Crusader_. Philadelphia: University of Pennsylvania Press, 1953.
Vickers, Hugo. _Gladys, Duchess of Marlborough_. New York: Holt, Rinehart and Winston, 1979.
Vidal, Gore. _1876: A Novel_. New York: Random House, 1976.
Vidal, Gore. _Lincoln_. New York: Random House, 1984.
Vidal, Gore, V. S. Pritchett, David Cante, Bruce Chatwin, Peter Conrad, Edward Jay Epstein. _Great American Families_. New York: W. W. Norton, 1977.
Vinson, John Chalmers. _Thomas Nast, Political Cartoonist_. Athens: University of Georgia Press, 1967.
Vithoulkas, George. _Homeopathy: Medicine of the New Man_. New York: Prentice Hall, 1979.
Wallace, Irving. _The Nympho and Other Maniacs_. London: Cassell, 1971.
Waller, Altina L. _Reverend Beecher and Mrs. Tilton: Sex and Class in Victorian America_. Amherst: University of Massachusetts Press, 1982.
Warren, John H. _The Rise of Urban America: Thirty Years of Battle with Crime_. New York: Arno Press, 1970.
Warren, Josiah. _Practical Applications of the Elementary Principles of "True Civilization._ "
Washington, Booker T. _Frederick Douglass_. 1906. Reprint, New York: Argosy-Antiquarian Ltd., 1969.
Wecter, Dixon. _The Saga of American Society_. New York: Charles Scribner's Sons, 1937.
Weideger, Paula. _Menstruation and Menopause: The Physiology and Psychology, The Myth and the Reality_. New York: Knopf, 1976.
Weintraub, Stanley. _Victoria: An Intimate Biography_. New York: Truman Talley Books, E. P. Dutton, 1987.
Werner, M. R. _Tammany Hall_. Garden City: Doubleday, Doran, 1928.
West, Richard S., Jr. _Lincoln's Scapegoat General: A Life of Benjamin F. Butler, 1818–1893_. Boston: Houghton Mifflin, 1965.
_Westminster Study Edition of The Holy Bible: Containing Old And New Testaments in the Authorized (King James) Version_. Philadelphia: Westminster Press.
Wheeler, Leslie. _Loving Warriors: Selected Letters of Lucy Stone and Henry B. Blackwell, 1853–1893_. New York: Dial, 1981.
Wheeler, T. C., ed. _The Immigrant Experience_. New York: Dial, 1971.
White, Howard A. _The Freedmen's Bureau in Louisiana_. Baton Rouge: Louisiana State University Press, 1970.
Wilson, Edmund. _Patriotic Gore: Studies in the Literature of the American Civil War_. Boston: Northeastern University Press, 1984.
Wingate, Charles. _How the Times Broke the Tweed Ring_. New York: Kilbourne Tompkins, 1875.
Woodhull, Victoria Claflin. _The Origin, Tendencies and Principles of Government: A Review of the Rise and Fall of Nations from Early Historic Time to the Present: with Special Consideration Regarding the Future of the United States as the Representative Government of the World_. New York: Woodhull & Claflin Co., 1871 (probably ghostwritten by Stephen Pearl Andrews).
Woodhull, Victoria Claflin (Mrs. John Biddulph Martin), and Tennessee C. Claflin (Lady Cook). _The Human Body, the Temple of God: or The Philosophy of Sociology_. London: Published by the authors, 1890. (Includes Woodhull's announcement for president, her memorial to Congress, the majority and minority committee reports, her major suffrage speeches.)
Young, G. M. _Victorian England_. New York: Oxford University Press, 1936.
Young, James Harvey. _The Medical Messiahs: A Social History of Health Quackery in Twentieth-Century America_. Princeton, NJ: Princeton University Press, 1967.
Younger, William Lee. _Old Brooklyn In Early Photographs 1865–1929_. New York: Dover, 1978.
## _Magazine and Journal Articles_
Andrist, Ralph K. "Paladin of Purity." _American Heritage_ , October 1973.
Hoff-Wilson, Joan. "The Unfinished Revolution: Changing Legal Status of U.S. Women." _Signs: Journal of Women in Culture and Society_ 13, no. 1 (1987):
Holmes, Jack. "The Underlying Causes of the Memphis Riot of 1866." _Tennessee Historical Quarterly_ 17 (1958).
Johnson, Howard P. "New Orleans Under General Butler." _Louisiana Historical Quarterly_ 24, (1941)
Lovett, Bobby L. "Memphis Riots: White Reaction to Blacks in Memphis, May 1865–July 1866." _Tennessee Historical Quarterly_ 9–33.
McPherson, James M. "Grant or Greeley? The Abolitionist Dilemma in the Election of 1872." _American Historical Review_ 71 (October 1965): 43–61.
Mott, Franz L. _"Woodhull & Claflin's Weekly." A History of American Magazines_ 3 (1938): 443–453.
_The Northeast Ohio Quarterly: A Journal of History and Civilization_ 44, no. 4 (fall 1972): 85–99.
_Report of the Mysterious Noises, heard in the House of Mr. John D. Fox, in Hydesville, Acadia, Wayne Co. Authenticated by the Certificates and Confirmed by the Statements of Citizens of that Place and Vicinity_. Undated, New York Public Library, Manuscript Collection.
Riddleberger, Patrick W. "The Radicals' Abandonment of the Negro During Reconstruction." _The Journal of Negro History_ 45, no. 2 (April 1960): 88–102.
Ryan, Gilbert James. "The Memphis Riots of 1866: Terror in a Black Community During Reconstruction." _The Journal of Negro History_ 62, no. 3 (July 1977): 243–257.
Seitz, Don. "Horace Greeley and His Wife." _McNaught's Monthly_ 4, no. 2 (August 1925): 40–42.
Smith-Rosenberg, Carol. "The Female World of Love and Ritual: Relations between Women in Nineteenth Century America." _Signs_ 1, no. 1 (autumn 1975): 1–29.
Smith-Rosenberg, Carol. "The Hysterical Women: Sex Roles and Role Conflict in Nineteenth-Century America." _Social Research_ 39, no. 4 (winter 1972): 652–678.
Strong, Byron. "Toward a History of the Experiential Family: Sex and Incest in the Nineteenth-Century Family." _Journal of Marriage and the Family_ 35 (August 1973): 457–466.
Waller, Altina L. "Community, Class and Race in the Memphis Riot of 1866." _Journal of Social History_ (winter 1984): 233–243.
## _Pamphlets, Articles, Theses_
Andrews, Stephen Pearl, Henry James, and Horace Greeley. _Love, Marriage and Divorce: A Discussion_. New York: Stringer and Townsend, 1853.
Basch, Norma. _Invisible Women: The Legal Fiction of Marital Unity in Nineteenth-Century America_.
Beecher, Henry Ward. "An Address Delivered Before the Long Island Historical Society," Brooklyn. March 8, 1897.
Beecher, Henry Ward. _Beecher to Special Committee of the Soldiers' and Sailors' Convention, August 30, 1866_. New York Public Library Tucker-Sachs Collection.
Beecher, Henry Ward. _Patriotic Addresses_. New York Public Library, Manuscript Collection.
Benson, Eugene. "New York Journalists—Theodore Tilton." _Galaxy_ 8 (September 1869): 355–359.
Boucher, Francois. _20,000 Years of Fashion: The History of Costume and Personal Adornment_. New York: Harry N. Abrams.
_The Beecher Trial: A Review of the Evidence_. Reprinted from the _New York Times_ , July 3, 1875, with some revision and additions. New York, 1875.
_Catalogue of Games and Amusements 1869–70_. Springfield, MA: Milton Bradley & Company.
Chused, Richard H. _Late Nineteenth-Century Married Women's Property Law: Reception of the Early Married Women's Property Acts by Courts and Legislatures_.
Clark, Edward H. G. _The Thunderbolt_. New York, Albany, and Troy: n.p., May 1873. Library of Congress.
Court of Appeals. Theodore Tilton, respondent, v. Henry Ward Beecher, appellant. Appellant's points. Statement. Brooklyn, NY, 1874.
Davis, Paulina Wright. _A History of the National Woman's Rights Movement for Twenty Years, with the Proceedings of the Decade Meeting held at Apollo Hall, October 20, 1870, from 1850 to 1870, with an Appendix Containing the History of the Movement during the winter of 1871, in the National Capitol_. New York: Journeymen Printers' Cooperative Association, 1871. Also, New York: Kraus, reprint, 1971.
Du Bois, W.E.B. _Black Reconstruction in America 1860–1880: An Essay Toward a History of the Part Which Black Folk Played in the Attempt to Reconstruct Democracy in America_. New York: Russell & Russell, 1966.
Farnam, Elsie Anne. _Isabella Beecher Hooker as Reformer: The Vote for Women or a Quest for Personal Power?_ New York University, 1967. (A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts at the University of Connecticut, 1970.)
Gerteis, Louis S. _From Contraband to Freedman: Federal Policy Toward Southern Blacks 1861–65_. Westport, CT, 1973.
Grisaffi, Philip E. _Beecher-Tilton Investigation of 1874_. Brooklyn Historical Studies, Spring 1878.
Holmes, Jack D. L. _The Effects of the Memphis Race Riot, 1866_. West Tennessee Historical Society, XII.
_Old Brooklyn Heights_. The Brooklyn Savings Bank, 1827–1927.
Rabkin, Margareth. _The Silent Feminist Revolution: Women and the Law (in New York State from Blackstone to the Beginnings of the American Woman's Rights Movement)_. Ph.D. dissertation, State University of New York, Buffalo, 1975.
Sachs, Albie, and Joan Hoff Wilson. _Sexism and the Law: A Study of Male Belief and Legal Bias in Britain and the United States_. Law in Society Series, edited by C. M. Campbell and P.N.P. Wiles.
_Star and Planet Locator_. Edmund Scientific Company, 1966.
Terry, Edwin W. _Theodore Tilton as Social Reformer, Radical Republican, Newspaper Editor, 1863–1872_. Ph.D. dissertation, St. John's University, 1971; University Microfilms International Dissertation Information Service, Ann Arbor, MI.
Tilton, Theodore. _Elizabeth Cady Stanton: Eminent Woman of the Age_. Hartford, CT: S. M. Batts, 1869.
Tilton, Theodore."Victoria C. Woodhull: A Biographical Sketch: Mr. Tilton's Account of Mrs. Woodhull." _Golden Age_ , New York, 1871, tract 3. Abbreviated as _GAT_.
Tilton, Theodore, et al. _The Enfranchisement of Women_. March 14, 1870, proof copy, New York Public Library.
Treat, Joseph, M.D. _Beecher, Tilton, Woodhull, The Creation of Society: All Four of Them Exposed, and if Possible Reformed and Forgiven, in Dr. Treat's Celebrated Letter to Victoria C. Woodhull_. Published by the author. New York, 1874.
_U.S. Report of the Select Committee on the Memphis Riots and Massacres_. 39th Congress, First Session, Report No. 101. Washington, 1866.
Woodhull, Victoria Claflin. "Tendencies of the Government." _New York Herald_ , April 16, 25; May 2, 9, 16, 27; June 4, 19; July 4, 11, 1870.
Woodhull, Victoria Claflin. _Victoria C. Woodhull's Complete and Detailed Version of the Beecher-Tilton Affair_. Washington, D.C.: J. Bradley Adams, 1876.
## _Newspapers_
_Banner of Light_
Boston _Globe_
_Boston Journal_
_Boston Post_
_Brooklyn Daily Argus_
_Brooklyn Daily Eagle_
_Brooklyn Daily Union_
_Brooklyn Eagle_ (Sunday)
Brooklyn _Review_
Brooklyn _Sunday Press_
_Brooklyn Union_
Burlington _Free Press_
_Chicago Mail_
_Chicago Post and Mail_
_Chicago Times_
_Chicago Tribune_
_Christian Union_
_Cincinnati Enquirer_
_Cincinnati Gazette_
_Cleveland Leader_
_Daily Morning Chronicle_
_Daily Patriot_
_Frank Leslie's Illustrated_
_Golden Age_
_Hartford Courant_
Hartford _Evening Post_
_Hartford Times_
_Independent_
_Indiana Times_
_Lancaster Gazette_
_Leavenworth Times_
_Lewiston Telegram_
_Liberator_
Louisville _Courier Journal_
_Memphis Argus_
Memphis _Daily Avalanche_
Memphis _Post_
New Orleans _Daily Picayune_
New Orleans _Tribune_
_Newark Sunday Call_
_New York Argus_
_New York Daily Graphic_
New York _Evening Post_
_New York Herald_
_New York Observer_
New York _Telegram_
New York _Star_
_New York Sun_
_New York Times_
_New York Tribune_
_New York World_
_North Star_
_Ottawa Free Trader_
_Ottawa Republican_
_Philadelphia Press_
_Rochester Democrat_
_Seneca County Courier_
Seymour _Times_
_Springfield Republican_
_St. Louis Times_
_Sunday Press_
_Thunderbolt_
Toledo _Sun_
_Train Ligue_
_Troy Daily Press_
_Weekly Argus_
## _Publications_
_American Heritage_
_The Arrow_
_Atlantic Monthly_
_Bonner's Weekly_
_Harper's Weekly_
_The Humanitarian_
_Ladies Home Journal_
_Macmillan's Magazine_
_The Masses_
_The Nation_
_The New Yorker_
_Omni_
_Putnam's Magazine_
_The Revolution_ —entire run, Jan. 8, 1868–May 1870. Sophia Smith Collection, Smith College (bound edition)
Syracuse University _Connections_ , April 1990
_Woman's Journal_
_The Word_
## _Library and Archive Collections_
_A Gentleman's Guide_. New York Historical Society, New York.
The American Society for Psychical Research, Inc., New York.
Beecher-Tilton Correspondence and trial, Lib Tilton Correspondence, Theodore Tilton Correspondence, Francis De Pau Moulton Correspondence, Mrs. Morse Correspondence, newspaper articles, clippings, pamphlets, church and civil trial—related material, etc., from the offices of the _Independent_. Tucker-Sachs Correspondence, Scrapbooks I–VIII, New York Public Library, Manuscript Collection. Abbreviated as _BTS_.
Colonel James H. Blood Correspondence. New York Historical Society.
The Boston Public Library.
Butler, Benjamin F., The Papers of. Library of Congress, Manuscript Division, Washington, D.C. Ref: Impeachment Trial.
Butler, Benjamin F., The Papers of. Massachusetts Historical Society, Boston Public Library. Columbia University, Rare Book Room, New York.
Davis, Paulina Wright, The Papers of. Alma Lutz Collection, Vassar College Library, Poughkeepsie, NY.
Dickinson, Anna E., The Papers of. Library of Congress, Manuscript Division, Washington, D.C. 25 reels, 10,000 items.
Garrison Family Papers, Garrison Collection, Sophia Smith Collection, Smith College Library. Also in Boston Public Library.
Horace Greeley Papers. New York Public Library.
Isabella Beecher Hooker Project. Letters and Papers of Isabella Beecher Hooker now at Harriet Beecher Stowe Center, Hartford, CT (formerly Stowe-Day Foundation). Material was at University of Massachusetts until 1990 and at the Connecticut Historical Society, Isabella Beecher Hooker (Mrs. John, 1822–1907, correspondence and papers); 1871–1903 Reel 80009. Abbreviated as _IBH_. Footnote citations are often taken from material at the University of Massachusetts and the Connecticut Historical Society supplied by Kathleen A. McDonough and may differ from later _ECS & SBA_, the guide to the microfilm, Ann D. Gordon and Patricia G. Holland, eds., and Kathleen A. McDonough and Gail K. Malmgreen, assoc. eds., _The Papers of Elizabeth Cady Stanton and Susan B. Anthony, Guide and Index_ , Vol. 1 (New Brunswick, NJ: Rutgers University. 110 Reels, microfilm edition, 1997). Whenever possible I have given both citations. Abbreviated as _ECS & SBA_.
"Investigation into the Causes of the Gold Panic." House Report No. 31, 41st Congress, Second Session, 1870. Library of Congress, Washington, D.C.
Moulton, Louise Chandler, Letters of. Library of Congress, Manuscript Division, Washington, D.C.
National American Woman Suffrage Association Collection. Library of Congress, Washington, D.C.
New York Historical Association.
New York Public Library, Manuscript Collection. Scrapbooks I-VIII, newspapers, bound court records, etc. All made available in the Frederick Lewis Allen Room.
New York University, Bobst Library, New York.
Plymouth Church Hearing of Henry Ward Beecher (July 28, 1874) transcript, New York Public Library, New York Historical Association.
Post, Amy, and Isaac Post, Family Papers of. University of Rochester Library, Department of Rare Books and Special Collections, Rochester, NY.
Whitelaw Reid Collection. Library of Congress, Washington, D.C.
Reid, Whitelaw, Letters of. New York Society Library.
Schlesinger Library, Radcliffe College, Cambridge, MA.
Smith, Gerrit, The Papers of. Syracuse University, Syracuse, NY.
Smith, Sophia, Collection. Smith College, Northampton, MA.
Stanton, Elizabeth Cady, and Susan B. Anthony, Papers of. University of Massachusetts, Amherst.
Stanton, Elizabeth Cady, and Susan B. Anthony, Papers of. The Stanton and Anthony Project. Ann D. Gordon and Patricia G. Holland, editors. New Brunswick, NJ: Rutgers University.
110 reels (including 7 reels Susan B. Anthony Scrapbooks and Diaries 1848–1900), microfilm edition. _Woodhull & Claflin's Weekly_, v. 1–3, Reel 4798. Abbreviated as _ECS & SBA_.
Elizabeth Cady Stanton Correspondence. New York Public Library, Manuscript Collection.
Stevens, Thaddeus, The Papers of. Library of Congress, Washington, D.C.
Transcript of the Plymouth Church Hearing of Henry Ward Beecher. New York Public Library, New York Historical Association. Abbreviated as _PCH_.
Vanderbilt Will Case Clippings, November 1877. New York Public Library, Room 315.
Weld-Crimke, Papers and Diaries. Willard L. Clements Library, University of Michigan.
Victoria Claflin Woodhull Correspondence. Butler Library, Columbia University, New York.
Victoria Claflin Woodhull Correspondence. Manuscript Collection, Historical Society of Pennsylvania, Philadelphia.
Victoria Claflin Woodhull Correspondence. Alma Lutz Collection, Vassar College Library, Poughkeepsie, NY.
Victoria Claflin Woodhull (Martin) Correspondence. New York Historical Society.
Victoria Claflin Woodhull (Martin) Papers. Boston Public Library, Manuscript Collection.
Victoria Claflin Woodhull (Martin) Papers. Southern Illinois University Special Collections, Morris Library, Carbondale, IL. Abbreviated as _VCWM_.
_Woodhull & Claflin's Weekly_. Reel 1, Vol. 1–3, May 14, 1870–Nov. 18, 1871. Reel 2, Vol. 4–7, Nov. 18, 1871–May 30, 1874. Reel 3, Vol. 8–12, June 6, 1874–June 10, 1876. Abbreviated as _WCW_.
Martha Coffin Wright Correspondence. Sophia Smith Collection, Smith College Library; 1,500 letters in William Lloyd Garrison Papers, Smith College, Northampton, MA. Abbreviated as _MCW_.
# _Acknowledgments_
IT HAS TAKEN ME over a decade to research and write this book. When I started out, I thought I would write a biography of Victoria Woodhull but as I researched further, I realized that her life and the age in which she lived were inextricably intertwined. This combination led me to many sources and many people whom I wish to acknowledge, for without them I could not have written the book that I did. Vartan Gregorian, the late Timothy Healy, and Paul LeClerc—all presidents of the New York Public Library—gave me access to material that I studied in the Frederick Lewis Allen Room of that institution, where much of this book was written. John Baker, Duane Watson, and Wayne Furman found rare materials (including eight scrapbooks compiled in the offices of the _Independent_ ). Mark Piel of the New York Society Library not only researched the answers to countless questions but requested books and papers from libraries the world over. Carlton Rochell provided access to research materials at the Bobst Library of New York University.
It was particularly difficult to penetrate the Victorian propriety that prevented me from seeing more than the prissy cardboard characters depicted in the books of the era. For that I needed primary sources: diaries, letters, newspaper accounts, advertisements, records of woman's rights meetings, church records, court transcripts, etc. I found an extraordinary group of people who helped me to find these documents and I feel that this infused life into my subjects.
Ann D. Gordon of the Elizabeth Cady Stanton and Susan B. Anthony Project worked with me, first at the University of Massachusetts with her colleague Patricia G. Holland, and then at Rutgers University. Kathleen McDonough was invaluable in finding Stanton and Anthony papers and the never-published diaries and letters of Isabella Beecher Hooker. Jo Blatti and Susan Zack at the Harriet Beecher Stowe Center (formerly the Stowe-Day Foundation) provided expertise and advice concerning the Beecher family.
Lynn Sherr gave me insight into the life of Susan B. Anthony, as well as the mixed blessing of a projector and 110 reels of microfilm from the Stanton and Anthony Project, which kept me researching nights as well as days. John B. Jentz of History Works, Inc., located the manslaughter indictment against Tennessee Celeste Claflin and affidavits covering the case. Karen Sturges meticulously researched Martha Coffin Wright, Lucretia Mott, and other members of the suffrage and Spiritualist movements. Edith Mayo, curator of Women's History at the Smithsonian, supplied the history of the McClintock table. Patrice Keane of the American Society for Psychical Research, Inc., as well as Maria Janis, provided both historical and present-day accounts of psychic powers.
My first researcher, Christopher Stevens, began the search for material well over a decade ago. Linda Amster has, over the years, continued that search and lent friendship and support. Vanessa Stark's computer knowledge, patience, and goodwill are appreciated. Elizabeth Newell, followed by Diana Stagner, have dealt with innumerable drafts of my manuscript, assisted me personally, and cheered me on. Joanne Arestou, Cinda Siler, Elizabeth Drorbaugh, Martin Timins, Elizabeth Berne, and Susan Carroll all contributed to the technical production of this book.
Several experts enriched the manuscript in its final stages, but I would particularly like to thank Sean Wilentz of Princeton University and Nicole Bisel, an expert on Anthony Comstock, of Northwestern University. Gloria Steinem provided innumerable insights into how Victoria's life resonates for women both in the past and present. The psychic and author Mary T. Browne explained Victoria and Tennie's powers in light of her own experience.
At Knopf, I thank Victoria Wilson, an editor with an incomparable eye who always believed in this book, as well as Sonny Mehta, Jane Friedman, Bob Gottlieb, Katherine Hourigan, Lee Buttala, Karen Deaver, and Carol Devine Carson. I also wish to thank Lynn Nesbit, Morton Janklow, Yves-André Istel, Jane Hitchcock, Jim Hoagland, Wayne Lawson, and Bernard Lewis.
Finally, my gratitude and admiration to the person who should have had first billing, Jason Epstein. You drove me back to the desk when I nearly abandoned this project and subsequently read successive drafts, demonstrating both your friendship and prodigious editorial skills. Yes, I know you would tell me to cut the word "prodigious."
# ILLUSTRATION CREDITS
Courtesy of The New York Public Library, Prints and Manuscripts Division: pp. ii, xii (from the _New York Times_ , January 20, 1871), 2.1 (left and right), 5.1, 28.1
Courtesy of The New York Historical Society: pp. 1.1, 3.1 (top), 8.1 (left and right), 9.1, 9.2, 9.3 (photograph by Matthew Brady), 11.1, 14.1 (left and right), 14.2, 16.1, 27.1, 29.1 (from the _Daily Graphic_ ), 33.1, 35.1
Courtesy of The Schlesinger Library, Radcliffe College: pp. 2.2, 7.1, 21.1, 22.1
Courtesy of The American Psychical Association: pp. 3.2 (bottom), 5.2
Courtesy of the Sophia Smith Collection, Smith College: pp. 4.1 (top), 18.1
Courtesy of Seneca Falls Historical Society: pp. 4.2 (bottom), 4.3
Courtesy of New Castle Historical Society: pp. 5.3 (left and right), 23.1
From the New York _Evening Telegraph:_ pp. 7.2, 16.1
Courtesy of The New York Public Library, Miriam and Ira D. Wallach Division of Art: pp. 9.4, 12.1, 13.1 (bottom), 17.1, 19.1, 20.1, 23.2, 25.1 (top, photograph by Brown Brothers), 25.2 (bottom), 27.2, 30.1
Courtesy of the Harriet Beecher Stowe Center: pp. 10.2, 10.3, 17.3 (top, center, and bottom), 34.1, 35.1
Courtesy of The New York Public Library, Rare Books and Manuscripts Division: pp. 13.1 (from the _National Police Gazette_ , March 13, 1847), 13.2
Courtesy of the Union Pacific System Archive: p. 15.1
Courtesy of the Library of Congress: pp. 17.3 (top and bottom), 24.1, 26.1
Courtesy of the University of Illinois at Carbondale: p. 18.2 (from _Woodhull & Claflin's Weekly_)
Courtesy of the American Antiquarian Society: pp. 20.2, 34.2
Courtesy of the New York Public Library: pp. 28.2 (from James McCabe's _Light and Shadows of New York Life_ ), 29.2 (from James McCabe's _Light and Shadows of New York Life_ )
Courtesy of the Boston Historical Society: p. 32.1
Courtesy of The New York Public Library, Schomberg Library Collection: p. 34.3
All illustrations not credited are from the collection of the author.
| {
"redpajama_set_name": "RedPajamaBook"
} | 500 |
package com.wtfrak.nixon.scanner.tokens;
public class Token {
private TokenType tokenType;
private String value;
public Token(TokenType tokenType) {
this.tokenType = tokenType;
this.value = null;
}
public Token(TokenType tokenType, String value) {
this.tokenType = tokenType;
this.value = value;
}
public Token(TokenType tokenType, Character value) {
this.tokenType = tokenType;
this.value = Character.toString(value);
}
public TokenType getTokenType() {
return this.tokenType;
}
public String getValue() {
return this.value;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,459 |
Image Title: Cal King Headboards Andymayberry Com Pertaining To California Headboard Only Decor 4. Post Title: California King Headboard Only. Filename: cal-king-headboards-andymayberry-com-pertaining-to-california-headboard-only-decor-4.jpg. Image Dimension: 450 x 450 pixels. Images Format: jpg/jpeg. Publisher/Author: kenstonpd. Uploaded Date: Sunday - October 14th. 2018 09:41:10 AM. Category: Architecture. Image Source: houzz.com. 11. Within King Platform White Regarding With Co Shop Remodel For Cal California Only Styles Intended Decor Grey 1. Stylish Designs In Point Upholstered 10. The Tommy Pertaining 15. Size Sofa Isle Headboards Headboard Metal To Ideas Bedroom Andymayberry Taupe Techthreads Aspen Prepare Steal 4. Ultimate Furniture Tufted 1 A BlogBeen Wood Bahama Cypress Home Throughout 17. 18. Black Complete 9. Plan Woven Driftwood 19. Cherry Your Accessory Com Rustic 14. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,677 |
export { SettingsView } from './SettingsView';
| {
"redpajama_set_name": "RedPajamaGithub"
} | 286 |
niug.calendari.afegirEsdeveniment = function(data)
{
require(["dojo/date"], function(date) {
if (undefined != data) {
data.setMinutes(0);
dijit.byId('desDeDate').set('value', data);
dijit.byId('desDeTime').set('value', data);
data = date.add(data, 'hour', 1);
dijit.byId('finsDate').set('value' , data);
dijit.byId('finsTime').set('value' , data);
}
});
niug.calendari.dialeg.show();
}
niug.calendari.eliminarEsdeveniment = function(url)
{
require(["dojo/request/xhr"], function(xhr){
var id = dijit.byId('esdeveniment').get('value');
xhr(url, {
data: { esdeveniment_id: id },
handleAs: "json",
method: "post"
}).then(function(data){
if (undefined != data) {
niug.calendari.dialeg.hide();
niug.calendari.esdeveniments.remove(data.id);
}
}, function(err){
alert('ERROR::Impossible eliminar l\'esdeveniment');
});
});
}
niug.calendari.guardarEsdeveniment = function(url)
{
require(["dojo/request/xhr"], function(xhr){
xhr(url, {
data: dijit.byId('formEsdeveniment').get('value'),
handleAs: "json",
method: "post"
}).then(function(data){
if (undefined != data) {
niug.calendari.dialeg.hide();
niug.calendari.esdeveniments.remove(data.id);
niug.calendari.esdeveniments.put(data);
}
}, function(err){
alert('ERROR::Impossible guardar l\'esdeveniment');
});
});
}
niug.calendari.modificarEsdeveniment = function(esdevenimentId)
{
// Introduir les dades del esdeveniment al formulari
var esdeveniment = niug.calendari.esdevenimentsStore.get(esdevenimentId);
if (!esdeveniment) {
return false;
}
dijit.byId('categoria').set('value', esdeveniment.categoriaId, false);
niug.calendari.mostrarOpcionsCategoria(esdevenimentId);
dijit.byId('esdeveniment').set('value', esdeveniment.id);
dijit.byId('desDeDate').set('value' , esdeveniment.startTime);
dijit.byId('desDeTime').set('value' , esdeveniment.startTime);
dijit.byId('finsDate').set('value' , esdeveniment.endTime);
dijit.byId('finsTime').set('value' , esdeveniment.endTime);
dijit.byId('lloc').set('value' , esdeveniment.lloc);
dijit.byId('motiu').set('value' , esdeveniment.lloc);
require(["dojo/dom-style"], function(style) {
var node = dijit.byId('botoEliminar').domNode;
style.set(node, 'display', '');
});
// Mostrar el Dialeg
niug.calendari.dialeg.show();
}
niug.calendari.mostrarOpcionsCategoria = function(esdevenimentId)
{
require(["dojo/dom-construct", "dojo/html", "dijit/form/ComboBox",
"dijit/form/Textarea", "dojo/dom", "dojo/query", "dojo/_base/lang",
"dojo/dom-style", "dijit/form/FilteringSelect", "dijit/form/Textarea",
"dijit/form/Form", "dijit/form/Button", "dijit/form/TextBox",
"dojo/parser" ],
function(domConstruct, hml, ComboBox, Textarea, dom,
query, lang, domStyle, FilteringSelect)
{
if (undefined == esdevenimentId) {
var widgetEsdeveniment = dijit.byId('esdeveniment');
esdevenimentId = widgetEsdeveniment.get('value');
}
// Eliminar les files variables
query('tr[data-niug-tipu="fila_variable"]').forEach(function(node) {
// Eliminar els widgets de les files
dijit.findWidgets(node).forEach(function(widget) {
widget.destroy();
});
// Eliminar la fila
domConstruct.empty(node);
});
// Si existeix obtenir les dades de l'esdeveniment
var clientId = '';
var resum = '';
if (undefined != esdevenimentId) {
var esdeveniment = niug.calendari.esdevenimentsStore.get(esdevenimentId);
if (esdeveniment) {
clientId = esdeveniment.clientId;
resum = esdeveniment.resum;
}
}
var widgetCategoria = dijit.byId('categoria');
if (!widgetCategoria) {
return false;
}
var categoriaId = widgetCategoria.get('value');
var categoria = niug.calendari.categoriesStore.get(categoriaId);
if (!categoria) {
return false;
}
if ('reserva' == categoria.codi || 'visita' == categoria.codi) {
// Mostrar camp client
var fila = domConstruct.toDom('<tr id="fila_client_id" data-niug-tipu="fila_variable">'
+ '<td><label for="client_id">Client:</td>'
+ '<td><div id="client_id"/></td>'
+'</tr>');
domConstruct.place(fila, 'fila_fins', 'after');
var filteringSelect = new FilteringSelect({
id: "client_id",
name: "client_id",
placeHolder: "Selecciona un client",
store: niug.calendari.clientsStore,
searchAttr: "name",
value: clientId
}, "client_id");
if ('visita' == categoria.codi) {
// Mostrar resum de la visita
var fila = domConstruct.toDom('<tr id="fila_resum_visita" data-niug-tipu="fila_variable">'
+ '<td><label for="resum_visita">Resum visita:</td>'
+ '<td><div id="resum_visita"/></td>'
+'</tr>');
domConstruct.place(fila, 'fila_motiu', 'after');
var textarea = new Textarea({
id: "resum_visita",
name: "resum_visita",
value: resum
}, "resum_visita");
}
}
});
}
niug.calendari.onHide = function()
{
dojo.byId('formEsdeveniment').reset();
require(["dojo/dom-style"], function(style) {
var node = dijit.byId('botoEliminar').domNode;
style.set(node, 'display', 'none');
});
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,223 |
\section{Introduction}
Game Theory is originally introduced by John von Neumann and Oskar
Morgenstern (Neumann J and Morgenstern O. 1944) in 1944. Later
John F. Nash (Nash JF. 1950) has made significant contribution to
Game Theory by introducing Nash Equilibrium and proving every
finite game has mixed Nash Equilibrium, which becomes the core
idea of Game Theory. Since then Game Theory has been widely
applied in various disciplines such as social sciences (most
notably economics), political science, international relations,
computer science and philosophy. So far eight game theorists have
won Nobel prizes in economics. John M. Smith in 1973 formalized
another central concept in Game Theory called the Evolutionary
Stable Strategy (ESS, Smith JM. 1973). Various work has been done
using ESS to investigate the behavior and evolutionary path of
animals ( Takada T and Kigami J. 1991, Crowley PH. 2000, Nakamaru
M and Sasaki A. 2003, Matsumura S and Hayden TJ. 2005, Wolf N and
Mangel M. 2007, Krivan V, Cressman R and Schneider C. 2008,
Hamblin S and Hurd PL. 2009).
\\
The simplest yet most common case of ESS game is $2 \times 2$
non-cooperative game. Multiple player game is more realistic but
according to Poincare-Bendixson theorem, multiplayer dynamic
system would result in chaos hence we only consider $2 \times 2$
games (Yi T et al. 1997). While the key idea in Game Theory is
Nash Equilibrium, we must define the payoff matrix very carefully
in order to compute Nash Equilibrium. However, most of the
research regarding ESS has arbitrarily assigned payoff matrix. To
overcome this problem, we want to use some realistic quantities to
define payoff matrix for the game. M-Gibbons has shown a neighbor
intervention model (Mesterton-Gibbons M, Sherratt TN. 2008) and
Luther further discussed whether food is worth fighting for
(Luther RM, Broom M and Ruxton GD. 2007). Just has studied the
aggressive loser in an ESS game (Just W, Morris MR and Sun X.
2006). Inspired by their ideas, we will use food source to compute
the payoff of aggressive and non-aggressive players in a $2 \times
2$ game. \\
While we assume the food source is finite and equivalent to any
member in the population, it is natural to use dynamic programming
(DP) to figure out the optimal foraging strategy as a resource
allocation problem. Animal growth rate is a logistic curve and we
divide the whole growth process into three distinct stages:
initialization, quasilinear growth and termination. In different
stages, the payoff is a linear function of food source with
different slope and our goal is to determine the maximum total
payoff at the end of growth using dynamic
programming. \\
Al-Tamimi has suggested using dynamic programming to implement
Game Theory model for designing (Al-Tamimi A, Abu-Khalaf M and
Lewis FL. 2007) but their model is zero-sum. We will first present
a more realistic general sum game framework, then discuss three
different types of aggressive players, calculate the numerical
payoff matrix for each case and determine the ESS for them. Our
work is the first of this kind to combine dynamic programming and
Game Theory, two different optimization tools together to solve
real biological problem.
\section{Defining the Model}
A typical $2 \times 2$ non-cooperative general sum game has the
following form where $P_{ji}$ defines the payoff of player \emph{\it{i}} in \emph{\it{j}}th strategy combination.\\
\begin{center}
\begin{tabular}{c c c}
\hline
Strategy & Non-aggressive & Aggressive \\
\hline
Non-aggressive & $(P_{111}, P_{112})$ & $(P_{121}, P_{122})$ \\
Aggressive & $(P_{211}, P_{212})$ & $(P_{221}, P_{222})$ \\
\hline
\end{tabular}
\\
Table 1. Payoff Matrix of non-cooperative general sum game
\end{center}
$P_{ijk}$ denotes the payoff of player \textit{k} when it uses
strategy \textit{i} and its components uses \textit{j}. Here we
have two types of strategies: aggressive (2) and non-aggressive
(1). Aggressive players would fight their neighbor and try to get
their resources. Non-aggressive players only concentrate on their
own food source and never fight back even when they are attacked.
However, if two aggressive players meet, it would result in a
severe fight and both players are terribly hurt. This definition
is similar to that of "Chicken-Dare" or "Hawk-Dove" game. The Nash
Equilibrium is defined as:\\
$\textbf{Definition 1. } x \in \Theta$ is a Nash Equilibrium if $x
\in \tilde{\beta} (x)$, where $\Theta$ is the mixed strategy space
and $\tilde {\beta}$ is
the mixed strategy best response correspondence.\\
Because this is a $2 \times 2$ finite symmetric game,
$\Delta^{NE}\neq \emptyset$ by Kakutani's Theorem. Next we switch
to a population perspective and define Evolutionary Stable
Strategy (ESS) as follows:\\
$\textbf{Definition 2. } x \in \Delta$ is an ESS if for every
strategy $y \neq x $ there exists some $\bar{\epsilon}_y \in
(0,1)$ such that $u[x,\epsilon y+(1-\epsilon)x]>u[y,\epsilon
y+(1-\epsilon)x]$ holds for all $\epsilon \in (0,
\bar{\epsilon}_y)$ where $\epsilon$ is the proportion of mutant
strategy.
\\
Basically, ESS is a subset of Nash Equilibrium. We use Maynard's
criterion to test whether a Nash Equilibrium is an ESS:\\
$\textbf{Theorem 1. }\Delta^{ESS}=\{x \in
\Delta^{NE}:u(y,y)<y(x,x), \forall y \in \beta^{*}(x), y \neq
x\}.$
\\
To perform all these analysis, we must first define the payoff
matrix of our original game. We will use DP to determine the
numerical payoff values for the four strategy combinations. Assume
each player has a total of \emph{\it{N}} food sources for the
entire development period and in each stage at least \emph{\it{1}}
resource should be consumed in order to maintain basal metabolism.
As we have discussed before, the development period is divided
into three stages: growth initialization, quasilinear growth and
growth termination, hence the player could consume ${1 \cdots
N-2}$ resources in each stage. While the growth is logistic and
nonlinear, we could use linear approximation in each stage as
follows where y is the payoff in each stage and x is the resource
consumed:
\begin{equation}
y=\begin{cases} ax, & \text{Growth Initialization},\\
bx, & \text{Quasilinear Growth},\\
cx, & \text{Growth Termination}.
\end{cases}
\end{equation}
Because logistic curve has a sigmoid shape and is usually
symmetric, it is reasonable to set $a=c$ to reduce computational
intensity. The coefficients $a$ and $b$ has biological meaning of
the efficiency of converting food sources into its own energy and
in our model $b>a$. The DP model is written as follows:
\begin{center}
$$\text{Maximize }z=\sum^{3}_{i=1}r_ix_i$$
$$\text{Subject to } \sum^{3}_{i=1}x_i=N$$
\end{center}
The backward DP Formulation for this model is:\\
OVF: $f_k(x)=$ optimal return for the allocation of x units of
resource to stage k $\cdots$ 3.\\
ARG: $(k,x) =$ (stage, units of resource consumed).\\
OPF: $P_k(x)=$ units of resource consumed at stage k.\\
RR: $f_k(x)=max_{x_k=0,1,\cdots x}(r_kx_k+f_{r+1}(x-x_k))$, $x=1 \cdots N-2$\\
BC: $f_N(x)=r_N(x)$\\
ANS: $f_1(x)$\\
For the non-aggressive and non-aggressive strategy
combination, we assume both players do not interfere each other.
In this case, we would only solve the DP for one of them and by
symmetry, the other player should adopt same strategy to maximize
its total payoff. The cost in each stage and state is shown in
Table 2 and we could
calculate the optimal value using DP.\\
For the non-aggressive and aggressive strategy combination, the
cost table is similar to Table 2. The difference is we must define
different $a$ and $b$ values for both strategies. Same thing
happens for the aggressive and aggressive combination. Once we
have figured out the payoff values for each combination we could
complete the payoff matrix and further investigate the ESS.
\begin{center}
\begin{tabular}{c c c c }
\hline
State/Stage & 1 & 2 & 3 \\
\hline
1 & a & b & a \\
2 & 2a & 2b & a \\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
N-2 & (N-2)a & (N-2)b & (N-2)a \\
\hline
\end{tabular}
\\
Table 2. Cost Table in Different Stages and States
\end{center}
\section{Model Results}
\subsection{Type I Model: Final Battle} In this simplest case, we
assume both aggressive player and non-aggressive player only fight
after they have depleted all their resources. In real ecosystem,
some animals don't fight while they are young. In fact, they may
even help each other (Taborsky M. 2001)! They fight only when they
are sexually matured. So here we don't even have to bother DP. We
assume the optimal payoffs of non-aggressive and non-aggressive
combination is $(1,1)$ and aggressive player could take advantage
of half of the non-aggressive player's payoff but lose 80\% of its
own payoff when it encounters another aggressive player. The
payoff matrix is:
\begin{center}
\begin{tabular}{c c c}
\hline
Strategy & Non-aggressive & Aggressive \\
\hline
Non-aggressive & (1,1) & (0.5,1.5) \\
Aggressive & (1.5,0.5) & (0.2,0.2) \\
\hline
\end{tabular}
\\
Table 3. Payoff Matrix of Type I Model
\end{center}
In this game there are two pure strategy Nash Equilibria:
nonaggressive- aggressive and aggressive- nonaggressive
combinations. There is another mixed strategy Nash Equilibrium
where both player use aggressive strategy with probability
$\dfrac{5}{8}$ and non-aggressive strategy with probability
$\dfrac{3}{8}$. All these three Nash Equilibria give aggressive
strategy 1.5 payoff and that of non-aggressive is 0.5.\\
According to Smith Maynard's criterion, both pure Nash Equilibria
are not pure ESS and the mixed Nash Equilibrium is the only ESS in
this game.
\subsection{Type II Model: Modified Final Battle} In this modified
final battle model, we assume the fight occurs at the end of
growth as well. Suppose $a=1$ and $b=2$ for non-aggressive player
and aggressive strategy player follows a different development
path. Because most of the development happens during the
quasilinear stage, the simplest case is to assign a smaller
\textit{b} coefficient for aggressive player, for instance,
$b=1.5$. To make our model more realistic, we will modify the
model assumption by taking development lag, development saturation
and both cases into account, thus the second development stage of
aggressive player is not linear. Development lag describes very
slow development when consuming less than certain amount of
resources (lower threshold). Development saturation describes no
development when consuming more than certain amount of resources
(upper threshold). Here we present some typical returns with
respect to different resources
allocation in different development conditions. \\
\begin{center}
\begin{tabular}{c c c c c c c c c}
\hline
& Stage 1 & & & & Stage 2 & & & Stage 3 \\
\hline
State/Case & Linear & Linear & Lag 1 & Lag 2 & Sat. 1 & Sat. 2 & Lag + Sat. & Linear \\
\hline
1 & 1 & 1.5 & 1 & 1 & 2 & 2 & 1 & 1 \\
2 & 2 & 3 & 1 & 1 & 4 & 4 & 2 & 2\\
3 & 3 & 4.5 & 1 & 1 & 6 & 6 & 4 & 3\\
4 & 4 & 6 & 2 & 1 & 8 & 6 & 8 & 4\\
5 & 5 & 7.5 & 4 & 2 & 8 & 6 & 8 & 5\\
6 & 6 & 9 & 8 & 4 & 8 & 6 & 8 & 6\\
7 & 7 & 10.5 & 8 & 8 & 8 & 6 & 8 & 7\\
8 & 8 & 12 & 8 & 8 & 8 & 6 & 8 & 8\\
\hline
\end{tabular}
\\
Table 4. Return Table in Different Development Conditions for
Aggressive Player
\end{center}
\begin{center}
\begin{tabular}{c c c c c c c c c}
\hline
& Stage 1 & Stage 2 & Stage 3 \\
\hline
State/Case & Linear & Linear & Linear \\
\hline
1 & 1 & 2 & 1 \\
2 & 2 & 4 & 2\\
3 & 3 & 6 & 3\\
4 & 4 & 8 & 4\\
5 & 5 & 10 & 5\\
6 & 6 & 12 & 6\\
7 & 7 & 14 & 7\\
8 & 8 & 16 & 8\\
\hline
\end{tabular}
\\
Table 5. Return Table for Non-Aggressive Player
\end{center}
Based on table 4, there are 6 different types of return in 2nd
stage for the aggressive player. While for the non-aggressive
player, its returns in different stages are provided in table 5.
So we could formulate a total of 7 DP problems for both players.
Solve these 7 DP problems we have got the optimal allocation
strategy and the corresponding maximum return, shown in table 6.
Please note in table 6 the condition title "linear", "Lag" and
"Saturation" is for the second stage of
entire process only. The notion is different from that of table 4 and 5.\\
\begin{center}
\begin{tabular}{c c c c c c c c}
\hline
& Non-aggressive & & & Aggressive & & \\
\hline
Condition & Linear & Linear& Lag 1 & Lag 2 & Sat. 1 & Sat. 2 & Lag + Sat. \\
\hline
$P_1(x)$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
$P_2(x)$ & 8 & 8 & 6 & 7 & 4 & 3 & 4 \\
$P_3(x)$ & 1 & 1 & 3 & 2 & 5 & 6 & 5 \\
$f_1(x)$ & 18 & 14 & 12 & 11 & 14 & 13 & 14 \\
\hline
\end{tabular}
\\
Table 6. Optimal Allocation Strategy and Returns
\end{center}
Since the entire development is symmetric, the resource allocated
in stage 1, $P_1(x)$ and in stage 3, $P_3(x)$ should be
interchangeable. For instance, in Lag 2 condition, the aggressive
player could either consumer 1 unit resource in stage 1 and 3
units of resource in stage 3, or 3 units in stage 1 and 1 unit in
stage 3; the final total returns are just identical.
\\
Now we have got the optimal return so we could calculate the
payoff matrix based on our previous definition. Depending on
different conditions, the total return of aggressive player varies
from 11 to 14. Here we use 12 as instance and calculate the payoff
matrix:\\
\begin{center}
\begin{tabular}{c c c}
\hline
Strategy & Non-aggressive & Aggressive \\
\hline
Non-aggressive & (18,18) & (9,21) \\
Aggressive & (21,9) & (2.4,2.4) \\
\hline
\end{tabular}
\\
Table 7. Payoff Matrix of One Condition of Type II Model
\end{center}
In this game there are three Nash Equilibria, almost the same as
in type I game except that mixed Nash Equilibrium requires
$\dfrac{11}{16}$ non-aggressive and $\dfrac{5}{16}$ aggressive
strategy. From a population point of view, by applying Maynard's
criterion, the mixed Nash Equilibrium is the only ESS in the
evolutionary game.
\subsection{Type III Model: Battles in Every Stage} In this model
the aggressive player will fight in all stages to maximize its
payoff. While it is difficult to model the interaction of fighting
for food source, instead we give the aggressive player larger
coefficients than in Model I and non-aggressive player smaller
coefficients when they encounter. For the aggressive-aggressive
strategy combination, we simply give both players 0 because of
fighting severity. Since they fight in each stage, there is no
final battle in this circumstance. In this model we also consider
two different conditions: $b=0.5$ for non-aggressive player and
$b=1.5$ for aggressive player; $b=1.5$ for non-aggressive player
and $b=2.5$ for aggressive player. The DP results are shown in the
following table:
\begin{center}
\begin{tabular}{c c c c c}
\hline
Condition & 1 & 1 & 2 & 2 \\
\hline
Player & Non-aggressive & Aggressive & Non-aggressive & Aggressive \\
\hline
$P_1(x)$ & 8 & 1 & 1 & 1\\
$P_2(x)$ & 1 & 8 & 8 & 8\\
$P_3(x)$ & 1 & 1 & 1 & 1\\
$f_1(x)$ & 9.5 & 14 & 14 & 22\\
\hline
\end{tabular}
\\
Table 8. Optimal Allocation Strategy and Returns
\end{center}
So the payoff matrix for condition one is:
\begin{center}
\begin{tabular}{c c c}
\hline
Strategy & Non-aggressive & Aggressive \\
\hline
Non-aggressive & (18,18) & (9.5,14) \\
Aggressive & (14,9.5) & (0,0) \\
\hline
\end{tabular}
\\
Table 9. Payoff Matrix of One Condition of Type III Model
\end{center}
The non-aggressive and non-aggressive strategy combination is the
only Nash Equilibrium in this game; there is no mixed strategy
Nash Equilibrium. This is also the ESS by Maynard's criterion.
\\
For condition two:
\begin{center}
\begin{tabular}{c c c}
\hline
Strategy & Non-aggressive & Aggressive \\
\hline
Non-aggressive & (18,18) & (14,22) \\
Aggressive & (22,14) & (0,0) \\
\hline
\end{tabular}
\\
Table 10. Payoff Matrix of One Condition of Type III Model
\end{center}
In this game there are three Nash Equilibria, almost the same as
in type I game except that mixed Nash Equilibrium requires
$\dfrac{7}{9}$ non-aggressive and $\dfrac{2}{9}$ aggressive
strategy. From a population point of view, by applying Maynard's
criterion, the mixed Nash Equilibrium is the only ESS in the
evolutionary game.
\section{Discussion}
Though we use DP to find out the optimal allocation strategy, we
have already found under certain circumstances, for instance if
$b>a$, most of our resources should be allocated to the second
stage, the quasilinear growth. In Type II Model we have also
realized the growth does not necessarily be a linear function.
Here we present a criterion to test if we could use the growth
function directly to allocate resources optimally:\\
Assume symmetric growth still holds and define $y=f(x)$ for both
growth initialization and growth termination and $y=g(x)$ for
quasilinear growth. Notice the term "quasilinear growth" here does
not mean the growth function is linear, it could be nonlinear
anyway. If the following is true then we should allocate most of
our resource in the quasilinear growth stage and minimum for
growth initialization and termination: $g(x)$ is not concave and
$g'(x)\geq f'(x), \forall x$.\\
However, this criterion is only sufficient but not necessary. It
is possible to investigate the sufficient and necessary condition
but the computational intensity is almost the same of using DP
because we must compute the first order partial derivative
(gradient) of $f(x)+f(y)+g(10-x-y)$ with respect to $x$ and $y$,
the resource allocated in stage 1 and 3, and determine the
structure of the gradient.\\
In this research project we focus on finite and equal resource
allocation problem for both players. However, our approach could
be extended to unequal resources because we use DP to determine
the optimal strategy for each player so it does not matter whether
the resources are equal for both players. In other words, the
player should not worry about the total amount of their resource
(and actually they cannot determine the amount of resource because
it is pre-specified.) but rather concentrate on how to optimize
the return from the resource (the optimal strategy). It is also
possible to assume infinite resource, however the consumption of
the player is bounded so infinite resource allocation problem
could be transformed to finite resource allocation problem. As we
have discussed before, saturation is a reasonable assumption to
deal with infinite resource. Therefore, we could use DP to solve
almost all types of resources allocation problem for 2 player
game.\\
Another possible improvement of our approach is to introduce
stochastic component into the model. Instead of assigning a
specific amount of resources, we could assume the food resource is
from a certain probability distribution, say, normal distribution.
In effect this is the extension of unequal resource allocation
problem for 2 players. Besides, it is reasonable to assign a
minimum threshold of development and if the player fail to reach
that threshold it then dies. The remaining resources are
transferred to its neighbor (its competitor). In this circumstance
DP could still be applied but we expect the formulation is much
more complicated. When we reach the optimal strategy of resource
allocation we could still apply Game Theory to determine Nash
Equilibrium for a given game but it is difficult to give a close
form representation of what ESS looks like in this scenario
because of stochasticity. We could use simulation to determine the
evolutionary path and this approach is more realistic and useful.
\nocite{Krivan_V} \nocite{Matsumura} \nocite{Nakamaru}
\nocite{Crowley} \nocite{Hardling} \nocite{Al-Tamimi}
\nocite{Hamblin} \nocite{Mesterton} \nocite{Luther} \nocite{Just}
\nocite{Yi} \nocite{Takada} \nocite{Taborsky} \nocite{Neumann}
\nocite{Nash} \nocite{Maynard}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,974 |
\section{Introduction}
It is well known that by removing or adding a nucleon from a magic nucleus the
self-consistent mean field solution breaks the rotational symmetry and time-odd
contributions also appear in the mean field. By adding or removing more nucleons
typically nuclei become more deformed in the mean field description and their excited
spectra evolve from vibrational to rotational spectra~\cite{Bohr:1969,Ring:2004}.
One can ask the question:
``How a large spin polarization of nuclei, neutron or nuclear matter would modify
the properties of their ground states and spectra and their response to various external
perturbations?" The interest in spin polarized nuclei, neutron, and nuclear
mater is extensive~\cite{Bauer:2020,Polls:2007,Urban:2020,Riz:2018,
Stein:2018,Stein:2016,Forbes:2014,Tews:2020,Torres-Patino:2019,
Behera:2016,Dexheimer:2017,Harutyunyan:2016,Rabhi:2015, Stein:2016a,
Sammarruca:2015,Behera:2015,Vidana:2016,Lacroix:2015,Kruger:2015,
Aguirre:2014,Bordbar:2013,Dong:2013,Isayev:2011,Isayev:2012,Isayev:2006}.
The odd or the odd-odd superfluid nuclei are the simplest examples
of spin polarized systems. The definition
of a spin polarized system depends on various circumstances. Polarization, in
particular the magnetization of condensed matter systems, can be due either to a
spontaneous symmetry breaking or it can be induced by
an external field. The magnitude of the polarization, or loosely speaking
of the difference between the particle numbers with spins up and down
$N_\uparrow-N_\downarrow$ is not as a rule an integer. Moreover,
$N_\uparrow-N_\downarrow\neq 0$
can occur in both normal and superfluid systems. Polarized systems can be also
artificially ``manufactured'' and this is routinely done in cold atom physics.
In a time-dependent framework, a major
difficulty in studying the dynamics of a superfluid fermionic system
in general or in particular of a nuclear system with an imbalance of
spin-up and spin-down fermions ($N_\uparrow \neq N_\downarrow$), is
the construction the initial stationary state needed in time-dependent
treatments. Using the simple blocking approximation for the initial
state could be problematic in the case of time-dependent
phenomena. Within the blocking approximation the odd fermion is often
described by a single-particle wave function and not by a Bogoliubov
quasiparticle wave function, and thus the orthogonality between the
two types of fermion wave functions cannot be enforced during the time
evolution, except within the BCS approximation. Using the equal
filling approximation, when the extra odd fermion occupation
probability is spread uniformly among several levels leads to
unphysical polarization properties of the system, which can have major
unphysical consequences when describing the nuclear fission of odd or
odd-odd nuclei.
Of particular interest is the microscopic description and the
understanding of the nuclear fission dynamics, particularly of the
stage where the fissioning nucleus emerges from below the potential
barrier near the saddle, to the scission configuration. This part
of the nuclear evolution is a
highly non-equilibrium process, where pairing correlations play
the role of a very efficient
``lubricant''~\cite{Bertsch:1997,Bulgac:2019b,Bulgac:2020}.
Experiments show that the fission of odd or odd-odd nuclei is hindered
significantly when compared to the fission of the neighboring
even-even nuclei~\cite{Vandenbosch:1973,Wagemans:1991}. The unpaired
fermion(s) can be found in a state with a relatively high total
orbital angular momentum projection on the reaction axis and in that
case the presence of an odd fermion can have a strong hindering
effect, as the pairing correlations, particularly within the
traditional theoretical nuclear approaches used so far, are
ineffective in the case of the extra nucleon. This is very easy to
understand. In the case of an odd or odd-odd heavy nucleus the odd
particle(s) can have a very large orbital angular momentum along the
symmetry axis $l_z\approx \pm p_FR\approx 9 \hbar$, where $p_F$ is the
Fermi momentum and $R\approx 1.1A^{1/3}$ is the waist radius of the
fissioning nucleus. Such single-particle states are populated for
example when studying the induced fission of $^{239}$U obtained in the
one-neutron transfer reaction, such as the recently published
experimental study
$^9$Be($^{238}$U,$^{239}$U)$^8$Be~\cite{Ramos:2020}. In the
fission fragments the maximum angular momentum is expected however to be
$l_z(H)\approx \pm 7.5\hbar$ and $l_z(L)\approx \pm 6.7\hbar $ for the
heavy (H) and light (L) fragments respectively, thus smaller by
roughly $15\pm5$\% than in the mother nucleus. Since nuclei in the
majority of theoretical models used so far are predominantly axially
symmetric along the fissioning
direction~\cite{Bender:2004,Ryssens:2015,Schunck:2016}, when they
fission the value of $l_z$ of the odd nucleon(s) should be conserved,
as the pairing correlations are ineffective in lowering
the angular momentum of the extra nucleon or
nucleons. This is unlike the case of an even-even nucleus where pairing is
very efficient in performing transitions of the type
$(j_z,-j_z)\rightarrow (j'_z,-j'_z)$ and can thus lower the orbital
angular momentum of nucleon
pairs~\cite{Bertsch:1997,Bulgac:2019b,Bulgac:2020}. Clearly the
maximum initial $l_z\approx \pm 1.1A^{1/3}p_F$ and the maximum final
values of the odd nucleon(s) in the heavy and light fragments
$l_z(H,L)\approx \pm 1.1(A_{H,L})^{1/3}p_F$ differ vastly, a tension
which is usually at the root of the argument why fission of odd and
odd-odd nuclei is hindered when compared with the fission of even-even
nuclei. The breaking of the axial symmetry obviously can help, but
that woulds require some deformation energy. Also collisions, which
are neglected in mean field treatments, can also help to transfer
several units of angular momentum, but they are Pauli suppressed at
low intrinsic excitation energies. Both of these processes are
expected to be less efficient that the pairing correlations, which are
can be enhanced due to the presence of a pairing
condensate~\cite{Bulgac:2019b,Bulgac:2020}.
One has to remember that the $s$-wave pairing type of matrix elements,
which are by design included within a time-dependent Density
Functional Theory (DFT) for superfluid
systems~\cite{Bulgac:2013a,Bulgac:2019,Bulgac:2019c}, account for a
significant portion of the collision integral at low excitation
energies, even in the absence of a true pairing condensate. The main
difference between balanced/unpolarized ($N_\uparrow =N_\downarrow$)
and imbalanced/polarized ($N_\uparrow \neq N_\downarrow$) superfluids
is that in the first case the Cooper pairs have zero momentum and the
Coopers pairs have a finite momentum in the other case
respectively~\cite{Bulgac:2008,Bulgac:2011a}, a fact known
theoretically for a long time in condensed matter
systems~\cite{Larkin:1964,Fulde:1964}
and in cold atom physics as well. The presence of Cooper pairs
with finite momentum will be instrumental in lowering the orbital
angular momentum of the extra nucleons in case of nuclear fission.
One theoretical method used to describe (static) pairing correlations
in a system with an odd number of fermions requires the use of two
different chemical potentials for the approximately time-reversed
single-particle states, for spin-up and spin-down respectively, and
was discussed in
Refs~\cite{Bulgac:2007,aslda,Sensarma:2007,Bulgac:2008}. The presence
of an odd fermion can lead to a relatively weak time-reversal mean
field symmetry breaking and to different mean fields for the partners
of the Cooper pair, induced by the polarization effects due to the odd
fermion. The two chemical potential approach has been used in
Refs.~\cite{Bulgac:2007,Sensarma:2007,Bulgac:2008,Bulgac:2011a,
Bertsch:2009,Wlazlowski:2018,Magierski:2019,Tuzemen:2019}.
\footnote{See also the actual MATLAB code used in those
calculations~\cite{aslda}, using the DVR method described in
Ref.~\cite{Bulgac:2013}, and where the simulated annealing method
was used along with the Broyden method~\cite{Baran:2008} for the
iterative process.}
The general approach used to describe odd fermion systems used in nuclear
physics~\cite{Dobaczewski:1997,Dobaczewski:2000a,Ring:2004,Schunck:2017,Bertsch:2009a}
has the major disadvantage that one needs to know {\it a priori} the
quantum numbers of the odd fermion. \textcite{Robledo:2011} however
have shown that a gradient technique approach~\cite{Ring:2004} is
apparently free of this difficulty. Sometimes the implementation of
the general approach is construed as gain, as one can determine at
once a slew of low lying excited states of the odd or of the odd-odd
nucleus, even though the computational price is high. While adding a
single extra fermion to an even fermion system can appear as a small
perturbation $~{\cal O}(1/A)$, since the low energy spectrum of odd
and odd-odd nuclei is relatively dense, a small perturbation can in
principle lead to significant changes of the nuclear mean field.
The question I raise here is: ``Can one devise a more transparent and
computationally faster framework to describe pairing correlations in
odd and odd-odd nuclei in particular and for arbitrary spin-polarizations
as well?'' Since in unpolarized fermion
systems, or in even-even nuclei one treats the spin-up and spin-down
fermions identically, naturally there is a need for only one
chemical potential for either neutrons or protons. It seems then that since in polarized or odd
fermion systems, or in odd and odd-odd nuclei, spin-ups and spin-down
clearly experience different mean fields, the introduction of two
chemical potentials does not appear to need a justification and it
seems like the most natural approach. In the end the framework I
describe can handle arbitrary spin-polarizations, not limited to the
$|N_\uparrow-N_\downarrow|=1$ case only, which can be of interest in a
number of situations.
In Section~\ref{sec:general} I briefly review the Bogoliubov transformation
and the definition of various densities for even and odd fermion
numbers. In Section~\ref{sec:nuclei} I review a previously suggested
framework for odd fermion systems with axial and parity
symmetry~\cite{Bertsch:2009}, I also introduce a couple of generalizations
applicable when an octupole deformation is present, and I discuss
their advantages and disadvantages. In Section~\ref{sec:cold_atoms} I
review the framework designed for cold atom systems, in which case the
spin-orbit interaction is absent. In Section~\ref{sec:best} I
introduce the optimal two chemical potential framework, which appears
to be free of any of the disadvantages of those previously suggested
in the literature frameworks for nuclear systems. In the
appendix~\ref{sec:Pauli} I discuss various representations of the
Pauli spin matrices and in the appendix~\ref{sec:numeric} I discuss
several aspects concerning the numerical implementation.
\section{Formulation of Bogoliubov transformations for even and odd fermion numbers}\label{sec:general}
First I review the case of a system with an even number of fermions,
such as even-even nuclei. The creation and annihilation quasiparticle
operators are represented as~\cite{Ring:2004}
\begin{align}
&\alpha_k^\dagger =
\int d\xi\left [ u_k(\xi) \psi^\dagger (\xi) + v_k(\xi) \psi (\xi)\right ], \label{eq:a0}\\
&\alpha_k=
\int d\xi\left [ v_k^*(\xi) \psi^\dagger (\xi ) + u_k^*(\xi) \psi (\xi)\right ], \label{eq:b0}
\end{align}
and the reverse relations are
\begin{align}
&\psi^\dagger (\xi) = \sum_k \left [ u^*_k(\xi) \alpha^\dagger _k
+ v_k(\xi)\alpha_k \right ], \label{eq:p1}\\
&\psi(\xi) = \sum_k \left [ v^*_k(\xi)\alpha^\dagger_k
+ u_k(\xi)\alpha_k \right ], \label{eq:p2}
\end{align}
where $\psi^\dagger (\xi )$ and $ \psi (\xi)$ are the field operators
for the creation and annihilation of a particle with coordinates
$\xi=({\bm r},\sigma)$. The normal number (Hermitian $n=n^\dagger$ )
and anomalous (skew symmetric $\kappa =-\kappa^T$) densities are
\begin{align}
&n(\xi,\xi') = \langle \Phi|\psi^\dagger(\xi')\psi(\xi)|\Phi\rangle \nonumber\\
&= \sum_k v_k^*(\xi) v_k(\xi')=\sum_{l=n,\bar{n}} v_l^2 \phi_l^*(\xi) \phi_l(\xi'),\label{eq:number} \\
&\kappa(\xi,\xi') = \langle \Phi|\psi(\xi')\psi(\xi)|\Phi\rangle \nonumber\\
& =\sum_k v_k^*(\xi)u_k(\xi') = \sum_{l=n,\bar{n}} u_lv_l \phi_l^*(\xi) \phi_{\bar{l}}^*(\xi'), \label{eq:anomal} \\
& \int d\xi \phi_k^*(\xi) \phi_l(\xi)=\delta_{kl},
\end{align}
with $u_l^2+v_l^2=1$, $0\le u_l=u_{\bar{l}}\le 1$, $0\le
v_l=-v_{\bar{l}}\le 1$, and $n$ and $\bar{n}$ label the time-reversed
states $\phi_l(\xi), \phi_{\bar{l}}(\xi) $ in the canonical
representation~\cite{Bloch:1962,Ring:2004}, and where
\begin{align} & \alpha_k|\Phi\rangle =0, \quad \langle
\Phi|\alpha_k^\dagger = 0, \quad \langle
\Phi|\alpha_k\alpha_l^\dagger|\Phi\rangle =\delta_{kl}.
\end{align} There is in general no rule on how to separate the
quasiparticle operators $\alpha^\dagger_k,\alpha_k$ into creation and
annihilation ones, and one can rename/interchange any number of them,
and declare a number of creation operators annihilation operators and
{\it vice versa}. This is unlike the field operators
$\psi^\dagger(\xi),\psi(\xi)$, which are defined with respect to the
true vacuum, $\psi(\xi)|0\rangle \equiv 0$. Only by requiring that the
quasiparticle vacuum $|\Phi\rangle$ corresponds to the lowest, or
often to a local minimum, of the total energy of an average even
number of fermions, one can clearly distinguish between creation and
annihilation quasiparticle operators.
In the case of an odd number of fermions the ground state is defined
as $\alpha^\dagger_\mu|\Phi\rangle$~\cite{Ring:2004}, where $\mu$ is
an appropriately chosen {\it a priori} quasiparticle state, and thus
\begin{align}
\alpha_l\alpha_\mu^\dagger|\Phi\rangle = \delta_{l\mu}, \quad \langle \Phi|\alpha_\mu \alpha_k^\dagger = \delta_{k\mu}.
\end{align}
Since by definition $|\Phi\rangle$ corresponds to an
average even number of fermions $N$, the state
$\alpha^\dagger_k|\Phi\rangle$ should describe an odd number of
fermions $N\pm1$, and their corresponding particle parity is given by
$(-1)^N=+1$ or $(-1)^{N\pm 1}=-1$ respectively. Note however, than
the state $\alpha^\dagger_k|\Phi\rangle$, where $|\Phi\rangle$ is defined in Eq.~\eqref{eq:phi},
does not automatically has an integer average odd number of fermions,
as the chemical potential, and therefore the quasiparticle
wavefunctions should be correspondingly adjusted. As
\begin{align}
|\Phi\rangle \propto \prod_k \alpha_k |0\rangle, \label{eq:phi}
\end{align}
(assuming that $\alpha_k|0\rangle \ne 0$ for any $k$, otherwise see
\textcite{Ring:2004}) \footnote{In the case of a finite dimensional
Hilbert space it could be problematic to establish if mathematically
$\alpha_k|\Phi\rangle\ne 0$, since when $\int d\xi |v_k(\xi)|^2$ is
smaller then the machine precision the corresponding annihilation
operators do not anti-commute anymore and the ordering of the terms in
product Eq.~\eqref{eq:phi} can lead to results differing by more than
just by a phase.} the ground state of an odd fermion system is
therefore defined as
\begin{align}
\alpha_l^\dagger|\Phi\rangle \propto \prod_{k\ne l} \alpha_k|0\rangle, \label{eq:odd}
\end{align}
where $|0\rangle$ is the particle vacuum, and thus $\psi(\xi)|0\rangle
\equiv 0$. The normal number and anomalous densities are in this case
\begin{align}
&n_l(\xi,\xi') = \langle \Phi|\alpha_l\psi^\dagger(\xi')\psi(\xi)\alpha_l^\dagger|\Phi\rangle \nonumber\\
&=\sum_{k \ne l} v_k^*(\xi) v_k(\xi')+u_l(\xi) u^*_l(\xi') ,\label{eq:number1} \\
&\kappa_l(\xi,\xi') = \langle \Phi|\alpha_l\psi(\xi')\psi(\xi)\alpha_l^\dagger|\Phi\rangle \nonumber \\
& =\sum_{k\ne l} v_k^*(\xi)u_k(\xi')+u_l(\xi)v^*_l(\xi'). \label{eq:anomal1}
\end{align}
Thus the major difference from Eqs.~(\ref{eq:number}-\ref{eq:anomal})
is the absence of the contribution of the chosen quasiparticle state
$l$ in the sum, which is replaced by the ``flipped'' quasiparticle
wavefunction $(u_l(x),v_l(x)) \leftrightarrow
(v^*_l(x),u^*_l(x))$. This quasiparticle state $l$ is chosen so
as to minimize the total energy of the system with a fixed average odd
fermion number. In the case of an odd-odd nucleus one has to naturally
chose two such quasiparticle states, one for the neutron and the other
for the proton subsystems respectively.
\section{Self-consistent equations for systems with an odd number of fermions}\label{sec:nuclei}
Here I describe the two chemical potential framework for a polarized
(odd) fermion system, assuming axial symmetry, which apparently
was first introduced in Ref.~\cite{Bulgac:2007} and in my argumentation here
I follow the line of reasoning of Ref.~\cite{Bertsch:2009}. I introduce a new
quantum number $\eta$, the sign of the expectation value of the
single-particle angular momentum operator along the axial symmetry
axis $Oz$ and the corresponding operator
\begin{align}
{\cal S}_z=\text{sign}( {j}_z)=\text{sign}\left ({l}_z+\frac{\hbar}{2}\sigma_z\right ).
\end{align}
If $Oz$ is the axial symmetry axis then the quasiparticle wavefunctions are eigenfunctions of
${\cal S}_z$ with eigenvalues
\begin{align}
\eta=\text{sign}(m)=\pm 1
\end{align}
where $\hbar m$ are the eigenvalues of $j_z$.
The self consistent equations are:
\begin{align}
\begin{pmatrix}
\text{H}_{\eta \uparrow \uparrow} & H_{ \uparrow \downarrow} & 0 & \Delta \\
H_{ \downarrow \uparrow} & \text{H}_{\eta \downarrow \downarrow}& -\Delta & 0 \\
0 & -\Delta^* & - \text{H}^*_{\eta \uparrow \uparrow} & -H^*_{\eta \uparrow \downarrow} \\
\Delta^* & 0 & -H^*_{\eta \downarrow \uparrow} & -\text{H}^*_{\eta \downarrow \downarrow}
\end{pmatrix}
&\begin{pmatrix}
u_{km,\uparrow} \\
u_{km,\downarrow} \\
v_{km,\uparrow} \\
v_{km,\downarrow}
\end{pmatrix}\nonumber \\
= {\cal H}_{km}\psi_{km}
&\begin{pmatrix}
u_{km,\uparrow} \\
u_{km,\downarrow} \\
v_{km,\uparrow} \\
v_{km,\downarrow}
\end{pmatrix}, \label{eq:eqp_static}
\end{align}
\begin{align}
&\text{H}_{\eta \sigma,\sigma} = H_{\sigma \sigma} -\mu-\mu_\eta {\cal S}_z -Q,\quad \text{with} \quad \sigma=\uparrow, \downarrow, \label{eq:hss}
\end{align}
where $\psi_{km}$ is the four component quasiparticle wave function
and where I suppressed the arguments $({\bm r},\sigma)$ for spatial
and spin coordinates (isospin is not explicitly displayed). $Q=\sum_l
\lambda_lQ_{l0} $ stands for all other necessary constraints,
including the corresponding Lagrange multipliers, and $k$ stands for
the rest of quantum numbers characterizing the quasiparticle states,
apart from $m$. I have also used a short hand notation for the
components of the quasiparticle wavefunctions $u_{km,\sigma}
=u_{km}({\bm r},\sigma), \; v_{km,\sigma} =s_{km}({\bm r},\sigma)$,
where $\sigma = \uparrow,\downarrow$. In this case there are two
chemical potentials $\mu+\eta\mu_\eta=\mu \pm \mu_\eta$.
The normal partial and total number (and other relevant) densities and
also anomalous densities are
\begin{align}
& n_\eta({\bm r},\sigma,\sigma')= \sum_{E_{km}>0} v_{k m}^*({\bm r},\sigma')v_{k m}({\bm r},\sigma)\delta_{\eta,\text{sign}(m)},\\
& n({\bm r},\sigma,\sigma')= n_+({\bm r},\sigma,\sigma') +n_-({\bm r},\sigma,\sigma') ,\\
& \nu({\bm r})= \sum_{E_{km}>0} v_{km}^*({\bm r},\downarrow)u_{km}({\bm r},\uparrow). \label{eq:dens}
\end{align}
The Hamiltonian $H_{\sigma,\sigma'}({\bm r})$ and the pairing
potential $ \Delta_{\sigma,-\sigma}({\bm r})$ are functional
derivatives of the energy density functional (EDF) ${\cal E}(n, \nu,
\tau, \ldots) $
\begin{align}
& H_{\sigma,\sigma'}({\bm r}) = \frac{ \delta {\cal E}(n, \nu, \tau, \ldots) }{ \delta n({\bm r},\sigma,\sigma') }, \\\
& \Delta_{\sigma,-\sigma}({\bm r})=\frac{\delta {\cal E}(n, \nu, \tau, \ldots)}{\delta \nu({\bm r},\sigma,-\sigma)}.
\end{align}
Note that $\Delta_{\sigma,\sigma}({\bm r})\equiv 0$ in case of
$s$-wave pairing and either proton-proton or neutron-neutron pairing.
The generalization of the formalism to either $s$-wave proton-neutron
pairing or higher partial wave nucleon-nucleon pairing is
straightforward. The EDF of nuclear systems typically depends on the
sum $n({\bm r},\sigma,\sigma')= n_+({\bm r},\sigma,\sigma') +n_-({\bm
r},\sigma,\sigma')$ alone. In the case of cold atoms the EDF depends
on both $n_\pm({\bm r},\sigma,\sigma')$ separately, as fermions of
various flavors can reside in different external potentials. This
situation is formally equivalent to placing the cold atom system in a
strong ``magnetic'' field, with large ``spin-up and spin-down magnetic
moments,'' see Section~\ref{sec:cold_atoms}. The two chemical
potentials $\mu\pm\mu_\eta$ are determined from the condition that the
total and partial particle numbers are
\begin{align}
&N=N_+ + N_-, \quad |N_+-N_-|=1,\\
&N_\pm = \int d^3{\bm r}\sum_{\sigma=\uparrow,\downarrow} n_\pm({\bm r},\sigma,\sigma).
\end{align}
Notice that the equations for the quasiparticle states with $m>0$
($\eta=+1$) and $m<0$ ($\eta = -1$) respectively have two different
chemical potentials $\mu_{\pm 1}= \mu\pm \mu_\eta$, and in both cases
the eigenvalues $E_{km}$ come also in pairs $(E_{km},-E_{km})$, see
also Section~\ref{sec:cold_atoms}.
The operator ${\cal S}_z$ can be used in the case of axial symmetry
for a nucleus with octupole deformation, unlike the hermitian
signature operator $i\exp(-i\pi{j}_x/\hbar)$ with eigenvalues $\pm 1$
suggested in Ref.~\cite{Bertsch:2009}.The operator
$R_x=\exp(-i\pi{j}_x/\hbar)=-R_x^\dagger$ is anti-Hermitian, since
$R_x^\dagger R_x=1$ and $R_x^2=-1$ for a single fermion state. One
can relatively easily use instead of the operator
$i\exp(-i\pi{j}_x/\hbar)$ the so called simplex operator
$i\exp(-i\pi{j}_x/\hbar)P$~\cite{Dobaczewski:2000a}, where $P$ is the
spatial parity operator, if both quadrupole and octupole deformations
are present. I find the use of the operator ${\cal S}_z$ however much
simpler to implement numerically, particularly if one uses a
coordinate representation of the quasiparticle wave functions on
spatial 3D lattice, see Apendix~\ref{sec:numeric}. If one uses the
operator ${\cal S}_z$, then quasiparticle states with
$\eta=\text{sign}(m)=\pm 1$ are assigned to particle numbers $N_\pm$
and densities $n_\pm({\bm r},\sigma, \sigma')$ respectively.
Another option would be to use the operator
$i\exp(-i\pi{j}_z/\hbar)=i\exp(-im\pi)=\pm 1$. In this case
quasiparticle states with $m=+1/2,-3/2,+5/2,\ldots$ are assigned to
particle number $N_+$ and $n_+({\bm r},\sigma, \sigma')$, and states
with $m=-1/2,+3/2,-5/2,\ldots$ are assigned to particle number $N_-$
and $n_-({\bm r},\sigma, \sigma')$, respectively. Time-reversed
partners are in both cases assigned to different groups, but using
different criteria.
This ambiguity in assigning the quasiparticle states to either one or
another partial number density in the case of an odd fermion number is
general. One can use any rule to separate quasiparticle states in two
groups, and there is no general prescription on how to assign them to
specific particle number (aka $``N_+$'' and ``$N_-$'') or number
densities. The best solution should always correspond to the lowest
total energy, which might not always favor the strongest pairing
correlations. The many-fermion self-consistent equations never have a
unique solution, even for even (in the case of electrons or cold
atoms) or even-even nuclei, though only one of them is the lowest
total energy. Multiple vacua however, can also correspond to
physically realizable states, separated by strong potential barrier, a
situation which is quite ubiquitous in quantum field theories or
infinite many-body systems, where symmetry is spontaneously broken.
The ambiguities I discuss here for odd fermion systems in this sense
should not come as a surprise, but merely as new examples of such
physically realizable ``ground states.'' However, since in the case
of odd fermion systems the density of low energy levels is relatively
high, any such prescription is to some extent arbitrary, and the true
ground state might emerge as an optimal superposition of many such
quasiparticle vacua, with well defined quantum numbers. One of the
simplest examples is that of a system with spontaneous parity
breaking, when the ground and the first excited states are separated
by an exponentially small energy difference, a phenomenon known in the
literature as parity doubling~\cite{Sheline:1989,Dobaczewski:2005}.
Another possibility is that of shape coexistence, see
e.g. Refs.~\cite{Andreyev:2000,Clement:2016} and references therein,
of which there are many examples of other types too.
There is still another ambiguity. If both quadrupole and octupole
deformations are present in a nucleus with axial symmetry, it is not
clear whether the ground state of the nucleus corresponds to either
$N_+-N_-=\pm 1$, and thus whether the projection of the total angular
momentum ${J}_z(n)=\sum_{n=1}^A{j}_z(n)\neq 0$ is along or
opposite to the fission direction. Such a situation can be
experimentally studied in fission induced by a nucleon transfer from
an impinging projectile to a nucleon state with relatively large
angular momentum~\cite{Ramos:2020}. The projectile will impact an
angular momenta perpendicular to the reaction plane equal to either
$l_z\approx \pm Rk_F$. The emergence of the fission fragments emitted
along the axis perpendicular to the reaction plane, might favor the
emission of the either the light or of the heavy fission fragment in
the direction of the impacted angular momentum. It would very
interesting to see if any asymmetry of the fission fragments
distribution along the direction perpendicular to the reaction plane
exists, as such a phenomenon has not been studied yet to my knowledge.
In order to illustrate more vividly the difficulties with the
prescriptions described in this section I will apply them to a very
simple case, non-interacting ``neutrons'' in a spherical harmonic
oscillator potential with a constant spin-orbit interaction and no
pairing field:
\begin{align}
H=-\frac{\hbar^2\Delta}{2m}+\frac{m\omega^2 {\bm r}^2}{2} + V{\bm l}\cdot{\bm s}, \label{eq:ho}
\end{align}
as spin-polarization is a property of both normal and superfluid fermionic
systems. Since there is no ``neutron-neutron'' interaction the mean
field is independent of the number of particles, and the ground state
is typically degenerate for even particle numbers.
In Fig.~\ref{fig:hj} I show the
quasi-particle spectra for the two types of constraints discussed in this section
\begin{align}
{H}-\mu_\eta {j}_z \quad \text{and} \quad {H}-\mu_\eta\;\text{sign}({j}_z).
\end{align}
Since the Hamiltonian has spherical symmetry the choice of constraining axis is arbitrary.
\begin{figure}[ht]
\includegraphics[width=0.85\columnwidth]{ejz}
\includegraphics[width=0.85\columnwidth]{esigma}
\caption{\label{fig:hj} (Color online) The colors for down-sloping
quasi-particle levels are black ($l=0$), blue ($l=1$), red ($l=2$),
green ($l=3$), magenta ($l-4$), cyan ($l=5$), and yellow ($l=6$)
respectively. The corresponding up-sloping levels are always
displayed with black dashed thin lines for the sake of the clarity of
the figure. In panel (a) are the energy levels for ${H}-\mu_\eta
{j}_x $ and the lower panel for $H-\mu_\eta\;\text{sign}({j}_x).$
Notice that in panel (b) the states with $j> 1/2$ are multiple
degenerate and for display purposes only have been artificially
perturbed. For example there are three down-sloping $1d_{5/2}$ levels
corresponding to $m=1/2, 3/2$ and $5/2$. The single-particle energies
$\epsilon_{njlm}$, the chemical potential $\mu_\eta$ are in units of
$\hbar \omega$ and the strength of the spin-orbit interaction was
chosen as $V= 0.12\hbar\omega$ in these examples.}
\end{figure}
The two spectra have distinctly different aspects. For small values
of $\mu_\eta$ one can hastily conclude that either type of constraint
apparently leads to desired outcomes. E.g. a system with $N=8$
particles at $\mu_\eta=0$ is a closed shell, but for small finite values
of $\eta$ a particle is promoted from the $1p_{1/2}, m=-1/2$ level to
the $1d_{5/2}$ level.
Since the cranking operator is ${j}_z$ the net result of such a
constraint is imparting the system a net total angular momentum
${J}_z=\sum_{i=1}^N{j}_z(i)$ however, which naturally also gives
raise to a spin-polarization. This is clearly seen in the upper panel of
Fig.~\ref{fig:hj} for $\mu_\eta \approx 1$, when mostly down-sloping
levels with $m>0$ are occupied. The same kind of ground state is
obtained also in the case of constraint $\text{sign}({j}_z).$ In
this case levels with $m>0$ are down-slopping and levels with $m<0$
are up-sloping, all with identical slopes. Nevertheless, the same
undesired characteristic of the system ground state is obtained for a
highly polarized system. In the case of real nucleus, when the depth
of the mean field potential is finite, the radial profiles of the
single-particle wave functions are affected in drastically different
manners for up-sloping (less bound) and down-sloping (more bound)
levels, as a result of a large finite total angular momentum of the
system. The system is polarized because it was forced to have a large
total angular momentum, as opposed to what one might expected to have
a finite total momentum due to the finite popularization of the
system. Basically in either of the prescriptions discussed in this
section and in Ref.~\cite{Bertsch:2009} the roles of the effect and
cause have been inadvertently switched. Since a finite spin-polarization of
a fermionic system corresponds in the mean field approximation to the
breaking of the time-reversal invariance, a finite spin-polarization is
always accompanied by a non-vanishing total angular momentum in {\it
vice versa}. In practice however, one has to clearly distinguish
between what kind of constraint one intends to impose, as different
constraints lead to different outcomes.
\section{The case of fermionic polarized cold atom systems}\label{sec:cold_atoms}
In the case of cold atoms one has two flavors of fermions, which I
shall denote with $a$ and $b$, and the Cooper pair is formed between
one fermion $a$ with another fermion $b$. Entangled states, when for
example a type $a$ fermion can coexist with a type $b$ fermion, in a
type of the Schr\"odinger cat single-particle state, have not been
studied yet, neither experimentally nor theoretically to my knowledge.
Only the formation of Copper pairs between a type $a$ fermion and a
type $b$ fermion have been considered so far in the literature. Such a
mixing is formally similar to the spin-orbit coupling of the nucleon
motion in nuclei, and it will be illustrated qualitatively in
Fig.~\ref{fig:Eqp}. In the absence of such mixing the mean field
equations
read~\cite{Bulgac:2007,Sensarma:2007,Bulgac:2008,Bulgac:2011a,aslda}:
\begin{align}
&\begin{pmatrix}
H_a -\mu_a & 0 & 0 & \Delta \\
0 & H_b-\mu_b & -\Delta & 0 \\
0 & -\Delta^* & -H^*_a +\mu_a& 0 \\
\Delta^* & 0 & 0 & -H^*_b +\mu_b
\end{pmatrix}
\begin{pmatrix}
u_{k}^{(a)} \\
u_{k}^{(b)} \\
v_{k}^{(a)} \\
v_{k}^{(b)}
\end{pmatrix} \nonumber\\
&= {\cal H}
\begin{pmatrix}
u_{k}^{(a)} \\
u_{k}^{(b)} \\
v_{k}^{(a)} \\
v_{k}^{(b)}
\end{pmatrix}
= E_{k}
\begin{pmatrix}
u_{k}^{(a)} \\
u_{k}^{(b)} \\
v_{k}^{(a)} \\
v_{k}^{(b)}
\end{pmatrix}. \label{eq:cold}
\end{align}
where the two chemical potentials $\mu_{a,b}$ are chosen by fixing the
particle numbers
\begin{align}
N_{a,b}=\sum_{E_k>0} \int d^3{\bm r} |v_k^{(a,b)}({\bm r})|^2.
\end{align}
\begin{figure}[ht]
\includegraphics[width=1\columnwidth]{Eqp}
\caption{\label{fig:Eqp} The quasiparticle spectrum
$E_{k,\pm}^{(a)}$~\eqref{eq:Ea} with blue and
$E_{k,\pm}^{(b)}$~\eqref{eq:Eb} with red lines respectively. The
densities are constructed from quasiparticle states with
$E_{k,\pm}^{(a,b)}> 0$ only. For a finite spin-polarization the
quasiparticle states corresponding to the red line with
$E_{k,+}^{(b)}> 0$. At the same time the fermions of kind $a$ occupy
quasiparticle states corresponding to the blue lines with
$E_{k,\pm}^{(a)}> 0$. The effect of a non-vanishing mixing between
the two flavors or of a spin-orbit coupling (characteristic to nuclear
systems) on the quasiparticle spectrum is illustrated with dashed
lines. In the case of cold atoms this type of mixing is equivalent to
the creation of a Schr\"odinger cat fermion state between type $a$ and
type $b$ flavors. }
\end{figure}
These equations obviously decouple and since in case of cold atoms one
typically has $H_{a,b}=H_{a,b}^*$, the equations simplify. By
introducing
\begin{align}
H_\pm = \frac{H_a\pm H_b}{2}
\end{align}
these equations can be re-written as
\begin{align}
&\begin{pmatrix}
H_+ -\mu & \Delta \\
\Delta^* & -H_+ +\mu
\end{pmatrix}
\begin{pmatrix}
u_{k}^{(a)} \\
v_{k}^{(b)}
\end{pmatrix}\label{eq:cold3}\\
&=
\begin{pmatrix}
E_k^{(a)} -(H_- -\mu_\eta) & 0 \\
0 & E_k^{(a)} -(H_- -\mu_\eta)
\end{pmatrix}
\begin{pmatrix}
u_{k}^{(a)} \\
v_{k-}^{(b)}
\end{pmatrix},\nonumber \\
&\begin{pmatrix}
H_+ -\mu & -\Delta \\
-\Delta^* & -H_+ +\mu
\end{pmatrix}
\begin{pmatrix}
u_{k}^{(b)} \\
v_{k}^{(a)}
\end{pmatrix} \label{eq:cold4}\\
&=
\begin{pmatrix}
E_k^{(b)} +(H_- -\mu_\eta) & 0 \\
0 & E_k^{(b)} +(H_- -\mu_\eta)
\end{pmatrix}
\begin{pmatrix}
u_{k}^{(b)} \\
v_{k}^{(a)}
\end{pmatrix},
\nonumber
\end{align}
and now one can disentangle the different roles operators $H_+-\mu$
and $H_- -\mu_\eta$ play on acting on quasiparticle wave functions.
The chemical potentials $\mu$ and $\mu_\eta$ are defined in a similar
manner
\begin{align}
\mu = \frac{\mu_a+\mu_b}{2},\quad \mu_\eta=\frac{\mu_a-\mu_b}{2}.
\end{align}
The self-consistent equations in the nuclear case can be brought to a similar form.
Assuming that $H_{a,b}=\varepsilon_{a,b}$ and $\Delta$ are diagonal one can show that
\begin{align}
& E_{k,\pm}^{(a)} = +e_- \pm \sqrt{ \epsilon_k^2+|\Delta |^2 }, \label{eq:Ea} \\
&E_{k,\pm}^{(b)} = -e_- \pm \sqrt{ \epsilon_k^2+|\Delta |^2 },\label{eq:Eb}\\
&e_-= \frac{\varepsilon_a-\varepsilon_b}{2} -\mu_\eta, \quad
\epsilon_k=\frac{\varepsilon_a+\varepsilon_b}{2} -\mu.
\end{align}
Clearly all eigenvalues come in pairs $E_{k,\pm}^{(a)} =
-E_{k,\mp}^{(b)}$. For each eigenvector $(u^{(a)}_k,v^{(b)}_k)$ and
corresponding eigenvalue $E_{k,\pm}^{(a)}$ of Eq.~\eqref{eq:cold3} the
Eq.~\eqref{eq:cold4} has a corresponding eigenvector $
(u^{(b)}_k,v^{(b)}_k)= (v^{(a)}_k,u^{(b)}_k)^*$ and a corresponding
eigenvalue $E_{k,\mp}^{(b)}=-E_{k,\pm}^{(a)}$.
As branches of the quasiparticle spectrum are displaced in opposite
directions, when part of the lower branch $E^{(a)}_{k,-}$ becomes
positive (with blue in Fig.\ref{fig:Eqp}) and the upper branch
$E^{(b)}_{k,+}$ becomes negative (with red in Fig.\ref{fig:Eqp}), the
roles of the components of the Bogoliubov quasiparticles change
exactly as discussed in Section~\ref{sec:general}, see
Eqs.~(\ref{eq:number1}, \ref{eq:anomal1}). If $N_a-N_b$ quasiparticle
energies $E_{k,-}^{(a)}$ change their signs, then the new
quasiparticle vacuum corresponds to
$\prod_{l=1}^{N_a-N_b}\alpha_{\mu_l}^\dagger|\Phi\rangle$ and a
particle parity $(-1)^{N_a-N_b}$, where $|\Phi\rangle$ is a total
fermion function for an unpolarized system, and $\mu_l$ are the
corresponding quantum numbers of the positive $E_{k,-}^{(a)}$
quasiparticle states.
When the branches of $E_{k,-}^{(a)}$ and $E_{k,+}^{(b)}$ cross zero,
some of the quasiparticles energies could vanish identically, as in
the case of bound states on a superfluid vortex line first discussed
by \textcite{Caroli:1964} and have a character similar to Majorana
particles. In such a case the fermion system is technically a
topological one, characterized by a Chern number associated with the
Berry connection and curvature~\cite{Thouless:1982}.
\section{My favorite two chemical potentials framework for a polarized superfluid Fermi system} \label{sec:best}
Likely the best option is to formulate the two chemical potentials
framework for nuclei along the same lines as for the scheme suggested
for cold atoms, see
Refs.~\cite{Bulgac:2007,Sensarma:2007,Bulgac:2008,Bulgac:2011a} and
Section~\ref{sec:cold_atoms}. The main difference between nuclei and
cold atom systems is in the presence of the spin-orbit interaction in
nuclei and the need to introduce the total single-particle angular
momentum ${\bm j}={\bm l}+{\bm s}$, where ${\bm l}$ is
the orbital angular momentum and ${\bm s}= \hbar{\bm
\sigma}/2$ is the nucleon spin, and the spin-orbit interaction, which
mixes the spin-up and spin-down states, which is qualitatively
different situation from the cold atom case discussed in the previous
Section~\ref{sec:cold_atoms}. (
The main problem with either the method suggested by
\textcite{Bertsch:2009} and briefly described in
Section~\ref{sec:nuclei} or the ``improvement'' suggested by me in the
same section is that either the constraining operators ${\cal
S}_{x,z}$ lead to polarization, exactly as any rotation with a finite
frequency would do or a strong magnetic field also induce. The goal
is not to bring the nucleus into rotation, and thus excite it, but
rather to keep the odd or odd-odd nucleus in its ground state at
a given spin-polarization in the
mean field approximation. The question arises then, what is the most
appropriate constraining operator?
When $\exp\left( -i2\pi {\bm
j}\cdot{\bm n}/\hbar\right )$ is acting on a $1/2$-spinor wave function
it changes its sign.
The rotation operator can be
factorized, since $[{{\bm l},{\bm s}}]=0$
\begin{align}
\exp\left ( -i\alpha \frac{{\bm j}\cdot{\bm n}}{\hbar}\right ) =
\exp\left ( -i\alpha \frac{{\bm l}\cdot{\bm n}}{\hbar} \right )
\exp \left ( -i\alpha \frac{{\bm s}\cdot{\bm n}}{\hbar} \right ),
\end{align}
where ${\bm n}=(\sin\beta\cos\gamma,\sin\beta\sin\gamma,\cos\beta)$ is an
arbitrary unit vector. If one considers now separately the action of
$\exp( -i2\pi {\bm l}\cdot{\bm n}/\hbar)$ on any component of the
spinor, that component does not change sign. However, when acting with
$ \exp \left ( -i2\pi {\bm s}\cdot{\bm n}/\hbar\right )$ alone on
the $1/2$-spinor wave function, the spatial part of the wave function
is obviously unaffected, but the whole spinor wave function changes
sign. That allows us to define the operator
\begin{align}
{\cal P}_N(\alpha) = \prod_{k=1}^N \exp \left [ -i\alpha\frac{ {\bm s}(k)\cdot{\bm n}}{\hbar}\right ],
\end{align}
where the product runs over all particles.
\footnote{ Note that the spin operator ${\bm s}$ in this expression
can be formed from arbitrary Pauli matrices $\tilde{ {\bm \sigma} }$,
which are defined through arbitrary angles $\kappa,\chi,\zeta$,
see \eqref{eq:Pauli}.}
The action of this operator leads to an apparently new and simple
representation of the particle parity operator for an $N$-fermion
system
\begin{align}
{\cal P}_N(2\pi)|\Phi\rangle =(-1)^N|\Phi\rangle .
\end{align}
It is particularly easy to check that $\langle\Phi | {\cal
P}_N(2\pi)|\Phi\rangle \equiv (-1)^N$ in the case of a Slater
determinant. Since the spin direction can be chosen arbitrarily, this
representation of the particle projection operator is clearly not
unique and one can use ${\bm n}=(0,0,1)$. Exactly as in the case
described in Section~\ref{sec:nuclei} the single-particle operator
${\cal P}_1(\alpha) = \exp(-i\alpha{\bm s}\cdot {n}/\hbar) $ has the
obvious properties
\begin{align}
&{\cal P}^\dagger_1(\alpha){\cal P}_1(\alpha)=1, \quad {\cal P}_1^2(\pi)={\cal P}_1(2\pi) =-1
\end{align}
and all the argumentation presented in Section~\ref{sec:nuclei} and in
Ref.~\cite{Bertsch:2009} follows, however, without some of the
limitations discussed in Section~\ref{sec:nuclei} and other
limitations discussed in Appendix~\ref{sec:numeric}.
A polarized Fermi system is spin polarized, similarly to the well
studied cold atom systems~\cite{Bulgac:2007,Bulgac:2008,Sensarma:2007,
Bulgac:2011a,Wlazlowski:2018,Tuzemen:2019,Magierski:2019}
or magnetized electron systems for example.
If there are no non-vanishing time-odd external or components of the
mean field one can show that the ground state of an even-even nucleus
has a vanishing total spin $\langle \Phi | \sum_{k=1}^N {\bm
s}(k)|\Phi \rangle\equiv 0$~\cite{Vautherin:1972}. Hence,
following the same argumentation presented in Ref.~\cite{Bertsch:2009}
and in Section~\ref{sec:nuclei},
for a polarized fermion superfluid system Eqs.~(\ref{eq:eqp_static},
\ref{eq:hss}) can be rewritten as follows:
\begin{align}
\begin{pmatrix}
\text{H}_{\eta \uparrow \uparrow} & H_{ \uparrow \downarrow} & 0 & \Delta \\
H_{\downarrow \uparrow} & \text{H}_{\eta \downarrow \downarrow}& -\Delta & 0 \\
0 & -\Delta^* & - \text{H}^*_{\eta \uparrow \uparrow} & -H^*_{\eta \uparrow \downarrow} \\
\Delta^* & 0 & -H^*_{\eta \downarrow \uparrow} & -\text{H}^*_{\eta \downarrow \downarrow}
\end{pmatrix}
&\begin{pmatrix}
u_{k,\uparrow} \\
u_{k,\downarrow} \\
v_{k,\uparrow} \\
v_{k,\downarrow}
\end{pmatrix}\nonumber \\
={\cal H}
\begin{pmatrix}
u_{k,\uparrow} \\
u_{k,\downarrow} \\
v_{k,\uparrow} \\
v_{k,\downarrow}
\end{pmatrix}
= E_{k}
&\begin{pmatrix}
u_{k,\uparrow} \\
u_{k,\downarrow} \\
v_{k,\uparrow} \\
v_{k,\downarrow}
\end{pmatrix}, \label{eq:nuclear}
\end{align}
\begin{align}
&\text{H}_{\eta \sigma,\sigma} = H_{\sigma \sigma} -\mu-\eta \mu_\eta
-Q,\label{eq:mu_nuc}
\end{align}
with $\eta = (\sigma_z)_{\sigma,\sigma}=\pm 1$ for
$\sigma=\uparrow,\downarrow$ respectively. Here I have replaced
either the operator ${\cal S}_z$ introduced in
Section~\ref{sec:nuclei}, or the operator $i\exp(-i\pi{j}_x/\hbar)$
introduced by ~\textcite{Bertsch:2009}, or the alternative simplex
operator $i\exp(-i\pi{j}_x/\hbar)P$~\cite{Dobaczewski:2000a}, both of
them discussed in Section~\ref{sec:nuclei}, with the much simpler
Hermitian operator $i{\cal P}_1(\pi)=i\exp( -i\pi\sigma_z/2)\equiv
\sigma_z$. Then the number densities, the particle numbers, and the
anomalous density are determined as follows
\begin{align}
& n_{\sigma}({\bm r})= \sum_{E_k>0} v_{k }^*({\bm r},\sigma)v_{k }({\bm r},\sigma),\quad \sigma=\uparrow,\downarrow, \label{eq:n_ud}\\
& n({\bm r})= n_{\uparrow}({\bm r})+ n_\downarrow({\bm r}),\label{eq:nnn} \\
& N=N_\uparrow + N_\downarrow, \quad N_{\uparrow,\downarrow} = \int d^3{\bm r} n_{\uparrow,\downarrow}({\bm r}), \label{eq:N_ud}\\
& \nu({\bm r})= \sum_{E_{k}>0} v_{k}^*({\bm r},\downarrow)u_{k}({\bm r},\uparrow).
\end{align}
The chemical potentials $\mu_{\uparrow,\downarrow}=\mu\pm \mu_\eta$
are chosen so as fix both the total particle
number $N$ and the degree of spin-polarization $N_\uparrow-N_\downarrow$.
Since the chemical potential $\mu_\eta$ enters in the self-consistent
equations Eq.~\eqref{eq:nuclear} as $\mu_\eta\sigma_z$ the
single-particle angular momentum ${ j}_z$ still commutes with the
quasiparticle Hamiltonian ${\cal H}$ for an axially symmetric
nucleus. However, a spin polarized odd or odd-odd nucleus strictly
speaking cannot be strictly spherical anymore in the mean field
approximation, as $[{\bm j}^2,\sigma_z]\ne 0$.
Naturally, the same procedure can be used for a nucleus without pairing
correlations. The chemical potential $\eta\mu_\eta$ act as a
fictitious magnetic field $ B|\mu_N|=\mu_\eta$ in case of neutral
particles. Consider for example the case of $N$ fermions occupying a
set of (ordered) single-particle levels $\varepsilon_k$, all with
Kramers degeneracies. By applying the ``external field'' $\mu_\eta
> (\varepsilon_{N}-\varepsilon_1)/2$ the system is completely
polarized and partially polarized for smaller values of $\mu_\eta$.
By placing a nucleus in an external magnetic field of arbitrary
magnitude one can also obtain non-vanishing spin-polarizations. However,
protons will respond more strongly to a magnetic field, due to their
finite orbital magnetic moment. Neutrons and protons cannot be
controlled independently in the present framework, as various applications might
require, for example when there is a multi-nucleon transfer reaction
and one can populate various proton and neutron single-particle
states.
The quasiparticle spectrum is not affected qualitatively by the
presence or absence of the spin-orbit interaction, see
Fig.~\eqref{fig:Eqp}. Notice that in these formulas I have dropped
the additional subscript $m$ for the energies and the quasiparticle
wave functions, as there is no need for singling out $m$. Various
branches of the quasiparticle spectrum are shifted upwards and
downwards, see Fig.~\eqref{fig:Eqp}, and the quasiparticle wave
functions corresponding to the lowest energies are automatically
``flipped'' when these quasiparticle energies change sign, see the
discussion in previous Sections~\ref{sec:general}, \ref{sec:nuclei},
and \ref{sec:cold_atoms}.
In the traditional approach used for odd fermion systems, see
Refs.~\cite{Ring:2004,Dobaczewski:1997,Schunck:2017,Bertsch:2009a} and
Section~\ref{sec:general} the quantum numbers for the singled out
quasiparticle state $l$, see Eq.~\eqref{eq:odd}, are {\it a priori}
unknown. In order to determine the ground state one needs to perform
many simulations with various choices of quantum numbers for
quasiparticle state $l$~\cite{Schunck:2017,Bertsch:2009a}. In this
latest formulation, c.f. Eqs.~\eqref{eq:nuclear}, one needs only to
specify the degree of spin-polarization $ N_\uparrow - N_\downarrow $ of
the system only. $| N_\uparrow - N_\downarrow |$ could be any integer
in principle and therefore in this framework one can generate two or
more quasiparticle excited states as well when needed. One can even
consider $N_\uparrow - N_\downarrow$ as any real number and consider
fractional total particle numbers~\cite{Dreizler:1990lr},
similarly to what has been done in the case of cold atom systems for
arbitrary spin-polarizations, and where the agreement of the EDF approach
with {\it ab initio} quantum Monte Carlo calculations of inhomogeneous
systems and also with experiments was
excellent~\cite{Bulgac:2007,Bulgac:2008,Bulgac:2011a,
Bulgac:2014,Wlazlowski:2015}. $N_\uparrow-N_\downarrow$ can take any
value in both even and odd nuclei, a situation which might be very
useful when analyzing nuclei in extremely strong magnetic fields,
e.g. in magnetars.
\begin{figure}[h]
\includegraphics[width=0.99\columnwidth]{en}
\includegraphics[width=0.83\columnwidth]{M}
\caption{\label{fig:en} (Color online)
The eigenvalues $\epsilon(\eta)$ of the constrained Hamiltonian
$H-\mu_\eta\sigma_z$, with $h$ given by Eq.~\eqref{eq:ho}, are shown
in panel (a). The corresponding projection of the total angular momentum
along the symmetry axis
$K_N(\mu_\eta)= \langle \Psi(\mu_\eta) |\sum_{i=1}^N {j}_x(i) |\Psi(\mu_\eta)\rangle$
for a ``neutron'' with $N=8, 12, 16$ and 20 particles system is shown
in panel (b). $\Psi(\mu_\eta)$ is the ground state Slater determinant.
For visual clarity $K_N(\mu_\eta)$ for different $N$ have
been slightly shifted in the vertical direction for better visualization.
The energies, $\mu_\eta$,
and the angular momentum are in units of $\hbar\omega$ and $\hbar$ respectively.
$N=8$ and 20 are magical numbers, and $N=16$ as zero spin-polarization corresponds to a
complete filling of the $1s_{1/2},1p_{3/2},1p_{1/2},1d_{5/2}, 2s_{1/2}$ levels and at
$\mu_\eta\equiv 0$ have $K_N(0)=0$ and also $\Sigma_N(0)\equiv 0$ as expected, see Fig~\ref{fig:sigma}. }
\end{figure}
I will present now a second argument in favor of the type of constraint discussed in this section.
For any nucleus one can consider the generalized number density for
one kind of nucleons
\begin{align}
n_{\sigma,\sigma'}({\bm r}) =\sum_{E_k>0}v_k({\bm r},\sigma)v_k^*({\bm r},\sigma').
\end{align}
\textcite{Vautherin:1972} observed that time-reversal invariance
implies that for an even-even nucleus in the ground state
\begin{align}
&n_{\sigma,\sigma'}({\bm r}) \equiv \frac{1}{2}\delta_{\sigma,\sigma'} n({\bm r}),\\
&\sum_{\sigma,\sigma'} n_{\sigma,\sigma'}({\bm r}) {\bm \sigma}_{\sigma',\sigma}\equiv 0.
\end{align}
For an odd-odd or odd nucleus that is not true anymore, as clearly that also holds true for excited states
of an even-even nucleus with a finite total spin-polarization, when
\begin{align}
{\bm \Sigma}_N =\int d^3r \sum_{\sigma,\sigma'} n_{\sigma,\sigma'}({\bm r}) {\bm \sigma}_{\sigma',\sigma} \ne 0 \label{eq:Sigma}
\end{align}
is non-vanishing. Obviously
\begin{align}
\int d^3r \sum_{\sigma} n_{\sigma,\sigma}({\bm r})=N
\end{align}
is the total number of either neutrons or protons. Without any loss
of generality one can choose ${\bm \Sigma}= (0,0,\Sigma)$ and then one
can easily see that for any nucleus
\begin{align}
N_{\sigma}= \int d^3r n_{\sigma,\sigma}({\bm r}), \quad \textrm{for}\quad \sigma=\uparrow,\downarrow,
\end{align}
in agreement with Eqs.~(\ref{eq:n_ud},\ref{eq:nnn},\ref{eq:N_ud}).
Obviously the spin-polarization $\Sigma_N$ has always values in a finite interval
\begin{align}
\Sigma_N\in [ -N, N]
\end{align}
and in case of nuclear systems one can consider only non-negative values for $\Sigma_N$.
\begin{figure}[h]
\includegraphics[width=0.85\columnwidth]{etot}
\includegraphics[width=0.85\columnwidth]{sigma}
\caption{\label{fig:sigma} (Color online) The total energy $E_N(\mu_\eta)$
of a normal polarized $N=8, 12, 16$ and $20$ non-interacting
``neutron'' system with Hamiltonian~\eqref{eq:ho} is shown in
panel (a) as a function of $\mu_\eta$ and the total spin-polarization ${\bm
\Sigma}_N (\mu_\eta)$, see Eq.~\eqref{eq:Sigma}, is shown in
panel (b). For visual clarity $\Sigma_N(\mu_\eta)$ for different $N$ have
been slightly shifted in the vertical direction for better visualization.
Energy and $\mu_\eta$ are in units of $\hbar\omega$. }
\end{figure}
I will compare now the results of imposing this type of constraint on
the system discussed at the end of Section~\eqref{sec:nuclei}, see
Eq.~\eqref{eq:ho}. Figs.~\ref{fig:en} and \ref{fig:sigma} summarize
the results for quasi-particle energies $\epsilon_n(\mu_\eta)$ for the constrained
Hamiltonian $H-\mu_\eta\sigma_z$ and for the projection of the total
angular momentum along the symmetry axis, energy, and spin-polarization respectively
$K_N(\mu_\eta), E_N(\mu_\eta)$ and $\Sigma_N(\mu_\eta)$
for the total particle numbers $N=8, 12, 16$ and 20.
In Fig.~\ref{fig:ES} for these
values of $N$ I display the dependence of $E_N(\Sigma)$, $K_N(\Sigma)$, and the ``yrast line''
$E(K)$ {\it versus } $K$.
The total angular momentum, the
total energy, and the total spin-polarization change by jumps at each
quasi-particle level crossing. The total spin-polarization
$\Sigma_z(\mu_\eta)$ changes in jumps of $\approx 2$ as expected at each
value of $\eta$ where a quasi-particle level crossing occurs, see Fig.~\ref{fig:en}. There
is a glaring difference between the maximum total angular momentum
obtained at maximum spin-polarization in Fig.~\ref{fig:hj} for the
constraints $ H-\mu_\eta {j}_z$ and $H-\mu_\eta \,\text{sign}({j}_z)$ and the
corresponding value in Fig.~\ref{fig:en} corresponding to the
constraint $H-\mu_\eta\sigma_z$. For $N=20$ one obtains at $\mu_\eta=1$ the
values 64 and 17, and 96 and 25 for $\mu_\eta=2$ respectively for the type
of constraints discussed in Section~\ref{sec:nuclei}. These values
should be compared to the maximum value $K_N(\mu_\eta) \approx 13$ and
lower values for the $N=20$ ``neutron'' system for large values of
$\eta$ in Fig.~\ref{fig:en}. Note that $K_N(\mu_\eta)$ is not a monotonous
function of $\mu_\eta$, see Fig.~\ref{fig:en}. One can now better appreciate
the difference between enforcing a constraint either on ${j}_z$ or
on $\text{sign}({j}_z)$, which imparts a relatively large angular
momentum to the system, while also by default leading to its
spin-polarization. The spin-polarization of the system obtained in this manner
is an effect caused by the finite angular momentum, per discussion in
Section~\ref{sec:nuclei}. For each $N$ the system reached it maximum
possible spin-polarization $\Sigma_\text{max} = |N_\uparrow-N_\downarrow| = N$.
The constraint on $\sigma_z$ is directly related to the generalized
density $n_{\sigma,\sigma'}({\bm r})$, which is naturally related to
$N_{\sigma}= \int d^3r\; n_{\sigma,\sigma}({\bm r})$ and thus to the
spin-polarization of the system $\Sigma=| N_\uparrow - N_\downarrow| $. In this case
the finite spin-polarization, namely the cause, induces a finite total angular
momentum, which is thus an effect.
While the many-body states thus obtained are characterized by axial symmetry,
and thus by a good value of the
projection of the total angular momentum on the symmetry axis, these states break
rotational invariance, as these states do not have a well defined total angular momentum.
But so do Slater determinants, formed with single-particle wave functions from an open shell
One can denote these states $K\Sigma$-isomers and after
angular momentum projection~\cite{Ring:2004} one can restore the rotational symmetry.
The spin asymmetry $\Sigma$ along the system symmetry axis is not affected by angular
momentum projection, similarly to the $K$ quantum number,
since $[j_z,\sigma_z]=0$ and $[\sum_{k=1}^Nj_z(k),\sum_{l=1}^N\sigma_z(l)]=0$.
Notice, see Figs.~\ref{fig:en},
\ref{fig:sigma}, and \ref{fig:ES},
that even though $\Sigma$ does not necessarily acquire integer values always, it is in a
matter of speaking ``quantized,'' as it changes by clearly defined jumps.
It would be interesting to establish to what extent these kind of
$\Sigma$-jumps survive upon the onset of pairing correlations.
These $K\Sigma$-isomers are by construction the lowest states
characterized by a good quantum number $K$ and a defined spin-asymmetry.
\begin{figure}[ht]
\includegraphics[width=0.85\columnwidth]{sigma_E}
\includegraphics[width=0.85\columnwidth]{sigma_J}
\includegraphics[width=0.82\columnwidth]{E_J}
\caption{\label{fig:ES} (Color online) The total energy $E_N(\Sigma)$ in panel (a),
the projection of the total angular momentum $K_N(\Sigma)$ along the symmetry axis,
and the `yrast lines'' $E(K)$ {\it versus} $K$, all for $N=8, 12, 16$, and 20 are shown in panel (c).
Energy is in units of $\hbar\omega$ and the angular momentum in units of $\hbar$.
For visual clarity $K_N(\mu_\eta)$ for different $N$ have
been slightly shifted in the vertical direction for better visualization. }
\end{figure}
\section{Conclusions}
I discussed here several frameworks suggested in the literature, as
well as a few new ones, designed to treat odd fermion superfluid
systems, their advantages and disadvantages, and I introduced a two
chemical potential
framework~\cite{Bulgac:2007,aslda,Sensarma:2007,Bulgac:2008}, which
appears to be much simpler to implement in comparison with other
frameworks suggested so far in the nuclear physics literature.
Unlike the general framework designed by \textcite{Dobaczewski:1997},
which requires an {\it a priori} knowledge of the quantum numbers
characterizing the extra add fermion, the two chemical potentials
framework outlined in Section~\ref{sec:best} and when properly
numerically formulated, see Appendix~\ref{sec:numeric}, eschews the
diagonalization of the quasiparticle Hamiltonian, used in order to
arrive at a self-consistent solution, and also the need to identify
{\it a priori} the quantum numbers of the extra fermion. The
contribution of the lowest energy quasiparticle state of the odd
fermion is automatically selected in this framework, particularly when
the simulated annealing method is also incorporated in the iterative
process. The gradient method advocated by \textcite{Robledo:2011}
apparently also does not require {\it a priori} identification of
the quantum numbers of the odd fermion, a method limited however only
to the ground states of odd or odd-odd nuclei. The two chemical
approach as described here can prove to be essential in constructing
the initial states in an induced fission process of a nucleus produced
in a many-nucleon transfer reaction, when the state of spin-polarization
can be arbitrary. One can also generalize the approach in a
straightforward manner to impose both a finite spin-polarization and a
total angular momentum as well, to obtained what I have called above \
$K\Sigma$-isomers. Since the spin-polarization $\Sigma$ can be
``quantized,'' the identification of $K\Sigma$-isomers can lead to a deeper
understanding of nuclear dynamics, e.g. in fission, and of the
properties of nuclear spectra, in a manner analogous to the role
played by $K$-isomers.
At this time it is not clear whether either the simulated annealing
method or the gradient method~\cite{Robledo:2011} for generating the
new mean field during the iterative process is superior
computationally, or even if a carefully combination of the simulated
annealing with the gradient method could prove to be the best choice.
Since there exists methods~\cite{jin:2017,Kashiwaba:2020} which eschew
the ubiquitous diagonalization of the quasiparticle Hamiltonian used
routinely in the literature, these methods can and should be used in
conjunction with simulated annealing and/or gradient methods, in the
mean field calculations of polarized fermion systems and then one can
achieve a significant numerical speed-up. The framework described
here is formally identical to describing nuclei in the presence of
strong magnetic fields, which one can encounter in magnetars for
example. In the presence of arbitrary strong magnetic fields one can
fully polarize a nucleus.
I have conjectured that by studying the asymmetry of the fission
fragments distributions, emitted along the direction of the angular
momentum of the fissioning odd or odd-odd nucleus one could shed new
light on whether time-odd mean field components play a qualitative new
role in fission. Compound states in fissioning nuclei with relatively
large angular momenta can be populated in one or many neutron transfer
reactions~\cite{Ramos:2020}.
Similar, but not identical, type of correlations have been recently
analyzed within the phenomenological model CGMF~\cite{Becker:2013}
based on the Hauser-Feshbach framework~\cite{Hauser:1952} by
\textcite{Lovell:2020} in neutron induced fission of actinides. These
authors, while pointing to quite a number of experimental results,
observe that with the increasing energy of the incident neutron the
anisotropy for the reaction U$^{238}$(n,f) is noticeably more
pronounced than for the reactions U$^{235}$(n,f) and
Pu$^{239}$(n,f). This analysis thus suggests that fission is favored
along the direction of the total angular momentum of the compound
nucleus. This type of anisotropy has been postulated by
\textcite{Bohr:1956}. It is not clear yet however, whether the
emission of the heavy fission fragment is favored or hindered over the
emission of the light fission fragment along the direction of the
total angular momentum of the compound nucleus.
\section {acknowledgement}
I thank G.F. Bertsch for a stimulating discussion and I. Stetcu for me
making me aware of the recent phenomenological study of fission
fragments anisotropy~\cite{Lovell:2020}. I also thank J.E. Drut for
making a number of useful suggestions on the manuscript. The work was
supported by U.S. Department of Energy, Office of Science, Grant
No. DE-FG02-97ER41014 and in part by NNSA cooperative agreement
DE-NA0003841.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,015 |
Q: How to sort specific rows in matlab I have a 15x100 matrix and I only want to sort the the first 10 rows in ascending order, what matlab code would do that?
edit: Sort the first 10 rows of each column
A: x(1:10, :) = sortrows(x(1:10, :), 1:size(x,2));
The second argument of sortrows tells it which columns in which order you want to sort. so 1:size(x, 2) will sort by each column in turn (in ascending order)
If you actually want all the columns to be perfectly sorted (rows 1 to 10) and not keep the row integrity (i.e. each row can no longer be found in the original) then (although this is weird):
for col = 1:size(B, 2)
B(1:10, col) = sort(B(1:10, col));
end
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,426 |
\section{Introduction}
\label{sec:intro}
Throughout this paper, $\mathbb E^n$ will denote Euclidean $n$-space and
$\mathbb H^n$ will denote hyperbolic $n$-space. Recall that (up to isometry)
these are the unique complete simply connected
Riemannian $n$-manifolds with constant
curvature $0$ and $-1$, respectively. We will use $d(x,y)$ for the
Riemannian distance between points $x$ and $y$ in either of these
geometries. We will sometimes identify
$\mathbb E^n$ with affine $n$-space $\mathbb A^n(\mathbb R)$ over the reals, which can
then be embedded as usual in projective $n$-space $\mathbb P^n(\mathbb R)$ (the
set of lines through the origin in $\mathbb A^{n+1}(\mathbb R)$).
\begin{definition}
\label{def:Euclconics}
One of the oldest notions in
geometry, going all the way back to Apollonius, is that of \emph{conic
sections} in $\mathbb E^2$. There are at least four equivalent
definitions of a conic section $C$:
\begin{enumerate}
\item a smooth irreducible algebraic curve in $\mathbb A^2(\mathbb R)$ of degree $2$;
\item the intersection of a right circular
cone in $\mathbb E^3$ (with vertex at the origin, say)
with a plane not passing through the origin, this plane in turn
identified with $\mathbb E^2$;
\item the \emph{two focus definition}: Fix two points $a_1, a_2\in
\mathbb E^2$. An \emph{ellipse} $C$ is the locus of points $x\in \mathbb E^2$ such that
$d(x,a_1) + d(x,a_2) = c$, where $c>0$ is a fixed
constant. Similarly, a \emph{hyperbola} $C$ is the locus of points
$x\in \mathbb E^2$ such that $\left\vert d(x,a_1) - d(x,a_2)\right\vert =
c$, where $c>0$ is a fixed constant. The points $a_1$ and $a_2$ are
called the \emph{foci} of the conic, and the line joining
them (assuming $a_1\ne a_2$) is called the \emph{major axis}. A
\emph{circle} is the
special case of an ellipse where $a_1=a_2$. A \emph{parabola} is the
limiting case of a one-parameter family of ellipses $C_t(a_1,a_2,c_t)$ where
$a_1$ is fixed and we let $a_2$ run off to infinity along the major
axis keeping $c_t-d(a_1,a_2)$ fixed.
\item the \emph{focus/directrix definition}: Fix a point $a_1\in
\mathbb E^2$, called the \emph{focus}, and a line $\ell$ not passing
through $a_1$, called the \emph{directrix}. A \emph{conic} $C$ is
the locus of points with $d(x,a_1)=\varepsilon d(x, \ell)$, where
$\varepsilon >0$ is a constant called the \emph{eccentricity}. If
$\varepsilon <1$ the conic is
called an \emph{ellipse}; if $\varepsilon=1$ the conic is
called a \emph{parabola}; if $\varepsilon>1$ the conic is
called a \emph{hyperbola}. A \emph{circle} is the limiting case of an
ellipse obtained by fixing $a_1$ and sending $\varepsilon\to 0$ and $d(a_1,
\ell)\to \infty$ while keeping $r= \varepsilon d(a_1, \ell)$ fixed.
\end{enumerate}
Note that these definitions come from totally different realms.
Definition (1) is from algebraic geometry. Definition (2) uses a
totally geodesic embedding of $\mathbb E^2$ into $\mathbb E^3$. Definitions (3)
and (4) use only the metric geometry of $\mathbb E^2$.
Since Definition (1) is phrased in terms of algebraic geometry, it
naturally leads to a definition of a \emph{conic in $\mathbb P^2(\mathbb R)$} as a smooth
irreducible algebraic curve of degree $2$. Such a curve must be given
(in homogeneous coordinates) by a homogeneous quadratic equation
$Q(x)=0$, where $Q$ is a nondegenerate indefinite quadratic form on
$\mathbb R^3$. This is the equation of a cone, and intersecting the cone
with an affine plane not passing through the origin (the vertex of the
cone) gives us back Definition 2.
\end{definition}
The topic of this paper is studying what happens to Definitions
\ref{def:Euclconics}(1)--(4) when we replace $\mathbb E^2$ by $\mathbb H^2$.
This is an old problem, and is discussed for example in
\cite{MR1505334,MR1628013,MR0095442,MR0171205,MR487028}. However, as
we will demonstrate, the analogues of Definitions
\ref{def:Euclconics}(1)--(4) are
no longer equivalent in $\mathbb H^2$. Thus there is some confusion in the
literature, and those who talk about conic sections in $\mathbb H^2$ (as
recently as \cite{MR3274526,2015arXiv150406450}) do not
always all mean the same thing. Our main results are
Theorems \ref{thm:twocircles}, \ref{thm:twoellipses},
\ref{thm:parabolas}, and \ref{thm:twohyps} in Section
\ref{sec:results}, which clarify the relationships among these
definitions (especially the two-focus and focus-directrix definitions)
in $\mathbb H^2$.
\section{Other Axiomatizations}
\label{sec:axioms}
Before discussing conics in $\mathbb H^2$, we first explain still another
definition of conic sections in $\mathbb P^2(\mathbb R)$, which is the definition
found in \cite[Ch.\ III]{MR1628013} and with a slight variation in
\cite{MR1505334}. This definition uses the notion of a \emph{polarity}
$p$ in $\mathbb P^2(\mathbb R)$. This is a particular type of
mapping of points to lines and lines to
points preserving the incidence relations of
projective geometry (or in the language of \cite[\S3.1]{MR1628013}, a
\emph{correlation}). It can be explained in
terms of algebraic geometry as follows. If $Q$ is a nondegenerate
quadratic form on $\mathbb R^3$, then there is an associated nondegenerate
symmetric bilinear form defined by $B(x,y)= (Q(x+y)-Q(x)-Q(y))/2$, and
if $V$ if a linear subspace of $\mathbb R^3$ of dimension $d=1$ or $2$, then
the orthogonal complement $V^{\perp, B}$ of $V$ with respect to $B$ is a linear
subspace of dimension $3-d$. Thus the process $p$ of taking orthogonal
complements with respect to $B$ sends points in $\mathbb P^2(\mathbb R)$, which
are $1$-dimensional linear subspaces of $\mathbb R^3$, to lines (copies of
$\mathbb P^1(\mathbb R)$), which are $2$-dimensional linear subspaces of $\mathbb R^3$,
and \emph{vice versa}. Given a polarity $p$, the associated
\emph{conic} is the set $C$ of points $x\in \mathbb P^2(\mathbb R)$ such that $x$
lies on the line $p(x)$, i.e., the set of $1$-dimensional linear
subspaces $V$ of $\mathbb R^3$ for which $V\subset V^{\perp, B}$, or in
other words, for which $V$ is $B$-isotropic. Thus if
we identify the point $x\in \mathbb P^2(\mathbb R)$ with its homogeneous
coordinates, or with a basis vector for $V$ up to rescaling, this
becomes the condition $B(x,x)=0$, or $Q(x)=0$, which is just
Definition \ref{def:Euclconics}(1). (Note that if $Q$ is definite,
the conic is empty, so we are forced to take $Q$ to be indefinite in
order to get anything interesting.) Conversely, it is well known
\cite[\S4.72]{MR1628013} that
every polarity arises from a nonsingular symmetric matrix or
equivalently from a nondegenerate quadratic form $Q$, so the polarity
definition of conics in \cite[Ch.\ III]{MR1628013} is equivalent to
Definition \ref{def:Euclconics}(1).
We now introduce several possible definitions of conic sections in
$\mathbb H^2$.
\begin{definition}[A metric circle]
\label{def:circ}
A \emph{circle} in $\mathbb H^2$ is the locus of points a fixed distance
$r>0$ from a \emph{center} $x_1\in \mathbb H^2$, i.e.,
$C=\{x\in \mathbb H^2: d(x,x_1)=r\}$.
\end{definition}
\begin{definition}[{Analogue of Definition \ref{def:Euclconics}(2)}]
\label{def:hypconicinH3}
A \emph{right circular cone} in $\mathbb H^3$ is defined as follows. Fix a point
$x_0\in \mathbb H^3$ (say the origin, if we are using the standard unit ball in
$\mathbb R^3$ as our model of $\mathbb H^3$) and fix a plane $P$ in $\mathbb H^3$
(a totally geodesic copy of $\mathbb H^2$) not passing through $x_0$.
There is a unique ray starting at $x_0$ and intersecting $P$
perpendicularly. Let $x_1$ be the intersection point (the closest
point on $P$ to $x_0$), and fix a radius $r>0$. We then have the circle
$C$ in $P$ centered at $x_1$ with radius $r$. The cone $c(x_0, C)$
through $x_0$ and $C$ is then the union of
the lines (geodesics) through $x_0$ passing through a point of $C$.
The point $x_0$ is called the \emph{vertex} of the cone.
A \emph{conic section} (in the literal sense!)
in $\mathbb H^2$ is then the intersection of a plane $P'$ in $\mathbb H^3$ (not
passing through $x_0$) with $c(x_0, C)$.
\end{definition}
Since we can take $P'=P$ in the above definition, it is obvious that a
circle (as in Definition \ref{def:circ}) is a special case of a conic
section in the sense of Definition \ref{def:hypconicinH3}.
In the Poincar\'e ball
model of $\mathbb H^3$ with $x_0$ the origin, geodesics through
$x_0$ are just straight lines for the Euclidean metric, so it's easy
to see that a right circular cone with vertex $x_0$ is also a right
circular cone in the Euclidean sense in $\mathbb R^3$. On the other hand,
planes in $\mathbb H^3$ not passing through $x_0$ correspond to Euclidean
spheres perpendicular to the unit sphere (the boundary of the model of
$\mathbb H^3$). Thus a conic section in the sense of Definition
\ref{def:hypconicinH3} is the intersection of a right circular cone
with a sphere, and is thus (in terms of the algebraic geometry of
$\mathbb A^3(\mathbb R)$) an algebraic curve of degree $\le 4$. To view this conic
in the usual Poincar\'e disk model of $\mathbb H^2$, we apply an
isometry (stereographic projection) from $P$ to the unit disk in
$\mathbb C$. Since this is a rational map, we see that any conic section in
the sense of Definition \ref{def:hypconicinH3} is an algebraic curve
(in fact of degree $\le 4$) when viewed in the disk model of $\mathbb H^2$.
Alternatively, if we use the Klein ball model of $\mathbb H^3$ with $x_0$ the
origin, then a right circular cone with vertex $x_0$ will again look
like a Euclidean right circular cone, while a $2$-plane in $\mathbb H^3$
will be the intersection of the ball with
a Euclidean $2$-plane, and any conic section in the
sense of Definition \ref{def:hypconicinH3} will also be a conic
section in the Euclidean sense of Definition \ref{def:Euclconics}.
Thus Definition \ref{def:hypconicinH3} is equivalent to the following
Definition \ref{def:alghypconic}.
\begin{definition}[{Analogue of Definition \ref{def:Euclconics}(1)}]
A \emph{conic in $\mathbb H^2$ in the algebraic sense} is the intersection of a
smooth irreducible algebraic curve of degree $2$
in $\mathbb A^2(\mathbb R)$ with the open unit
disk, viewed as the Klein disk model for $\mathbb H^2$. (This is a
non-conformal model in which points of $\mathbb H^2$ are points of the open
unit disk, and the straight lines are intersections with
the open disk of straight lines in the plane.) Such a conic is
closed (compact) if and only if it is a circle or ellipse not
intersecting the unit circle (the \emph{absolute} in the terminology
of \cite{MR1505334} and \cite{MR1628013}).
\label{def:alghypconic}
\end{definition}
Definition \ref{def:alghypconic} is the definition of conics used in
\cite{MR1505334} and \cite{MR1628013}.
Still another approach to defining conics may be found in \cite{MR487028},
based on the axiom system for $\mathbb E^2$ and $\mathbb H^2$ developed in
\cite{MR0346643}. First we need to discuss Bachmann's approach to
metric geometry. Bachmann observes that in either $\mathbb E^2$ and $\mathbb H^2$, there
is a unique isometry which is reflection in a given line $a$ or around
a given point $A$. Thus we can identify lines and points with certain
distinguished involutory elements $\mathcal S$ (the reflections in lines)
and $\mathcal P$ (the reflections around points) of the isometry group
$G$. More is true: every element of $G$ is a product of at most three
elements of $\mathcal S$. Elements of $\mathcal S$ are orientation-reversing; elements
of $\mathcal P$ are orientation-preserving. The product of two elements
$a,\,b\in \mathcal S$ is a non-trivial involution if and only if $a\ne b$ and
$ab=ba$; in this case, the lines associated to $a$ and $b$ are
perpendicular (we write $a\perp b$)
and $ab\in \mathcal P$ is the reflection around the unique intersection point
of $a$ and $b$. Furthermore, every element of $G$ of order $2$ belongs
to $\mathcal S$ or to $\mathcal P$, but not to both. A point $A\in \mathcal P$ lies on a
line $a\in \mathcal S$ exactly when there exists $b\in \mathcal S$ commuting with
$a$ such that $A = ab$. Thus a \emph{metric plane} $\mathcal M$
can be identified with a
group $G$ together with a distinguished generating set $\mathcal S$
consisting of involutions and the set $\mathcal P$ of non-trivial products of
commuting elements of $\mathcal S$, satisfying certain axioms.
We won't need the axioms here, since they will be evident in the
cases we are interested in. In the case of $\mathbb E^2$, $G=\mathbb R^2\rtimes
O(2)$, the usual Euclidean
motion group, and in the case of $\mathbb H^2$, $G=PGL(2,\mathbb R)\cong
O^+(2,1)$. (If $\begin{pmatrix} a&b\\c&d\end{pmatrix}$ has determinant
$+1$, then it operates on the upper half-plane by linear fractional
transformations, which are orientation-preserving, and if it has
determinant $-1$, then it operates on the upper half-plane by
\[
z\mapsto \frac{a\bar z + b}{c\bar z + d},
\]
and this conjugate-linear map is an orientation-reversing isometry of
$\mathbb H^2$.)
Bachmann also points out that the metric plane $(\mathcal M,
\mathcal S, \mathcal P)$ corresponding to $\mathbb E^2$ or $\mathbb H^2$ can be embedded naturally in a
\emph{projective-metric plane} $(\mathcal P\mathcal M, \mathcal S', \mathcal P')$, in such a way
that $\mathcal S\subseteq \mathcal S'$ and $\mathcal P\subseteq \mathcal P'$.
In the case of $\mathbb E^2$, this is just the usual embedding of
$\mathbb A^2(\mathbb R)$ in $\mathbb P^2(\mathbb R)$ by adjoining a copy of $\mathbb P^1(\mathbb R)$ at
$\infty$, and the associated group is $PGL(3,\mathbb R)$. In the case of
$\mathbb H^2$, $\mathcal P\mathcal M$ is again a copy of $\mathbb P^2(\mathbb R)$, but its points
and lines consist of \emph{ideal points} and \emph{ideal lines} of
$\mathbb H^2$. A simple way to visualize the embedding of $\mathbb H^2$ in
$\mathbb P^2(\mathbb R)$ is to use the (non-conformal) Klein model of $\mathbb H^2$, in
which points are points in the interior of the unit disk in $\mathbb R^2$,
and lines are the intersections of ordinary straight lines in
$\mathbb A^2(\mathbb R)$ with the unit disk. Then each point or line of $\mathbb H^2$
obviously corresponds to a unique point or line of $\mathbb P^2(\mathbb R)$.
When viewed as $\mathcal P\mathcal M$ in this way, $\mathbb P^2(\mathbb R)$ carries a canonical
polarity, namely the one associated to the unit circle in
$\mathbb A^2(\mathbb R)$, viewed as a conic in the sense of the polarity
definition at the beginning of this Section. When we embed
$\mathbb A^2(\mathbb R)$ in $\mathbb P^2(\mathbb R)$ as usual via $(x,y)\mapsto [x,y,1]$
(homogeneous coordinates denoted by square brackets), this polarity is
associated to the quadratic form $Q\colon\, (x,y,z)\mapsto x^2+y^2-z^2$, since
$Q(\cos\theta,\sin\theta,1)=0$ for any real angle $\theta$.
\begin{definition}[{\cite[Definition 4.1]{MR487028}}]
\label{def:Molnar}
A \emph{conic $C$ in the sense of Moln\'ar}, with foci $A,\,B\in
\mathbb P^2(\mathbb R)$, is defined by choosing a line $x_1$ in $\mathbb P^2(\mathbb R)$ which is not a
boundary line (i.e., $x_1$ is not tangent to the unit circle) and not
passing through either $A$ or $B$, and with $A$ and $B$ not each
other's reflections across $x_1$. Then $C$ consists of points $X_{11}$
and $X$ chosen as follows. $X_{11}$ is the intersection of the lines
$a_{11}$ through $A$ and $B^{x_1}$ (the reflection of $B$ across
$x_1$) and $b_{11}$ through $B$ and $A^{x_1}$. (The line $x_1$ is
chosen so that $a_{11}$ and $b_{11}$ are not boundary lines.) The other
points $X$ are defined by fixing a point $Y$ on $x_1$ and
taking the lines $a$ through $Y$ and $A$ and $b$ though $Y$ and $B$,
and then if neither $a$ nor $b$ is a boundary line, letting $X$ be the
intersection of $a_{11}^a$ and $b_{11}^b$ (the reflections of $a_{11}$
and $b_{11}$ across $a$ and $b$, respectively). Appropriate
modifications are made if $a$ or $b$ is a boundary line.
\end{definition}
As is quite evident, Moln\'ar's definition is quite complicated but
results in a conic section in $\mathbb H^2$ being the intersection of a conic in
$\mathbb P^2(\mathbb R)$ with the unit disk (in the Klein model). We will not
consider this definition further, but it's closely related to
Definition \ref{def:alghypconic}. A picture of the construction with
$A=(0,0)$, $B=(.5,0)$, $x_1=\{y=.5\}$ is shown in Figure
\ref{fig:MolnarConic}.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=2in]{MolnarConicPic.pdf}
\end{center}
\caption{Moln\'ar's construction of a conic}
\label{fig:MolnarConic}
\end{figure}
\section{Main Results}
\label{sec:results}
\begin{definition}[{Analogue of Definition \ref{def:Euclconics}(3)}]
\label{def:twofocushypconic}
The definition of \emph{two focus conics} in Definition
\ref{def:Euclconics}(3) immediately goes over to $\mathbb H^2$, simply by
replacing the Euclidean distance by the hyperbolic distance. Note
that the case of a circle was already mentioned in
Definition \ref{def:circ}.
\end{definition}
The last definition is the only one that is not immediately
obvious. However, were we to carry Definition \ref{def:Euclconics}(4)
over to $\mathbb H^2$ without change, then since in the upper half-plane
or disk models of $\mathbb H^2$, the distance function is the log of an
algebraic expression, in the case of irrational eccentricity $\varepsilon$ we
would effectively get the equation
\[
(\text{algebraic expression}) = (\text{algebraic expression})^\varepsilon,
\]
which is a transcendental equation, and could not possibly agree with
the other definitions of conic sections. This explains the
modification made in \cite{MR1505334}. The use of the hyperbolic sine
comes from its role in hyperbolic geometry via the solution of the
Jacobi equation.
\begin{definition}
\label{def:focdirhypconic}[{Analogue of Definition
\ref{def:Euclconics}(4)}]
Fix a point $a_1\in \mathbb H^2$, called the \emph{focus}, and a line
(geodesic) $\ell$ not passing through $a_1$, called the
\emph{directrix}. A \emph{conic} $C$ is the locus of
points $x\in \mathbb H^2$ with $\sinh d(x,a_1)=\varepsilon \sinh d(x,
\ell)$, where $\varepsilon>0$
is a constant called the \emph{eccentricity}. If $\varepsilon<1$ the conic is
called an \emph{ellipse}; if $\varepsilon=1$ the conic is
called a \emph{parabola}; if $\varepsilon>1$ the conic is
called a \emph{hyperbola}. (Note: in the case of the parabola, but
only in this case, the hyperbolic sines cancel and can be removed
from the definition.) A \emph{circle} is the limiting case of an
ellipse obtained by fixing $a_1$ and sending $\varepsilon\to 0$ and $d(a_1,
\ell)\to \infty$ while keeping $r=\varepsilon \sinh d(a_1, \ell)$ fixed.
\end{definition}
\subsection{Circles}
\label{sec:circle}
We begin now to compare the various definitions. We start with the
circle, which is the most straightforward. Definition \ref{def:circ}
clearly coincides with Definition \ref{def:hypconicinH3}, in the sense
that if we intersect a right circular cone with a plane perpendicular
to the axis, the result is a circle in the sense of
Definition \ref{def:circ}. We also have the following.
\begin{proposition}
\label{prop:twocircles}
Definition \ref{def:circ} coincides with the case of circles
in Definition \ref{def:alghypconic}, but with the Klein model replaced
by the Poincar\'e model. In other words, an ordinary
circle in $\mathbb A^2(\mathbb R)$, contained in the open unit disk,
when viewed as a curve in the Poincar\'e disk model of $\mathbb H^2$
is a metric circle in $\mathbb H^2$, and \emph{vice versa}. Similarly, an
ordinary circle contained in the upper half-plane, when viewed as a
curve in the Poincar\'e upper half-plane model of $\mathbb H^2$,
is a metric circle in $\mathbb H^2$, and \emph{vice versa}.
\end{proposition}
\begin{proof}
First consider the disk model.
If the center is the origin, this is clear since the hyperbolic
distance from $0$ to $z$ in $\{z: |z|<1\}$
in $\mathbb C$ is a (nonlinear) function $\tanh^{-1}(|z|)=
\frac{1}{2}\log\bigl(\frac{1+|z|}{1-|z|}\bigr)$ of the Euclidean distance
$|z|$ from $0$ to $z$, so that each Euclidean circle centered at $0$ is
also a hyperbolic circle (of a different radius), and \emph{vice
versa}. However, any circle in $\mathbb H^2$ can be mapped to a
circle centered at $0$ via an isometry of $\mathbb H^2$, and since linear
fractional transformations send circles to circles
\cite[Ch.\ 3, \S3.2, Theorem 14]{MR510197}, the general case
follows. The case of the half-plane model also follows since there is a linear
fractional transformation relating this model to the disk model.
\end{proof}
\begin{remark}
\label{rem:circcenter}
However, one should note that the center of a circle in the unit disk
or the upper half-plane may differ, depending on whether one considers
it as a Euclidean circle or a metric circle in $\mathbb H^2$. For example,
the metric circle in $\mathbb H^2$ (in the upper-halfplane model) around the
point $i$ with hyperbolic metric radius $\log 2$ has Euclidean equation
\[
\frac{\vert z - i\vert}{\vert z + i\vert} = \tanh\left(\frac12 \,\log 2\right)
= \frac{1}{3} \text{ or }
\left\vert z - \frac{5}{4}i\right\vert = \frac{3}{4},
\]
so its center as a Euclidean circle is $\frac{5}{4}\, i$.
Metric circles in $\mathbb H^2$, when drawn in the Klein disk model,
only appear to be circles when centered at the origin. Otherwise, they
are ellipses.
\end{remark}
However, the focus/directrix definition of circles is quite different.
\begin{theorem}
\label{thm:twocircles}
The definition of circle in Definition \ref{def:focdirhypconic} does
\emph{not} agree with the definition of circle in Definitions
\ref{def:circ}, \ref{def:hypconicinH3},
\ref{def:alghypconic}, and \ref{def:twofocushypconic}.
\end{theorem}
\begin{proof}
Consider a circle in the sense of Definition
\ref{def:focdirhypconic}. Without loss of generality, we work in the
upper half-plane model of $\mathbb H^2$ and set $a_1=i$, $\ell = \{z\in
\mathbb H^2 : |z| = R\}$, where we let $R\to +\infty$. In this case
$d(a_1,\ell) = \log R$ and we want to keep $r=\varepsilon \sinh(\log
R) = \frac{\varepsilon(R^2-1)}{2R}$ constant, so we take
$\varepsilon = \frac{2rR}{R^2-1}$. For $z\in \mathbb H^2$,
\[
d(z,\ell)=\tfrac12 \,d(z,w),\text{ where }w=\frac{R^2}{\overline z}
=\text{reflection of }z\text{ across }\ell.
\]
Then the equation $\sinh d(z,a_1)=\varepsilon \sinh d(z, \ell)$
becomes
\[
\sinh\left(2\tanh^{-1}\left\vert\frac{z-i}{z+i}\right\vert\right)
= \frac{2rR}{R^2-1}\sinh\left(
\tanh^{-1}\left\vert\frac{z-\frac{R^2}{\overline z}}{z-\frac{R^2}{z}}
\right\vert\right) .
\]
The left-hand side simplifies to
\[
\frac{2\vert z+i\vert\,\vert z-i\vert}{\vert z+i\vert^2 -
\vert z-i\vert^2} = \frac{\vert z^2+1\vert}{2\Im z}.
\]
On the right-hand side,
\[
\left\vert\frac{z-\frac{R^2}{\overline z}}{z-\frac{R^2}{z}}
\right\vert = \frac{R^2 - \vert z\vert^2}{\left\vert R^2 - \vert z\vert^2
\frac{z}{\overline z}\right\vert} = \frac{R^2-a}{\sqrt{R^4
+a^2 -2R^2a\cos(\theta)}}
\]
where $a= \vert z\vert^2$ and $\theta = 2\arg z$. Then
\[
\lim_{R\to \infty}
\frac{2rR}{R^2-1}\sinh\left(\tanh^{-1}\left(\frac{R^2-a}{\sqrt{R^4
+a^2 -2R^2\cos(\theta)}}\right)\right)
=\frac{\sqrt{2}r}{|z|\sqrt{1-\cos(\theta)}}.
\]
Thus Definition \ref{def:focdirhypconic} gives for our circle
the equation
\[
\frac{\vert z^2+1\vert}{2\Im
z}=\frac{\sqrt{2}r}{|z|\sqrt{1-\cos(\theta)}}
= \frac{r}{|z|}\csc(\theta/2)= \frac{r}{|z|}\frac{|z|}{\Im z} =
\frac{r}{\Im z},
\]
or
\begin{equation}
\vert z^2+1\vert = {2r}.
\label{eq:focdircircle}
\end {equation}
in the upper half-plane. This is an algebraic curve but not a metric
circle. Figure \ref{fig:circ} shows the case of $r=.25$ (in solid
color) as drawn with
\texttt{Mathematica}. This curve passes through the points
$i\sqrt{\frac{3}{2}}$, $i\sqrt{\frac{1}{2}}$, and $i \pm
\sqrt{\frac{\sqrt{17}-4}{2}}$; the circle centered on the imaginary axis
tangent to it at $i\sqrt{\frac{3}{2}}$ and $i\sqrt{\frac{1}{2}}$
is shown with a dashed line in the same figure. The curves are close
but do not coincide.
\end{proof}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=2in]{focusdirectrixcircle.pdf}
\end{center}
\caption{A ``circle'' with focus $i$ and $r=.25$ (solid color) and a
tangent metric circle (dashed) in the upper half-plane}
\label{fig:circ}
\end{figure}
Aside from circles, there are various other circle-like curves that
play a role in hyperbolic geometry. These may be considered to be
conics according to certain definitions. Note also that they are
distinct from the circles of Definition \ref{def:focdirhypconic}.
\begin{definition}
\label{def:cycle}
A \emph{horocycle} (occasionally called a \emph{paracycle})
in the Poincar\'e disk model of $\mathbb H^2$ is the
intersection of the disk with a circle tangent to the unit circle (and
lying inside the circle). A \emph{hypercycle} in the Poincar\'e disk
model of $\mathbb H^2$ is the intersection of the disk with a circle meeting
the unit circle in exactly two points. These have well-known
intrinsic definitions. A horocycle is the limit of a sequence
of circles $C_n$ (in the sense of Definition \ref{def:circ}) all
passing through a fixed point $x_0$, with
centers $x_n$ all lying on a fixed ray through $x_0$
and with radii $d(x_n,x_0)=r_n\to \infty$. See Figure \ref{fig:horo}(a).
A hypercycle is a curve on one side of a given line $\ell$
whose points all have the same orthogonal distance from $\ell$.
See Figure \ref{fig:horo}(b).
Note that horocycles and hypercycles are clearly conics
in the sense of Definition \ref{def:alghypconic}. But they are not
covered by Definitions \ref{def:twofocushypconic} and
\ref{def:focdirhypconic}. Moln\'ar observes in \cite{MR487028} that
metric circles (Definition \ref{def:circ}), horocycles, and
hypercycles are all special cases of
Definition \ref{def:Molnar} when the two foci coincide.
\end{definition}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=2in]{horocycle.pdf}\hspace{1in}
\includegraphics[width=2in]{hypercycle.pdf}
\end{center}
\caption{(a) (left) A horocycle (solid red) as a limit of circles (black)
through $x_0$ with radii going to infinity. (b) (right) A hypercycle
(solid blue) and a straight line $\ell$ (orange)
with the same ideal limits at infinity}
\label{fig:horo}
\end{figure}
\subsection{Ellipses}
\label{sec:ellipse}
Next, we consider the case of the (noncircular) ellipse. There are two
main competing definitions: Definition \ref{def:twofocushypconic} and
Definition \ref{def:focdirhypconic}.
\begin{theorem}
\label{thm:twoellipses}
The definition of ellipse in Definition \ref{def:focdirhypconic} does
\emph{not always} agree with the definition of ellipse in Definition
\ref{def:twofocushypconic}. However, there are cases where they
coincide. More precisely, when Definition \ref{def:focdirhypconic}
gives a closed curve in $\mathbb H^2$, this curve is also a two-focus ellipse.
\end{theorem}
\begin{proof}
We will work in the upper half-plane model of $\mathbb H^2$ and, without
loss of generality, put one focus at $i$ and let the imaginary axis be
an axis of the ellipse. For an ellipse with the ``two-focus
definition'' and foci at $i$ and $bi$, $b>0$, the equation is
\[
2\tanh^{-1}\left({\frac{|z-i|}{|z+i|}}\right) +
2\tanh^{-1}\left({\frac{|z-bi|}{|z+bi|}}\right) = c,
\]
which can be rewritten as the algebraic equation
\begin{multline}
\label{eq:2focell}
\left(x^2+y^2+1+\sqrt{(x^2 - y^2 + 1)^2 + 4 x^2 y^2}\right)\\
\times \left(x^2+y^2+b^2+\sqrt{(x^2 - y^2 + b^2)^2 + 4 x^2 y^2}\right)
=4be^c\,y^2
\end{multline}
with $c>0$. Plots of this equation for $b= \frac34$ and for
various values of $c$ are shown in Figure \ref{fig:2focell}.
The minimal value of $c$ to have the foci inside the ellipse is
the hyperbolic distance between the foci, or $\vert\log b\vert$.
As $c$ increases, the curves get bigger and bigger and look more like
circles. Now that since \eqref{eq:2focell} implies that $d(z,i)\le c$,
any ellipse in the sense of Definition \ref{def:twofocushypconic} is
automatically compact (closed) in $\mathbb H^2$.
\begin{figure}
\begin{center}
\includegraphics[width=3in]{twofocusellipse.pdf}
\end{center}
\caption{Two-focus ellipses in the upper half-plane with foci at $i$
and $\frac{3i}{4}$, as drawn with \texttt{Mathematica}}
\label{fig:2focell}
\end{figure}
Now consider the focus-directrix definition for an ellipse in the
upper half-plane, with a focus at $i$ and directrix $|z|=r$, $r>1$ (this
choice makes the imaginary axis an axis of the ellipse). The distance
from $z$ to the directrix is half the distance to the reflection of
$z$ across the directrix, which is $\frac{r^2}{\overline z}$. Thus
the equation becomes
\[
\sinh\left(2\tanh^{-1}\left({\frac{|z-i|}{|z+i|}}\right)\right)
=\varepsilon\,\sinh \left(\tanh^{-1}
\left(\frac{\left\vert z-\frac{r^2}{\overline z}\right\vert}
{\left\vert z-\frac{r^2}{z}\right\vert}\right)\right),
\]
which simplifies (after squaring both sides) to
\begin{multline}
\label{eq:focdirell}
r^2 \left(1 + x^4 + y^4 + 2 y^2 \left(-1 + \varepsilon^2\right) +
2 x^2 \left(1 + y^2 + \varepsilon^2\right)\right) \\=
\varepsilon^2\, \left(r^4 + \left(x^2 +
y^2\right)^2\right).
\end{multline}
This is a relatively simple quartic equation in $x$ and $y$, basically
the Cassini oval equation, and has
some interesting features. For example, if one sets
$\varepsilon=\frac{1}{r}$, this reduces to a lemniscate passing
through $(0,0)$ (an ideal boundary point of $\mathbb H^2$). When
$\varepsilon > \frac{1}{r}$, the curve (viewed in $\mathbb H^2$) is not
closed and approaches two distinct ideal boundary points.
Pictures of this behavior appear in Figure \ref{fig:focdirell}.
As a check that having two distinct ideal boundary points is not just
an artifact of the calculation,
one can check that upon substituting $r=3$ and $\varepsilon =
\frac12$ into \eqref{eq:focdirell}, one gets two points with $y=0$,
namely $x=\pm \sqrt{\frac37}$.
\begin{figure}
\begin{center}
\includegraphics[width=2in]{closedfocusdirellipse.pdf}\qquad\qquad
\includegraphics[width=2in]{openfocusdirellipse.pdf}
\end{center}
\caption{Focus/directrix ellipses in the upper half-plane with focus at $i$
and directrix $|z|=2$, as drawn with \texttt{Mathematica}. On the
left, cases with $\varepsilon \le
.5$. The case $\varepsilon =.5$ is a lemniscate. On the right,
cases with $\varepsilon$ from $.6$ to $.9$.}
\label{fig:focdirell}
\end{figure}
To illustrate another difference between the two definitions, consider the
case of the two-focus definition
when the foci coincide, i.e., $b=1$ in equation
\eqref{eq:2focell}. Then equation \eqref{eq:2focell} reduces to
\[
x^2+y^2+1+\sqrt{(x^2 - y^2 + 1)^2 + 4 x^2 y^2}= 2e^{c/2}y
\]
or
\[
(x^2 - y^2 + 1)^2 + 4 x^2 y^2 - (2e^{c/2}y-x^2-y^2-1)^2 = 0,
\]
which simplifies to the equation of a circle:
\begin{equation}
x^2+\left(y-\cosh(c/2)\right)^2 = \sinh^2(c/2).
\label{eq:2foccirc}
\end{equation}
However, the focus/directrix equation \eqref{eq:focdirell} never
reduces to a circle.
However, perhaps rather surprisingly, focus/directrix ellipses with
$\varepsilon < \frac{1}{r}$ (this is the case where the curve is closed)
turn out to be special cases of two-focus ellipses.
A rather horrendous calculation
with \texttt{Mathematica} or \texttt{MuPAD}
shows for example that \eqref{eq:2focell}
with $b=2$ and $c=\log\left(\frac52\right)$ is equivalent to
\eqref{eq:focdirell} with
\[
\varepsilon=\frac{\sqrt{209}}{21}, \quad r=\sqrt{\frac{11}{19}}.
\]
To see this, rewrite \eqref{eq:focdirell} in the form
\[
\begin{aligned}
x^2+y^2+1&+\sqrt{(x^2 - y^2 + 1)^2 + 4 x^2 y^2}\\ &=
\frac{20y^2}{x^2+y^2+4+\sqrt{(x^2 - y^2 + 4)^2 + 4 x^2 y^2}}\\
&=\frac{20y^2\left(x^2+y^2+4-\sqrt{(x^2 - y^2 + 4)^2 + 4 x^2 y^2}\right)}
{(x^2+y^2+4)^2-((x^2 - y^2 + 4)^2 + 4 x^2 y^2)},
\end{aligned}
\]
simplify, and rewrite in the form $E + \sqrt{B} = F\sqrt{D}$,
where $B=(x^2 - y^2 + 1)^2 + 4 x^2 y^2$ and $D=(x^2 - y^2 + 4)^2 + 4 x^2 y^2$.
Square both sides, again simplify and regroup to get the
term with $\sqrt{B}$ by itself, and finally square again.
After factoring out $y^2$, one finally ends up with the equation
\[
20x^4 + 40x^2y^2 + 325x^2 + 20y^4 - 116y^2 + 80 = 0,
\]
which is equivalent to \eqref{eq:focdirell} for the given parameters.
Other values of $r$ and $\varepsilon$
(with $r\varepsilon < 1$) can be handled similarly; one just
needs to solve for the values of $b$ and $c$ giving the same $y$-intercepts.
\end{proof}
\subsection{Parabolas}
\label{sec:parabola}
Next, we consider the case of the parabola. Here the result is rather simple:
\begin{theorem}
\label{thm:parabolas}
The definitions of parabolas in Definition \ref{def:focdirhypconic}
and in Definition \ref{def:twofocushypconic}
never agree. In all cases, however, a parabola in $\mathbb H^2$ is not
closed.
\end{theorem}
\begin{proof}
Without loss of generality, we can again use the Poincar\'e upper
half-plane model of $\mathbb H^2$ and put one focus at $i$ and take
the axis of the parabola to be the imaginary axis. The two-focus
definition of Definition \ref{def:twofocushypconic} is the limiting
case of \eqref{eq:2focell} as we keep $be^c = C/2$ fixed and
let $b\to 0_+$. (This is because $d(i,ib)=\vert\log b\vert = - \log b$
for $0<b<1$ and we want $c-d(i,ib)= c + \log b$ to be held constant.)
Then \eqref{eq:2focell} reduces to
\begin{equation}
\label{eq:2fopar}
\left(x^2+y^2+1+\sqrt{(x^2 - y^2 + 1)^2 + 4 x^2 y^2}\right)
\left(x^2+y^2\right) = C\,y^2,
\end{equation}
or equivalently (after regrouping and squaring to get rid of the radical,
then factoring out a $y^2$)
\begin{equation}
\label{eq:2fopara}
2(C - 2)(x^2 + y^2)^2 + 2C(x^2 + y^2) - C^2 y^2 =0 .
\end{equation}
This is the equation of a lemniscate through the origin. (Remember
that $0$, however, is only an ideal boundary point of $\mathbb H^2$.)
Definition \ref{def:focdirhypconic} simply gives \eqref{eq:focdirell}
with $\varepsilon = 1$, which reduces to
\begin{equation}
\label{eq:fodirpar}
1 - r^2 + 4 x^2 + \left(1 - \tfrac{1}{r^2}\right) (x^2 + y^2)^2 = 0 ,
\end{equation}
which is a Cassini oval equation. Note that \eqref{eq:2fopara} and
\eqref{eq:fodirpar} never agree, since for $r\ne 1$ (we don't want
the directrix of the parabola to pass through the focus), the curve
given by \eqref{eq:fodirpar} doesn't pass through the origin. Pictures
of the various kinds of parabolas, plotted by \texttt{Mathematica},
are shown in Figures \ref{fig:parabolas1} and \ref{fig:parabolas2}.
\end{proof}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=2in]{focdirparabolas1.pdf}\qquad\qquad
\includegraphics[width=2in]{focdirparabolas2.pdf}
\end{center}
\caption{Focus/directrix parabolas in the upper half-plane with focus at $i$
and directrix $|z|=r$, as drawn with \texttt{Mathematica}. On the
left, cases with $r < 1$. These are Cassini ovals. On the right,
cases with $r>1$. Of course, if one were wearing ``hyperbolic glasses,''
all would look roughly the same.}
\label{fig:parabolas1}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=2in]{twofocusparabolas.pdf}
\end{center}
\caption{Two-focus parabolas in the upper half-plane with focus at $i$,
as drawn with \texttt{Mathematica}. Note the lemniscate shape.}
\label{fig:parabolas2}
\end{figure}
\subsection{Hyperbolas}
\label{sec:hyperbola}
Finally, we consider the case of the hyperbola.
\begin{theorem}
\label{thm:twohyps}
The definition of hyperbola in Definition \ref{def:focdirhypconic} does
\emph{not always} agree with the definition of hyperbola in Definition
\ref{def:twofocushypconic}. However, the two-focus hyperbola
from Definition \ref{def:twofocushypconic} is a special case of
the focus-directrix hyperbola of Definition \ref{def:focdirhypconic}.
\end{theorem}
\begin{proof}
Consider the two-focus hyperbola. Fix $c>0$.
(When $c=0$, the definition degenerates to the bisector of the line
segment joining the two foci, which is a straight line (i.e., a geodesic).)
We will work in the upper half-plane model of $\mathbb H^2$ and, without
loss of generality, put one focus at $i$ and the other focus at $ib$, $b>1$.
The equation of the two-focus hyperbola is then
\[
2\tanh^{-1}\left({\frac{|z-i|}{|z+i|}}\right) -
2\tanh^{-1}\left({\frac{|z-bi|}{|z+bi|}}\right) = \pm c,
\]
which can be rewritten as the algebraic equation
\begin{multline}
\label{eq:2fochyp}
b^2 + x^2 + y^2 + \sqrt{b^4 + 2 b^2 (x^2 - y^2) + (x^2 + y^2)^2} =\\
b e^{\pm c} \left(1 + x^2 + y^2 + \sqrt{1 + 2 (x^2 - y^2)
+ (x^2 + y^2)^2}\right)
\end{multline}
with $c>0$. Note that the hyperbola should intersect its axis
(here the imaginary axis) at two points of the form $iy$, $1< y < b$,
so we want $0 < c < \log b$, and the two $y$-intercepts are at
$i\sqrt{be^{\pm c}}$. Comparing this with the $y$-intercepts for the
focus-directrix hyperbola \eqref{eq:focdirell} (the equation is the
same as for the ellipse --- the only difference is the value of the
eccentricity $\varepsilon$), we see that this agrees with a
focus-directrix hyperbola with parameters satisfying
\[
\frac{\sqrt{r + r^2 \varepsilon}}{\sqrt{r + \varepsilon}} =
\frac{\sqrt{b}}{e^{c/2}}, \quad \frac{\sqrt{r - r^2\varepsilon}}
{\sqrt{r - \varepsilon}} = \sqrt{b}\,e^{c/2},
\]
or
\begin{equation}
r =\sqrt{\frac{-b + 2 b^2 e^c - b e^{2 c}}
{b - 2 e^c + b e^{2 c}}},\quad \varepsilon =
\frac{\sqrt{b (-1 + 2 b e^c - e^{2 c})(b - 2 e^c + b e^{2 c})}}
{b(e^{2c}-1)}.
\label{eq:revals}
\end{equation}
Note that since $c < \log b$, the value of $\varepsilon$ is $>1$.
Just as an example, if $b=2$ and $c = \log(3/2)$, after removing some
superfluous factors, equation \eqref{eq:2fochyp} reduces to
$24 + 6 x^4 - 26 y^2 + 6 y^4 + 3 x^2 (-17 + 4 y^2) = 0$, which
agrees with the focus-directrix hyperbola with focus $i$, directrix
$|z|=\sqrt{\frac{11}{7}}$, and eccentricity $\varepsilon=\frac{\sqrt{77}}{5}$.
A graph of this hyperbola, drawn with \texttt{Mathematica}, appears in Figure
\ref{fig:hyp}.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=3in]{hyperbola.pdf}
\end{center}
\caption{A hyperbola in the upper half-plane with foci at $i$ and $2i$,
$c=\log(3/2)$, as drawn with \texttt{Mathematica}.}
\label{fig:hyp}
\end{figure}
So this analysis shows that every two-focus hyperbola is also a
focus-directrix hyperbola. The converse fails, however. Indeed,
one can see from \eqref{eq:focdirell} that the focus-directrix hyperbola
with $r=\varepsilon>1$ degenerates to the equation
\[
(r^2 + 1) x^2 + (r^2 - 1) y^2 = \frac{r^4 - 1}{2},
\]
which, surprisingly, is an \emph{ellipse} in Cartesian coordinates.
This has only one $y$-intercept in the upper half-plane, at
the point $i\sqrt{\frac{r^2 + 1}{2}}$. So this ``hyperbola'' has only
one vertex, the other vertex having gone to $+\infty i$, and this
cannot be written as a two-focus hyperbola.
\end{proof}
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,748 |
Jim was born in New York City and raised on Long Island. He started playing acoustic guitar at age 12, inspired by the British Invasion. By high school, his tastes ran to country/rock music such as Buffalo Springfield, Poco, New Riders of the Purple Sage, Linda Ronstadt, etc. Through this music, Jim became exposed to, and fell hopelessly in love with, the sound of the pedal steel guitar which has led him into a wide range of music incuding jazz, pop, country, rock and folk. Jim has recorded 5 CDs as leader and 2 with his Philadelphia-based Texas swing band, Beats Walkin'.
After Jim moved to St. Louis, MO to attend college at Washington University, he somehow stumbled into the late DeWitt Scott ("Scotty"), who ran a tiny, but influential steel guitar studio on the outskirts of town. Scotty became Jim's first steel guitar teacher and sold him his first steel guitar (a 1940's-model Gibson Electraharp, with 8 strings and 6-pedals coming off the side: what a contraption!) While in college, Jim earned some pocket money by playing steel for a country-rock band called "Whitehorse". Among other places, they used to play regularly at Chuck Berry's private club on his home grounds in Wentzville, MO.
Throughout the 1980s and early 1990s, Jim continued to play, on and off, working in a variety of jazz, country and swing bands in Missouri, Florida and Pennsylvania. During that time, he was fortunate to have the opportunity to study steel guitar with many of the world's finest players, including Buddy Emmons, Jeff Newman, Paul Franklin, Maurice Anderson, Bobby Black, Winnie Winston, and others.
In 1994, influenced by hearing a Junior Brown concert while on a visit to Austin, Jim decided to form his own western swing band, Beats Walkin'. That group remains together to this date and performs widely around the Philadelphia, PA area. Their debut album, Western Omelette was released in 1997. In 2002, they released their second CD, a live recording entitled Bop-a-billy Swing!, to critical acclaim.
In recent years, Jim has been much in demand for recording sessions around the Philadelphia area, and as a solo performer at numerous steel guitar shows around the country and in Europe where he continues to thrill audiences with his unique style and arrangements of jazz, pop and classic rock tunes.
In 2000, Jim teamed up with producer, arranger and keyboardist extraordinaire, Dan Kleiman, to produce his first solo CD. The disc, aptly entitled "Finally Here", was released to critical acclaim and showcases the many styles and genres of music that Jim loves to play on steel guitar. With Dan's influence, however, this is much more than just another "steel guitar recording". We think you'll find it to be wonderful listening, no matter what kind of music you enjoy.
On his second CD, "Four to the Bar", Jim continues his tradition of cherry-picking great melodies from his youth, as well as wonderful jazz standards, and uses some of the top Philadelphia-area jazz musicians on this disc. Once again, producer Dan Kleiman has "set the table", creating a wonderful backdrop for Jim's unique steel guitar stylings. If you liked "Finally Here", you're sure to love "Four to the Bar".
In 2006, Jim released "Home, James", his tribute to the British Rock invasion of the 1960s. As with his previous discs, "Home, James" has enjoyed wide popular appeal and was reviewed in the Winter 2007 issue of "Fretboard Journal" where it was written, "In steel guitar circles, Cohen is noted for his bebop chops, but 'Home, James' largely forgoes the fireworks in favor of arrangements that focus on the strong melodies of a string of British Invasion classics." Check out 'Home, James' for great tunes like, 'A Whiter Shade of Pale', 'Bus Stop', 'As Tears Go By', 'Time of the Season' and more.
In 2008, Jim released his first CD devoted to exclusively to country music - a genre he had studiously avoided in the past - only because, as a steel guitarist, it was expected of him to play country music. Entitled "Cohen Goin' Country", Jim explored the classic country repertoire with songs written or popularized by artists including Merle Haggard, George Jones, Hank Williams, Willie Nelson, The Everly Brothers, Skeeter Davis, Linda Ronstadt and others.
Jim's latest release, "Pedal Steel Jazz" delves into the jazz standards repertoire and features some of the very best Philadelphia-area jazz musicians, including Frank Strauss (keys), Chico Huff (electric bass), Steve Beskrone (upright bass), and Grant MacAvoy (drums). In addition, legendary jazz guitarist, Jimmy Bruno, appears on two tunes. To sweeten the pot even further, six of the world's greatest steel guitarists appear on the final track, an original jazz-blues Jim wrote, called "Jimbeaux's Blues"; guests soloists include the late Maurice Anderson (formerly with Bob Wills' Texas Playboys), Lucky Oceans (founding member/steel guitarist, Asleep at the Wheel), Randy Beavers (Leann Womack), Buck Reid (Lyle Lovett), Doug Jernigan (Nashville steel guitar legend), and Chuck Campbell (The Campbell Brothers). This album is sure to change the perception of pedal steel as a complete musical instrument, capable of jazz or any other genre of music, when in the right hands.
Since 2017, Jim has been a founding member of the premier Linda Ronstadt tribute band in the world, Ronstadt Revue, fronted by the remarkable vocalist, Gesenia Erolin. This band brings him full circle to his original country-rock roots and allows for an expanded role for the steel guitar into other genres as well, such as jazz, Mariachi, Cajun, and Great American Songbook standards. More info and the band's schedule can be found at www.RonstadtRevue.com
© 2020 Jim Cohen | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,587 |
Infection of rhesus and cynomolgus macaques with a rapidly fatal SIV (SIVSMM/PBj) isolate from sooty mangabeys.
Integrin alpha4beta7 is downregulated on the surfaces of simian immunodeficiency virus SIVmac239-infected cells.
Classic AIDS in a sooty mangabey after an 18-year natural infection. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,816 |
{"url":"https:\/\/gateoverflow.in\/324653\/andrew-tanenbaum-os-edition-exercise-question-14-page-no-255","text":"A machine has a $32-bit$ address space and an $8-KB$ page. The page table is entirely in hardware, with one $32-bit$ word per entry. When a process starts, the page table is copied to the hardware from memory, at one word every $100\\: nsec.$ If each process runs for $100\\: msec$ (including the time to load the page table), what fraction of the $CPU$ time is devoted to loading the page tables?","date":"2020-01-21 23:32:38","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2132418304681778, \"perplexity\": 1001.3551075196708}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579250606226.29\/warc\/CC-MAIN-20200121222429-20200122011429-00239.warc.gz\"}"} | null | null |
Eric "Fish" Clarke is a Jamaican drummer who has been a member of the Roots Radics and Prince Far I's backing band, The Arabs.
Biography
Born c.1960 in Kingston, Jamaica, Clarke studied at the renowned Alpha Boys School, and began his career as a drummer in the 1970s, playing in Prince Far I's backing band The Arabs and recording with artists such as B. B. Seaton, Bunny Wailer, Keith Hudson, and The Morwells. Clarke toured the UK with Prince Far I and joined Adrian Sherwood's Creation Rebel band, playing on the 1977 album Dub from Creation. He went on to join the Roots Radics, and also recorded solo material such as "Nice In Jamaica" and "Need Someone To Help Me" (the latter produced by Hudson). In the 1980s he also recorded with Sugar Minott, and in the 1990s recorded with Ras Sam Brown.
He is the younger brother of singer Johnny Clarke.
References
Jamaican reggae musicians
Living people
Musicians from Kingston, Jamaica
Year of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,843 |
require "formula"
class Pianobar < Formula
homepage "https://github.com/PromyLOPh/pianobar/"
url "https://github.com/PromyLOPh/pianobar/archive/2013.09.15.tar.gz"
sha256 "4b18582eb794def5bf4e7d5de211d1f6c79295edac344928e09072aa9386796c"
revision 2
head do
url "https://github.com/PromyLOPh/pianobar.git"
depends_on "ffmpeg"
end
depends_on "pkg-config" => :build
depends_on "libao"
depends_on "mad"
depends_on "faad2"
depends_on "gnutls"
depends_on "libgcrypt"
depends_on "json-c"
fails_with :llvm do
build 2334
cause "Reports of this not compiling on Xcode 4"
end
def install
# Discard Homebrew's CFLAGS as Pianobar reportedly doesn't like them
ENV['CFLAGS'] = "-O2 -DNDEBUG " +
# Or it doesn't build at all
"-std=c99 " +
# build if we aren't /usr/local'
"#{ENV.cppflags} #{ENV.ldflags}"
system "make", "PREFIX=#{prefix}"
system "make", "install", "PREFIX=#{prefix}"
prefix.install "contrib"
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 789 |
Don Allen "Donnie" Robinson (born June 8, 1957) is a former Major League Baseball pitcher who played for the Pittsburgh Pirates, San Francisco Giants, California Angels, and Philadelphia Phillies, from 1978 through 1992. Nicknamed "The Caveman", Robinson's career record was 109–106 with a 3.79 ERA.
In 1978, as a 21-year-old rookie, Robinson was 14–6 with a 3.47 ERA. His won-loss percentage was second to NL Cy Young Award winner Gaylord Perry. Robinson was named The Sporting News NL Rookie Pitcher of the Year and was third in overall NL Rookie of the Year. He finished eighth in the Cy Young Award contest.
The next year, he went 8–8 with a 3.87 ERA. He also pitched in six postseason games, winning two and helping the Pirates to the 1979 World Series championship.
Robinson was regarded as one of the best-hitting pitchers during his time, winning three Silver Slugger Awards in 1982, 1989, and 1990. In 1990, Robinson became the first pitcher to hit a pinch-hit home run since 1971 against the San Diego Padres. He hit 13 home runs in his career. He compiled a .231 batting average (146-631) with 47 runs and 69 RBI.
Robinson was the recipient of the Hutch Award in 1984. On April 18, 1987, he gave up Mike Schmidt's 500th career home run. In 1990, pitching for the Giants, he lost the first ever no-hitter at Veteran's Stadium to the Phillies' Terry Mulholland, 6–0.
He made the World Series again in 1989, winning Game 3 of the NLCS against the Chicago Cubs to help the Giants get there. Robinson enjoyed a renaissance with the Giants: from 1987 to 1991, he was 42–33 with a 3.56 ERA, and started a majority of his 170 games with them.
Robinson's parents, Donald and Priscilla Robinson, reside in the city of Kenova, West Virginia, where Robinson served on the city council. Currently, Don Robinson is the pitching coach for the State College of Florida baseball team in Bradenton, Florida.
References
External links
1957 births
Baseball players from Kentucky
Baseball players from West Virginia
California Angels players
Charleston Patriots players
Columbus Clippers players
Gulf Coast Pirates players
Living people
Lynn Pirates (1983) players
Major League Baseball pitchers
Sportspeople from Ashland, Kentucky
People from Kenova, West Virginia
Philadelphia Phillies players
Pittsburgh Pirates players
Prince William Pirates players
San Francisco Giants players
San Jose Giants players
West Virginia city council members
Silver Slugger Award winners | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3 |
{"url":"https:\/\/www.clutchprep.com\/chemistry\/practice-problems\/98439\/enter-a-chemical-formula-for-the-following-molecular-model-see-appendix-ii-a-in--1","text":"# Problem: Enter a chemical formula for the following molecular model. (See Appendix II.A in the textbook for color codes.)\n\n###### FREE Expert Solution\n\n1 S \u00a0+ 2 Hs \u2192 SH2\n\n93% (405 ratings)\n###### Problem Details\n\nEnter a chemical formula for the following molecular model. (See Appendix II.A in the textbook for color codes.)","date":"2021-04-17 05:35:57","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\": 0.8059987425804138, \"perplexity\": 11880.404778595832}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038101485.44\/warc\/CC-MAIN-20210417041730-20210417071730-00034.warc.gz\"}"} | null | null |
Матильда Скотт Хауэлл (; 28 августа 1859 — 20 декабря 1939) — американская лучница, трёхкратная чемпионка летних Олимпийских игр 1904.
На Играх 1904 в Сент-Луисе Хауэлл участвовала во всех женских дисциплинах. Она стала обладательницей трёх золотых медалей, выиграв два индивидуальных и одно командное первенство.
Ссылки
Матильда Хауэлл на sports-reference.com
Персоналии по алфавиту
Родившиеся в 1859 году
Родившиеся 28 августа
Умершие в 1939 году
Умершие 20 декабря
Чемпионы летних Олимпийских игр 1904 года
Стрелки из лука США
Стрелки из лука на летних Олимпийских играх 1904 года
Олимпийские чемпионы от США
Олимпийские чемпионы по стрельбе из лука | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,764 |
{"url":"https:\/\/gmatclub.com\/forum\/in-how-many-ways-can-a-coach-select-a-university-team-from-a-pool-of-260895.html","text":"It is currently 24 Mar 2018, 01:25\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# In how many ways can a coach select a university team from a pool of\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 44421\nIn how many ways can a coach select a university team from a pool of\u00a0[#permalink]\n\n### Show Tags\n\n06 Mar 2018, 07:18\nExpert's post\n2\nThis post was\nBOOKMARKED\n00:00\n\nDifficulty:\n\n25% (medium)\n\nQuestion Stats:\n\n80% (00:39) correct 20% (01:48) wrong based on 54 sessions\n\n### HideShow timer Statistics\n\nIn how many ways can a coach select a university team from a pool of eligible candidates?\n\n(1) The number of eligible candidates is three times greater than the number of slots on the team.\n(2) 60% of the 20 athletes are eligible to play on the four-person university team.\n[Reveal] Spoiler: OA\n\n_________________\nVP\nStatus: Learning\nJoined: 20 Dec 2015\nPosts: 1150\nLocation: India\nConcentration: Operations, Marketing\nGMAT 1: 670 Q48 V36\nGRE 1: 314 Q157 V157\nGPA: 3.4\nWE: Manufacturing and Production (Manufacturing)\nIn how many ways can a coach select a university team from a pool of\u00a0[#permalink]\n\n### Show Tags\n\n06 Mar 2018, 09:29\nFrom 1\nThe number of eligible candidates is three times greater than the number of slots on the team.\nlet slots be n\nLet the number of eligible candidates be x then x=3n\nthen also we can not calculate no of ways\n\nFrom 2\n60% of the 20 athletes are eligible to play on the four-person university team.\n\n.6*20=12 so no of ways to form team 12C4\n_________________\n\nLast edited by arvind910619 on 06 Mar 2018, 09:43, edited 1 time in total.\nPS Forum Moderator\nJoined: 25 Feb 2013\nPosts: 1012\nLocation: India\nGPA: 3.82\nIn how many ways can a coach select a university team from a pool of\u00a0[#permalink]\n\n### Show Tags\n\n06 Mar 2018, 09:34\nBunuel wrote:\nIn how many ways can a coach select a university team from a pool of eligible candidates?\n\n(1) The number of eligible candidates is three times greater than the number of slots on the team.\n(2) 60% of the 20 athletes are eligible to play on the four-person university team.\n\nStatement 1: if number of team slots is $$n$$, then number number of eligible candidates is $$3n$$\n\nHence total selection = $$3n_C_n$$. but we don't know the value of $$n$$. Hence Insufficient\n\nStatement 2: Total eligible candidate $$= 60$$%$$*20=12$$\n\nTotal members in a team $$=4$$. Hence number of combinations possible = $$12_C_4$$. Sufficient\n\nOption B\nIn how many ways can a coach select a university team from a pool of \u00a0 [#permalink] 06 Mar 2018, 09:34\nDisplay posts from previous: Sort by","date":"2018-03-24 08:25:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.2604029178619385, \"perplexity\": 3822.5403868632716}, \"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-2018-13\/segments\/1521257649961.11\/warc\/CC-MAIN-20180324073738-20180324093738-00250.warc.gz\"}"} | null | null |
package com.florianf.gwtcordova.client.plugin.devicemotion;
import jsinterop.annotations.JsProperty;
import jsinterop.annotations.JsType;
@JsType(isNative=true)
public interface Acceleration {
@JsProperty(name = "x")
double getX();
@JsProperty(name = "y")
double getY();
@JsProperty(name = "z")
double getZ();
@JsProperty(name = "timestamp")
String getTimeStamp();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,265 |
{"url":"https:\/\/tobydriscoll.net\/fnc-julia\/localapprox\/interpolation.html","text":"# 5.1. The interpolation problem\u00b6\n\nDefinition 5.1.1 : \u00a0Interpolation problem\n\nGiven $$n+1$$ distinct points $$(t_0,y_0)$$, $$(t_1,y_1),\\ldots,(t_n,y_n)$$, with $$t_0<t_1<\\ldots <t_n$$ called nodes, the interpolation problem is to find a function $$p(x)$$, called the interpolant, such that $$p(t_k)=y_k$$ for $$k=0,\\dots,n$$.\n\nIn this chapter, we use $$t_k$$ for the nodes and $$x$$ to denote the continuous independent variable.\n\nAttention\n\nThe interpolation nodes are numbered from 0 to $$n$$. This is convenient for our mathematical statements, but less so in a language such as Julia in which vector indices start with 1. Our approach is that indices in a computer code have the same meaning as those identically named in the mathematical formulas, and therefore must be incremented by one whenever used in an indexing context.\n\n## Polynomials\u00b6\n\nPolynomials are the obvious first candidate to serve as interpolating functions. They are easy to work with, and in Section 2.1 we saw that a linear system of equations can be used to determine the coefficients of a polynomial that passes through every member of a set of given points in the plane. However, it\u2019s not hard to find examples for which polynomial interpolation leads to unusable results.\n\nDemo 5.1.2\n\nHere are some points that we could consider to be observations of an unknown function on $$[-1,1]$$.\n\nn = 5\nt = range(-1,1,length=n+1)\ny = @. t^2 + t + 0.05*sin(20*t)\n\nscatter(t,y,label=\"data\",leg=:top)\n\n\nThe polynomial interpolant, as computed using fit, looks very sensible. It\u2019s the kind of function you\u2019d take home to meet your parents.\n\np = Polynomials.fit(t,y,n) # interpolating polynomial\nplot!(p,-1,1,label=\"interpolant\")\n\n\nBut now consider a different set of points generated in almost exactly the same way.\n\nn = 18\nt = range(-1,1,length=n+1)\ny = @. t^2 + t + 0.05*sin(20*t)\n\nscatter(t,y,label=\"data\",leg=:top)\n\n\nThe points themselves are unremarkable. But take a look at what happens to the polynomial interpolant.\n\np = Polynomials.fit(t,y,n)\nx = range(-1,1,length=1000) # use a lot of points\nplot!(x,p.(x),label=\"interpolant\")\n\n\nSurely there must be functions that are more intuitively representative of those points!\n\nObservation 5.1.3\n\nInterpolation by a polynomial at equally spaced nodes is ill-conditioned as the degree of the polynomial grows.\n\nIn Chapter 9 we explore the large oscillations in the last figure of Demo 5.1.2; it turns out that one must abandon either equally spaced nodes or $$n\\to\\infty$$ for polynomials. In the rest of this chapter we will keep $$n$$ fairly small and let the nodes be unrestricted.\n\n## Piecewise polynomials\u00b6\n\nIn order to keep polynomial degrees small while interpolating large data sets, we will choose interpolants from the piecewise polynomials. Specifically, the interpolant $$p$$ must be a polynomial on each subinterval $$[t_{k-1},t_k]$$ for $$k=1,\\ldots,n$$.\n\nExample 5.1.4\n\nSome examples of piecewise polynomials for the nodes $$t_0=-2$$, $$t_1=0$$, $$t_2=1$$, and $$t_3=4$$ are $$p_1(x)=x+1$$, $$p_2(x)=\\operatorname{sign}(x)$$, $$p_3(x)=|x-1|^{3}$$, and $$p_4(x)=(\\max\\{0,x\\})^{4}$$. Note that $$p_{1}$$, $$p_{2}$$, and $$p_4$$ would also be piecewise polynomial on the node set $$\\{t_0,t_1,t_3\\}$$, but $$p_3$$ would not.\n\nUsually we designate in advance a maximum degree $$d$$ for each polynomial piece of $$p(x)$$. An important property of the piecewise polynomials of degree $$d$$ is that they form a vector space: that is, any linear combination of piecewise polynomials of degree $$d$$ is another piecewise polynomial of degree $$d$$. If $$p$$ and $$q$$ share the same node set, then the combination is piecewise polynomial on that node set.\n\nDemo 5.1.5\n\nLet us recall the data from Demo 5.1.2.\n\nn = 12\nt = range(-1,1,length=n+1)\ny = @. t^2 + t + 0.5*sin(20*t)\n\nscatter(t,y,label=\"data\",leg=:top)\n\n\nHere is an interpolant that is linear between each consecutive pair of nodes, using plinterp from Section 5.2.\n\np = FNC.plinterp(t,y)\nplot!(p,-1,1,label=\"piecewise linear\")\n\n\nWe may prefer a smoother interpolant that is piecewise cubic, generated using Spline1D from the Dierckx package.\n\np = Spline1D(t,y)\nplot!(x->p(x),-1,1,label=\"piecewise cubic\")\n\n\nWe will consider piecewise linear interpolation in more detail in Section 5.2, and we look at piecewise cubic interpolation in Section 5.3.\n\n## Conditioning of interpolation\u00b6\n\nIn the interpolation problem we are given the values $$(t_k,y_k)$$ for $$k=0,\\ldots,n$$. Let us consider the nodes $$t_k$$ of the problem to be fixed, and let $$a=t_0$$, $$b=t_n$$. Then the data for the interpolation problem consists of a vector $$\\mathbf{y}$$, and the result of the problem is a function on $$[a,b]$$.\n\nLet $$\\mathcal{I}$$ be a prescription for producing the interpolant from a data vector. That is, $$\\mathcal{I}(\\mathbf{y})=p$$, where $$p(t_k)=y_k$$ for all $$k$$. The interpolation methods we will consider are all linear, in the sense that\n\n(5.1.1)$\\cI(\\alpha\\mathbf{y} + \\beta\\mathbf{z}) = \\alpha \\cI(\\mathbf{y}) + \\beta \\cI(\\mathbf{z})$\n\nfor all vectors $$\\mathbf{y},\\mathbf{z}$$ and scalars $$\\alpha,\\beta$$.\n\nLinearity greatly simplifies the analysis of interpolation. To begin with, for any data vector $$\\mathbf{y}$$ we have the standard expression $$\\mathbf{y}=\\sum y_k \\mathbf{e}_k$$, where as always $$\\mathbf{e}_k$$ is a column of an identity matrix.1 Hence by linearity,\n\n(5.1.2)$\\cI( \\mathbf{y} ) = \\cI \\left( \\sum_{k=0}^n y_k \\mathbf{e}_k \\right) = \\sum_{k=0}^n y_k \\cI( \\mathbf{e}_k ).$\n\nThe functions appearing within the sum above have particular significance.\n\nDefinition 5.1.6 : \u00a0Cardinal function\n\nA cardinal function $$\\phi_k$$ for a node set $$t_0,\\ldots,t_n$$ is the function that interpolates the value $$(t_k,1)$$ and $$(t_j,0)$$ for all $$j\\neq k$$.\n\nFor any set of $$n+1$$ nodes, there are $$n+1$$ cardinal functions $$\\phi_0,\\ldots,\\phi_n$$, each singling out a different interpolation node in the set. We finish (5.1.2) by writing\n\n(5.1.3)$\\cI( \\mathbf{y} ) = \\sum_{k=0}^n y_k \\phi_k.$\n\nIn the following result we use the function infinity-norm or max-norm defined by\n\n(5.1.4)$\\| f\\|_{\\infty} = \\max_{x \\in [a,b]} |f(x)|.$\nTheorem 5.1.7 : \u00a0Conditioning of interpolation\n\nSuppose that $$\\cI$$ is a linear interpolation method on nodes $$t_0,\\ldots,t_n$$. Then with respect to the infinity norm, the absolute condition number of $$\\cI$$ satisfies\n\n(5.1.5)$\\max_{0\\le k \\le n}\\, \\bigl\\| \\phi_k \\bigr\\|_\\infty \\le \\kappa(\\mathbf{y}) \\le \\sum_{k=0}^n \\, \\bigl\\| \\phi_k \\bigr\\|_\\infty,$\n\nwhere the $$\\phi_k$$ are cardinal interpolating functions.\n\nProof\n\nSuppose the data vector is perturbed from $$\\mathbf{y}$$ to $$\\mathbf{y}+ \\mathbf{d}$$. Then\n\n(5.1.6)$\\cI(\\mathbf{y} + \\mathbf{d}) - \\cI(\\mathbf{y}) = \\cI(\\mathbf{d}) = \\sum_{k=0}^n d_k \\phi_k.$\n\nHence\n\n$\\frac{\\bigl\\|\\cI(\\mathbf{y} + \\mathbf{d}) - \\cI(\\mathbf{y}) \\bigr\\|_{\\infty}}{\\| \\mathbf{d} \\|_{\\infty}} = \\left\\|\\, \\sum_{k=0}^{n} \\frac{d_k}{\\|\\mathbf{d} \\|_{\\infty}} \\phi_k \\, \\right\\|_{\\infty}.$\n\nThe absolute condition number maximizes this quantity over all $$\\mathbf{d}$$. Suppose $$j$$ is such that $$\\|\\phi_j\\|_\\infty$$ is maximal. Then let $$\\mathbf{d}=\\mathbf{e}_j$$ and the first inequality in (5.1.5) follows. The other inequality follows from the triangle inequality:\n\n$\\left\\| \\, \\sum_{k=0}^{n} \\frac{d_k}{\\|\\mathbf{d} \\|_{\\infty}} \\phi_k \\, \\right\\|_{\\infty} \\le \\sum_{k=0}^{n} \\frac{|d_k|}{\\|\\mathbf{d} \\|_{\\infty}} \\| \\phi_k \\|_\\infty.$\n\nSince $$|d_k|\\le \\|\\mathbf{d}\\|_\\infty$$ for all $$k$$, this finishes (5.1.5).\n\nDemo 5.1.8\n\nIn Demo 5.1.2 and Demo 5.1.5 we saw a big difference between polynomial interpolation and piecewise polynomial interpolation of some arbitrarily chosen data. The same effects can be seen clearly in the cardinal functions, which are closely tied to the condition numbers.\n\nn = 18\nt = range(-1,stop=1,length=n+1)\ny = [zeros(9);1;zeros(n-9)]; # data for 10th cardinal function\n\nscatter(t,y,label=\"data\")\n\n\u03d5 = Spline1D(t,y)\nplot!(x->\u03d5(x),-1,1,label=\"spline\",\nxlabel=L\"x\",ylabel=L\"\\phi(x)\",\ntitle=\"Piecewise cubic cardinal function\")\n\n\nThe piecewise cubic cardinal function is nowhere greater than one in absolute value. This happens to be true for all the cardinal functions, ensuring a good condition number for any interpolation with these functions. But the story for global polynomials is very different.\n\nscatter(t,y,label=\"data\")\n\n\u03d5 = Polynomials.fit(t,y,n)\nplot!(x->\u03d5(x),-1,1,label=\"polynomial\",\nxlabel=L\"x\",ylabel=L\"\\phi(x)\",legend=:top,\ntitle=\"Polynomial cardinal function\")\n\n\nFrom the figure we can see that the condition number for polynomial interpolation on these nodes is at least 500.\n\n## Exercises\u00b6\n\n1. \u2328 Create data by entering\n\nt = -2:4; y = tanh.(t);\n\n\n(a) Use fit to construct and plot the polynomial interpolant of the data, superimposed on a scatter plot of the data.\n\n(b) Use Spline1D to construct and plot a piecewise cubic interpolant of the data, superimposed on a scatter plot of the data.\n\n2. \u2328 The following table gives the life expectancy in the U.S. by year of birth.\n\n1980\n\n1985\n\n1990\n\n1995\n\n2000\n\n2005\n\n2010\n\n73.7\n\n74.7\n\n75.4\n\n75.8\n\n77.0\n\n77.8\n\n78.7\n\n(a) Defining \u201cyear since 1980\u201d as the independent variable, use fit to construct and plot the polynomial interpolant of the data.\n\n(b) Use Spline1D to construct and plot a piecewise cubic interpolant of the data.\n\n(c) Use both methods to estimate the life expectancy for a person born in 2007. Which value is more believable?\n\n3. \u2328 The following two vectors define a flying saucer shape.\n\nx = [ 0,0.51,0.96,1.06,1.29,1.55,1.73,2.13,2.61,\n2.19,1.76,1.56,1.25,1.04,0.58,0 ]\ny = [ 0,0.16,0.16,0.43,0.62,0.48,0.19,0.18,0,\n-0.12,-0.12,-0.29,-0.30,-0.15,-0.16,0 ]\n\n\nWe can regard both $$x$$ and $$y$$ as functions of a parameter $$s$$, with the points being values given at $$s=0,1,\\ldots,15$$.\n\n(a) Use Spline1D once on each coordinate as functions of $$s$$, and make a picture of the flying saucer.\n\n(b) One drawback of the result in part (a) is the noticeable corner at the left side, which corresponds to $$s=0$$ from above and $$s=15$$ from below. There is a periodic variation on cubic spline interpolation that you can invoke by adding the keyword periodic=true to the Spline1D call. Use this to re-plot the flying saucer.\n\n4. \u270d Define\n\n$q(s) = a\\frac{s(s-1)}{2} - b (s-1)(s+1) + c \\frac{s(s+1)}{2}.$\n\n(a) Show that $$q$$ is a polynomial interpolant of the points $$(-1,a)$$, $$(0,b)$$, $$(1,c)$$.\n\n(b) Find a change of variable $$s=Ax+B$$ so that the values $$s=-1,0,1$$ correspond to $$x=x_0-h,x_0,x_0+h$$.\n\n(c) Find a quadratic polynomial interpolant $$\\tilde{q}(x)$$ for the points $$(x_0-h,a)$$, $$(x_0,b)$$, $$(x_0+h,c)$$.\n\n5. \u270d (continuation) Use the result of the previous exercise and Theorem 5.1.7 to derive bounds on the condition number of quadratic polynomial interpolation at the nodes $$x_0-h$$, $$x_0$$, $$x_0+h$$.\n\n1\n\nTo be precise, we are using $$\\mathbf{e}_k$$ to mean column number $$k+1$$ from an $$(n+1)\\times (n+1)$$ identity matrix, since in linear algebra we start indexing at 1.","date":"2022-05-29 00:03:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8278868198394775, \"perplexity\": 414.7019927816032}, \"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-2022-21\/segments\/1652663021405.92\/warc\/CC-MAIN-20220528220030-20220529010030-00353.warc.gz\"}"} | null | null |
\section{Introduction}
The successful construction of Galileon theories \cite{Nicolis:2008in} has changed our perspective on standard effective field theories (EFTs). Even though the resulting Lagrangian contains higher derivative terms, the equations of motion remain second order and hence avoid the Ostrogradski instability. For astrophysical applications these higher order operators generically need to be large. From a standard EFT point of view, this would be disastrous as it would rely on `irrelevant operators'' becoming large and thus going beyond the regime of validity of the theory. For the Galileon theories, however, this is different. The operators are rearranged in a way that higher order derivative operators with second order equations of motion can become relevant whereas operators with even more derivatives per field giving rise to higher order equations of motion are treated as irrelevant. Furthermore, this reorganization of the operators is stable under quantum corrections \cite{Luty:2003vm,Nicolis:2004qq,Heisenberg:2019udf,Heisenberg:2019wjv}. This is the non-renormalization theorem of the Galileon. For this, it is crucial that the Galileon symmetry is only realized up to total derivatives \cite{deRham:2012ew}.
A similar attempt to construct such Galileon-like Lagrangians for arbitrary p-forms immediately met a no-go theorem in four dimensions \cite{Deffayet:2010zh}. This includes massless 1-form. Hence, derivatives acting on the field strength of a Maxwell field do not permit the construction of Lagrangians with second order equations of motion and gauge invariance. However, this obstruction does not apply to the case of massive spin-1 fields. The removal of gauge invariance allows the construction of non-trivial Galileon-like derivative self-interactions of the massive vector field with three propagating degrees of freedom, the generalized Proca theories \cite{Heisenberg:2014rta,Allys:2015sht,Jimenez:2016isa} (see also \cite{Heisenberg:2018vsk}). They represent rich phenomenology in cosmological and astrophysical applications \cite{Tasinato:2014eka,DeFelice:2016yws,DeFelice:2016cri,DeFelice:2016uil,Heisenberg:2016wtr,deFelice:2017paw,Heisenberg:2017hwb,Chagoya:2017fyl}. Concerning Galileon-like Lagrangians for a Kalb-Rammond field, a massless 2-form, it is only possible to construct such derivative self-interactions starting from seven dimensions. Even if the gauge symmetry is removed, the difficulty persists. It was shown in \cite{Heisenberg:2019akx}, that only interactions belonging to the quadratic and quartic Lagrangians can successfully be constructed. Within the quartic Lagrangians, a unique interaction $\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma}_{\;\;\;\;\;\;\sigma}\partial_{\mu}B_{\alpha\rho}\partial_{\nu}B_{\beta\gamma}$ manifests itself as a total derivative, which becomes non-trivial with an overall general function of the 2-form norm. We show that this special interaction corresponds to the interaction $B\wedge F$ of BF theories \cite{Blau:1989dh,Blau:1989bq,Horowitz:1989ng} which gives rise to ``topological mass generation'' \cite{Allen:1990gb,Quevedo:1996uu,Dvali:2005ws}. This ``topological mass generation'' refers to the fact that the single degree of freedom propagated by a massless $2-$form is absorbed by a $1-$form, generating a massive mode for the $1-$form. This mechanism was revisited in \cite{Almeida:2018fwe,Almeida:2019xzt} when looking for particular models of coupled $p-$forms suitable for cosmological applications such as inflation and dark energy. \\
This paper is organized as follows. In section \ref{sec:syspf} we revisit the results from \cite{Almeida:2018fwe,Almeida:2019xzt} about interacting $p-$forms paying special attention to the interaction between the $1$ and $2$-forms through the topological term $B\wedge F$. We show its direct relation to the unique Galileon-like term of the systematic construction of massive $2-$form interactions. This allows us to build a direct duality between BF theories and Galileon-like theories. In section \ref{g_construction} we recall the systematic Galileon-like construction carried out in \cite{Heisenberg:2019akx} where a new kind of interaction for the massive $2-$form is found. Then, in section \ref{decoupling} we discuss the decoupling limit of the system, that gives supporting arguments for the uniqueness of such a topological mass term and absence of the Galileon-like interactions. Finally, in section \ref{capp} we show some simple cosmological applications aiming to emphasize the relevance and to boost the interest for $2-$forms models applied in cosmological setups.
\section{ $B\wedge F$ term and the topological generation of mass }
\label{sec:syspf}
In this section we briefly recall the motivations and results from \cite{Almeida:2018fwe,Almeida:2019xzt}. In that reference, the authors discussed general models of interacting $p-$form Lagrangians subject to the following restrictions: 1) $U(1)$ gauge invariance, 2) up to first order derivatives of the $p-$forms in the Lagrangian, 3) up to cubic terms in the derivatives of the $p-$forms, 4) having a Hamiltonian bounded from below, 5) hyperbolicity of the equations of motion \cite{Fleury:2014qfa}. With the restrictions mentioned before, it was found that, in four dimensions, the more general action involving general interactions between the $p-$forms is:
\begin{equation}
\mL{S} = \int \mr{d}^4 x \sqrt{-g} \left[ \frac{M_{\mr{pl}}^2}{2}R -\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - V_{\rm eff}(\phi) - \frac{1}{4} f_{1}(\phi) F^2 - \frac{1}{12} f_{2}(\phi) H^2 - \frac{1}{4} g_{1}(\phi) F \tilde{F} - \frac{1}{2} m B\tilde{F} \right]\,,
\label{eq:LT}
\end{equation}
where the $1-$ and $2-$forms are $A_{\mu}$ and $B_{\mu \nu}$ respectively, and their field strengths are
\begin{equation}
F_{\mu \nu} \equiv 2 \partial_{[\mu } A_{\nu]}\,, \qquad
H_{\mu \nu \alpha} \equiv 3 \partial_{[\mu} B_{\nu \alpha]} \,.
\end{equation}
Additionally, we used the shorthand notation
\begin{equation}
F^2 \equiv F_{\mu \nu}F^{ \mu \nu}\,,\quad
H^2 \equiv H_{\mu \nu \alpha}H^{\mu \nu \alpha}\,,\quad F \tilde{F} \equiv F_{\,\mu \nu}\tilde{F}^{\mu \nu} \, ,\quad
B\wedge F = B \tilde{F} \equiv B_{\,\mu \nu }\tilde{F}^{\mu \nu}\,.
\end{equation}
In the previous action, the functions $f_{i}(\phi)$ are arbitrary functions of the fields, only restricted to be positive definite $f_i >0$, and $m$ is constant in order to preserve the gauge invariance. The coupling function $g_1(\phi)$ is not restricted by any of the conditions mentioned before. The effective potential $V_{\rm eff}$ is induced by the coupling of the $3-$form field and the scalar field \cite{Kaloper:2008qs,Kaloper:2008fb, Bielleman:2015ina,Ibanez:2015fcv,Valenzuela:2016yny,Farakos:2017jme}. The term $B \wedge{F}$ \cite{Blau:1989dh,Blau:1989bq,Horowitz:1989ng} is responsible for the mechanism of topological generation of mass described in detail in \cite{Allen:1990gb,Quevedo:1996uu,Dvali:2005ws}.
\subsection{Relating BF with Galileon-like interactions}\label{sec_relBFgal}
Let's focus now in a model involving only $1$ and $2-$forms without introducing an extra scalar field degree of freedom, and let's restrict our analysis to flat space (some discussion about curved space and non minimal coupling with gravity can be found in \cite{Heisenberg:2019akx}. For complementary discussion on this topic see also \cite{Yoshida:2019dxu,Takahashi:2019vax,Koivisto:2009sd}). As found in \cite{Almeida:2018fwe}, the only coupling term involving $1$ and $2-$forms consistent with the five restrictions mentioned before, particularly with the condition of being gauge invariant for $A_{\mu}$ and $B_{\mu \nu}$ is the term $B_{\,\mu \nu }\tilde{F}^{\mu \nu}$. With this, the model we will consider is written with the Lagrangian
\begin{equation}
\mL{S}_{AB} = -\int \mr{d}^4 x \left[ \frac{1}{4} F^2 + \frac{1}{12} H^2 + \frac{1}{2} m B\tilde{F} \right]\,,
\label{ABm}
\end{equation}
where $m$ is a constant in order to preserve gauge invariance. From this, we derive the equation of motion for the $2-$form
\begin{equation}
\label{A2eq}
\partial^{\mu} H_{\mu\nu \alpha} -m \tilde{F}_{\nu\alpha} =0 \,,
\end{equation}
which can be formally solved for $\tilde{F}^{\mu\nu}$ and $F^{\mu\nu}$ obtaining
\begin{equation}\label{solft}
\tilde{F}_{\mu\nu} = \frac{1}{m } \partial^{\alpha} H_{\alpha\mu \nu}\,, \quad F^{\mu\nu} = -\frac{1}{6 m }\partial^{[\mu} \tilde{H}^{\nu ]}\,.
\end{equation}
Using this formal solution we can rewrite the Lagrangian \eqref{ABm} only in terms of the $2-$form.
The structure of the interacting term $B\wedge F$ is of particular interest. Using \eqref{solft} we can write the term $B\wedge F$ as follows
\begin{equation}
B\wedge F = B^{\mu\nu} \tilde{F}_{\mu\nu} = \frac{1}{m } B_{\mu\nu} \partial^{\alpha}\left( H_{\alpha\mu \nu} \right) = \frac{ 1 }{ m } B^{\mu\nu} \left[ \partial^{\alpha} \partial_{\alpha} B_{\mu\nu} + \partial^{\alpha} \partial_{\mu} B_{\nu \alpha} + \partial_{\nu } \partial^{\alpha}B_{ \alpha \mu} \right] \,,
\end{equation}
where we exchanged the order of the derivatives in the last term. Now, integrating by parts the terms with second derivatives of the $2-$form we obtain
\begin{equation}
B\wedge F = - \frac{ 1 }{m } \left[ \partial^{\alpha} B^{\mu\nu} \partial_{\alpha} B_{\mu\nu} + \partial^{\alpha} B^{\mu\nu} \partial_{\mu} B_{\nu \alpha} + \partial_{\nu} B^{\mu\nu} \partial^{\alpha} B_{\alpha \mu} \right]\,,
\end{equation}
which can be reordered in the form
\begin{equation}
B\wedge F = - \frac{ 1 }{m } \left[ \partial^{\alpha} B^{\mu\nu} \partial_{\alpha} B_{\mu\nu} + \partial^{\alpha} B^{\mu\nu} \partial_{\mu} B_{\nu \alpha} - \partial^{\alpha} B^{\mu\nu} \partial_{\nu} B_{\mu \alpha} \right] - \frac{ 1 }{m } \left[ \partial^{\alpha} B^{\nu\mu} \partial_{\mu} B_{\nu \alpha} + \partial_{\nu} B^{\nu\mu} \partial^{\alpha} B_{ \mu \alpha } \right]\,.
\end{equation}
We can identify the term inside the first brackets as $H^2/3$, so, it is absorbed in the kinetic Maxwell like term for the $2-$form. On the other hand, the term in the second brackets can be recognized as the novel interaction term
\begin{equation}
{\cal L}_4^{T} = \partial^{\alpha} B^{\nu\mu} \partial_{\mu} B_{\nu \alpha} + \partial_{\nu} B^{\nu\mu} \partial^{\alpha} B_{ \mu \alpha }
\end{equation}
found in the systematic construction carried out in reference \cite{Heisenberg:2019akx} (see \cref{modKinL4} in next section). To summarize, we can write the topological term $ B\wedge F$ as follows
\begin{equation}
B\wedge F = - \frac{ 1 }{m } {\cal L}_4^T - \frac{ 1 }{3 m } H^{\alpha \mu \nu} H_{\alpha \mu\nu}\,.
\end{equation}
As warned in \cite{Heisenberg:2019akx}, the term ${\cal L}_4^{T}$ is a total derivative as can be checked after integration by parts twice. Nevertheless, ${\cal L}_4^{T}$ is not a total derivative anymore when it is multiplied by an arbitrary function $f_4(B^2)$ (see \cref{modKinL4} below) where $B^2=B_{\mu\nu}B^{\mu\nu}$.
The identification of ${\cal L}_4^{T}$ as part of the topological term $B\wedge F$ provides a link between the construction of Galileon-like derivative self interactions developed in \cite{Heisenberg:2019akx} and briefly recalled in \ref{g_construction}, and BF theories (specially the approach followed in \cite{Almeida:2018fwe, Almeida:2019xzt}).
\subsection{ Including the duals in the systematic construction }
\label{topological}
If we allow the possibility of including parity breaking terms, we could also consider the inclusion of the duals of the $2-$form and its field strength, this is $*B$ and $*H$, in the systematic construction. We follow here closely the discussion of the example of ``compact QED'' presented in \cite{Quevedo:1996uu}. Beside the topological term $B\wedge F$ that we discussed before, the possible non vanishing contributions that can be constructed with those objects are
\begin{equation}\label{LB}
\mL{L}_{B}= \int \left[ a_1 H\wedge *H + a_2 B \wedge *B + a_3 B\wedge B \right] = \int \mr{d}^4 x \left[ a_1 H_{\mu\nu\sigma} H^{\mu\nu\sigma} + a_2 B_{\mu\nu } B^{\mu\nu} + a_3 B_{\mu\nu } \tilde{B}^{\mu\nu} \right]\,,
\end{equation}
where $\tilde{B}^{\mu\nu} = \epsilon^{ \mu\nu\sigma\rho} B_{\sigma\rho }/2 $ and we consider constant coefficients $a_1, a_2, a_3$.
The equations of motion derived from this action are
\begin{equation}\label{eom2form}
6 a_1 \partial_{\mu} H^{\mu\nu\sigma} - 2 a_2 B^{ \nu \sigma} - 2 a_3 \frac{ \epsilon^{ \nu\sigma \mu \rho} }{2 } B_{ \mu \rho} = 0\,.
\end{equation}
We can derive the previous expression with respect to $ \partial_{\nu} $ and obtain
\begin{equation}
a_2 \partial_{\nu} B^{ \nu \sigma} + a_3 \frac{ \epsilon^{ \nu\sigma \mu \rho} }{2 } \partial_{\nu} B_{ \mu \rho} = 0\,,
\end{equation}
which can be arranged in the form
\begin{equation}\label{BTH}
a_2 \partial_{\nu} B^{ \nu \sigma} - a_3 \tilde{H}^{\sigma} = 0,\quad \mbox{with} \quad \tilde{H}^{\sigma} \equiv \frac{ \epsilon^{ \sigma \nu \mu \rho} }{6 } \partial_{[\nu} B_{ \mu \rho]}= \frac{ \epsilon^{ \sigma \nu \mu \rho} }{6 } H_{\nu \mu \rho}\,.
\end{equation}
We further apply the exterior derivative ${\mr{d} }\wedge = \epsilon_{\alpha\nu\sigma\rho} \partial^{\alpha}$ to the equation of motion \cref{eom2form} and use \cref{BTH} to obtain
\begin{equation}
12 a_1\left( \partial_{\beta} \partial^{\mu} \tilde{H}_{\mu} - \partial_{\mu} \partial^{\mu} \tilde{H}_{\beta} \right) + 4 a_2 \tilde{H}_{\beta} + 4 \frac{a_3^2}{a_2} \tilde{H}_{\beta} = 0\,.
\end{equation}
The term $ \partial^{\mu} \tilde{H}_{\mu}$ vanishes due to the Bianchi identity and we are left with
\begin{equation}\label{masstH}
\left( \partial_{\mu} \partial^{\mu} + m^2 \right) \tilde{H}_{\beta} = 0\,, \quad \mbox{with} \quad m^2 = - \frac{ a_2}{3 a_1} \left( \frac{a_3^2}{ a_2^2} + 1\right)\,,
\end{equation}
which we recognize as the equation of motion for a massive vector field $\tilde{H}_{\beta}$. In this sense, the particle content of the Lagrangian \cref{LB} is the same as the content of a model with a massive vector field $*H$. Despite the fact that this vector is defined as an axial vector, and the topological mass term $ B\wedge B$ seems to be a source of parity breaking, the theory remains parity conserving. Then, the presence of the term $B\wedge B$ provides another source of a mass term into the theory just like the topological $B\wedge F$ term does. This situation would change if we consider general non linear functions $F(U, V, W)$ where $U=H_{\mu\nu\sigma} H^{\mu\nu\sigma}$, $V= B_{\mu\nu } B^{\mu\nu} $ and $W=B_{\mu\nu } \tilde{B}^{\mu\nu}$. In such case, the equation of motion and the Bianchi identity leads to
\begin{align}\label{eq:nonlineartB}
& 3 \left[ \partial_{\beta} \partial^{\gamma} \left( F_{U} \tilde{H}_{\gamma} \right)- \partial^{\gamma} \partial_{\gamma} \left( F_{U} \tilde{H}_{\beta} \right) \right]
+ \left( F_{V} + \frac{F_W^2}{F_{V}} \right) \tilde{H}_{\beta} \nonumber \\
& - \left( \partial^{\gamma}F_{V} + \frac{F_W}{F_{V}} \partial^{\gamma}F_{W} \right) \tilde{B}_{\gamma \beta} + \left( \partial^{\gamma}F_{W} - \frac{F_W}{F_{V}} \partial^{\gamma}F_{V} \right) {B}_{\gamma \beta} =0\,,
\end{align}
where $F_{U} = \partial_{U} F$, etc. As can be checked, the previous equation reduces to \cref{masstH} when the function $F$ is linear on their arguments, and then, leads to a parity conserving system. For non linear $F$ we have a parity breaking situation as evidenced from the presence of the third and fourth terms in \cref{eq:nonlineartB}. Nevertheless, as we will not consider parity breaking models, in the following we will neglect non linear Lagrangians in the argument $W$. \\
In the next section, we will revisit the Galileon-like construction for massive $2-$forms performed in \cite{Heisenberg:2019akx}, to which we had linked the coupled 1 and $2-$form discussed in the subsection \ref{sec_relBFgal}.
\section{Systematical construction}
\label{g_construction}
The fact that one cannot construct derivative Galileon interactions beyond the trivial cases can be shown in two complementary and independent ways. The first one is using the systematical construction in terms of the Levi-Civita tensor and the second one is the decoupling limit analysis. This was already studied in detail in \cite{Heisenberg:2019akx}. We summarize the argument based on the systematical construction in this section and give more detail on the decoupling limit analysis in the next section.
In \cite{Heisenberg:2019akx} the antisymmetry properties of the Levi-Civita tensor was used to investigate the possible construction of Galileon-type interactions for the massive 2-form with 3 propagating degrees of freedom. It was found, that besides the $\mathcal{L}_2$ and $\mathcal{L}_4$ Lagrangians, it is not possible to construct derivative self-interactions for the massive 2-form in four dimensions. The quadratic Lagrangian is simply a combination of potential-like and gauge-invariant quantities
\begin{equation} \label{LagL2}
\mathcal{L}_{2} = f_{2}(B_{\mu\nu}, H_{\mu\nu\rho}, \bar{H}_{\mu})\,.
\end{equation}
The higher order Lagrangians are constructed systematically in terms of powers of the fundamental object $\partial_{\alpha}B_{\mu\nu}$ together with two Levi-Civita tensors $ f(B^2)\epsilon\epsilon(\partial B)^{m}B^{n}$. Since $\partial B$ carries 3 but the Levi-Civita tensor 4 indices, the Lagrangian $\mathcal{L}_{3}$ cannot be constructed
\begin{equation} \label{LagL3}
\mathcal{L}_{3} = 0\,.
\end{equation}
This problem of even versus an odd number of indices is avoided in the quartic Lagrangian $\mathcal{L}_{4}$. Ignoring contributions that belong to $\mathcal{L}_2$ (like the kinetic term), the only non-trivial term that we can construct at this order is
\begin{align}\label{LagL4}
\mathcal{L}_{4}^T&=\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma}_{\;\;\;\;\;\;\sigma}\partial_{\mu}B_{\alpha\rho}\partial_{\nu}B_{\beta\gamma} \, , \nonumber\\
&=\partial_\mu B^{\mu\nu}\partial_\alpha B_\nu{}^{\alpha}+\partial_\nu B_{\mu\alpha}\partial^\alpha B^{\mu\nu}\,.
\end{align}
Without multiplying this contraction with an overall function $f_4(B^2)$, it corresponds to a total derivative and can be directly related to $B\wedge F$, as shown in the section \ref{sec_relBFgal}.
Hence, at this order this is the only non-trivial genuinely new term, that cannot be absorbed into $\mathcal{L}_2$
\begin{equation}\label{modKinL4}
\mathcal{L}_{4}^{0B}=f_4(B^2)\Big(\partial_\mu B^{\mu\nu}\partial_\alpha B_\nu{}^{\alpha}+\partial_\nu B_{\mu\alpha}\partial^\alpha B^{\mu\nu} \Big)\,.
\end{equation}
This term is quite special. It looks like a modified kinetic term without gauge invariance but as we saw above can be directly related to the topological mass term. We could also construct contractions higher in $n$. For instance,
\begin{align}
\mathcal{L}_{4}^{(1B)}&=
\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}\partial_{\mu}B_{\alpha\rho}\partial_{\nu}B_{\beta\gamma} B_{\sigma\delta}\,, \\
\mathcal{L}_{4}^{(2B)}&=
\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma\delta}\partial_{\mu}B_{\alpha\rho}\partial_{\nu}B_{\beta\gamma} B_{\sigma\lambda}B^\lambda{}_\delta \, ,
\end{align}
multiplied by a general function of the 2-form norm, respectively. Going to higher order contributions in $(\partial B)$ is not possible beyond this order since the two Levi-Civita tensors
contain eight indices and $(\partial B)^3$ would require 9 and $(\partial B)^4$ 12 indices and so on. Therefore we have
\begin{equation}
\mathcal{L}_{i}=0 \qquad \text{for} \qquad i\geqq5 \, .
\end{equation}
Hence, the systematical construction stops at $\mathcal{ L}_{4}$ and we cannot construct Galileon interactions for the massive 2-form beyond $\mL{L}_{2}$ and $\mL{L}_{4}$.
Additional support for this difficulty of constructing Galileon-type derivative interactions also comes from the decoupling limit analysis, which we will discuss in the next section.
\section{Decoupling limit analysis}
\label{decoupling}
The decoupling limit analysis already reveals important conditions about the allowed interactions once the gauge symmetry is restored using the Stueckelberg trick. For this purpose, we perform the following change of variables
\begin{equation}\label{TrafoStuck}
B_{\mu\nu} \to B_{\mu\nu}+\frac{1}{m}\partial_{[\mu}A_{\nu]}\,,
\end{equation}
where the massless spin-1 field $A_\mu$ represents the Stueckelberg field. The original massive 2-form propagates 3 propagating degrees of freedom. After reintroducing the Stueckelberg field, the massive 2-form decomposes into a massless 1-form and a massless 2-form, still propagating $1+2$ degrees of freedom. In the decoupling limit, where we sent the mass of the 2-form to zero, we obtain two decoupled massless 1- and 2-form. In order to illustrate that, we take the standard Lagrangian of a massive 2-form
\begin{equation}
\mathcal{L}=-\frac1{12} H_{\mu\nu\rho} H^{\mu\nu\rho} -\frac{m^2}4 B_{\mu\nu}B^{\mu\nu} \,,
\end{equation}
where $H$ is the field strength of the massive 2-form, and perform the transformation in \eqref{TrafoStuck} to it. The kinetic term is immun to it but the mass term changes into
\begin{equation}
\mathcal{L}=-\frac1{12} H_{\mu\nu\rho} H^{\mu\nu\rho} -\frac{m^2}4 \left(B_{\mu\nu}+\frac{1}{m}\partial_{[\mu}A_{\nu]}\right)^2 \,.
\end{equation}
In the limit $m\to0$, we obtain a massless 1-form and a massless 2-form, decoupled from each other and each of them is invariant under gauge symmetries
\begin{equation}
\mathcal{L}=-\frac1{12} H_{\mu\nu\rho} H^{\mu\nu\rho} -\frac14 F_{\mu\nu}F^{\mu\nu} \,,
\end{equation}
where $F$ is the field strength of the massless 1-form. If there are genuinely new Galileon interactions for the original massive 2-form, we would see their presence in the decoupling limit. We would need to construct Galileon interactions for the massless 2-form and massless 1-form sectors in the decoupling limit. Since the 1-form and the 2-form have gauge symmetries, they can manifest themselves only through the gauge invariant field strengths $F$ and $H$. For the massless 1-form we would be after interactions of the form
\begin{equation}\label{Galmassless1form}
\mathcal{L}=\epsilon^{\mu_1\mu_2\cdots}\epsilon^{\nu_1\nu_2\cdots}F_{\mu_1\mu_2} F_{\nu_1\nu_2}\left( \partial_{\mu_k}F_{\nu_l\nu_{l+1}}\cdots \right)\left( \partial_{\nu_j}F_{\mu_m\mu_{m+1}}\cdots\right) \,.
\end{equation}
Since $F$ comes in with 2 indices, we can only start constructing such terms for dimensions $D\ge 5$. However, they do not correspond to any genuinely new interaction, since they contribute in the form of a total derivative. Hence, the interactions in \cref{Galmassless1form} can be rewritten as
\begin{equation}
\mathcal{L}=\frac12\partial_{\nu_j}\left\{\epsilon^{\mu_1\mu_2\cdots}\epsilon^{\nu_1\nu_2\cdots}F_{\mu_1\mu_2} F_{\nu_1\nu_2}F_{\nu_m\nu_{m+1}}\left( \partial_{\mu_k}F_{\nu_l\nu_{l+1}}\cdots \right)\left( \partial_{\nu_i}F_{\mu_n\mu_{n+1}}\cdots\right) \right\} \,.
\end{equation}
Therefore, this constitutes a no-go theorem for a massless 1-form to have Galileon-type derivative self-interactions in any dimensions \cite{Deffayet:2013tca}. Hence, there is no way to construct Galileon interactions for the massless 1-form in our decoupling limit. A similar no-go also exists for the massless 2-form, in four dimensions. Similarly, this time we are after following type of interactions for the massless 2-form
\begin{equation}\label{Galmassless2form}
\mathcal{L}=\epsilon^{\mu_1\mu_2\mu_3\cdots}\epsilon^{\nu_1\nu_2\nu_3\cdots}H_{\mu_1\mu_2\mu_3} H_{\nu_1\nu_2\nu_3}\left( \partial_{\mu_k}H_{\nu_{l-1}\nu_l\nu_{l+1}}\cdots \right)\left( \partial_{\nu_j}H_{\mu_{m-1}\mu_m\mu_{m+1}}\cdots\right) \,.
\end{equation}
Since $H$ comes in with 3 indices, such construction of interactions is only possible for $D\ge7$ \cite{Deffayet:2010zh}. For instance, in seven dimensions we can construct $\mathcal{L}^{(D=7)}=\epsilon^{\mu\nu\rho\sigma\tau\phi\chi}\epsilon^{\alpha\beta\gamma\delta\epsilon\xi\eta}H_{\mu\nu\rho}H_{\alpha\beta\gamma}\partial_\sigma H_{\delta\epsilon\xi}\partial_\eta H_{\tau\phi\chi} $. This time they do correspond to genuinely new interactions and are not total derivatives, in difference to the case for the massless 1-form. However, for our case in $D=4$ dimensions this means that we cannot construct Galileon-type of interactions for the massless 2-form either. Hence, in four dimensions we can neither have Galileon interactions $f(F^2)\epsilon\epsilon(\partial F)^mF^n$ for the massless 1-form nor $f(H^2)\epsilon\epsilon(\partial H)^mH^n$ for the massless 2-form.
Another way how we could construct interactions are via mixings between the massless 2-form and the massless spin-1 field. Thus, one could construct terms of the form $H^mF^n$. Since $H$ carries three but $F$ two indices, the first contribution starts at $m=2$ and $n=2$. This type of construction will nevertheless not generate Galileon interactions but only contribute to the quadratic Lagrangian $\mathcal{L}_2=f_2(B_{\mu\nu}, H_{\mu\nu\rho}, \bar{H}_{\mu})$ in the original formulation in terms of the massive 2-form. The attempt to construct derivative mixings like $\epsilon\epsilon H^m(\partial F)^n$ or $\epsilon\epsilon F^m(\partial H)^n$ face the same difficulty in four dimensions as the Galileon construction for the pure sectors, since one can construct them only starting from $D\ge6$ dimensions. Thus, it is not possible to construct Galileon-type of derivative self-interactions in the decoupling limit while keeping the gauge invariance for the 2- and 1-forms in four dimensions.
\section{First cosmological application}
\label{capp}
In this section we highlight some features of the $2-$form model and discuss the background evolution in some particular simple models interesting for cosmology. Previous interesting studies of $p-$forms with and without non-minimal couplings to gravity in the context of inflation and dark energy scenarios can be found in \cite{Germani:2009iq,Koivisto:2009sd,Koivisto:2009fb,Koivisto:2012xm,Thorsrud:2012mu,Mulryne:2012ax,Ohashi:2013qba, Ohashi:2013mka,Ito:2015sxj,Kumar:2016tdn,Farakos:2017jme,Obata:2018ilf,Almeida:2019xzt,Almeida:2019iqp,Guarnizo:2019mwf,Takahashi:2019vax}.
\subsection{Gauge invariant}
The simplest example that we can consider is a theory based on a non-linear kinetic term. Since the kinetic term only depends on the gauge invariant field strength $H_{\alpha\beta\mu}$, the gauge symmetry will be intact. Such non-linear kinetic terms are very interesting since they are the simplest extension that one can consider and they also provide a K-mouflage screening mechanism together with a promising quantum behaviour. We consider in this subsection a Lagrangian of a $2-$form minimally coupled to gravity: $\mathcal{L}_2=\sqrt{-g}\left[ \frac{M_{\rm pl}^2}{2}R + f_2(X)\right]$, where $X=-\frac1{12}H^2$. We use the FLRW Ansatz
\begin{equation}
\mr{d} s^2=-N(t)^2\mr{d} t^2+a(t)^2\mr{d}\boldsymbol{x}^2\, ,
\end{equation}
for the metric and for the 2-form the background field configuration $B^{ij}=\frac13C\epsilon^{ijk}x_k$, where the remaining components all vanish. The associated energy density and pressure are given by
\begin{equation}
\rho_B=-f_2 \qquad \text{and} \qquad p_B=f_2-2Xf_{2,X} \,.
\end{equation}
The Einstein's field equations are then simply given by
\begin{equation}
3H^2=\rho_B \qquad \text{and} \qquad 2\dot{H}+3H^2=-p_B \,.
\end{equation}
The slow roll parameter $\epsilon=\frac32(w+1)=-\frac{\dot{H}}{H^2}$ indicates a regime with quasi-de Sitter solution when $\log f_{2,\log X}\ll1$. The covariant expression for the stress energy tensor $T_{\alpha\beta}=f_2g_{\alpha\beta}+\frac12H_{\alpha\rho\sigma}H_{\beta}{}^{\rho\sigma}f_{2,X}$ helps us to obtain quickly the propagation speed of scalar perturbations after introducing them in the metric and the 2-form. It is simply given by
\begin{equation}
c_s^2=1+2X\frac{f_{2,XX}}{f_{2,X}}\,.
\end{equation}
Similarly to the scalar counterpart of k-essence theories, the quasi-de Sitter regime would suffer from gradient instabilities.
\subsection{Non-minimal coupling}
In \cite{Heisenberg:2019akx} non-minimal couplings for the massive 2-form were investigated and it was shown that a unique coupling via the double dual Riemann tensor
arises
\begin{equation}\label{2formLBB}
\mathcal{L}^{\rm non-min}=\sqrt{-g}L^{\mu\nu\alpha\beta}B_{\mu\nu}B_{\alpha\beta}\,,
\end{equation}
where $L^{\mu\nu\alpha\beta}$ represents the double dual Riemann tensor $L^{\mu\nu\alpha\beta}=\frac14\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\kappa\delta}R_{\rho\sigma\kappa\delta}$.
Additional support for this unique non-minimal coupling of the 2-form comes from the decoupling limit. After introducing the Stueckelberg field in \eqref{TrafoStuck} we have a massless 2-form and a massless spin-1 field, and the potential non-minimal couplings have to be valid couplings for these separate sectors. Since the massless spin-1 field has a unique non-minimally coupling to gravity via $L^{\alpha\beta\gamma\delta}F_{\alpha\beta}F_{\gamma\delta}$, this translates back to having \eqref{2formLBB} as the unique possible non-minimal coupling for the original massive 2-form.
Let us consider the following action
\begin{equation}\label{eq:act1}
\mathcal{S}=\int \mr{d}^4x \sqrt{-g}\left( \frac{M^2_{\rm{pl}}}{2}R-\frac1{12} H_{\mu\nu\rho} H^{\mu\nu\rho}-V[B_{\mu\nu}B^{\mu\nu}]+\gamma L^{\mu\nu\alpha\beta}B_{\mu\nu}B_{\alpha\beta} \right)\,,
\end{equation}
where $V[B_{\mu\nu}B^{\mu\nu}]$ is a general potential function for the $2-$form field. It is important to emphasize that the previous Lagrangian includes the topological $B\wedge F$ term (or, equivalently the term ${\cal L}_4$), through the mechanism of topological generation of mass \cite{Allen:1990gb,Quevedo:1996uu,Dvali:2005ws} (see also \cite{Almeida:2018fwe, Almeida:2019xzt} for the discussion in curved background) reviewed in section \ref{sec:syspf}. \\
On top of a FLRW background the massive 2-form shall admit the following background field configuration
\begin{eqnarray}
B_{\mu\nu} =
\begin{pmatrix}
0&0&0&0 \\
0&0 &a^2 b&-a^2 b \\
0&-a^2 b&0 &a^2 b \\
0&a^2 b&-a^2 b&0
\end{pmatrix}\,,
\end{eqnarray}
where the splitting of variables $a^2b=a(t)^2 b(t)$ is solely chosen for convenience.\\
The action in \cref{eq:act1} can then be brought up to total derivatives into the following symmetry reduced form
\begin{equation}\label{eq:act2}
\mathcal{S}=\int \mr{d}^4x \,\frac{3 a}{N} \left(
\frac{a^2b'^2}{2}
+2\Gamma\,a a' b b'
+\left(2\Gamma b^2-M_{\rm{pl}}^2\right)a'^2
\right)
- a^3 N V[6b^2]\,,
\end{equation}
where primes denote derivatives with respect to the time coordinate $t$ and we redefine the coupling constant $\gamma$ through $\Gamma \equiv 1 +4 \gamma$.
Using the invariance of the reduced action under reparametrization of $t$, one can absorb the lapse function and rewrite the action and the equations of motion by defining the proper time $\tau$ as $\mr{d}\tau=N \mr{d}t$. Introducing the notation $\dot{a}=\mr{d}a/\mr{d}\tau$ and $\dot{b}=\mr{d}b/\mr{d}\tau$ the resulting background equations of motion are given by:
\begin{equation}
\begin{aligned}\label{eq:eombH}
\mathcal{E}_b=\; &\ddot{b}+3 H \dot{b}+2 b \left\{2 V'[6 b^2]+\Gamma
\left(\dot{H}+H^2\right)\right\}=0 \,,\\
\mathcal{E}_N=\; &
H^2 \left(12 \Gamma b^2-6 M_{\rm{pl}}^2\right)+2
V[6 b^2]+3 \dot{b}^2+12 \Gamma H b \dot{b} =0\,,\\
\mathcal{E}_a=\; &
\dot{H} \left(8 \Gamma b^2-4 M_{\rm{pl}}^2\right)
+H^2 \left(12 \Gamma b^2-6 M_{\rm{pl}}^2\right)
+2 V[6 b^2]
+4 \Gamma b\ddot{b}
+(4 \Gamma -3) \dot{b}^2
+16 \Gamma H b \dot{b}=0\, .
\end{aligned}
\end{equation}
These can be brought into an autonomous form
\begin{equation}\label{auto}
\dot{H}=f_1 \qquad \text{and} \qquad \dot{b}=f_2 \; ,
\end{equation}
where we have defined:
\begin{equation}\label{autodef}
\begin{aligned}
f_1=\; &\frac{4 \Gamma\, H\,b \,f_2+(4 \Gamma -3) \,f_2^2-4 \Gamma\, b^2
\left(4 V'+(2 \Gamma -3)\, H^2\right)+2 V-6
H^2 M_{\rm{pl}}^2}{4 \left((2\Gamma-2)\Gamma\, b^2+M_{\rm{pl}}^2\right)} \, ,\\
f_2=\; &-2 \Gamma\, H\, b \pm\sqrt{2 H^2 \left((2\Gamma-2)\Gamma b^2+M_{\rm{pl}}^2\right)-\frac{2}{3} V}\,.
\end{aligned}
\end{equation}
The critical points correspond to $\dot{H}=0$ and $\dot{b}=0$. For the stability analysis of the fixed points, we can consider small perturbations $\delta H$ and $\delta b$.
Defining the perturbation vector $v=\{\delta H, \delta b\}$ we can write the perturbations equations as $v'=\mathcal{M}v$, where the matrix $\mathcal{M}$ is given by
\begin{equation}\label{eq:Jacob}
\mathcal{M} =
\begin{pmatrix}
\frac{\partial f_1}{\partial H}&\frac{\partial f_1}{\partial b} \\[2mm]
\frac{\partial f_2}{\partial H}&\frac{\partial f_2}{\partial b}
\end{pmatrix}\, ,
\end{equation}
evaluated at the critical point. In order for the critical point to be an attractor, all the eigenvalues $\lambda_i$ of the matrix $\mathcal{M}$ have to be negative, since the
perturbations in the environment of the critical point evolve as $e^{\lambda_i}t$.
One can find non trivial critical points of the autonomous system \eqref{auto} by considering for example an interacting potential of the form $V[x]=-g_b\,x^2$.\footnote{Such that $V[6b^2]=-36g_b\,b^4$ and $V'[6 b^2]=-12g_b\,b^2$.} In this case, the system admits the five distinct critical points:
\begin{equation}\label{critP}
\{b_c,H_c\}=\{0,0\},\,\left\{\pm\frac{ \sqrt{\tfrac{2}{3}}\,M_{\rm{pl}}}{\sqrt{\Gamma}},\,\pm \frac{4\sqrt{g_b}\,M_{\rm{pl}}}{\Gamma}\right\},\,\left\{\mp\frac{ \sqrt{\tfrac{2}{3}}\,M_{\rm{pl}}}{\sqrt{\Gamma}},\,\pm \frac{4\sqrt{g_b}\,M_{\rm{pl}}}{\Gamma}\right\}\,,
\end{equation}
which are well defined as long as $\Gamma > 0$. Choosing for concreteness a unit coupling $g_b=1$ together with $\Gamma=2$ ($\gamma=\tfrac{1}{4}$) and using units in which $M_{\rm{pl}}=1$ one can analyze a concrete phase portrait of the dynamical system in the $\{b,H\}$ phase plane, depicted in Fig.\ref{fig:SLP1} for the $+$ sign choice in Fig.\eqref{autodef} and \ref{fig:SLP2} in the case of a $-$ sign with the five critical points represented as colored dots.
Focusing on the $+$ case, that is the left graph of Fig.\ref{fig:SLP}, one immediately observes that the positive critical point $\{b_c,H_c\}=\{\tfrac{1}{\sqrt{3}},2\}$ plays the role of a global attractor. This means that the theory admits a stable de Sitter solution which is essential for a possible application of the model to late time cosmology. The negative critical point, on the other hand, is a repeller, while the mixed ones together with the trivial fixed point are saddle points. In that sense, the phase portrait also indicates promising applications of the theory to singularity-free alternative early universe scenarios, in which trajectories starting at the global repeller $\{b_c,H_c\}=\{-\tfrac{1}{\sqrt{3}},-2\}$ initially possess a negative value of the Hubble parameter and thus describe a contracting phase of the universe. Without any exception, the trajectories will then at some point cross the value $H=0$ and inevitably end up in an expanding phase at the stable de Sitter critical point.
\begin{figure}
\centering
\subfloat[\label{fig:SLP1}]{%
\includegraphics[width=0.48\columnwidth]{Plot1}%
}\hfill
\subfloat[\label{fig:SLP2}]{%
\includegraphics[width=0.48\columnwidth]{Plot2}%
}
\caption{These plots show the dynamical phase portrait of the autonomous system of equations \eqref{auto} with an interacting potential term $V[x]=-g_b\,x^2$ for the case $g_b=1$, $\Gamma=2$ ($\gamma=\tfrac{1}{4}$) and $M_{\rm{pl}}=1$ in the phase plane $\{b,H\}$. The colored dots represent the five critical points \eqref{critP}. (a) The phase portrait of the $+$ sign choice in \eqref{autodef}. The positive critical point $\{b_c,H_c\}=\{\tfrac{1}{\sqrt{3}},2\}$ is a global attractor, such that the theory admits a stable critical de Sitter point. This means that the model as such is a successful dark energy theory candidate. Moreover, trajectories evolving from the repeller $\{b_c,H_c\}=\{-\tfrac{1}{\sqrt{3}},-2\}$, thus starting with $H<0$ which could model a contracting phase of the universe will all cross $H=0$ and end up in an expanding phase at the stable de Sitter attractor. This shows that the theory in principle as well represents a possible model of early universe scenarios without any initial singularity. (b) The phase portrait of the $-$ sign choice in \eqref{autodef} essentially shares the same characteristics in a mirrored manner.}
\label{fig:SLP}
\end{figure}
The above statements can readily be verified by solving the equations \eqref{auto} numerically. A parametric plot of three different solution branches which all start close to the global repeller ($\mL{A}$) is shown in Fig.\ref{fig:PP}. Every branch after very distinct trajectories winds up as expected at the global attractor ($\mL{D}$). A comparison with Fig.\ref{fig:SLP1} confirms the phase portrait results.
The $-$ sign case, that is \cref{fig:SLP2} shows qualitatively very similar results. The only difference is that the mixed critical points $\{b_c,H_c\}=\{\pm\tfrac{1}{\sqrt{3}},\mp2\}$ now play the role of the global attractor and repeller respectively.
Choosing more involved potentials $V[6b^2]$ leads to even richer structures of solution space. For example, one can add a mass term $V[x]=\frac{1}{4}m^2\,x-g_b\,x^2$, such that $V[6b^2]=\frac{3}{2}m^2\,b^2-36g_b\,b^4$. This leads to additional four critical points which modify the phase portrait of the theory in a non trivial manner. As an illustrative example we will again choose specific coefficients $m=2$, $g_b=1$, $\Gamma=2$ ($\gamma=\tfrac{1}{4}$) and $M_{\rm{pl}}=1$. The corresponding phase protrait for a $+$ sign choice in the equation for the field $b$ is depicted in \cref{fig:WM}.
\begin{figure}
\centering
\includegraphics[width=0.55\linewidth]{Plot3}
\caption{This plot shows numerically integrated branches of solutions to the autonomous equations \eqref{auto} in the $+$ sign choice and choosing $g_b=1$, $\Gamma=2$ ($\gamma=\tfrac{1}{4}$) and $M_{\rm{pl}}=1$. The initial values of all three branches were chosen in the vicinity of the repeller $\{b_c,H_c\}=\{-\tfrac{1}{\sqrt{3}},-2\}$ ($\mL{A}$). All branches then evolve in distinct trajectories in the phase plane towards the stable de Sitter attractor at $\{b_c,H_c\}=\{\tfrac{1}{\sqrt{3}},2\}$ ($\mL{D}$). This represents a numerical check of the phase portrait consideration.}
\label{fig:PP}
\end{figure}
\begin{figure}
\centering
\subfloat[\label{fig:WM1}]{%
\includegraphics[width=0.48\columnwidth]{Plot4}%
}\hfill
\subfloat[\label{fig:WM2}]{%
\includegraphics[width=0.48\columnwidth]{Plot5}%
}
\caption{These plots show the dynamical phase portrait of the autonomous system of equations \eqref{auto} in the $+$ sign choice in \eqref{autodef} and an interacting potential with additional mass term $V[x]=\frac{1}{4}m^2\,x-g_b\,x^2$, with choices $m=2$, $g_b=1$, $\Gamma=2$ ($\gamma=\tfrac{1}{4}$) and $M_{\rm{pl}}=1$ in the phase plane $\{b,H\}$. The colored dots represent the nine critical points. This shows that additional terms in the choice of the potential have direct implications on the numbers of critical points and the shape of the solution space. (a) The overall picture far away from the central trivial critical point essentially remains the same and still admits stable as well as unstable critical de Sitter points. (b) An enlarged section of the phase portrait around the trivial critical point shows the direct effect of the additional mass term.}
\label{fig:WM}
\end{figure}
Far from the trivial critical point $\{b_c,H_c\}=\{0,0\}$, the picture remains qualitatively the same, with an attractor and a repeller as stable and unstable de Sitter solutions as shown in \cref{fig:WM1}. Near the null critical point, however, the picture changes significantly as seen in the enlarged phase portrait section in \cref{fig:WM2}. In particular, there are two spherical regions for which no real solution can be found.
Again a $-$ sign choice in \eqref{autodef} would lead to a mirrored picture of the above.
Of course these considerations should be viewed as preliminary checks for possible cosmological applications of the theory. We leave a more rigorous analysis including the matter sector for future work.
\subsubsection{The conformal coupling case $\gamma=-1/4$}
We end this section mentioning the special case of the coupling $\Gamma = 0 \; (\gamma = -1/4)$. From the system \eqref{eq:eombH} we see that this particular value leads to several simplifications. The simplified system reads:
\begin{equation}\label{eq:eombH14}
\begin{aligned}
\mathcal{E}_b=\; &\ddot{b}+3 H \dot{b}+4 b V'[6 b^2] =0,\\
\mathcal{E}_N=\; &
-6 M_{\mr{pl}}^2 H^2 +2
V[6 b^2]+3 \dot{b}^2 =0,\\
\mathcal{E}_a=\; &
-4 M_{\mr{pl}}^2 \dot{H}
-6 M_{\mr{pl}}^2 H^2 +2 V[6 b^2] -3 \dot{b}^2
=0\, .
\end{aligned}
\end{equation}
It is worth to notice that for this value of $\Gamma$, the equation for the $2-$form is identical to the equation for a minimally coupled scalar field with a potential $V$ and identical to the vector inflation model with non-minimal coupling of the form $RA_{\mu} A^{\mu}/6$ studied in \cite{Golovnev:2008cf}. A closely related case of a $2-$form non-minimally coupled to gravity was also studied in \cite{Germani:2009iq}. \\
IN order to see that such model can be relevant for the discussion of inflationary dynamics, we can compute the slow roll parameter $\epsilon$ for this system. Combine the second and the third equations we obtain
\begin{equation}\label{eq:srepsilon}
\epsilon = - \frac{\dot{H}}{H^2} = \frac{9 \dot{b}^2}{2V + 3\dot{b}^2} \approx \frac{3 \dot{b}^2}{2M_p^2 H^2 + \dot{b}^2},
\end{equation}
which tell us that for suitable potentials with $V \gg \dot{b}^2$ it is possible to sustain slow roll inflation. However, as extensively discussed in the literature \cite{Himmetoglu:2008zp,Himmetoglu:2009qi} this particular choice of coupling suffers from ghost instabilities in the longitudinal mode. A way out of instability problems relies on the inclusion of general kinetic couplings of the form $f(B^2) H_{\mu \nu\rho} H^{\mu \nu\rho}$ and general potential $V(B^2)$ as explored here.
\section{Conclusion}
The construction of effective field theories is straightforward after determining the involved symmetries and the field content. In standard field theories, the usage of representations of the Lorentz group enables us to categorize the number of physical propagating degrees of freedom. A crucial difference arises between representations of massless and massive particles. A mass term generically breaks existing gauge symmetries of the massless limit and introduces additional propagating modes. A massless 1-form possesses, for instance, tow physical degrees of freedom, whereas its massive generalization introduces one additional degree of freedom due to broken $U(1)$ symmetry. There are different ways how these modes could be represented in alternative formulations.
2-forms naturally arise in the low energy effective field theories of string theory. In this work, we investigated the topological mass generation of 2-forms and connected such a topological term to the recently proposed unique derivative coupling arising in the quartic Lagrangian of the systematic construction of massive $2-$form interactions. The massive 2-form finds a dual description in terms of a massless $1-$form and a massless $2-$form via a topological mass term $B\wedge F$. In this dual description the single degree of freedom propagated by the $2-$form is absorbed by the $1-$form, generating a massive mode for the $1-$form. There is a non-trivial correspondence between such a topological mass generation term and the massive 2-form interaction $\epsilon^{\mu\nu\rho\sigma}\epsilon^{\alpha\beta\gamma}_{\;\;\;\;\;\;\sigma}\partial_{\mu}B_{\alpha\rho}\partial_{\nu}B_{\beta\gamma}$ arising from the systematical construction in terms of the Levi-Civita tensor. This interaction is unique in the sense, that it represents a total derivative on its own but becomes a non-trivial interaction once an overall general function is introduced. Based on the decoupling limit analysis, we showed the uniqueness of such a topological mass term and absence of the Galileon-like interactions, in support of the arguments represented in \cite{Heisenberg:2019akx}. We also presented some preliminary applications in cosmology.
\section*{Acknowledgments}
This work was partly supported by COLCIENCIAS -- DAAD grant 110278258747. JPBA and AG acknowledge support from Universidad Antonio Nari\~no grant 2019248. AG also aknowledges support from Universidad Antonio Nari\~no grant 2019101. LH is supported by funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme grant agreement No 801781 and by the Swiss National Science Foundation grant 179740. C.A.V.-T. acknowledges financial support from Vicerrector\'ia de Investigaciones (Univalle) grant 71220.
\bibliographystyle{utcaps}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 330 |
{"url":"http:\/\/opentradingsystem.com\/quantNotes\/Random_walk_.html","text":"I. Basic math.\n II. Pricing and Hedging.\n III. Explicit techniques.\n IV. Data Analysis.\n V. Implementation tools.\n VI. Basic Math II.\n 1 Real Variable.\n 2 Laws of large numbers.\n 3 Characteristic function.\n 4 Central limit theorem (CLT) II.\n 5 Random walk.\n A. Zero-or-one laws.\n B. Optional random variable. Stopping time.\n C. Recurrence of random walk.\n D. Fine structure of stopping time.\n E. Maximal value of random walk.\n 6 Conditional probability II.\n 7 Martingales and stopping times.\n 8 Markov process.\n 9 Levy process.\n 10 Weak derivative. Fundamental solution. Calculus of distributions.\n 11 Functional Analysis.\n 12 Fourier analysis.\n 13 Sobolev spaces.\n 14 Elliptic PDE.\n 15 Parabolic PDE.\n VII. Implementation tools II.\n VIII. Bibliography\n Notation. Index. Contents.\n\n## Random walk.\n\ne consider the sequence of r.v. and a triple . The is the event space of and is the minimal -algebra that makes all measurable. We think of as a consecutive sequence of trials as increases. Hence, is a \"process\" and is a time parameter.\n\nWe use the notation to represent the -algebra containing the information available up to time . Hence, is the minimal -algebra that makes the family measurable.\n\nWe use the notation to represent the -algebra containing the information that comes after the time . Hence, is the minimal -algebra that makes the the family measurable.\n\nWe use the notation to represent the minimal -algebra that contains all of the , . By the minimality of we have\n\nProposition\n\n(Random walk space approximation). Given and there exists such that\n\nWe use the notation for the intersection\n\n (Remote field)\n\nSometimes it is convenient to think of as the product space\n\n (Random walk space)\nwhere each is the probability space for . The is the product measure consistent with the d.f. on each . The is a collection of infinite sequences of real numbers: The denotes the \"shift\":\n (Shift)\nThe is the applied times.\n\nThe denotes the set of permutations of integers from the range . A permutation (and similarly ) produces a mapping on and on according to the rules\n\n (Permutation)\n\nDefinition\n\n(Invariant set) The set is called \"invariant\" if .\n\nDefinition\n\n(Permutable set) The set is called \"permutable\" if for every permutation of finite number of positions. A function is \"permutable\" if .\n\nDefinition\n\n(Remote event). Any set (see the formula ( Remote field )) is called \"remote event\".\n\nClearly, an invariant set is remote and a remote set is permutable.\n\nDefinition\n\n(Independent process) The family is an \"independent\" process if are independent r.v. The family is \"stationary independent\" process if are iid.\n\nDefinition\n\n(Random walk) For a stationary independent process we defined the \"random walk\" as the process , where and for .\n\n A. Zero-or-one laws.\n B. Optional random variable. Stopping time.\n C. Recurrence of random walk.\n D. Fine structure of stopping time.\n E. Maximal value of random walk.\n Notation. Index. Contents.","date":"2017-03-30 12:36:23","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\": 0.9211063981056213, \"perplexity\": 1704.6014149425757}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218194600.27\/warc\/CC-MAIN-20170322212954-00197-ip-10-233-31-227.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/ncatlab.org\/nlab\/show\/singular+point+of+an+algebraic+variety","text":"Contents\n\nIdea\n\nAn algebraic variety $X \\hookrightarrow \\mathbb{A}^n$ defined as the zero locus of polynomials $\\{f^i \\in R[x_1, \\cdots, x_n]\\}$ is singular at a point $p$, if the Jacobian matrix of first derivatives at this point has rank lower than at other points.\n\nA variety is called singular if it has at least one singular point and non-singular if it has none.\n\nReferences\n\nLast revised on December 27, 2014 at 01:09:59. See the history of this page for a list of all contributions to it.","date":"2023-01-31 07:33:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 3, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6049043536186218, \"perplexity\": 168.4127875591961}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2023-06\/segments\/1674764499845.10\/warc\/CC-MAIN-20230131055533-20230131085533-00364.warc.gz\"}"} | null | null |
Il Monte Tarino è un rilievo dei monti Simbruini, nel Lazio, nella provincia di Frosinone, nel comune di Filettino.
Il Monte Tarino si erge imponente, con la sua maestosa cresta panoramica sino all'altezza di 1961 mt. slm. Fa parte dello stesso sistema montuoso anche il Monte Tarinello (1846 mt. slm.), da cui il Tarino è separato dal "Pozzo della neve" (depressione carsica ove si raccoglie la neve fino in primavera inoltrata).
Note
Montagne della provincia di Frosinone
Tarino
Siti di interesse comunitario del Lazio | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,383 |
Tribute to Community Builders moving on to pastures new – Community Matters
By Christine Durrant, Community Builder for Preston, Paignton
Today, I would like to pay tribute to three of our community builders who will be leaving us this month for pastures new.
We wish them well in their new roles, knowing that they will bring the same passion and energy to their future as they have over the past years here in Torbay.
Marianne Parker has been working in Wellswood for five-and-a-half years.
She was the mastermind behind the flash mobs that have been popping up everywhere from the seafront to the town centres and all because a lady of mature years said that she would love to be part of a flash mob.
Marianne thought it a great idea and set about making it happen.
Working with exercise and dance groups across the Bay, she found brilliant dance leaders who choreographed the routines, led rehearsals and made YouTube videos, so everyone had the opportunity to learn them.
Suddenly the over-50s had a way of showing their community that life does not stop at 40. We can still have fun and dance at whatever age we are.
I hope a volunteer will come forward and keep the flash mob going.
Tara Acton has been working for five-and-a-half years in Chelston and has been instrumental in much of what we have seen happen there.
One legacy I think will remain visible for many years to come is the transformed Goshen Triangle.
Once a derelict piece of land covered in rubbish and rusty cars, it is now a wonderful community garden with a play area for the children.
Sally Farrant started working with us just before Covid and the lockdowns.
Despite barely having time to find her feet, she has forged strong relationships with the community groups in Shiphay and The Willows.
She hopes people will continue to meet for exercise and socials in Kitson Park, Shiphay and local residents will enjoy getting to know each other through 'Community Clean Ups' in The Willows.
They have all made countless other less visible differences in individuals' lives, as I am sure many members of their communities will testify.
Our Ageing Well funding has been extended to March 2022 but after that, if we cannot find new means of paying our community builders, we will have to say goodbye to them all.
I would hate to see that happen, so let's put our thinking caps on and come up with a plan.
You can help us. I invite you to contact our communications officer David Gledhill with your story of the difference your community builder has made in your life.
This can be your way of thanking them for the passion and commitment they have given.
Every story you tell us can be used by us as evidence that the work of the community builders transforms both individual lives and our communities.
Please send them in to davidgledhill@torbaycdt.org.uk
Thank you, Marianne, Tara, and Sally. We are going to miss you!
If you have what it takes to be a community builder, we will be looking to fill these roles so keep an eye on the Torbay Community Development Trust website www.torbaycdt.org.uk for new job opportunities, or on Facebook and Twitter @TorbayCDT
mattcdt July 9, 2021 July 12, 2021 community matters
Previous Previous post: Stronger Together – A Co-production Toolkit goes live
Next Next post: What comes after Freedom Day? – Lockdown easing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,701 |
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<!--NewPage-->
<HTML>
<HEAD>
<!-- Generated by javadoc (build 1.6.0_65) on Fri Jan 10 13:24:37 PST 2014 -->
<TITLE>
N-Index
</TITLE>
<META NAME="date" CONTENT="2014-01-10">
<LINK REL ="stylesheet" TYPE="text/css" HREF="../stylesheet.css" TITLE="Style">
<SCRIPT type="text/javascript">
function windowTitle()
{
if (location.href.indexOf('is-external=true') == -1) {
parent.document.title="N-Index";
}
}
</SCRIPT>
<NOSCRIPT>
</NOSCRIPT>
<script>
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
})(window,document,'script','//www.google-analytics.com/analytics.js','ga');
ga('create', 'UA-48664735-1', 'kylelk.github.io');
ga('send', 'pageview');
</script>
</HEAD>
<BODY BGCOLOR="white" onload="windowTitle();">
<HR>
<!-- ========= START OF TOP NAVBAR ======= -->
<A NAME="navbar_top"><!-- --></A>
<A HREF="#skip-navbar_top" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=2 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_top_firstrow"><!-- --></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
<TR ALIGN="center" VALIGN="top">
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../overview-summary.html"><FONT CLASS="NavBarFont1"><B>Overview</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Package</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Class</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Use</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../overview-tree.html"><FONT CLASS="NavBarFont1"><B>Tree</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../deprecated-list.html"><FONT CLASS="NavBarFont1"><B>Deprecated</B></FONT></A> </TD>
<TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> <FONT CLASS="NavBarFont1Rev"><B>Index</B></FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD>
</TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM>
</EM>
</TD>
</TR>
<TR>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="index-11.html"><B>PREV LETTER</B></A>
<A HREF="index-13.html"><B>NEXT LETTER</B></A></FONT></TD>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../index.html?index-filesindex-12.html" target="_top"><B>FRAMES</B></A>
<A HREF="index-12.html" target="_top"><B>NO FRAMES</B></A>
<SCRIPT type="text/javascript">
<!--
if(window==top) {
document.writeln('<A HREF="../allclasses-noframe.html"><B>All Classes</B></A>');
}
//-->
</SCRIPT>
<NOSCRIPT>
<A HREF="../allclasses-noframe.html"><B>All Classes</B></A>
</NOSCRIPT>
</FONT></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_top"></A>
<!-- ========= END OF TOP NAVBAR ========= -->
<A HREF="index-1.html">A</A> <A HREF="index-2.html">B</A> <A HREF="index-3.html">C</A> <A HREF="index-4.html">D</A> <A HREF="index-5.html">E</A> <A HREF="index-6.html">F</A> <A HREF="index-7.html">G</A> <A HREF="index-8.html">H</A> <A HREF="index-9.html">I</A> <A HREF="index-10.html">L</A> <A HREF="index-11.html">M</A> <A HREF="index-12.html">N</A> <A HREF="index-13.html">O</A> <A HREF="index-14.html">P</A> <A HREF="index-15.html">Q</A> <A HREF="index-16.html">R</A> <A HREF="index-17.html">S</A> <A HREF="index-18.html">T</A> <A HREF="index-19.html">U</A> <A HREF="index-20.html">V</A> <A HREF="index-21.html">W</A> <A HREF="index-22.html">_</A> <HR>
<A NAME="_N_"><!-- --></A><H2>
<B>N</B></H2>
<DL>
<DT><A HREF="../com/hipipal/texteditor/TedActivity.html#NewLineIndex(int)"><B>NewLineIndex(int)</B></A> -
Method in class com.hipipal.texteditor.<A HREF="../com/hipipal/texteditor/TedActivity.html" title="class in com.hipipal.texteditor">TedActivity</A>
<DD>Calculate the index of the Nth new line
<DT><A HREF="../com/hipipal/texteditor/CONF.html#NEWS_LINK"><B>NEWS_LINK</B></A> -
Static variable in class com.hipipal.texteditor.<A HREF="../com/hipipal/texteditor/CONF.html" title="class in com.hipipal.texteditor">CONF</A>
<DD>
</DL>
<HR>
<!-- ======= START OF BOTTOM NAVBAR ====== -->
<A NAME="navbar_bottom"><!-- --></A>
<A HREF="#skip-navbar_bottom" title="Skip navigation links"></A>
<TABLE BORDER="0" WIDTH="100%" CELLPADDING="1" CELLSPACING="0" SUMMARY="">
<TR>
<TD COLSPAN=2 BGCOLOR="#EEEEFF" CLASS="NavBarCell1">
<A NAME="navbar_bottom_firstrow"><!-- --></A>
<TABLE BORDER="0" CELLPADDING="0" CELLSPACING="3" SUMMARY="">
<TR ALIGN="center" VALIGN="top">
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../overview-summary.html"><FONT CLASS="NavBarFont1"><B>Overview</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Package</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Class</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <FONT CLASS="NavBarFont1">Use</FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../overview-tree.html"><FONT CLASS="NavBarFont1"><B>Tree</B></FONT></A> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../deprecated-list.html"><FONT CLASS="NavBarFont1"><B>Deprecated</B></FONT></A> </TD>
<TD BGCOLOR="#FFFFFF" CLASS="NavBarCell1Rev"> <FONT CLASS="NavBarFont1Rev"><B>Index</B></FONT> </TD>
<TD BGCOLOR="#EEEEFF" CLASS="NavBarCell1"> <A HREF="../help-doc.html"><FONT CLASS="NavBarFont1"><B>Help</B></FONT></A> </TD>
</TR>
</TABLE>
</TD>
<TD ALIGN="right" VALIGN="top" ROWSPAN=3><EM>
</EM>
</TD>
</TR>
<TR>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="index-11.html"><B>PREV LETTER</B></A>
<A HREF="index-13.html"><B>NEXT LETTER</B></A></FONT></TD>
<TD BGCOLOR="white" CLASS="NavBarCell2"><FONT SIZE="-2">
<A HREF="../index.html?index-filesindex-12.html" target="_top"><B>FRAMES</B></A>
<A HREF="index-12.html" target="_top"><B>NO FRAMES</B></A>
<SCRIPT type="text/javascript">
<!--
if(window==top) {
document.writeln('<A HREF="../allclasses-noframe.html"><B>All Classes</B></A>');
}
//-->
</SCRIPT>
<NOSCRIPT>
<A HREF="../allclasses-noframe.html"><B>All Classes</B></A>
</NOSCRIPT>
</FONT></TD>
</TR>
</TABLE>
<A NAME="skip-navbar_bottom"></A>
<!-- ======== END OF BOTTOM NAVBAR ======= -->
<A HREF="index-1.html">A</A> <A HREF="index-2.html">B</A> <A HREF="index-3.html">C</A> <A HREF="index-4.html">D</A> <A HREF="index-5.html">E</A> <A HREF="index-6.html">F</A> <A HREF="index-7.html">G</A> <A HREF="index-8.html">H</A> <A HREF="index-9.html">I</A> <A HREF="index-10.html">L</A> <A HREF="index-11.html">M</A> <A HREF="index-12.html">N</A> <A HREF="index-13.html">O</A> <A HREF="index-14.html">P</A> <A HREF="index-15.html">Q</A> <A HREF="index-16.html">R</A> <A HREF="index-17.html">S</A> <A HREF="index-18.html">T</A> <A HREF="index-19.html">U</A> <A HREF="index-20.html">V</A> <A HREF="index-21.html">W</A> <A HREF="index-22.html">_</A> <HR>
</BODY>
</HTML>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,008 |
package de.errorcraftlp.sugarplus.util;
import net.minecraft.item.EnumRarity;
import net.minecraft.util.text.TextFormatting;
import net.minecraftforge.common.util.EnumHelper;
public class Constants {
// General constants
public static final String ID = "sugarplus";
public static final String NAME = "Sugar+ Mod";
public static final String VERSION = "GRADLE_VERSION"; // This is set by the buildscript
// Sugar types
public static final EnumRarity POWDERED = EnumHelper.addRarity("POWDERED", TextFormatting.BOLD, "Powdered");
public static final EnumRarity REFINED = EnumHelper.addRarity("REFINED", TextFormatting.WHITE, "Refined");
public static final EnumRarity WHITE = EnumHelper.addRarity("WHITE", TextFormatting.GRAY, "White");
public static final EnumRarity BASTERD = EnumHelper.addRarity("BASTERD", TextFormatting.DARK_GRAY, "Basterd");
public static final EnumRarity FARIN = EnumHelper.addRarity("FARIN", TextFormatting.YELLOW, "Farin");
public static final EnumRarity RAW = EnumHelper.addRarity("RAW", TextFormatting.RED, "Raw");
public static final EnumRarity[] SUGAR_TYPES = {POWDERED, REFINED, WHITE, BASTERD, FARIN, RAW};
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,147 |
{"url":"http:\/\/reader.feedshow.com\/show_items-feed=5190feae4c9ebcec8e2c7c60d906f712","text":"\u2022\u00a0Shortcuts : 'n' next unread feed - 'p' previous unread feed \u2022\u00a0Styles\u00a0:\u00a01\u00a02\n\nDate: Friday, 14 Dec 2012 08:55\n\n## Introduction\n\nREST services has been steadily gaining popularity over the past years.\u00a0 While SOAP web services are still being used, (and will not go away), it is no longer the preferred method of providing an online API (application programming interface) to web sites and\/or services.\u00a0 Google for example have stopped supporting their SOAP-based API and is almost exclusively using REST based services. Several companies like , especially startups (or previous startups) such as Reddit, Imgur only provide REST APIs\n\nWith this tutorial I'd like to show you how easy it is to test REST services using a tool called WCFStorm.REST. With just 2 clicks you can already start interacting with your service\n\n## Before we begin\n\nWCFStorm.REST can be downloaded here at http:\/\/www.wcfstorm.com, or directly from here.\u00a0 If you go into the web site, you probably noticed that WCFStorm.REST is a commercial tool.\u00a0 But don't worry a LITE version of it is available for free and has enough features especially for developers who just needs to simply call REST services.\u00a0 The FULL version is aimed mainly at Software Testers who needs to prepare test cases, compare actual and expected responses, etc.\n\nSo now go download WCFStorm.Rest.\u00a0 The first time you run it, it will open in 15-day trial mode.\u00a0 That is OK.\u00a0 After the trial period is over it will convert to LITE mode and you'll still be able to use the tool.\n\nNote:\u00a0 You can use discount code RESTPROMO to get a 30% off on WcfStorm.Rest\n\nIn order to get the content of the programming subreddit, we need to use this url, http:\/\/www.reddit.com\/r\/programming.json.\u00a0 Notice that it ends with \".json\"?\u00a0 That's because that URL returns JSON.\u00a0 If you want to get back an XML instead, just change the \".json\" to \".xml\"\n\nReddit's full API is described here http:\/\/www.reddit.com\/dev\/api\n\n## Let's get started\n\nAssuming, you've already downloaded the tool,\u00a0 go double click on WcfStorm.Rest.exe to open it up.\u00a0\u00a0 You'll get the screen below.\u00a0 It's a welcome screen.\u00a0 It shows you the operations\/functionalities of the tool.\n\n## Click #1 : Click on the QuickTest Tab\n\nOr you can also click on the \"Test\" tile.\u00a0 It doesn't matter you will still get the window below.\u00a0 Enter the text \"http:\/\/www.reddit.com\/r\/programming.json\" on the url textbox.\u00a0 Reddit expects an HTTP GET request on that URL, so leave the \"Verb\" dropdown list unchanged. By default, the it should have the \"GET\" option selected.\n\n## Click #2: Click the Send button. Done.\n\nClick the send button (the one with the Green Play icon) to send the HTTP GET request to reddit's service.\u00a0 Wait a second or two and if all goes well you'll get back the response below.\u00a0 The HTTP headers are displayed on the upper right while the response body is shown below it.\u00a0 Notice that the JSON response has syntax highlighting ( a nice little touch, if I may say so)\n\nSo there you go.\u00a0 All it took was 2 clicks.\u00a0 And finally, as the JSON response says, the top submission is about a security vulnerability in internet explorer, titled \"Any page loaded in IE can track your mouse movements, anywhere, any time\",\".\u00a0 Not a very comforting thought.\n\n## Points to consider\n\nWcfStorm.Rest runs on .NET 4.0 so you need to install the 4.0 framework if you want to use this tool.\u00a0 If you only have .NET 4.5, you have to delete WcfStorm.Rest.exe.config so that it will run, otherwise it will crash.\u00a0 For some reason the config points to the 4.0 Fsharp assemblies.\u00a0 Deleting the config file, forces it to use the 4.5 assemblies.\n\nAuthor: \"Erik Araojo\"\n\nDate: Monday, 20 Sep 2010 05:29\n\nyou must be a C++ programmer.\n\nAuthor: \"Erik Araojo\"\n\nDate: Wednesday, 04 Aug 2010 19:36\n\nOne of the benefits of using WCF\u00a0is that you're not\u00a0 required to host it in IIS.\u00a0 Most people create console hosts during development and then\u00a0just\u00a0use windows service in production.\u00a0 This is a quick and very easy way of hosting WCF services especially if you just have 2 or 3 services.\u00a0 However,\u00a0when the number of wcf services gets too high, controlling them using the mmc snap-in (services.msc) can get a bit tedious (maybe not if your powershell or WMI\u00a0kung fu skills are quite good).\n\nHowever, if you're tired of writing these repetetive console\/windows service hosts and is\u00a0looking for another way of managing WCF services,\u00a0then\u00a0try out WcfStormHost.\u00a0 It will let you\u00a0dynamically host WCF Services without ever writing a single code.\u00a0\u00a0Each WCF service is hosted in its own AppDomain so you wont have to worry about wcf services interfering with each other.\n\nHere's a quick rundown of its features\n\n1. Auto-discover and host WCF services\n- WcfStormHost can read a folder containing assemblies (*.dll and *.exe) and *.config and automatically determine the service type to host.\n\n2. It monitors changes in the *.dll and *.config files.\u00a0 If it detects changes, it automatically re-hosts the service!\u00a0 You won't have to manually restart the service, unlike a windows service host. It doesnt lock the assembly file which means\u00a0deployment is much simplified.\u00a0 You\u00a0just simple need to copy over the dll or config file.\n\nDuring development this is also\u00a0very useful\u00a0 because you can setup WcfStormHost to host a service\u00a0located in\u00a0your $Project\/bin\/Debug folder. This means that everytime you make a code change and recompile, the service automatically gets re-hosted. 3. It has a simple, intuitive UI for managing services. WcfStormHost (specifically the system tray host) can be started as an application that sits on your system stray (as in the screenshot above) which means it is easy to find. If you need to do something on a wcf service, just click on the system tray icon and the main UI will popup over it. 4. It can auto-load wcf services. WcfStormHost has an optional config file called Host.config. This file can be created by clicking the \"Save\" icon in the menu. The host.config file can contain a list of WCF services which are automatically loaded at startup. Shown below is a sample service configured to be auto-started. The assembly files + the config file (<ConfigFile>WcfService.dll.config<\/ConfigFile>) are located in the value specified by the <Path> node. <WcfService id=\"MySvc\" type=\"WcfService.MyTestSvc, WcfService\"> <Status>Start<\/Status> <BinFolder> <ConfigFile>WcfService.dll.config<\/ConfigFile> <Path>D:\\Work\\Source-Mine\\WcfStormHost\\WcfService\\bin\\Debug<\/Path> <\/BinFolder> <Activity> <ShutDownifIdle>false<\/ShutDownifIdle> <MaxIdleTime>10000<\/MaxIdleTime> <\/Activity> <\/WcfService> 5. For quick debugging, it allows you to trace WCF messages being sent and received by the hosted wcf service. Just select a service and then click on the \"Trace\" button on the right-hand side. Shown below is the trace window. 6. It creates log files for each of the hosted services. The log files are created daily and can be configured not to exceed a certain size. By default a log file can not exceed 0.5 MB. Old log files and log files that exceed the size limit are moved into the \"archive\" folder. Try it out and see for yourself. Get Started Download WcfStormHost (blue download icon on the right hand-side) These are the current features of WcfStormHost and of course suggestions for more features are most welcome! Please post it in the comments or send a mail to erik-dot-araojo@wcfstorm.com Author: \"Erik Araojo\" Date: Friday, 04 Dec 2009 16:27 One of the more common design patterns used in object oriented languages like C# is the Composite Pattern. The main benefit of using this pattern is that it allows us to treat individual objects and composite of objects in the same way. In the UML below, the type Component is abstract and defines the operations (method1 and method2) which are implemented by the concrete types Leaf and Composite. The type Composite contains a collection of Component objects. Due to the fact that it inherits from Component we can write code that invokes method1 and method2 without knowing the specific type. For example, suppose we work at a fast food chain and we'd like to write code for computing the price of fries, burger, coke or a combo meal. Following the composite pattern, we could create an abstract class called Product which has the GetPrice() method and then one class each for Burger, Fries, Coke and ComboMeal. Component --- maps to --> Product Leaf --- maps to --> Burger, Fries, Coke Composite --- maps to ---> ComboMeal Our C# code would look like public abstract class Product { public abstract double GetPrice(); } public class Fries : Product { public override double GetPrice() { return 1.0; } } public class Burger : Product { public override double GetPrice() { return 5.0; } } public class Coke : Product { public override double GetPrice() { return 2.0; } } public class ComboMeal : Product { List<Product> _products = new List<Product>(); public void Add(Product p) { this._products.Add(p); } public override double GetPrice() { var sum = 0.0; foreach (var p in _products) sum += p.GetPrice(); return sum; } } we can then write a helper method that Computes a price. This method does not need to know whether its computing the price of a single item (Fries, Burger ) or that of a combo meal. public static double ComputePrice(Product p) { return p.GetPrice(); } Suppose, someone ordered 1 fries and 1 Burger separately, our code will be, var totalFriesAndBurger = ComputePrice(new Fries()) + ComputePrice(new Burger()); Now if another one ordered a Combo meal (Fries + Coke + Burger), our code will be var combo = new ComboMeal(); combo.Add( new Fries()); combo.Add( new Burger()); combo.Add( new Coke()); var totalCombo = ComputePrice(combo); That's it. For our C# implementation (not counting the helper method ComputePrice() ), we have 5 classes and approximately 40 lines of code. #### Now lets see how we can implement the same pattern in F#. Knowing that the types Fries, Burger, Coke and ComboMeal are just different representations of the Product type, we can model this relationship cleanly in F# using discriminated unions. To represent the the combo meal, we make it into a tupleof 2 products. An item of this tuple can be a single product (Coke, Fries or Burger) or any combination (i.e a combo) type Product = | Fries of double | Coke of double | Burger of double | Combo of Product * Product And how about the GetPrice method, you'd ask? That's easy enough; F# allows us to add methods to union types. Implementing that, our code now looks like this. Notice the \"with member w.GetPrice()\" part? type Product = | Fries of double | Coke of double | Burger of double | Combo of Product * Product with member w.GetPrice() = match w with | Fries(p) | Coke(p) | Burger(p) -> p | Combo(p1,p2) -> p1.GetPrice() + p2.GetPrice() Again, if someone ordered a combo meal our code will look like let combomeal = Combo(fries, Combo(burger, coke)) Console.WriteLine( \"Combo meal Total = \" + combomeal.GetPrice().ToString()) That's just cool isn't it? With F# we ony 1 Union type and about 10 lines of code Author: \"Erik Araojo\" Date: Thursday, 03 Dec 2009 12:33 I bought myself an early christmas gift and purchased the book \"Functional Programming in the Real World\" by Tomas Petricek. The book is available through Manning's early access program. I've been drooling over this book for a long time and finally I have it now! (cue music: Jingle bells, jingle bells jingle all the way..) Sadly though I have an exam coming (I'm taking my masters in Software Engineering) in 2 weeks time and now I need to force myself to study the lecture notes and not read Tomas's book. Arrgh. P.S I was able to buy the book using donations from Storm. Thanks guys! Author: \"Erik Araojo\" Date: Friday, 31 Jul 2009 10:44 A few months ago I've promised that version 1.1 will be released in July. Today I make good on that promise and am happy to announce that version 1.1 is ready for download! A very big thanks goes out to those who not only have purchased WCFStorm but also provided valuable feedback in terms of feature requests and bug reports. Thank you very much guys. Below are the main features of version 1.1 1. Loading of existing client App.config files into WCFStorm 2. One-click execution of all functional test cases under a method or of the whole WCF service. 3. A plugin interface for custom validations of functional test cases. 4. A plugin interface for randomizing requests or re-using the previous response as input to the next request. 5. A plugin interface for programmatically modifying the ServiceEndpoint instance 6. A plugin interface to support Duplex services Here's the full list of what's new in 1.1 Author: \"Erik Araojo\" Date: Wednesday, 08 Apr 2009 09:48 If you're not yet familiar on how to add a service and invoke a method, please see Getting Started The short video below shows how to create and execute a functional test case in WCFStorm. Towards the end of the video, it shows one of the main features of functional testing with WCFStorm - which is the ability to visually view the difference between the expected and actual responses in a test. Author: \"Erik Araojo\" Date: Wednesday, 08 Apr 2009 04:52 If you're not yet familiar on how to add a service and invoke a method, please see Getting Started The video below shows the steps on how to create and run performance test cases. When a performance test case is executed, WCFStorm captures the responses and displays a graph in real-time the actual and average response times as well the rate at which the service is responding the method invocations. ### The following are the performance test parameters \u2022 Number of agents \u2022 This is the number of background workers that will repeatedly invoke the service method until the test is stopped or the test duration has completed. \u2022 Rampup \u2022 This is the interval in seconds at which the agents are created. For example, if the rampup value is \"1\", every second, WCFStorm will create an agent until it has reached the specified number of agents \u2022 Test duration \u2022 This is the test duration in seconds \u2022 Invoke interval \u2022 This is the interval in milliseconds at which an agent repeatedly invokes a service method Sample Perf Test setup 1. Constant Load \u2022 This can be achieved by setting the \"Rampup\" value to zero 2. Incrementing number of users \u2022 Set \"Rampup\" and \"Number of Agents\" to a non-zero value. ### Some screenshots Author: \"Erik Araojo\" Date: Sunday, 05 Apr 2009 02:57 In order for WCFStorm to invoke a service having a netTcp endpoint, the service must be configured to expose the WSDL via http. A sample config for the WCF service will be, In above configuration, the service is using 3 bindings. mexHttpBindings is used to expose the WSDL, while netTcpBinding and wsHttpBinding are for the actual methods calls. To test the serivice using WCFStorm, use the mexHttpBinding url to add the service. As can be seen from the log, the tool detected both the netTcpBinding and WsHttpBindings. If we select a method and invoke it ( (please see getting started on the steps to invoke a method) , the tool will select the endpoint that was declared first (which is NetTcpBinding). If you'd like to use the other endoint, wsHttpBinding, right-click the service and select \"ModifyEndpoint\" This will bring up a form which lets you choose the binding that you'd like to use. Click on the dropdown to select a new binding. You can also specify a new endpoint address in this form. Click OK to use the selected binding. This will trigger WCFStorm to regenerate a proxy class that uses the new binding. All subsequent method invocations will now use the new proxy (and hence the new binding). Author: \"Erik Araojo\" Date: Sunday, 05 Apr 2009 02:19 EDIT : Please visit www.wcfstorm.com for th updated tutorials. This is a first in a series for post describing the tasks and the features available inWCF Storm. 1. Getting Started 4. How to create and execute a functional test Ok, so lets get started.... ## Getting Started : (Invoking a method of a WCF\/Web service) 1. ### Add a service. This can be done by either clicking \"File --> Service --> Add\" or by clicking the Add button in the menu bar. This will bring up a form where you can type in the metadata exchange endpoint (i.e. WSDL endpoint). WcfStorm will then read the wsdl, generate the client code and compile it. If all goes well, the service and its method will be displayed on the left hand side. Now go ahead and select on method ### 2. Select a method If a method is selected, its parameters will be read and displayed in the \"RequestPane'. In the screenshot below I've selected the method \"GetDataUsingContract\" which takes parameter of type \"CompositeType\" The structure of CompositeType is shown below. As you can see, what's displayed in the request pane matches exactly the defined type. Now, select a field of the composite type parameter. ### 3. Edit a field in the parameter This will bring up the Object Editor which will let you edit the field. If the type is not primitive, the \"Set to null' checkbox will be enabled. If the tye of the selected field is polymorphic, the object editor will display the subtypes that can be assigned to the field. Here I've selected the string field. Clicking OK will assign the value \"my string\" to the StringValue field of the CompositeType parameter. Now Click Send (Green arrow) ### 4. Invoking the service method Click on send to invoke the web method. The test service we are invoking merely echoes back the input parameters it has received ( a convenient way of checking that the object editor in the previous step worked). This is shown in the screenshot below Looking at the log, we can see that WCFStorm used the NetTcpBinding endpoint. Recall that when we added the service, we used http:\/\/localhost:8080\/httpEndpoint?wsdl - obvoiusly an endpoint that uses http. So how come the tool is now using NetTcpBinding? In this case, the service that we are using declared 3 endpoints in its config. An http endpoint for the metadata exchange (to expose the wsdl) a NetTcpBinding and a WsHttpBinding endpoint for the actual method calls. By default WCFStorm uses 1st endpoint that was declared by the service (which was the NetTcpBinding.) That's it! we have now successfully invoked the WCF service. Author: \"Erik Araojo\" Date: Friday, 03 Apr 2009 01:16 During the last few months I've been working on a commercial tool for testing WCF services. The knowledge I learned building STORM really helped alot in building this tool. So here it is my fellow coders, WCF Storm. It has more features than its open source cousin, STORM and the best part is it works on both WCF and Web services! EDIT : www.wcfstorm.com is up! Please visit the site for updates and to purchase WCFStorm online ### System Requirement: \u2022 .NET Framework 3.0 or higher. ### Features at a glance: General: \u2022 Dynamically invoke and test WCF Services \u2022 Dynamically invoke and test Web services written in any language \u2022 Save your opened service and its test cases into a \"project\" which can be reloaded anytime \u2022 Dynamically invoke service methods even those containing complex data types \u2022 UI-based, dynamic editing of complex data types \u2022 Test multiple WCF and Web services within a single UI. \u2022 Multiple-tabbed interface \u2022 Basic and windows authentication \u2022 Test services sitting behind a proxy \u2022 Dynamically modify the URL endpoint of a WCF or Web service. \u2022 Dynamically edit the service binding. Functional Testing: \u2022 Create functional test cases and save it as part of a project \u2022 Create and save functional test cases containing Expected results. \u2022 Graphically compare (side-by-side) the expected results with the actual response of a service. Performance Testing: \u2022 Create performance test cases and save it as part of a project \u2022 Graphically display charts in real-time as the performance test progresses. \u2022 Configurable test parameters (# of Agents, Test duration, interval etc.). You can stress out your service as much as you want. What all these features mean is that testing WCF\/Web services becomes a breeze and you'll have plenty of time to spend on writing the logic of your service (which is what you should be really doing). Performance testing screenshot: Functional testing screenshot: Check out the gallery for more screenshots. Try it out! Download the trial version below. ### or Buy Now orLearn more ### So how much does it cost? Short answer is ... Not so much! I've priced the tool for the budget conscious. Consider this, I used to work for a company that bought a license for a soap testing tool that costs almost 700 USD (that's right7 hundred!) annually per seat. It was a great tool. It had tons of features but the problem was, we rarely used them. We mainly used it to invoke web service methods and verify that the results are correct. All the other features that the company paid (dearly) for, were rarely or sometimes never used. Why pay so much for unused features? So how much does WCF Storm cost? \u2022 Personal Edition : 14.99 USD only (annually\/seat ) \u2022 Enterprise Edition : 49.99 USD only (annually for 10 seats!) \u2022 (EDIT : I've finalized the pricing structure for WCFStorm. Please visit http:\/\/www.wcfstorm.com\/wcf\/buynow.aspx to view the offers) Not bad, huh? :) For such a low price not only can you test WCF services, it even works on Web services. if you buy now and be part of the first 100 to purchase WCF Storm, I'd extend your license for 2 years! However, if you think that the price is expensive, drop me an email and we can discuss the price we're both comfortable with. ### I love the price, but why didn't you just open source it like STORM? I think these words are sufficient to answer that question: \u2022 #### Recession This recession that we all are experiencing right now has also affected myself and my family. I still have my job but I can't tell whether next month I'd still have it. Some people had already been let go and management has been awfully quiet on when the next round of layoffs will be. I have a wife and a lovely 5 year-old daughter to care for. I also have a mortgage and credit card bills to pay. My job, which is our main source of income is in an unstable state right now. \u2022 #### 9 millimeter My wife is 2 months pregnant. During our last visit to the doctor, we found out our baby is about 9 mm in length already. In programmer-speak, our baby is in the Alpha or Beta state right now. But he\/she has already achieved a major project milestone- he\/she has a heartbeat! Oh I am the happiest person in the world right now! We can't wait for him\/her to come join us in November later this year. I had a DONATE button on the codeplex site of STORM and even added it to the Storm application itself. Its been there for more than 6 months and inspite of the 6000+ downloads, I have yet to receive a single donation. I know, you'd say, open source doesn't work like that. One cannot expect donations on open source projects. It's the spirit of giving and sharing code that matters. Yes, that is true, I believe that as well. But as you can see, I simply can't afford to give out WCF Storm for free right now. If my family is to weather this difficult time we need to have another source of income. I hope people understand. ### or Buy Now or Learn more Author: \"Erik Araojo\" Date: Friday, 10 Oct 2008 04:29 The Goal Simulate a class using the Record type How we'll do this Create a tool that can translate a number into another format. In particular, we'll convert a number into 1. Roman Numerals or 2. multiply it by 10 Why are you doing this? No particular reason. I was just curious. I was just thinking that if F# did not support OOP, how can I still achieve the same things I was used to doing in C#? Ok so here we go.. F# record types are simple named types. We can pattern match over it and it is also constructed quite easily. For example The record Person contains 2 named fields, which are both strings. Most of the time records are used to hold data but since F# is an FP language, we can also have the fields of the record carry a function. For example, we can modify the Person record type to instead of having the \"LastName\" as a string, we'll turn it into a function that computes the last name. Something like, As you can see we have changed the signature of LastName from \"string\" to \"unit->string\". The function \"fun -> Guid.NewGuid().ToString() \" matches this new signature. (Side note: as a C# coder, this was a big mind shift for me) Going back to our example, lets define a Converter record type as this type holds a string field \"x\" and \"ToNewFormat\" field with the signature unit->string. The function does not take any parameter because we'll have it use the value assigned to \"x\" i.e. it will convert the value in \"x\" into a different format. If we were using a class, our code (C#) will be something like and to use it we'd write So how do we contsruct our F# Converter record so that it would behave essentially like the C# Converter class? like this. We create a converter function that takes 2 parameters, \"numStr\" and \"myConverterFunction\". NumStr gets assigned to the field \"x\" while the myconverterFunction is wrapped inside the \"intToNewFormat\" function which has the signature unit->string. MyConverterFunction was wrapped inside intToNewFormat function because we want this function to work on the value of field \"x\". Well actually myConverter function uses the value of \"positiveNum\" (an int) which was derived from the string value of temp.x. The \"temp\" value is of course of type Converter. The important part here is that in order for us to acces the instance value of \"x\" we need add the \"rec\" keyword in the definition of the value temp. If we didn't do that we won't be able to access temp.x! To call this createConverter function, all we need to do is pass a string and a function that has the signature int->string In the code above, we create a romanConverter function that takes a string and creates the instance of the Converter record by passing the string and \"RomanConverterFunction.romanConverter\" function. The \"RomanConverterFunction.romanConverter\" function takes an integer and converts it to Roman numerals. For example, if you pass it \"1980\" it will output \"MCMLXXX\" That's it! whenever we call, romanNumeralConverter.ToNewFormat(), it is converting its instance value of \"x\" into roman numerals exactly the same way the C# Converter class is working. Note that because we are passing around functions we can easily create different kinds converters. In the above code we created another converter, \"multiplier10converter\" which merely multiplies a number by 10 (yes, this example is not very good. :-p ). If we needed a different converter all we need to do is write the code for that converter and pass the new conversion function to \"createConverter\". Simple. If we had to do this in OO, we have to through the inheritance-override route. Here's the tool in action Author: \"Erik Araojo\" Date: Thursday, 18 Sep 2008 02:05 F# is an awesome language. I'd choose to write F# code any day over any language if my work allows it. The way you are able to concisely and cleanly write code with it is what brought me over. Sure there is an initial overhead (if you we are an OO programmer) in learning all the functional constructs but in the end its all worth it. Even though you might not be able to use it at work (read : management only wants C#\/VB.NET), the way you are writing OO code will definitely be improved. I for example have come to appreciate Generics and delegates more because of F#. Microsoft Research has done such a great job with it and I'm quite sure they will continue to do so. There is no question that \"F# as a language\" is great. People do however have a gripe about \"F# as a Microsoft product\". It is not open source nor is it standard like C#. There is a technology-lockin which I presume some fear will later on will translate into a vendor-lockin. F# targets mainly the research and financial institutions. I can not say much about financial companies but research instituions and technology companies are big Linux users. There is very high likelihood that these people are also open-source advocates. It will be in their best interest to ensure that the language works perfectly on both Windows and non-windows systems. This is both a win for the users and for Microsoft. It would also propel the language forward at a much quicker pace. It might be too early to ask this given that the F# CTP was only recently released, but should Microsoft open source F# and get the community involved? Author: \"Erik Araojo\" Date: Friday, 23 May 2008 03:36 One of the commonly used design pattern is the Observer Pattern because its just so easy to use and implement. The classic OO way of implementing it is to have a Subject class having the methods \"Attach\", \"Detach\" and \"Notify\". The Subject class usually stores the observers in an ArrayList and then when it needs to update the observers then iterates on the list and invokes the \"Update\" method of each observer. See diagram below. In C#, a more elegant way of implementing this is with Events and Delegates. Here's a nice sample implementation. By using using events and delegates, the code became a lot shorter and straight-forward In F#, implementation of the observer pattern became even more concise and elegant. How come? Take a look at the code below. line 11 : A clean and simple (one-liner!) way of creating events and triggers. A call to IEvent.create is all it takes lines 16 & 17 : calling \"trigger\" and passing the correct parameters notifies all listeners of the changeEvent. We don't even need to check if there are listeners attached. If this is in C# we need to make changeEvent != null otherwise a NullReferenceException will be raised. lines 28 & 30. In F# Events are first-class values , that means we can pass it around and more importantly we can do all sorts of wonderful things with it using the IEvent module. For example, in line 28, we filtered the event, basically saying that we are only interested in events of the Remove operation. Now imagine if the Operation type (an enum actually, lines 5-7 ) contains more items such as Update, Delete, AddedThenRemoved, AddedThenUpdated, RemovedAddedAgainThenFinallyUpdated ... (you get what I mean). If we want to a handle only the Remove operation then we'd have no choice but to break it down using an if-elseif-else or switch-case construct. It would work but its not going to be pretty. And this is where IEvent.filter shines. With just a single line of code (line 28, please ignore the comment above it.), we were able to filter out the event that we are only interested in. Apart from filtering, IEvent also allows us to map, partition, fold, split etc. events. lines 25 & 30 : 2 ways of attaching to an Event. I'm not very sure what's the difference between these 2. (TODO : Investigate) Here's the output! Notice that the \"observerForRemove\" printed out only the message related to the Remove operation even though the Subject class raised 2 events (one for Add, another one for Remove) in the Notify() method at line 15. Sweet. Author: \"Erik Araojo\" Date: Sunday, 11 May 2008 11:12 To everyone who've either sent me an email or posted a comment asking whether SoapBits can be made into an open source project, well here I am pleased to let you all know that finally it has happened! SoapBits is now open source and is available at codeplex! In line with this change, it have given it a new name,STORM. (I meant \"STORM\" to be an acronym but I can only come up with Soap Testing for \"ST\" . The \"ORM\" part I have yet to figure out, so please do suggest if you think of a clever meaning. ;) ) Another major change is that the tool is now written mostly in F#, a language which I believe is a lot more expressive and powerful than C# because of the ease with which it has combined Object Oriented and Functional programming. The user interface part though is still written in C# mainly because F# is not yet fully integrated into Visual Studio. STORM site : http:\/\/codeplex.com\/storm As you all can see below, the UI has also changed a bit but the old functionalities are still there Enjoy the tool everyone! And don't forget to give back and contribute to the project. Author: \"Erik Araojo\" Date: Wednesday, 02 Apr 2008 08:36 Here's a sample template that lets you have the String.Replace() functionality in XSLT 1.0. The template \"string-replace-all\" takes 3 parameters and recursively processes the input text string. \u2022 text : main string \u2022 replace : the string fragment to be replaced \u2022 by : the replacement string <xsl:template name=\"string-replace-all\"> < xsl:param name=\"text\" \/> < xsl:param name=\"replace\" \/> < xsl:param name=\"by\" \/> < xsl:choose> < xsl:when test=\"contains($text,\u00a0$replace)\"> < xsl:value-of select=\"substring-before($text,$replace)\" \/> < xsl:value-of select=\"$by\"\u00a0\/>\n<\nxsl:call-template\u00a0name=\"string-replace-all\">\n<\nxsl:with-param\u00a0name=\"text\"\nselect=\"substring-after($text,$replace)\"\u00a0\/>\n<\nxsl:with-param\u00a0name=\"replace\"\u00a0select=\"$replace\" \/> < xsl:with-param name=\"by\" select=\"$by\"\u00a0\/>\n<\/\nxsl:call-template>\n<\/\nxsl:when>\n<\nxsl:otherwise>\n<\nxsl:value-of\u00a0select=\"$text\" \/> <\/ xsl:otherwise> <\/ xsl:choose> <\/ xsl:template> Here's how it is called: < xsl:variable name=\"myVar\"> < xsl:call-template name=\"string-replace-all\"> < xsl:with-param name=\"text\" select=\"'This is a sample text : {ReplaceMe} and {ReplaceMe}'\" \/> < xsl:with-param name=\"replace\" select=\"'{ReplaceMe}'\" \/> < xsl:with-param name=\"by\" select=\"'String.Replace() in XSLT'\" \/> <\/ xsl:call-template> <\/ xsl:variable> (Edit : Thanks to Marky and granadaCoder for typing out the xslt code in the comments.) The resulting value of$myVar after {ReplaceMe} is replaced is \"This is a sample text : String.Replace() in XSLT and String.Replace() in XSLT\"\n\nFor those who are not familiar with XSLT syntax and here's the C# equivalent.\u00a0 An excellent\u00a0material for the\u00a0thedailywtf!\u00a0:)\n\n(Note: I'm not so sure, but I think in XSL 2.0 there is already a built-in replace function on strings)\n\nAuthor: \"Erik Araojo\"\n\nDate: Monday, 18 Feb 2008 17:15\n\nProblem:\n\n2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.\n\nWhat is the smallest number that is evenly divisible by all of the numbers from 1 to 20?\n\nSolution:\n\nFinding the LCM can be done via factorization.\u00a0 Below is an excerpt from the wikipedia article.\n\nThe unique factorization theorem says that every positive integer number greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined together, make up a composite number.\n\nFor example:\n\n$90 = 2^1 \\cdot 3^2 \\cdot 5^1 = 2 \\cdot 9 \\cdot 5. \\,\\!$\n\nHere we have the composite number 90 made up of one atom of the prime number 2, two atoms of the prime number 3 and one atom of the prime number 5.\n\nThis knowledge can be used to find the lcm of a set of numbers.\n\nExample: Find the value of lcm(8,9,21).\n\nFirst, factor out each number and express it as a product of prime number powers.\n\n$8\\; \\, \\; \\,= 2^3 \\cdot 3^0 \\cdot 5^0 \\cdot 7^0 \\,\\!$\n$9\\; \\, \\; \\,= 2^0 \\cdot 3^2 \\cdot 5^0 \\cdot 7^0 \\,\\!$\n$21\\; \\,= 2^0 \\cdot 3^1 \\cdot 5^0 \\cdot 7^1. \\,\\!$\n\nThe lcm will be the product of multiplying the highest power in each prime factor category together. Out of the 4 prime factor categories 2, 3, 5, and 7, the highest powers from each are 23, 32, 50, and 71. Thus,\n\n$\\operatorname{lcm}(8,9,21) = 2^3 \\cdot 3^2 \\cdot 5^0 \\cdot 7^1 = 8 \\cdot 9 \\cdot 1 \\cdot 7 = 504. \\,\\!$\n\nCode:\n\nmodule\u00a0Prob5\u00a0=\u00a0begin\nopen\u00a0System\nopen\u00a0Microsoft.FSharp.Math\n\nlet\u00a0primeFactors\u00a0(num:float)\u00a0=\n\/\/let\u00a0num\u00a0=\u00a050.0\nlet\u00a0divSeq\u00a0=\u00a0num\u00a0|>\u00a0Seq.unfold\u00a0(fun\u00a0x\u00a0->\nlet\u00a0rec\u00a0get\u00a0num2\u00a0\u00a0=\nlet\u00a0sq\u00a0=\u00a0Math.Sqrt\u00a0(num2)\nlet\u00a0div\u00a0=\u00a0ref\u00a02.0\nwhile(\u00a0(not(num2\u00a0%\u00a0!div\u00a0=\u00a00.0))\u00a0&&\u00a0(!div\u00a0<\u00a0sq)\u00a0)\u00a0do\nif\u00a0(!div\u00a0=\u00a02.0)\u00a0then\ndiv\u00a0\u00a0:=\u00a0!div\u00a0+\u00a01.0\nelse\ndiv\u00a0\u00a0:=\u00a0!div\u00a0+\u00a02.0\ndiv\nlet\u00a0sq\u00a0=\u00a0Math.Sqrt\u00a0(x)\nlet\u00a0divisor\u00a0=\u00a0get\u00a0x\nif\u00a0(x\u00a0=\u00a01.0)\u00a0then\nNone\nelse\u00a0if\u00a0(sq\u00a0<\u00a0!divisor)\u00a0then\nSome\u00a0(x,\u00a01.0)\u00a0\u00a0\/\/\u00a0x\u00a0is\u00a0prime!\nelse\nSome(!divisor,\u00a0x\/(!divisor))\n)\ndivSeq\nlet\u00a0pFactors\u00a0=\u00a0(2,20)\u00a0|>\u00a0Seq.unfold\u00a0(fun\u00a0(x,limit)\u00a0->\nlet\u00a0y\u00a0=\u00a0float_of_int\u00a0x\nlet\u00a0s\u00a0=\u00a0\u00a0(primeFactors\u00a0y)\u00a0|>\u00a0Seq.map\u00a0(fun\u00a0x\u00a0->\u00a0Convert.ToInt32(x))\nif\u00a0x\u00a0<=limit\u00a0then\nSome\u00a0(\u00a0(x,s),\u00a0(x+1,\u00a0limit))\nelse\nNone\n)\n\nlet\u00a0factorSort\u00a0x\u00a0y\u00a0=\nlet\u00a0(num1,\u00a0_)\u00a0=\u00a0x\nlet\u00a0(num2,_)\u00a0=\u00a0y\n\nif\u00a0(num1\u00a0<\u00a0num2)\u00a0\u00a0then\n-1\nelse\u00a0if\u00a0num1\u00a0>\u00a0num2\u00a0then\n1\nelse\n0\nlet\u00a0powSort\u00a0x\u00a0y\u00a0=\nlet\u00a0(_,num1)\u00a0=\u00a0x\nlet\u00a0(_,num2)\u00a0=\u00a0y\n\nif\u00a0num1\u00a0<\u00a0num2\u00a0then\n1\nelse\u00a0if\u00a0num1\u00a0>\u00a0num2\u00a0then \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0-1\nelse\n0\n\nlet\u00a0all\u00a0=\u00a0\u00a0\u00a0pFactors\u00a0|>\u00a0Seq.map\u00a0(fun\u00a0(num,s)\u00a0->\u00a0s\u00a0|>\u00a0\u00a0Seq.countBy\u00a0(fun\u00a0x\u00a0->\u00a0x\u00a0)\u00a0)\u00a0\u00a0|>\u00a0Seq.concat\nlet\u00a0allList\u00a0=\u00a0\u00a0(Seq.to_list\u00a0all)\u00a0|>\u00a0List.sort\u00a0factorSort\u00a0|>\u00a0List.sort\u00a0powSort\n\nlet\u00a0a\u00a0=\u00a0ref\u00a00\nlet\u00a0b\u00a0=\u00a0ref\u00a00\nlet\u00a0filtered\u00a0=\u00a0allList\u00a0|>\u00a0List.filter\u00a0(fun\u00a0(x,y)\u00a0->\nlet\u00a0(factor,pow)\u00a0=\u00a0allList\u00a0|>\u00a0\u00a0List.find\u00a0(fun\u00a0(r,s)\u00a0->\u00a0r=x\u00a0)\n\nif\u00a0(factor\u00a0=\u00a0x\u00a0)\u00a0&&\u00a0(pow\u00a0=\u00a0y)\u00a0&&\u00a0(\u00a0(!a\u00a0<>\u00a0x)\u00a0||\u00a0(!b\u00a0<>\u00a0y)\u00a0)\u00a0then\na\u00a0:=\u00a0factor\nb\u00a0:=\u00a0pow\ntrue\nelse\nfalse\n\n)\n\nlet\u00a0main()\u00a0=\nlet\u00a0lcm\u00a0=\u00a0filtered\u00a0|>\u00a0List.fold_left\u00a0(fun\u00a0a\u00a0(x,y)\u00a0->\u00a0a\u00a0*\u00a0Math.Pow(float_of_int\u00a0x,\u00a0float_of_int\u00a0y\u00a0)\u00a0)\u00a01.0\nprintfn\u00a0\"\\n\\nLCM\u00a0=\u00a0%f\"\u00a0lcm\n\nend\n\nMy\u00a0solution for this one follows the steps outlined in the wikipedia excerpt above. It is a bit long but essentially it can be broken down into the following stages...\n\n1 - Factor out the number 1 to 20.\n\nThis is done on function pFactors.\u00a0 pFactors outputs a sequenct of tuples wherein the first value is the number being factored while the second value is a sequence of its prime factors. For example, the output of this function when applied to the numbers 2 to 20\n\n{ (2, {2}); (3, {3}; (4, {2;2})\u00a0; (5, {5}) ...,<snip> .. (12, {2;2;3}) ...<snip> .. (20, {2;2;5})\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0}\n\ntype\u00a0: seq< int * seq<int>>\n\n2 - Count the number of occurrences of each prime factor and consolidate into just one sequence\n\nThis is done on line\u00a0 let all = pFactors |> Seq.map (fun (num,s) -> s |> Seq.countBy (fun x -> x ) ) |> Seq.concat\n\nSeq.countBy counts the number of occurences of an item in a sequence.\u00a0 For example, if we apply to the sequence of prime factors of the number 20, the output will be\n\n(20, {2;2;5} ) -> (2,2),(5,1) : where the 2nd value of the tuple is the number of occurences\n\nSo for the number 2 to 20 the output is\n\n(2, 1)(3, 1)(2, 2)(5, 1)(2, 1)(3, 1)(7, 1)(2, 3)(3, 2)(2, 1)(5, 1)(11, 1)(2, 2)(3, 1)(13, 1)(2, 1)(7, 1)(3, 1)(5, 1)(2, 4)(17, 1)\n\n(2, 1)(3, 2)(19, 1)(2, 2)(5, 1)\n\ntype : seq<int * int >\n\n3 - Sort the output of stage 2.\n\nThe output of stage 2 obviously contains duplicates.\u00a0\u00a0Referring back to the\u00a0wikipedia excerpt above, what we need only are those factors with the\u00a0highest power (or most number of occurences).\u00a0 So\u00a0in order for us to identify those factors,\u00a0first we\u00a0sort the sequence.\u00a0 This is done on line\n\nlet allList = (Seq.to_list all) |> List.sort factorSort |> List.sort powSort\n\nThe output (though a bit confusing) guarantees that the\u00a0for any number, the one with the highest\u00a0number of occurences comes out on top.\u00a0 For example (2,4) is on top\u00a0of all the other tuple entries with first value of 2. And so is (3,2) for tuple values with first value of\u00a03.\n\n(2, 4)\n(2, 3)\n(2, 2)\n(2, 2)\n(2, 2)\n(3, 2)\n(3, 2)\n(2, 1)\n(2, 1)\n(2, 1)\n(2, 1)\n(2, 1)\n(3, 1)\n(3, 1)\n(3, 1)\n(3, 1)\n(5, 1)\n(5, 1)\n(5, 1)\n(5, 1)\n(7, 1)\n(7, 1)\n(11, 1)\n(13, 1)\n(17, 1)\n(19, 1)\n\ntype : seq<int * int>\n\n4.\u00a0 Filter the resulting sequence of stage 3 and take only those with the highest 2nd element.\u00a0 This is done on line\n\nlet filtered = allList |> List.filter (fun (x,y) -> ...\u00a0 <snip>...\n\noutput is :\u00a0 seq<int*int>\n\n(2, 4)\n(3, 2)\n(5, 1)\n(7, 1)\n(11, 1)\n(13, 1)\n(17, 1)\n(19, 1)\n\n5. And finally we can compute the LCM.\u00a0 This is done on the\u00a0line\n\nlet\u00a0lcm = filtered |> List.fold_left (fun a (x,y) -> a * Math.Pow(float_of_int x, float_of_int y ) ) 1.0\n\nTotal execution time on my machine is 0.17 seconds\u00a0 which a great improvement from the 25 seconds using the brute force approach described here\n\nAuthor: \"Erik Araojo\"\n\nDate: Monday, 18 Feb 2008 15:36\n\nThese 2 problems have very short solutions so I'll put them together\n\nProblem 3:\n\nA palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 \u00d7 99.\n\nFind the largest palindrome made from the product of two 3-digit numbers.\n\nSolution:\n\nlet\u00a0prodOf3digit\u00a0=\n(100,100)\u00a0|>\u00a0Seq.unfold\u00a0(fun\u00a0(x,y)\u00a0->\nmatch\u00a0y\u00a0with\n|\u00a0y\u00a0when\u00a0y\u00a0<\u00a00\u00a0->\u00a0None\n|\u00a0y\u00a0when\u00a0x\u00a0<=\u00a0999->\nif\u00a0(y\u00a0<=\u00a0999)\u00a0then\nSome(x*y,\u00a0(x,y+1))\nelse\u00a0if\u00a0(x+1)\u00a0<=\u00a0999\u00a0then\nSome\u00a0(\u00a0(x+1)*(x+1),\u00a0(x+1,\u00a0x+2)\u00a0)\nelse\nNone\n|\u00a0y\u00a0->\u00a0None\u00a0)\n\nlet\u00a0palindromes\u00a0=\nprodOf3digit\n|>\u00a0Seq.filter\u00a0(fun\u00a0x\u00a0->\nlet\u00a0intStr\u00a0=\u00a0Int32.to_string\u00a0x\nlet\u00a0rev\u00a0(s:string)\u00a0=\nlet\u00a0chunks\u00a0=s.ToCharArray()\nlet\u00a0reversed\u00a0=\u00a0Array.rev\u00a0chunks\nlet\u00a0r\u00a0=\u00a0new\u00a0string(reversed)\nr\nlet\u00a0newNum\u00a0=\u00a0System.Convert.ToInt32\u00a0(\u00a0rev\u00a0intStr\u00a0)\n\nnewNum\u00a0=\u00a0x\n)\nlet largest =\npalindromes |> Seq.fold (fun a x -> if x > a then x else a) 0\n\nThe problem asked for the largest\u00a0palindrome of 3-digit numbers, hence\u00a0the magic number 100 and 999.\u00a0 The rest of the code is pretty much self-explanatory.\n\nProblem 6:\n\nThe sum of the squares of the first ten natural numbers is,12 + 2 + ... + 102 = 385\n\nThe square of the sum of the first ten natural numbers is, (1 + 2 + ... + 10) = 552 = 3025\n\nHence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 \u2212 385 = 2640.\n\nFind the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.\n\nSolution:\n\nlet\u00a0sumOfSq\u00a0(n:bigint)\u00a0=\nlet\u00a0rec\u00a0getSum\u00a0(x:bigint)\u00a0=\nmatch\u00a0x\u00a0with\n|\u00a0x\u00a0when\u00a0x\u00a0=\u00a01I\u00a0->\u00a01I\n|\u00a0x\u00a0->\u00a0x*x\u00a0+\u00a0getSum\u00a0(x\u00a0-\u00a01I)\ngetSum\u00a0n\n\nlet\u00a0sqOfSum\u00a0n\u00a0=\nlet\u00a0s\u00a0=\u00a0[\u00a01I\u00a0..\u00a0n\u00a0]\u00a0|>\u00a0Seq.fold\u00a0(fun\u00a0a\u00a0x\u00a0->\u00a0a+x)\u00a00I\ns*s\n\nlet\u00a0main2()\u00a0=\nlet\u00a0diff\u00a0=\u00a0(sqOfSum\u00a0100I)\u00a0-\u00a0(sumOfSq\u00a0100I)\nprint_any\u00a0diff\n\nTo compute the sum of squares, I've decided to use recursion,\u00a0 which apart from lists and sequences, is another\u00a0workhorse of functional programming.\u00a0\u00a0\u00a0One thing to remeber when using recursion is to always make sure you cover the base case in order to avoid looping indefinitely.\u00a0 In this case it is the line x when x = 1I -> 1I\n\nBecause of the large resulting values, the type used\u00a0was BigInt.\n\nAuthor: \"Erik Araojo\"\n\nDate: Monday, 11 Feb 2008 15:58\n\nProblem 1 :\n\nIf we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.\n\nSolution :\n\nlet mySeqP1 = [ 1 .. 999 ] |> Seq.filter (fun x -> ( (x % 5 = 0) || (x % 3 = 0)) ) |> Seq.fold(+) 0\nprintfn \"sum = %i\" mySeqP1\n\nProblem 2:\n\nFind the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.\n\nSolution :\n\nlet sumofEven = \u00a0\u00a0(1,1) |> Seq.unfold (fun (x,y) ->\nmatch y with\n| y when y < 0 -> None\n| _ -> Some(y, (y, x+y)))\n|> Seq.filter ( fun x -> \u00a0\u00a0\u00a0\u00a0 x > 0 && x < 1000000 && (x % 2 = 0))\n|> Seq.fold (+) 0\n\nprintfn \"Sum of even = %i\" sumofEven\n\nFor this solution I had to put the guard statement \"when y < \u00a00\" so that negative numbers won't be included in the generated sequence.\u00a0 If that guard istatement s not there, negative numbers will be included when it goes over the\u00a0maximum value\u00a0of Int32.\n\nProblem 3:\n\nFind the largest prime factor of 317584931803.\n\nSolution:\n\nlet\u00a0primefactorsOf\u00a0(num:float)=\nlet\u00a0divSeq\u00a0=\u00a0num\u00a0|>\u00a0Seq.unfold\u00a0(fun\u00a0x\u00a0->\nlet\u00a0rec\u00a0get\u00a0num2\u00a0\u00a0=\nlet\u00a0sq\u00a0=\u00a0Math.Sqrt\u00a0(num2)\nlet\u00a0div\u00a0=\u00a0ref\u00a02.0\nwhile(\u00a0(not(num2\u00a0%\u00a0!div\u00a0=\u00a00.0))\u00a0&&\u00a0(!div\u00a0<\u00a0sq)\u00a0)\u00a0do\nif\u00a0(!div\u00a0=\u00a02.0)\u00a0then\ndiv\u00a0\u00a0:=\u00a0!div\u00a0+\u00a01.0\nelse\ndiv\u00a0\u00a0:=\u00a0!div\u00a0+\u00a02.0\ndiv\n\nlet\u00a0sq\u00a0=\u00a0Math.Sqrt\u00a0(x)\nlet\u00a0divisor\u00a0=\u00a0get\u00a0x\n\nif\u00a0(Int32.of_float(x)\u00a0=\u00a01)\u00a0then\nNone\nelse\u00a0if\u00a0(Int32.of_float(\u00a0sq\u00a0)\u00a0<\u00a0Int32.of_float(\u00a0!divisor\u00a0))\u00a0then\nSome\u00a0(\u00a0Int32.of_float(\u00a0x\u00a0)\u00a0,\u00a01.0)\u00a0\u00a0\/\/\u00a0x\u00a0is\u00a0prime!\nelse\nSome(Int32.of_float\u00a0!divisor,\u00a0x\/(!divisor))\n)\n\ndivSeq\n\nlet\u00a0primefactors\u00a0=\u00a0\u00a0(primefactorsOf\u00a0317584931803.0)\n\nlet\u00a0mainp3()\u00a0=\nprimefactors\u00a0|>\u00a0Seq.iter\u00a0(fun\u00a0x\u00a0->\u00a0printfn\u00a0\"%i\u00a0\"\u00a0x)\n\nThe function \"primefactorsOf\" takes\u00a0a number and\u00a0returns a sequence containing the prime factors of that number.\u00a0 The prime factors are determined using Trial Division.\u00a0 This of course is not very fast for ver large numbers but is\u00a0sufficient enough for 317584931803.0 If it's not clear why the square root of the number being factored out was used instead of directly using\u00a0the number itself, the wikipedia article explaining\u00a0Trial Division is here\n\nIf you want to output only the largest primefactor you can just pass the value primefactors to Seq.fold like so\n\nlet largestFactor = primefactors |> Seq.fold (fun a x -> if x > a then x else a ) 0\n\nNext time I'll post my solution to problems 4-6.\n\nAuthor: \"Erik Araojo\"\n\nDate: Friday, 18 Jan 2008 09:24\n\nThe Problem :\u00a0 What is the first term in the Fibonacci sequence to contain 1000 digits?\n\nThe sweet solution :\n\n#light\n\nopen System\n\nlet fib =\n(1I,1I) |> Seq.unfold ( fun (sqEntry, acc) -> Some (acc, (acc, acc + sqEntry)) )\n|> Seq.filter (fun x -> x.ToString().Length = 1000)\n|> Seq.nth 0\n\nprint_any fib\n\nahh... such\u00a0power and elegance....\n\nLet me breakdown the solution\n\n(1I,1I) |> Seq.unfold ( fun (sqEntry, acc) -> Some (acc, (acc, acc + sqEntry)) )\n\nThis line creates an inifinite list of fibonacci numbers using Seq.unfold.\u00a0 Let me say that again, it creates a in INFINITE list.\u00a0\u00a0How is that possible, you'd ask\u00a0? The reason is that F# performs Lazy Evaluation. By that, I mean the program will\u00a0perform\u00a0only the\u00a0computation\u00a0it needs to do.\u00a0So in our case, the program will actually compute only the parts of the list that it will need.\u00a0\u00a0 Here we used Seq.unfold to perform the computation.\u00a0\u00a0The unfold function creates a list and uses an\u00a0accumulator\u00a0to\u00a0maintain the state between computations.\u00a0 The accumulator in this case is the tuple that contains the last 2 entries of the sequence\n\nSeq.filter( fun x -> x.ToString().Length = 1000\n\nThis line applies\u00a0the filter\u00a0function to the sequence.\u00a0 This function returns a subset of the original list\u00a0that\u00a0matches the condition.\u00a0 In this case, the resulting list will contain only those numbers that is made up 1000 digits.\n\nSeq.nth 0\n\nThis line retrieves the nth element in the sequence.\u00a0 Since the problem ask for the 1st term, naturally we retrieved the 0th element.\n\nAnd finally the construct that ties everything together and makes the code such a wonder to look at is the\n\n|>\u00a0 (the Pass-forward operator)\n\nAs its name implies, it passes forward the argument behind it to the function in front of it.\u00a0 So the solution in essence means\n\nPass the tuple (1I, 1I) to Seq.Unfold, then pass the resulting infinite list to Seq.filter, then pass the filtered list to Seq.nth which finally retrieves the 1st element!\n\nAuthor: \"Erik Araojo\"\n\nNext page\n\u00bb You can also retrieve older items : Read","date":"2013-05-24 08:58:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 5, \"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\": 0.37911108136177063, \"perplexity\": 544.2877791575248}, \"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-2013-20\/segments\/1368704392896\/warc\/CC-MAIN-20130516113952-00064-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
José Conrado Roza est un peintre portugais de la seconde moitié du , fils et élève de Domingo da Rosa, auquel il succéda dans la charge de maître de dessin et de peinture des princes de la Maison royale, et spécialement des princesses.
Biographie
José Conrado Roza travaille à la décoration des appartements du pavillon Robillon au palais royal de Queluz, comme les panneaux de la Sala das Merendas (salle à manger), présentant des scènes de poésie pastorale, ou les bas-reliefs du boudoir de la Reine.
En 1788, il peint l'huile sur toile Portrait des nains de la reine Marie de Portugal, dite aujourd'hui La Mascarade nuptiale. Le tableau représente huit personnages, dont des nains envoyés depuis le Brésil à la cour de la reine portugaise, Marie , entre 1784 et 1787, friande de curiosités exotiques.
Notes et références
Bibliographie
La Mascarade nuptiale, catalogue d'exposition de Laurent Vidal, Annick Notter et Roger Little, musée du Nouveau Monde de La Rochelle, 34 p., 2011
Peintre portugais du XVIIIe siècle | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,671 |
{"url":"http:\/\/tcpc.me\/projects\/6s191-lecture-notes.html","text":"# 6.S191 IAP 2021 Lecture Notes\n\nMonday, April 26, 2021, 04:00 PM\n\nThese lecture notes are based on the course 6.S191 Introduction to Deep Learning during IAP 2021 semester at MIT. Only the first five lectures are available as I dropped the course afterward. These lectures include:\n\n1. Intro to Deep Learning\n2. Deep Sequence Modeling\n3. Deep Computer Vision\n4. Deep Generative Modeling\n5. Deep Reinforcement Modeling\n\nOfficial recordings from this class are available at the official course website.\n\nPlease feel free to email me at tcpc at mit dot edu if you find any typos or factual errors.","date":"2022-09-25 01:38:12","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\": 0.23420347273349762, \"perplexity\": 5045.0728883091015}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2022-40\/segments\/1664030334332.96\/warc\/CC-MAIN-20220925004536-20220925034536-00481.warc.gz\"}"} | null | null |
Having a pool with low Total Alkalinity can cause corrosion, plaster stains or etching, and irritation to swimmers. Leslie's Alkalinity Up quickly balances water chemicals, creating a safe swimming environment.
The ideal ranges for any pool, regardless of size, should be between 7.4 and 7.6 for pH, and between 80 and 140 ppm for Total Alkalinity. We you spend time and effort to maintain a balanced pH level, and do not want to add chemicals that can disrupt this balance. Alkalinity Rise will correct problems with Total Alkalinity deficiency while preserving your pH balance.
Keeping a Total Alkalinity level between the recommended 80 and 140 ppm will make it very easy to maintain a constant pH balance. Total Alkalinity is necessary to keep your pH levels balanced, and low Total Alkalinity can also cause damage to pools and irritation for swimmers. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,706 |
Earthly Body located: Alex Hendry, 19 - NY college baseball player
With sadness, earthly body located. Our prayers for his family, friends and all who love and will miss him.
Alex Hendry, a 19-year-old freshman at Onondaga Community College, New York, was reported missing Sunday by his roommate after he failed to return home the previous night, the Onondaga County Sheriff's Department . His body was found around 1 a.m. Monday morning.
The sheriff's office is investigating the cause of death, but foul play is not suspected.
Temperatures in the Syracuse area dipped below -20 degrees Fahrenheit overnight Saturday into Sunday. They were still below zero overnight Sunday into Monday. He was last seen at an off campus event and seen leaving on foot.
http://www.democratandchronicle.com/story/news/2016/02/15/hilton-high-school-graduate-found-dead/80396562/
Posted by LostNMissing, Inc at 7:00 AM No comments:
Earthly body located: Matthew Genovese, 24 - NJ - missing Jan 23,2016
Earthly body located: Matthew Genovese, 24 - NJ - was missing Jan 23,2016
Funeral held early February, 2016. Our prayers.
A 24-year-old New Jersey man who went missing Saturday night after visiting a bar with friends during the weekend's blizzard has been found dead, police said Tuesday.
Matthew Genovese was last seen leaving McSwiggan's Pub at First Street and Bloomfield Avenue in Hoboken between 10:30 and 11:30 p.m. He told three friends at the bar that he was walking home, a 10-minute trek away, Hoboken police said.
Genovese, an associate with a Manhattan tax consulting firm for companies that do business overseas, had lived on Garden Street in Hoboken for about a year before his disappearance.
He was originally from the Throgs Neck section of the Bronx, where he was remembered Monday during a funeral service at St. Benedict's Church, according to the Sisto Funeral Home, which handled the arrangements.
His remains were then interred in a family tomb at Ferncliffe Cemetery in Hartsdale, N.Y., in Westchester County, according to Sisto.
Police said Genovese's trip home should have been a familiar 10-minute walk, a block west and several blocks uptown. Instead, police said, Genovese left the bar and headed east, toward the River, three blocks away.
His wallet, with cash and credit cards still in it, was found along with his keys, in the snow near a waterfront railing at Pier A Park. City surveillance cameras were not trained on the spot. http://www.nj.com/hudson/index.ssf/2016/02/bronx_funeral_held_monday_for_matthew_genovese.html
Posted by LostNMissing, Inc at 12:00 AM No comments:
Earthly Body: Elizabeth Luebke, 21 - Oshkosh,WI - Jan. 17, 2016
Luebke, Elizabeth K. "Lizzy"
Elizabeth K. "Lizzy" Luebke, age 21, of Oshkosh, passed away unexpectedly on January 17, 2016. She was born on July 23, 1994 the daughter of Gary and Kathryn (Griese) Luebke. She attended and graduated from Omro High School where she had been active in band and drama. She had been an active member of the Pickett 4-H Club. She loved horses and was a member of the Waukau Flying Mane riding club and also greatly enjoyed art, music, reading, time spent with her family and friends and cats, Tilly Monster and Honey, and of course "Augie Doggie". She worked at Griese & Sons in Pickett as the Safety Director and had also worked at Crown Trophy in Oshkosh in graphic design which she loved. Lizzy was an amazing young woman (smart, talented and witty) with a beautiful soul and she was loved and will be missed by many.
Milwaukee County Medical Examiner's Office has identified the 21-year-old woman found dead near Keefe and Bremen in Milwaukee as Elizabeth Luebke of Oshkosh.
Officials say Luebke was found frozen in front of a home.
Elizabeth Luebke
According to the medical examiner's report, Luebke "was reportedly in the area for a concert, and following the concert, a house party near the location from where she was found."
The report indicates the woman "was not dressed for the elements and noted to be frozen."
The report also says just before 4:30 a.m., a surveillance camera in the neighborhood "noted the decedent to collapse" at the address where she was found.
Luebke was found by a passer-by around 9:00 a.m.
The medical examiner's report indicates the temperature around the time 911 was called was "recorded as -5.8 degrees F with the wind chill recorded as -27.5 degrees F."
A friend shared a photo of Luebke with WITI. He said he knew Luebke as a sweet woman who went by the name of Lana Kane. She was at his home with friends from Oshkosh before she died.
"When she got here, she was really, really drunk and she got mad at her other friend and she stormed out," the man told WITI.
When Luebke didn't return, the medical examiner's report indicates someone called Luebke's mom, who drove to Milwaukee and filed a missing person's report.
"She's a really sweet person. I hung out with her at another show when I went to Oshkosh. She's a really nice person -- a really sweet person. It's really sad. She's only 21. Such a young age," the man said.
The report indicates Luebke had a history of binge drinking, but friends said she most recently had been sober for several weeks.
Police say no one should go out amid sub-zero, double-digit wind chills alone.
"If you can stay in groups -- at least with somebody -- and that's just general personal safety," Joel Dhein with the Glendale Police Department said.
Luebke's body was found frozen just around the corner from the home where the party was taking place.
The medical examiner's report indicates Luebke was not dressed for the bitterly cold weather -- as she was wearing party-type clothing, including shorts, stockings and a tank top. A sweater was found on the ground near her body.
Above credit to: cw39.com/2016/01/19/wisconsin-woman-found-frozen-to-death-after-leaving-party-in-shorts/
Earthly Body: Elizabeth Luebke, 21 - Oshkosh,WI - ...
Earthly body located: Matthew Genovese, 24 - NJ - ...
Earthly Body located: Alex Hendry, 19 - NY college... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,070 |
Q: How to create multiple clones of a VM in KVM/libvirt? I (will) have a VM in libvirt (using QEMU/KVM) that I will set up and consider the "gold master". I would then like to be able to clone it 'n' times to create multiple systems. What is the best way to do this, and how best to "individualize" the cloned systems after they are instantiated? The hypervisor platform is:
OS: Ubuntu 12.04.2
Hypervisor pkg: qemu-kvm 1.0+noroms-0ubuntu14.8
Libvirt: 1.0.4 (compiled from source)
I am managing the VMs from another system (Fedora 18 OS) using virt-manager 0.9.5
I know I can shut down the VM and then use virt-manager's "Clone..." function, but am wondering if there is a better way (especially if I could do multiple clones at once, not sure if that's possible or not...)
A: To massively clone (baseline or even customized) installations I would say that the best method is to set up a PXE+TFTP server and provide a preseed/kickstart file.
Add your config management tool of choice (puppet/chef/cfengine/other) to the mix and you will have a powerful framework to deploy as many clones as you need. Moreover, add local mirrored repositories to increase speed.
The upper limit for simultaneously deployed clones will be the ethernet and I/O throughput you can sustain.
A: First prepare the master vm (not sure about ubuntu but sys-unconfig works great in RHEL
Then simply run virt-clone as many times as you need, using an ampersand at the end of the command to send it to background
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,087 |
import rightHelixPrime from "./rightHelixPrime";
export function run() {
return import(/* webpackChunkName: "left" */ "./leftHelix");
}
export default {
rightHelixPrime: () => rightHelixPrime
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,382 |
Q: Postfix handling Amazon SES Maximum Send Rate error We have a postfix server which we were using to send emails. This server is used by many services. Thus for using Amazon SES, I've integrated our postfix server with SES SMTP interface ( using "http://docs.aws.amazon.com/ses/latest/DeveloperGuide/postfix.html" ). The configuration is working fine and mails are getting delivered properly.
Now, there is a limit of 5 email/sec, imposed by SES and it raises error '454 Throttling failure: Maximum sending rate exceeded ' if limit exceeds.
I'm a newbie to Postfix.
Kindly guide me with the configuration settings in Postfix to make postfix resend the mail when error '454 Throttling failure: Maximum sending rate exceeded' occurs.
Also, how to resend email, when occasionally 'Connection timed out' error occurs with relay server(Amazon SES).
A: This is not the actual answer you are asked,but you can bypass the issue with this,
You can add this lines in main.cf file
default_destination_concurrency_limit=1
default_destination_rate_delay=10s
This will increase the time delay but you don't get the error message.
You can also verify this link to learn more about postfix performance turning
A: I was wondering the same thing so I tried it out on a fresh postfix install. I found that no additional configuration was required and postfix did indeed retry sending the messages about 5 minutes after original throttling error was reported in the log file.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,157 |
<resources>
<string name="google_maps_key" translatable="false" templateMergeStrategy="preserve">
AIzaSyAHOX1eQrjMdyAfSyWu0YfnQtl6n4a0TXo
</string>
</resources>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,193 |
Q: I can't delete cascade with Sqlalchemy Database : Mysql with InnoDB
I have 2 table : MACHINE and FAMILY. A machine belongs to a family.
FAMILY :
#The family of a machine. For exemple, the machine "AMAT 50" belong to the family "AMAT"
class FAMILY(Base):
__tablename__ = 'Family'
id_family = Column(Integer, primary_key = True)
name_family = Column(String(100), nullable=False, unique = True)
machine = relationship('MACHINE', cascade='all,delete', backref='Family')
MACHINE :
#The machine on which the tests are executed
class MACHINE(Base):
__tablename__ = 'Machine'
id_machine = Column(Integer, primary_key = True)
number_machine = Column(Integer)
id_family = Column(Integer, ForeignKey("Family.id_family"), nullable=False)
__table_args__ = (
UniqueConstraint(number_machine, id_family),
)
I would like that when a family is deleted, the machines linked to this family are deleted.
The way I showed above doesn't work. So I also tried :
FAMILY :
#The family of a machine. For exemple, the machine "AMAT 50" belong to the family "AMAT"
class FAMILY(Base):
__tablename__ = 'Family'
id_family = Column(Integer, primary_key = True)
name_family = Column(String(100), nullable=False, unique = True)
MACHINE :
#The machine on which the tests are executed
class MACHINE(Base):
__tablename__ = 'Machine'
id_machine = Column(Integer, primary_key = True)
number_machine = Column(Integer)
id_family = Column(Integer, ForeignKey("Family.id_family", ondelete='CASCADE'), nullable=False)
__table_args__ = (
UniqueConstraint(number_machine, id_family),
)
family = db.relationship('FAMILY', backref=backref('Machine', passive_deletes=True))
But it didn't work either.
Thank you for your help
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,299 |
\section{Introduction}
The space of complex rational maps $\mathrm{Rat}_d$ of degree $d \ge 2$ admits a natural compactification by $\mathbb{P}^{2d+1}$. For each $n \ge 2$,
the iterate map
$\Psi_n:\mathrm{Rat}_d\to\mathrm{Rat}_{d^n}$ defined by sending $f$ to $f^n$ extends to a rational map $\Psi_n: \mathbb{P}^{2d+1}\dasharrow\mathbb{P}^{2d^n+1}$.
According to DeMarco \cite[Theorem 0.2]{DeMarco05} the map $\Psi_n$ has an indeterminacy locus $I(d)$ independent of $n \ge 2$.
The group $\mathrm{PSL}_2(\mathbb{C})$ acts on the space
$\mathrm{Rat}_d$ by conjugacy. The induced quotient space
$\mathrm{rat}_d$ is the moduli space of degree $d$ rational
maps. Moduli space is a complex orbifold of dimension
$2d-2$. Geometric invariant theory (GIT) provides a compactification
$\overline{\mathrm{rat}}_d$ of the moduli space $\mathrm{rat}_d$ by
considering the action of $\mathrm{PSL}_2(\mathbb{C})$ on the
semistable loci $\mathrm{Rat}_d^{ss}\subset\mathbb{P}^{2d+1}$
\cite{Silverman98}. The iterate map $\Psi_n$ induces a regular map
$\Phi_n:\mathrm{rat}_d\to\mathrm{rat}_{d^n}$ that sends the conjugacy
class $[f]$ to $[f^n]$, see \cite[Proposition 4.1]{DeMarco07}.
However, $\Phi_n$ does not extend continuously to the compactification
$\overline{\mathrm{rat}}_d$ \cite[Theorem 10.1]{DeMarco07}. That is,
$\Phi_n:\overline{\mathrm{rat}}_d\dasharrow\overline{\mathrm{rat}}_{d^n}$ has
a non-trivial indeterminacy locus denoted $I(\Phi_n)$.
Our main result gives a complete description of the indeterminacy
locus $I(\Phi_n)$. Our work answers a question posed by DeMarco
in~\cite{DeMarco07}.
\medskip
We say that $f \in \mathrm{Rat}_d^{ss} \setminus I(d)$ is \emph{$n$-unstable} if $f^n \notin \mathrm{Rat}_{d^n}^{ss}$. The subset of $\mathrm{Rat}_d^{ss}$ formed by the $n$-unstable maps is denoted by ${\mathcal{U}}_n$.
\begin{introtheorem}\label{main}
For $d\ge 2$ and $n\ge 2$, let $\Phi_n:\overline{\mathrm{rat}}_d\dasharrow\overline{\mathrm{rat}}_{d^n}$ be the iterate map and denote by $I(\Phi_n)$ its indeterminacy locus.
For all $f \in \Rat^{ss}_d$, we have that $[f] \in I(\Phi_n)$ if and only if $f \in I(d)\cup{\mathcal{U}}_n$.
\end{introtheorem}
As discussed in Section~\ref{upper-bound}, in our setting the indeterminacy locus coincides with the points where $\Phi_n$ has no continuous extension (in the analytic topology). Thus,
to prove our main theorem, we
only need to establish that for $f\in \Rat_d^{ss}$, the map $\Phi_n$ has no continuous extension at $[f]$ if and only if $f\in I(d)\cup{\mathcal{U}}_n$.
For quadratic rational maps, the indeterminacy locus $I(\Phi_n) \subset \overline{\mathrm{rat}}_2 $ was explicitly described by DeMarco in~\cite[Theorem 5.1]{DeMarco07}.
Theorem~\ref{main} is an easy consequence of this description in the case of quadratic maps. Hence, in this paper, we focus on the case that $d\ge 3$.
In the same work, DeMarco~\cite[Lemma 4.2]{DeMarco07} also proved that if $[f] \in I(\Phi_n)$ then
$f \in I(d) \cup {\mathcal{U}}_n$, for even degrees.
{Then she asked for the veracity of the converse, see \cite[Question]{DeMarco07}. Our Theorem \ref{main} answers her question in the affirmative.}
One direction of Theorem~\ref{main}: if $[f] \in I(\Phi_n)$ then $f \in I(d)\cup{\mathcal{U}}_n$ is easily obtained by extending a result by DeMarco~ \cite[Lemma 4.2]{DeMarco07} to all degrees (see Proposition~\ref{semistable}). The reverse implication, if $f\in I(d)\cup{\mathcal{U}}_n$ then $[f] \in I(\Phi_n)$ requires substantial work.
When $f\in I(d)\cup{\mathcal{U}}_n$ we have that
$f$ is a ``degenerate rational map''.
A degenerate rational map $f$
is defined by a pair of polynomials with shared zeros called the \emph{holes of $f$}.
To show that the map $\Phi_n$ is indeterminate at certain $[f] \in \overline{\rat}_d$, we construct holomorphic families $f_t$ and $g_t$ of (possibly degenerate) degree $d$ rational maps parametrized by a neighborhood of $t=0$ in $\mathbb{C}$. These families are carefully chosen to
materialize an indeterminacy of $\Phi_n$ at $[f]$. More precisely, the constructions are such that both $[f_t]$ and $[g_t]$ converge to $[f]$ in $\overline{\rat}_d$ while the iterates $[f^n_t]$ and $[g^n_t]$ converge to different elements of
$\overline{\rat}_{d^n}$. In almost all the cases, it is useful to employ techniques from non-Archimedean rational dynamics.
Namely, the holomorphic families $f_t$ and $g_t$ act on the Berkovich projective line $\mathbf{P}^1$ over a suitable non-Archimedean field.
In fact, the construction itself will
take place in Berkovich space, and Berkovich dynamics will allow us to tailor the construction so that $[f^n_t]$ and $[g^n_t]$ converge as $t \to 0$ to distinct elements $[F]$ and $[G]$ of $\overline{\rat}_{d^n}$. To certify that these conjugacy classes are distinct, in most of the cases, we show that the holes of $F$ and $G$ give rise to non-equivalent markings of $\P^1$.
This paper is mostly devoted to introduce techniques suitable to exploit the interplay between GIT (semi)stability of complex rational maps and dynamics on the Berkovich projective line.
Thus the dynamical content of our constructions is better understood in Berkovich space.
An important relation between Berkovich dynamics of families as $f^n_t$ and convergence of $[f^n_t]$ in $\overline{\rat}_{d^n}$ is
addressed in Rumely's work on \emph{(semi)stable reductions} (see \cite[Theorem C]{Rumely17}).
In Rumely's language, for any family $f_t$ such that $f=f_0$ is semistable but $f_0^n$ is unstable,
the Gauss point is a point of semistable reduction for $f_t$ but not for $f_t^n$. According to Rumely, there exists a type II point in Berkovich space
such that $f_t^n$ has semistable reduction there.
Moreover, if the reduction of $f^n_t$
at a point is stable, then it is the unique point in
Berkovich space with semistable reduction.
However we were unable to apply directly Rumely's work to gain the required control at points of stable reduction.
Our constructions can be rephrased in this language by saying that given $f\in {\mathcal{U}}_n$ we construct several
families $g_t$ with reduction $f$ at the Gauss point
such that $g^n_t$ has stable reduction at a point $\zeta_0\in\mathbf{P}^1$. The construction
is such that we know where in Berkovich space is the point $\zeta_0$ and
have some control on the reduction of $g^n_t$ at $\zeta_0$.
{More precisely, we have the following result. For the definition of induced maps, see Section \ref{subsection:stable}.}
\begin{introtheorem}\label{Thm:perturbation}
Let $d\ge 4$. Suppose $f\in{\mathcal{U}}_n$ and the induced map $\hat f$ is nonconstant.
Then there exists $g_{\lambda,t} \in \mathbb{C}[\lambda, t] (z)$ such that for
$\lambda$ in the complement of a finite subset of $\mathbb{C}$ the following holds:
\begin{enumerate}
\item
$g_{\lambda,t}$ has semistable reduction $f$ at the Gauss point $\xi_g$.
\item
There exists a type II point $\zeta_0 \neq \xi_g$, independent of $\lambda$, in ${\mathbf{P}^1}$ such that
$g^n_{\lambda,t}$ has stable reduction $G_\lambda$ at $\zeta_0$.
\item
$[G_\lambda]\in \overline{\rat}_{d^n}$ is not a constant function of $\lambda$.
\item
The action of $g_{\lambda,t}$ in the convex hull of
$\{ \zeta_0, \dots, g^n_{\lambda,t}(\zeta_0) \}$ is independent of $\lambda$.
\end{enumerate}
\end{introtheorem}
We point out that in the case that $d=3$, if $\deg\hat f=1$, we also obtain the same conclusion as the above theorem. If $\deg\hat f=2$, we construct two families $f_t$ and $g_t$ satisfying the property stated in the previous paragraph, see Sections \ref{cubic-poly} and \ref{cubic-monomial}.
\medskip
This paper is organized as follows:
In Section \ref{background}, we introduce the relevant preliminaries about degenerate rational maps and Berkovich spaces.
Not all of the material here is standard. In particular, Sections~\ref{reductions} and~\ref{depths-and-multiplicities} establish a bridge between
Berkovich dynamics and degenerate rational maps which is exploited throughout the paper.
In Section \ref{bad-hole-s}, we identify a distinguished hole of $n$-unstable maps which we call the \emph{bad hole} and establish a basic depth-multiplicity inequality for this hole.
The orbit, depth and multiplicity of the bad hole will organize the proof of Theorem~\ref{main} in cases.
A degenerate rational map $f$ of degree $d$ induces a rational map $\hat{f}$ of lower degree acting on $\P^1$.
Section~\ref{deep}, which concentrates most of the work of the paper, contains the proof of Theorem \ref{Thm:perturbation} and that the GIT-class $[f]$ of any $n$-unstable map $f$
with non-constant induced map $\hat{f}$ lies in $I(\Phi_n)$. To evidence the indeterminacy of $\Phi_n$ at $[f]$, we organize our argument into cases according to
the degree of $\hat{f}$, the depth of the bad hole and the dynamics of the bad hole under $\hat{f}$.
In Section~\ref{constant}, we show that $[f] \in I(\Phi_n)$ provided $f$ is a degenerate semistable rational map with constant induced map and $f \in I(d)\cup {\mathcal{U}}_n$. These includes all
the cases not covered in~Section~\ref{deep} and finishes the proof of Theorem~\ref{main}.
\medskip
\textbf{Acknowledgements.}
The work was initiated during the visit of the second author to the Facultad de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile in 2017. He thanks the Facultad de Matem\'aticas for its hospitality.
\section{Preliminaries}\label{background}
In this section we discuss background material and stablish some useful
results about degenerate rational maps, the (GIT) stable and semistable loci of rational maps and Berkovich dynamics.
In Section~\ref{git-ss}, following DeMarco, we focus on degenerate rational maps $f$, their induced map $\hat{f}: \P^1 \to \P^1$, the holes of $f$ and their depths, as well as the numerical criteria for (semi)stability in terms of holes and depths. In Section~\ref{upper-bound}, we establish Proposition~\ref{semistable} which states that if $[f] \in I(\Phi_n)$ then $f\in I(d)\cup{\mathcal{U}}_n$. We introduce the basic background on Berkovich dynamics with emphasis on the behavior of the surplus multiplicity in Section~\ref{berkovich}. After discussing reductions in Section~\ref{reductions}, the fundamental relations between Berkovich dynamics and degenerate complex rational maps are established in Section~\ref{depths-and-multiplicities}. Namely we
relate the depths and holes of reductions with surplus multiplicities and Berkovich dynamics. In Section~\ref{perturbation}, we state and prove a simple perturbation lemma for rational maps in Berkovich space which plays a key role in our constructions. Finally, in Section~\ref{complex-action} we briefly discuss the action of complex rational maps on the Berkovich projective line.
\subsection{Stable and semistable rational maps}\label{subsection:stable}
\label{git-ss}
We identify the elements of $\P^{2d+1}$, via coefficients,
with pairs of degree $d$ homogeneous polynomials in two variables modulo
scalar multiplication. That is,
we regard $[P:Q]$ as elements $[a_d:\cdots:a_0:b_d:\cdots:b_0]$ of $\P^{2d+1}$, where $P$ and $Q$ are the degree $d$ homogeneous polynomials
\begin{eqnarray*}
P(X,Y) & = & a_d X^d + a_{d-1} X^{d-1} Y + \cdots + a_0 Y^d,\\
Q(X,Y) & = & b_d X^d + b_{d-1} X^{d-1} Y + \cdots + b_0 Y^d.
\end{eqnarray*}
The space $\Rat_d$ of degree $d$ rational maps corresponds to
all $f=[P:Q] \in \P^{2d+1}$ such that $P$ and $Q$ are relatively prime.
Equivalently, the resultant of $P$ and $Q$, denoted by $\mathrm{Res}(P,Q)$,
does not vanish. Via the identification of $[X:Y] \in \P^1$ with
$z = X/Y \in \mathbb{C} \cup\{\infty\}$ we work, according to convenience, in homogenous or non-homogeneous coordinates.
For $f = [P:Q] \in \P^{2d+1}$, following DeMarco~\cite{DeMarco05}, we will consistently write
\begin{equation}
\label{hat map}
f= H_f \cdot \hat{f} = H_f \cdot [\widehat{P}: \widehat{Q}]
\end{equation}
where $H_f = \gcd(P,Q)$ and $P = H_f \widehat{P}$, $Q = H_f \widehat{Q}$.
Note that the rational map $\hat{f} = [\widehat{P}: \widehat{Q}]$, called the \textit{induced map of $f$}, may have any degree
$\hat{d}$ with $0 \le \hat{d} \le d$.
It has degree $d$ exactly when $f \in \Rat_d$.
If $\mathrm{Res}(P,Q)=0$ or equivalently $\hat{d} \le d-1$, then we say that
$f$ is a \emph{degenerate rational map}. In this case,
the zeros of $H_f$ are called the \emph{holes of $f$}. The set of holes of $f$ is denoted by $\mathrm{Hole}(f)$.
The multiplicity $d_z(f)$ of $z \in \P^1$ as a zero of $H_f$ is called the \emph{depth of $z$}. So $z\in\mathrm{Hole}(f)$ if and only if $d_z(f)\ge 1$.
The action of $\mathrm{PSL}_2(\mathbb{C})$ by conjugation on $\mathrm{Rat}_d$ extends to $\mathbb{P}^{2d+1}$. Geometric invariant theory provides us
with the \emph{stable and semistable loci}, denoted by $\mathrm{Rat}_d^s$ and $\mathrm{Rat}_d^{ss}$, respectively. Both, the stable and
the semistable locus are $\mathrm{PSL}_2(\mathbb{C})$-invariant. Moreover,
$\mathrm{Rat}_d \subset \mathrm{Rat}_d^s \subset \mathrm{Rat}_d^{ss}
\subset \mathbb{P}^{2d+1}$. The quotient of $\mathrm{Rat}_d^s$ by the $\mathrm{PSL}_2(\mathbb{C})$-action is a quasiprojective variety where $\rat_d$ embeds naturally. However, in order to obtain a (compact) projective variety containing $\rat_d$ the semistable locus is taken into account.
That is, we say that two semistable rational maps $f,g\in\mathrm{Rat}_d^{ss}$ are GIT conjugate if the Zariski closures of their $\mathrm{PSL}_2(\mathbb{C})$-orbits have common points. For $f\in\mathrm{Rat}_d^s$ the GIT conjugacy coincides with $\mathrm{PSL}_2(\mathbb{C})$-conjugacy. The categorical quotient $\overline{\mathrm{rat}}_d:=\mathrm{Rat}_d^{ss}//\mathrm{PSL}_2(\mathbb{C})$, which set theoretically is formed by GIT conjugacy classes, is a projective variety that gives us a natural compactification $\overline{\mathrm{rat}}_d$ of the moduli space $\mathrm{rat}_d:=\mathrm{Rat}_d/\mathrm{PSL}_2(\mathbb{C})$. We simply say that \emph{ $\overline{\mathrm{rat}}_d$ is the GIT compactification of $\mathrm{rat}_d$}.
The following equivalent stability criteria are due to Silverman and DeMarco, respectively.
\begin{proposition}[{\cite[Proposition 2.2]{Silverman98}}]
Let $f\in\mathbb{P}^{2d+1}$. Then
\begin{enumerate}
\item $f\not\in\mathrm{Rat}_d^{ss}$ if and only if there exists $M\in\mathrm{PSL}_2(\mathbb{C})$ such that $a_i=0$ for all $i\ge (d+1)/2$ and $b_i=0$ for all $i\ge (d-1)/2$ where $M^{-1}\circ f\circ M=[a_d:\cdots:a_0:b_d:\cdots:b_0]$.
\item $f\not\in\mathrm{Rat}_d^s$ if and only if there exists $M\in\mathrm{PSL}_2(\mathbb{C})$ such that $a_i=0$ for all $i>(d+1)/2$ and for all $b_i=0$ for $i>(d-1)/2$ if we write $M^{-1}\circ f\circ M=[a_d:\cdots:a_0:b_d:\cdots:b_0]$.
\end{enumerate}
\end{proposition}
\begin{proposition}[{\cite[Section 3]{DeMarco07}}]\label{stability-depth}
Let $f=H_f\hat f\in\mathbb{P}^{2d+1}$. Then
\begin{enumerate}
\item $f \in\mathrm{Rat}_d^{ss}$ if and only if the depth $d_z(f)\le (d+1)/2$ for all $z\in\mathbb{P}^1$, and if $d_h(f)\ge d/2$ for some $h\in\mathbb{P}^1$, then $\hat f(h)\neq h$.
\item $f \in\mathrm{Rat}_d^s$ if and only if the depth $d_z(f)\le d/2$ for all $z\in\mathbb{P}^1$, and if $d_h(f)\ge (d-1)/2$ for some $h\in\mathbb{P}^1$, then $\hat f(h) \neq h$.
\end{enumerate}
\end{proposition}
It follows that the behavior of the depths of the holes under
iteration is relevant to the study the indeterminacy locus of $\Phi_n$.
According to DeMarco~\cite{DeMarco05}, for all $n \ge 2$, the indeterminacy locus $I(d)$
of the iteration map $\Psi_n : \mathbb{P}^{2d+1}\dasharrow\mathbb{P}^{2d^n+1}$ defined by $\Psi_n (f) = f^n$ is independent of $n$ and characterized as
$$I(d)=\{ f\in\P^{2d+1}: \hat f \equiv c\in\P^1, c \in \operatorname{Hole}(f)\}.$$
A formula for the iterates of a map outside $I(d)$ as
well as for the depths of its holes is the content of the next lemma. In the sequel, given a complex rational map $g$, we denote by $m_z(g)$ the
\emph{multiplicity of $g$ at $z\in\mathbb{P}^1$}. {That is, $m_z(g)$ is the number of
preimages in a neighborhood of $z$ of a generic
point $w$ close to $g(z)$.}
\begin{lemma}[{\cite[Lemma 2.2]{DeMarco05} and \cite[Lemma 2.4]{DeMarco07}}]
\label{depth-iteration}
If $f\in\mathbb{P}^{2d+1}\setminus I(d)$, then
$$f^n=(\prod_{k=0}^{n-1}(H_f\circ\hat f^k)^{d^{n-k-1}})\hat f^n.$$
Moreover, for all $z\in\mathbb{P}^1$,
\begin{equation}
\label{depth-iterate}
d_z(f^n)=d^{n-1}d_z(f)+\sum_{k=1}^{n-1}d^{n-1-k} m_z(\hat f^k) \, d_{\hat f^k(z)}(f).
\end{equation}
\end{lemma}
The above lemma and the stability criteria suggest that it is useful to work with the \textit{proportional depths}
$$\overline{d}_z(f) = \dfrac{d_z(f)}{\deg f}$$
and the \textit{proportional multiplicities}
$$\overline{m}_z(\hat f) = \dfrac{m_z(\hat f)}{\deg f}.$$
It follows that
$$\overline{d}_z(f^n)=\overline{d}_z(f)+\sum_{k=1}^{n-1} \overline{m}_z(\hat f^k)\, \overline{d}_{\hat f^k(z)}(f).$$
After remarking that for all $z$,
$$\overline{m}_z(\hat{f}^0 = \mathrm{id}) =1,$$
the above Formula (\ref{depth-iterate}) simply becomes
\begin{equation}
\label{proportional-depth-iterate}
\overline{d}_z(f^n)=\sum_{k=0}^{n-1} \overline{m}_z(\hat f^k) \, \overline{d}_{\hat f^k(z)}(f).
\end{equation}\par
It is also convenient to introduce a notation for the proportional depths thresholds for stability and semistability. That is, for $d\ge 2$, define
$$\mu^-(d):=
\begin{cases}
\dfrac{1}{2} & \text{if}\ d\ \text{is even},\\
\\
\dfrac{d-1}{2d} & \text{if}\ d\ \text{is odd},
\end{cases}$$
and
$$\mu^+(d):=
\begin{cases}
\dfrac{1}{2} & \text{if}\ d\ \text{is even},\\
\\
\dfrac{d+1}{2d} & \text{if}\ d\ \text{is odd}.
\end{cases}$$
Then we may write the stability criteria in terms of
the proportional depths as follows:
\begin{proposition}\label{stability-proportional-depth}
Let $d\ge 2$, $n\ge 1$ and $f \in \mathbb{P}^{2d+1}$ with induced map $\hat{f}$. Then
\begin{enumerate}
\item $f \in\mathrm{Rat}_{d}^{ss}$ if and only if the proportional depth $\overline d_z(f)\le\mu^+(d)$ for all $z\in\mathbb{P}^1$, and if $\overline d_h(f)= \mu^+(d)$
for some $h\in\mathbb{P}^1$, then $\hat f(h)\neq h$.
\item $f \in\mathrm{Rat}_{d}^{s}$ if and only if the proportional depth $\overline d_z(f)\le\mu^-(d)$ for all $z\in\mathbb{P}^1$, and if $\overline d_h(f)=\mu^-(d)$ for some $h\in\mathbb{P}^1$, then $\hat f(h)\neq h$
\end{enumerate}
\end{proposition}
\begin{proof}
It immediately follows Proposition \ref{stability-depth} since $\deg f=d$.
\end{proof}
\subsection{Upper bound for $I(\Phi_n)$}
\label{upper-bound}
As mentioned in the introduction, $[f] \in I(\Phi_n)$ if and only if
$\Phi_n$ has no continuous extension to $[f]$. In fact, by
definition of the indeterminacy locus, if $[f] \in I(\Phi_n)$, then there is
no regular map $\widetilde{\Phi}_n$ defined on a neighborhood $U$ of $[f]$ which agrees with
$\Phi_n$ in the open set where $\Phi_n$ is naturally defined.
Then obviously the lack of a continuous extension
of $\Phi_n$ at $[f]$
implies $[f]\in I(\Phi_n)$. Conversely, noting that $\overline{\rat}_d$ is
a normal variety (see \cite[Theorem 2.1]{Silverman98}) and applying the Zariski's main theorem \cite[Section III.9]{Mumford88}, we have that
$[f]\in I(\Phi_n)$ implies that $\Phi_n$ has no continuous extension
at $[f]$. Indeed, by contradiction,
suppose $\Phi_n$ extends continuously at $[f]$. Using the graph of $\Phi_n$ and the
projection onto the first coordinate, we conclude that the graph has
an isolated point above $[f]$. By Zariski's Main Theorem, it follows
that there is a local isomorphism between a neighborhood of $[f]$ and
the graph. The projection $\widetilde{\Phi}_n$ of the local isomorphism onto the
second coordinate coincides with $\Phi_n$, which implies $[f]\not\in
I(\Phi_n)$, that is a contradiction.
The following result extends Lemma 4.2~in~\cite{DeMarco07} and implies that if $[f]\in I(\Phi_n)$ then $f\in I(d)\cup{\mathcal{U}}_n$ in Theorem~\ref{main}.
\begin{proposition}\label{semistable}
Suppose $f\in\mathbb{P}^{2d+1}\setminus I(d)$. If $f^n\in\mathrm{Rat}_{d^n}^{ss}$ for some $n>1$, then $f\in\mathrm{Rat}_d^{ss}$ and the iterate map $\Phi_n$ is continuous at $[f]$.
\end{proposition}
\begin{proof}
From \cite[Lemma 4.2]{DeMarco07}, we may assume $d$ is odd and $f^n\in\mathrm{Rat}_{d^n}^{ss}\setminus\mathrm{Rat}_{d^n}^s$. By contradiction, suppose that $f\not\in\mathrm{Rat}_d^{ss}$. According to Proposition \ref{stability-depth}, there would exist $z\in\mathbb{P}^1$ such that $d_z(f)\ge (d+1)/2$. By Lemma \ref{depth-iteration}, we would have $d_{z}(f^n)\ge (d^{n}+d^{n-1})/2$ which is a contradiction with $f^n\in\mathrm{Rat}_{d^n}^{ss}$.\par
The continuity of $\Phi_n$ at $[f]$ is a direct consequence of the continuity of $\Psi_n : f \mapsto f^n$ at $f\not\in I(d)$ together with
the fact that the semistable loci are open.
\end{proof}
As an immediate consequence we have:
\begin{corollary}
\label{upper-bound-c}
If $[f] \in I(\Phi_n)$, then $f \in I(d)\cup{\mathcal{U}}_n$.
\end{corollary}
\subsection{Berkovich spaces}
\label{berkovich}
In this section we briefly summarize some notions and notations regarding
the Berkovich projective line. For more details, we refer the reader to \cite{Baker10, Benedetto19, Berkovich90, Faber13I, Jonsson15, Kiwi15}.
The algebraic closure of the field of formal Laurent series
$\mathbb{C} ((t))$ with coefficients in $\mathbb{C}$ is
the field $\overline{\mathbb{C} ((t))}$ of formal Puiseux series.
It is naturally endowed with a valuation $\mathrm{ord}(\cdot)$ given by the order
of vanishing at $t=0$ and with its associated
non-Archimedean absolute value $|z| = \mathrm{e}^{-{\mathrm{ord}(z)}}$.
Let $\mathbb{L}$ be the completion of the field of Puiseux series.
Write ${\mathcal{O}}_\L$ for the ring of integers and $\mathfrak{M}_\L$ for the maximal ideal.
Then the residue field ${\mathcal{O}}_\L/ \mathfrak{M}_\L$ is canonically identified with $\mathbb{C}$.
For $ r \ge 0$ and $z \in \L$, let $B_r (z) = \{ w \in \L : |w -z| \le r \}$ and
$B_r^- (z) = \{ w \in \L : |w -z| < r \}$. When $r \notin \mathrm{e}^\mathbb{Q} = |\L^\times|$
these balls coincide: $B_r (z) = B_r^- (z)$. Although both are clopen sets in the metric topology, we say that $B_r (z)$ is a \emph{closed disk} and $B_r^- (z)$ is an \emph{open disk}.
The Berkovich projective line $\mathbf{P}^1$ is a connected compact Hausdorff topological space which contains $\P^1_\L$ as a dense subset \cite[Proposition 2.6 and Lemma 2.9]{Baker10}.
It consists of $4$ types of points. After identification of
$\mathbb{L} \cup \{\infty\}$ with $\P^1_\L$ these types can be described as follows. The points of the projective space $\mathbb{P}^1_{\mathbb{L}}$ are the type I points. The type II (resp. type III) points correspond to closed disks in $\mathbb{L}$ with radii in (resp. not in) the value group $|\mathbb{L}^\times|$.
The type IV points are related to a decreasing sequence of closed disks in $\mathbb{L}$ with empty intersection.
Given a disk $B = B_r (z)$ we will denote the associated point either by
$\xi_B$ or $\xi_{z,r}$ according to convenience.
The type II point $\xi_{0,1}$ associated to the closed unit disk containing $z=0$
is called the \emph{Gauss point} and simply denoted by $\xi_g$.
The space $\mathbb{H}_\mathbb{L}:=\mathbf{P}^1\setminus\mathbb{P}^1_{\mathbb{L}}$ admits a natural hyperbolic metric, see \cite[Section 2.7]{Baker10}. We denote by $\rho(\xi_1, \xi_2)$ the hyperbolic distance of two points $\xi_1,\xi_2\in\mathbb{H}_\mathbb{L}$. With this metric $\mathbb{H}_\mathbb{L}$
is a metric $\mathbb{R}$-tree with endpoints at infinity parametrized by $\P^1_\L$. However, the metric topology of $\mathbb{H}_\mathbb{L}$ is stronger than the subspace topology of ${\mathbf{P}^1}$.
In fact, $\mathbb{H}_\mathbb{L}$ is not locally compact in the metric topology.
For $\xi\in\mathbf{P}^1$, the tangent space $T_{\xi}\mathbf{P}^1$ is the set of connected components of $\mathbf{P}^1\setminus\{\xi\}$. Each element $\vec{v}\in T_{\xi}\mathbf{P}^1$ is called a \emph{tangent vector at $\xi$} and the corresponding connected component is denoted by $\mathbf{B}_{\xi}^-(\vec{v})$. At each type II point $\xi$, the tangent space $T_\xi\mathbf{P}^1$ can be identified with the complex projective line $\mathbb{P}^1$ \cite[Section 3.8.7]{Jonsson15}. At the Gauss point $\xi_g$, this identification is canonical. Namely,
each direction at $\xi_g$ contains a unique point $z \in \P^1 \subset \P^1_\L$.
Now consider a rational map $\phi \in \L(z)$. Then $\phi: \P_\L \to \P_\L$ has a unique continuous extension to Berkovich space
$\phi:\mathbf{P}^1\to\mathbf{P}^1$ \cite[Section 2.3]{Baker10}. At each point $\xi\in\mathbf{P}^1$, the map $\phi$ has a well defined local degree $\deg_\xi\phi$ \cite[Proposition 9.28]{Baker10}.
Moreover, if $\xi$ is a type II point, it induces a \emph{tangent map}
$T_{\xi}\phi:T_{\xi}\mathbf{P}^1\to T_{\phi(\xi)}\mathbf{P}^1$ which is a rational map of degree $\deg_\xi \phi$ in the corresponding $\P^1$-structures, see \cite[Theorem 9.26]{Baker10}.
For each point $\xi\in\mathbf{P}^1$ and each tangent vector $\vec{v}\in T_{\xi}\mathbf{P}^1$, there exist two well defined multiplicities:
the \emph{directional multiplicity $m_\phi ( \vec{v}) \ge 1$} and the
\emph{surplus multiplicity $s_\phi(\vec{v}) \ge 0$} characterized as follows. A point in $\mathbf{B}_{\phi(\xi)}^-(T_{\xi}\phi(\vec{v}))$ has exactly $m_\phi( \vec{v})+s_\phi(\vec{v})$ preimages, counting multiplicities, in $\mathbf{B}_{\xi}^-(\vec{v})$ and a point in the complement of $\mathbf{B}_{\phi(\xi)}^-(T_{\xi}\phi(\vec{v}))$ has exactly $s_\phi( \vec{v})$ preimages, counting multiplicities, in $\mathbf{B}_{\xi}^-(\vec{v})$, see \cite[Proposition 9.41]{Baker10},\cite[Proposition 3.10]{Faber13I} and \cite[Lemma 2.1]{Rivera03II}.
If $\xi$ is a type II point, then $m_\phi(\vec{v})$ coincides with the multiplicity of $T_\xi \phi$ at $\vec{v}$.
Moreover, for all $\xi \in {\mathbf{P}^1}$,
\begin{equation}
\label{surplus-sum}
d = \deg_\xi \phi + \sum_{\vec{v} \in T_{\xi}\mathbf{P}^1} s_\phi(\vec{v}).
\end{equation}
\begin{lemma}\label{surplus-composition}
Let $\phi, \psi \in\mathbb{L}(z)$ be non-constant rational maps. Then for any $\xi\in\mathbf{P}^1$ and $\vec{v}\in T_{\xi}\mathbf{P}^1$,
$$s_{\psi \circ \phi} (\vec{v}) = \deg \psi \cdot s_\phi (\vec{v}) + s_\psi (T_\xi\phi (\vec{v})) \cdot m_\phi (\vec{v}).$$
\end{lemma}
\begin{proof}
Let $\xi_0 = \xi, \xi_1 = \phi(\xi)$ and $\xi_2 = \psi \circ \phi (\xi)$.
Similarly, let $\vec{v}_0 = \vec{v}, \vec{v}_1 = T_\xi\phi(\vec{v})$ and $
\vec{v}_2 = T_{\xi_1}\psi(\vec{v}_1)$.
Given $x \notin \mathbf{B}_{\xi_2}^-(\vec{v}_2)$, out of the $\deg \psi$ preimages under $\psi$
of $x$ there are exactly $s_\psi(\vec{v}_2)$ in $\mathbf{B}_{\xi_1}^-(\vec{v}_1)$.
Each of these $s_\psi(\vec{v}_2)$ points has
$m_\phi( \vec{v})+s_\phi(\vec{v})$ preimages under $\phi$ in
$\mathbf{B}_{\xi}^-(\vec{v})$.
Each of the $\deg \psi - s_\psi(\vec{v}_2)$ preimages under $\psi$
of $x$ which are not in $\mathbf{B}_{\xi_1}^-(\vec{v}_1)$ has
exactly $s_\phi(\vec{v})$ preimages in $\mathbf{B}_{\xi}^-(\vec{v})$.
Thus the total number of preimages of $x$ in $\mathbf{B}_{\xi}^-(\vec{v})$ is
$$s_\psi(\vec{v}_2) \cdot (m_\phi( \vec{v})+s_\phi(\vec{v})) +
(\deg \psi - s_\psi(\vec{v}_2)) \cdot s_\phi(\vec{v}).$$
\end{proof}
Observe that the previous lemma suggests that it is also nicer in this context to work with the \emph{proportional multiplicities} defined as follows:
$$\overline{s}_\phi(\vec{v}) := \dfrac{s_\phi (\vec{v})}{\deg \phi},$$
and
$$\overline{m}_\phi(\vec{v}) := \dfrac{m_\phi (\vec{v})}{\deg \phi}.$$
With this notation the formula of the lemma becomes:
$$\overline s_{\psi \circ \phi} (\vec{v}) = \overline{s}_\phi (\vec{v}) + \overline{s}_\psi (T_\xi\phi (\vec{v})) \cdot \overline{m}_\phi (\vec{v}).$$
Now we consider the behavior of surplus multiplicities under iteration.
When the map $\phi$ is clear from context we lighten notation and simply write $s(\vec{v})$ for $s_\phi(\vec{v})$ and $m( \vec{v})$ for $m_\phi( \vec{v})$. Moreover, for $k \ge 1$,
we write
\begin{eqnarray*}
s^k (\vec{v}) & := & s_{\phi^k} (\vec{v}), \\
\overline{s}^k (\vec{v}) & := & \overline{s}_{\phi^k} (\vec{v}),\\
m^k (\vec{v}) & := & m_{\phi^k} (\vec{v}),\\
\overline{m}^k (\vec{v}) & := & \overline{m}_{\phi^k} (\vec{v}).
\end{eqnarray*}
For $k=0$ we agree that $m^0 = \overline{m}^0 = 1$.
\begin{lemma}\label{surplus-iterate}
Let $\phi \in\mathbb{L}(z)$ be a rational map of degree $d \ge 1$.Then for any $\xi\in\mathbf{P}^1$ and $\vec{v}\in T_{\xi}\mathbf{P}^1$,
$$s^n ( \vec{v})=d^{n-1}s(\vec{v})+\sum_{k=1}^{n-1}m^k(\vec{v}) \cdot s( T_\xi\phi^k(\vec{v}))d^{n-1-k},$$
Equivalently,
\begin{equation}
\label{surplus-iteration}
\overline{s}^n ( \vec{v})= \sum_{k=0}^{n-1} \overline{m}^k(\vec{v}) \cdot \overline{s}( T_\xi\phi^k(\vec{v})).
\end{equation}
\end{lemma}
\begin{proof}
Apply induction after observing that from the previous lemma we have
$$s^n (\vec{v}) = d s^{n-1} (\vec{v}) + m^{n-1} (\vec{v}) s( T_\xi\phi^{n-1}(\vec{v})).$$
\end{proof}
\subsection{Reductions}
\label{reductions}
Under the canonical identification of the residue field ${\mathcal{O}}_\L/\mathfrak{M}_\L$ with $\mathbb{C}$, given
$a \in {\mathcal{O}}_\L$ we denote by $\tilde{a} \in \mathbb{C}$ its reduction $\mod \mathfrak{M}_\L$.
A rational map $\phi$ in $\mathbb{L}(z)$ of degree $d$ is naturally identified with
an element of $\P^{2d+1}_\L$ via its coefficients. In homogenous coordinates we may write
$\phi([X:Y]) = [F(X,Y): G(X,Y)]$ where
\begin{eqnarray*}
F(X,Y)& =& \sum a_j X^j Y^{d-j},\\
G(X,Y) &= &\sum b_j X^j Y^{d-j}
\end{eqnarray*}
for some $a_j,b_j \in \L$.
We identify $\phi$ with
$[a_d : \dots :a_0 : b_d: \dots :b_0] \in \P^{2d+1}_\L$ and also write
$\phi=[F:G] \in \P^{2d+1}_\L$.
There are two related notions of ``reductions''
of $\phi$, one as a \emph{map} which we will denote by $\tilde{\phi}$,
and the other by its \emph{coefficients} which will be denoted by $\phi_0$.
To introduce both reductions we first consider a \emph{normalized representation of $\phi$}
which, in homogenous coordinates, consists on scaling $\phi$ by a suitable element
of $\L$ so that we have $\phi = [F: G]$ where $F, G \in {\mathcal{O}}_\L [X,Y]$ with at least one coefficient being a unit. Equivalently, it consists on scaling in order to write
$\phi= [a_d : \dots :a_0 : b_d: \dots :b_0]$ where $a_j,b_j \in {\mathcal{O}}_\L$ with at least one entry of absolute value $1$.
We say that $$\phi_0 =[\widetilde{F}: \widetilde{G}] = [\widetilde{a_d} : \dots :\widetilde{a_0} : \widetilde{b_d}: \dots :\widetilde{b_0}]
\in \P^{2d+1}$$ is the \emph{coefficient reduction} of $\phi$. The coefficient reduction is independent of the normalized representation of $\phi$.
Note that the coefficient reduction is just the one induced by reduction on
parameter space, that is, the natural reduction from $\P^{2d+1}_\L$ onto $\P^{2d+1}$.
Following Rumely, our notion of coefficient
reduction is a particular case
of a more general notion of reduction at type II points.
Indeed, given $\psi \in \L(z)$, a type II point $\zeta_0 \in \mathbf{P}^1$ and a M\"obius transformation $M \in \L(z)$ such that $M(\xi_g) = \zeta_0$, we say that the coefficient reduction $f$ of $M \circ \psi \circ M^{-1}$ is a \emph{reduction of $\psi$ at $\zeta_0$}. This reduction is unique up to conjugacy by a M\"obius transformation in $\mathbb{C}(z)$. The coefficient reduction introduced above corresponds to reduction at the Gauss point. It follows that a reduction of $\psi$ at
a point $\zeta_0$ is stable, semistable or unstable independently of the choice of $M$.
Let $H_{\phi_0} (X,Y) = \gcd (\widetilde{F}(X,Y) , \widetilde{G} (X,Y) )$ and consider
$\widehat{F} ,\widehat{G} \in \mathbb{C}[X,Y]$ such that $\widetilde{F}
= H_{\phi_0} \cdot \widehat{F}$ and $\widetilde{G} = H_{\phi_0} \cdot \widehat{G}$. Then we say that $\tilde{\phi} =
[\widehat{F} :\widehat{G}]$ is the \emph{reduction of $\phi$}.
Note that $\tilde{\phi}$ is induced by reduction on dynamical space, that is, by the natural projection $\P^{1}_\L \to \P^{1}$.
With the notation of Section~\ref{git-ss}, we have that
$$\phi_0 = H_{\phi_0} \cdot \tilde{\phi}.$$
Thus, the induced map of the coefficient reduction is the reduction map:
$$\widehat{\phi}_0 = \tilde{\phi}.$$
\subsection{Depths and multiplicities}
\label{depths-and-multiplicities}
Depths of holes and surplus multiplicities are closely related when we consider holomorphic families of rational maps as dynamical systems acting on the Berkovich projective line.
Given a neighborhood $V$ of $t=0$ in $\mathbb{C}$, we say that a family $\{f_t\}\subset\mathbb{P}^{2d+1}$ parametrized by $t \in V$ is a \emph{holomorphic family of rational maps} if the map $V \to\mathbb{P}^{2d+1}$, sending $t$ to $f_t$, is holomorphic and $f_t\in\mathrm{Rat}_d$ for all $t\not=0$. If $f_0 \notin \mathrm{Rat}_d$, we say that the family $\{f_t\}$ is a \emph{degenerate} holomorphic family of rational maps.
A holomorphic family $\{f_t\}$ of degree $d\ge 1$ complex rational maps induces a rational map $\mathbf{f}:\mathbf{P}^1\to\mathbf{P}^1$
since $f_t = [F_t:G_t]$ where $F_t$ and $G_t$ are homogenous polynomials in two variables with coefficients given by holomorphic functions in $t$. In particular, the coefficients
are Taylor series in $t$ and thus we may regard the family $\{f_t\}$ as a rational map
$\mathbf{f} \in \L(z)$.
The coefficient reduction of $\mathbf{f}$
is precisely $f_0$. We will systematically abuse of notation also writing $f_t$ for
the rational map $\mathbf{f}$ with coefficients in $\L$ and
$f_t : {\mathbf{P}^1} \to {\mathbf{P}^1}$ for its
action on Berkovich space.
\begin{lemma}[{\cite[Lemma 2.17]{Baker10} and \cite[Lemma 3.17]{Faber13I}}] \label{depth-surplus}
Let $\phi \in\mathbb{L}(z)$ be a rational map.
Then the reduction map $\tilde{\phi}$ is non-constant if and only if $\phi(\xi_g) = \xi_g$.
In this case, under the canonical identification $z \leftrightarrow \vec{v}_z$ of
$\P^1$ with $T_{\xi_g} \mathbf{P}^1$ we have that
$T_{\xi_g} \phi = \tilde{\phi}$. Moreover,
\begin{eqnarray*}
d_z (\phi_0) &= & s_\phi (\vec{v}_z),\\
m_z (\tilde{\phi}) &= & m_\phi (\vec{v}_z).
\end{eqnarray*}
\end{lemma}
If the reduction map is constant, we have
\begin{lemma}\label{main-lemma}
Consider a degree $d$ rational map $\phi \in\mathbb{L}(z)$ such that
$\phi(\xi_g) = \xi \neq \xi_g$.
Let $\vec{w}$ be the direction at $T_\xi {\mathbf{P}^1}$ containing the Gauss point.
Then
$$
d_z(\phi_0)=
\begin{cases}
s_\phi(\vec{v}_z) & \mbox{if }\,\, T_{\xi_g}\phi(\vec{v}_z)\not=\vec{w},\\
s_\phi(\vec{v}_z) + m_\phi (\vec{v}_z) & \mbox{if }\,\, T_{\xi_g}\phi(\vec{v}_z) =\vec{w}.
\end{cases}
$$
\end{lemma}
\begin{proof}
Consider a degree $1$ map $M \in \L(z)$ such that $\xi = M(\xi_g)$.
We claim that $u \in \P^1$ is a hole of $M_0$ if and only if the corresponding
direction $\vec{v}_u$ is such that $\xi_g \in M(\vec{v}_u)$.
Namely we claim the assertion of the lemma for degree $1$ maps.
We assume that the hole $h$ of $M$ is not $\infty$ and proceed
using non-homogenous coordinates. For $h = \infty$ the claim follows along similar lines.
Since $h \neq \infty$, there exists $a, b \in {\mathcal{O}}_\L$ and $c \in \L$
such that
$$M (z) = c \cdot \dfrac{z-a}{z-b}$$
with
$$M_0 (z) = H_\mu(z) \cdot \alpha,$$
where $H_\mu(z) = z -h$ and $\alpha = \tilde{c}$, $h = \tilde{a} = \tilde{b}$.
It follows that $M$ maps every direction $\vec{v}_z$ with $z \neq h$,
into the direction $\vec{v}_\alpha$ and therefore $\xi_g \in M(\vec{v}_h)$. The claim easily
follows.
Let $\phi$ be as in the statement of the Lemma and consider
$\psi (z) = M^{-1} \circ \phi (z) \in \L(z)$
with non-constant reduction map $\tilde{\psi}$.
It follows that $$\phi_0 (z) = (M \circ \psi)_0 (z) = H_{\psi_0} (z) \cdot H_\mu ( \tilde{\psi} (z)) \cdot \alpha,$$
where $\psi_0 = H_{\psi_0} \cdot \tilde{\psi}$.
Therefore, if $H_\mu ( \tilde{\psi} (z)) \neq 0$, we have $d_z(\phi_0)=d_z(\psi_0) = s_\psi(\vec{v}_z)$, otherwise, if $H_\mu ( \tilde{\psi} (z)) = 0$, we have
$d_z(\phi_0) = d_z(\psi_0) + m_z (\tilde{\psi})$.
Since $s_\phi (\vec{v}) = s_\psi (\vec{v})$ for all $\vec{v}$ and
$m_z (\tilde{\psi}) = m_\psi (\vec{v}_z) = m_\phi(\vec{v}_z)$, it only remains to observe that the above claim says that
$H_\mu ( \tilde{\psi} (z)) = 0$ if and only if $T_{\xi_g}\phi(\vec{v}_z) =\vec{w}$.
\end{proof}
We apply the previous lemma to study the depths of the holes of the reduction at a general type II point $\zeta_0$:
\begin{corollary}\label{relative-position}
Consider a rational map $\phi \in \L(z)$ and a type II point $\xi_0 \in \mathbf{P}^1$. Let $\xi_n = \phi^n (\xi_0)$
and let $L$ in $\L(z)$ be an affine map such that $\xi_g = L(\xi_0)$. Set
$$f=(L \circ \phi^n \circ L^{-1})_0.$$
Given $z \in\P^1$, let $\vec{v} \in T_{\xi_0} {\mathbf{P}^1} $ be such that $\vec{v}_z= T_{\xi_0}L(\vec{v})$. Then
$$d_z(f)=
\begin{cases}
s^n_\phi (\vec{v}) & \mbox{if}\,\, \xi_0\ \mbox{is not in}\,\, T_{\xi_0} \phi^n (\vec{v}), \\
s^n_\phi (\vec{v}) + m^n_\phi (\vec{v}) & \mbox{if}\,\, \xi_0\ \mbox{is in}\,\, T_{\xi_0} \phi^n (\vec{v}).
\end{cases}$$
\end{corollary}
\begin{proof}
Set $\varphi=L \circ \phi^n \circ L^{-1}$ and $\xi=\varphi(\xi_g)$. If $\xi=\xi_g$, then the conclusion follows form Lemma \ref{depth-surplus}. If $\xi\not=\xi_g$, let $\vec{w}$ be the direction at $\xi$ containing $\xi_g$. By Lemma ~\ref{main-lemma}, for any direction $\vec{v}_z$ at $\xi_g$, we have
$$
d_z(\varphi_0)=
\begin{cases}
s_\varphi(\vec{v}_z) & \mbox{if }\,\, T_{\xi_g}\varphi(\vec{v}_z)\not=\vec{w},\\
s_\varphi(\vec{v}_z) + m_\varphi (\vec{v}_z) & \mbox{if }\,\, T_{\xi_g}\varphi(\vec{v}_z) =\vec{w}.
\end{cases}
$$
Since $\xi=L(\xi_n)$, the lemma follows after taking the preimages of the relevant directions under the affine map $L$.
\end{proof}
\subsection{Perturbation of rational maps in Berkovich space}
\label{perturbation}
Our constructions rely on starting with a map $\phi \in \L(z)$ and conveniently increasing its degree by strategically placing new zeros and poles. We may perform this ``perturbation'' without changing the action of $\phi$ nearby the Gauss point provided that the new zeros and poles are sufficiently close:
\begin{lemma}\label{construction}
Let $\phi\in\mathbb{L}(z)$ be a {nonconstant} rational map. Let $\xi=\xi_{z_0,|t|^\alpha}$ and let $\phi(\xi)=\xi_{w_0,|t|^\beta}$. Suppose $N>\max\{\alpha,\beta,\alpha+\beta\}$ is an integer and $p\in\mathbb{L}$. Consider
$$\psi(z):=\left(1+\frac{t^N}{z-p}\right)\phi(z).$$
Then $\psi(\xi)=\phi(\xi)$ and $T_{\xi}\psi=T_\xi\phi$.
Moreover, provided that $\phi(p) \neq0$, for any $\vec{v} \in T_\xi {\mathbf{P}^1}$, if $p$ is in the direction $\vec{v}$
then $s_\psi (\vec{v}) = s_\phi (\vec{v}) +1$, otherwise $s_\psi (\vec{v}) = s_\phi (\vec{v})$.
\end{lemma}
\begin{proof}
Identifying $T_{\xi}\mathbf{P}^1$ with $\P^1$, for all but finitely many $c\in\P^1$, we have
$$\phi(z_0+ct^\alpha+h.o.t.)=w_0+T_{\xi}\phi(c)t^\beta+h.o.t.$$
For such $c$, we have
$$\psi(z_0+ct^\alpha+h.o.t.)=(1+\frac{t^N}{z_0+ct^\alpha+h.o.t.-p})(w_0+T_{\xi}\phi(c)t^\beta+h.o.t.).$$
If $|p-z_0|\le |t|^\alpha$,
\begin{align*}
\psi(z_0+ct^\alpha+h.o.t.) &=(1+\frac{t^N}{c't^\alpha+h.o.t.})(w_0+T_{\xi}\phi(c)t^\beta+h.o.t.)\\
&=w_0+T_{\xi}\phi(c)t^\beta+\frac{w_0}{c'}t^{N-\alpha}+\frac{T_{\xi}\phi(c)}{c'}t^{N-\alpha+\beta}+h.o.t..
\end{align*}
If $|p-z_0|>|t|^\alpha$,
\begin{align*}
\psi(z_0+ct^\alpha+h.o.t.)&=(1+\frac{t^N}{(z_0-p)+ct^\alpha})(w_0+T_{\xi}\phi(c)t^\beta+h.o.t.)\\
&=w_0+T_{\xi}\phi(c)t^\beta+\frac{w_0}{z_0-p}t^{N}+\frac{T_{\xi}\phi(c)}{z_0-p}t^{N+\beta}+h.o.t.
\end{align*}
Since $N>\max\{\alpha,\beta,\alpha+\beta\}$, we have $\psi(\xi)=\phi(\xi)$ and $T_{\xi}\psi=T_\xi\phi$.
Finally, counting the number of preimages of $\infty$ in the direction $\vec{v}$, the lemma follows.
\end{proof}
An immediate corollary of Lemma \ref{construction} is the following.
\begin{corollary}
\label{perturbation-c}
Let $\phi\in\mathbb{L}(z)$ be a non-constant rational map. Suppose $\Gamma$ is a graph contained in a bounded set in $\mathbb{H}_\mathbb{L}$ with respect to the hyperbolic metric $\rho$. For $p_1,\cdots p_k\in \mathbb{L}$, consider
$$\psi(z):=\phi(z)\prod_{i=1}^k\left(1+\frac{t^N}{z-p_i}\right).$$
Then for sufficiently large $N>0$, we have $\psi(\xi)=\phi(\xi)$ and $T_{\xi}\psi=T_\xi\phi$ for all $\xi\in\Gamma$.
Moreover, suppose that $\phi(p_i) \neq0$ for all $1\le i\le k$. Then
for any $\xi\in\Gamma$ and $\vec{v} \in T_\xi {\mathbf{P}^1}$, we have
$$s_\psi (\vec{v}) = s_\phi (\vec{v}) +\#\{i: p_i\in\mathbf{B}_\xi^-(\vec{v})\}.$$
\end{corollary}
\subsection{Action of complex rational maps on Berkovich space}
\label{complex-action}
The starting point of our constructions are complex rational maps $g \in \mathbb{C}(z)$ of degree at least $1$ which we
may regard as elements of $\L(z)$. The action of such maps on ${\mathbf{P}^1}$ is not difficult to describe. In fact, elementary arguments omitted here show that
for all $z \in \mathbb{C} \subset \P^1_\L$ and $\alpha >0$, we have that
$$\xi= \xi_{z, |t|^\alpha} \mapsto g(\xi) =\xi_{g(z), |t|^{\alpha m}},$$
where $m= m_z(g)$ is the (complex)
multiplicity of $z$. That is, $g$ is linear in the interval $[\xi_g, z[$ (with respect to the hyperbolic length) with ``slope'' $m_z(g)$. For all $0 < \alpha \in \mathbb{Q}$, the point $\xi$ is of type II, and for all $w \in \mathbb{C}$, the direction in $T_\xi {\mathbf{P}^1}$ containing
$z + w t^\alpha$ is mapped by $T_\xi g$ to the direction containing $g(z) + w^m t^{m \alpha}$ with zero surplus multiplicity. The direction containing the Gauss point is mapped by
$T_\xi g$ to the direction at $g(\xi)$ containing the Gauss point with surplus multiplicity $\deg g - m$.
The multiplicity of a direction in $T_\xi{\mathbf{P}^1}$ is $m$ if it contains $z$ or the Gauss point, and $1$ otherwise. A similar description holds for $z = \infty$.
\section{The bad hole of $n$-unstable maps}
\label{bad-hole-s}
In this section we show that each $n$-unstable map $f$ has a distinguished hole $\mathtt{h}$ where semistability of the $n$-th iterate fails.
The dynamics and depth of this distinguished hole $\mathtt{h}$, which we call the ``bad hole of $f$'', will organize the proof of Theorem~\ref{main}.
In fact, the next section is devoted to prove that if a $n$-unstable map has non-constant induced map $\hat f$, then $[f] \in I(\Phi_n)$.
Our proof relies on the construction of holomorphic families through $f$ that confirm the indeterminacy of $\Phi_n$ at $[f]$.
The constructions are organized in five cases according to the dynamics and depth of the bad hole.
\medskip
Recall that the set of $n$-unstable maps is denoted by ${\mathcal{U}_n}$.
\subsection{Maps in ${\mathcal{U}}_n$ and the bad hole}
For $f\in{\mathcal{U}}_n$, the semistability condition in Proposition \ref{stability-depth} for $f^n$ breaks down at a unique hole:
\begin{lemma}\label{bad hole}
If $f\in{\mathcal{U}}_n$, then there is a unique $h\in\mathrm{Hole}(f)$ such that $d_h(f^n)\ge d^n/2$.
\end{lemma}
\begin{definition}[Bad Hole]
Given $f \in {\mathcal{U}_n}$, we say that the hole given by Lemma \ref{bad hole} is the \emph{bad hole of $f$ for the $n$-th iterate} and we denote it by $\mathtt{h}$.\par
\end{definition}
\begin{proof}
Consider $f \in {\mathcal{U}_n}$. If $d$ is odd, then there is a unique hole $h\in\mathrm{Hole}(f^n)$ such that $d_h(f^n)\ge (d^n+1)/2$. Indeed, since the sum of the depth of the holes is at most $d^n$, we have that no other hole has depth at least $d^n/2$.
Now we consider the even degree case. Since $f\not\in I(d)$, then $f^n\not\in I(d^n)$. For otherwise, there exists $h\in\mathrm{Hole}(f^n)$ such that $\widehat{f^n}=h$. By Lemma \ref{depth-iteration}, we have that $h\in\mathrm{Hole}(f)$ and $\hat f=h$. It follows that $f\in I(d)$, which is a contradiction. We claim that $f^n$ has a unique hole $h\in\mathrm{Hole}(f^n)$ with depth $d_h(f^n)\ge d^n/2$. By contradiction, suppose there are two distinct holes in $\mathrm{Hole}(f^n)$ with depths at least $d^n/2$. Then up to conjugacy,
$$f^n([X:Y])=X^{d^n/2}Y^{d^n/2}[1:1].$$
Hence $f^n\in\mathrm{Rat}_{d^n}^s$. So $f\not\in{\mathcal{U}}_n$ which is a contradiction. \par
Now we show $h\in\mathrm{Hole}(f)$. Note
$$\mathrm{Hole}(f^n)=\bigcup_{i=0}^{n-1}\hat f^{-i}(\mathrm{Hole}(f)).$$
In particular, if $\deg\hat f=0$, then $\mathrm{Hole}(f^n)=\mathrm{Hole}(f)$. So we may assume that $\hat{f}$ is not constant.
We proceed by contradiction. Suppose $h\not\in\mathrm{Hole}(f)$. Then
$$\dfrac{d^n}{2} \le d_h ( f^n ) = m_h (\hat{f}) \, d_{\hat{f}(h)} (f^{n-1}) \le d \cdot d_{\hat{f}(h)} (f^{n-1}) \le d_{\hat{f}(h)} (f^{n}).$$
By the already proven uniqueness of the bad hole, $\hat{f}(h) = h$, and hence
$h$ is a hole of $f$ which contradicts $h\not\in\mathrm{Hole}(f)$.
\end{proof}
\begin{corollary}
\label{increasing}
For all $n \ge 2$, we have that $\mathcal{U}_n \subset \mathcal{U}_{n+1}$. Moreover,
if $\mathtt{h}$ is the bad hole for the $n$-th iterate of $f \in \mathcal{U}_n$, then
$\mathtt{h}$ is the bad hole for the $n+1$-th iterate of $f$.
\end{corollary}
\begin{proof}
Consider $f \in \mathcal{U}_n$.
If $f^n$ has a hole $h$ of depth $d_h (f^n) > (d^n+1)/2 $ then
$d_h (f^{n+1}) > (d^{n+1}+d)/2$. Hence $f^{n+1}\not\in\mathrm{Rat}_{d^{n+1}}^{ss}$.
If $f^n$ has a hole $h$ with $ (d^n+1)/2 \ge d_h (f^n) \ge d^n/2 $ such that
$\hat f^n (h) = h$, then $h$ is a hole of $f$ and $\hat f$ is non-constant, since $f \notin I(d)$. Moreover, $$d_h (f^{n+1} ) = d_h (f^n) \cdot d +
m_h (\hat f^n) \cdot d_h (f) \ge d^{n+1}/2 + 1.$$
Hence, $f^{n+1}\not\in\mathrm{Rat}_{d^{n+1}}^{ss}$ (i.e. $f \in \mathcal{U}_{n+1}$).
\end{proof}
The following result asserts that the forward orbit of the bad hole meets a hole of $f$ (maybe the bad itself) provided the depth is at most $d/2$.
\begin{proposition}\label{orbit-hole}
Suppose that $f\in\mathcal{U}_n$ has non-constant induced map $\hat f$. If the forward $\hat f$-orbit of the bad hole $\mathtt{h}$ does not intersect $\mathrm{Hole}(f)$, then $d_\mathtt{h}(f)=(d+1)/2$.
\end{proposition}
\begin{proof}
Since $f\in\mathrm{Rat}_d^{ss}$, by Proposition \ref{stability-depth}, we have that $d_\mathtt{h}(f)\le (d+1)/2$. By contradiction, suppose that the forward orbit of $\mathtt{h}$ is hole free and $d_\mathtt{h}(f)<(d+1)/2$. Then $d_\mathtt{h}(f)\le d/2$. Since $f^n\not\in\mathrm{Rat}^{ss}_{d^n}$ and $\hat f^n(\mathtt{h})\not=\mathtt{h}$, again by Proposition \ref{stability-depth}, we have that $d_\mathtt{h}(f^n)> d^n/2$. However, by Formula (2), we obtain that $d_{\mathtt{h}}(f^n)=d^{n-1}d_\mathtt{h}(f)\le d^n/2$, which is a contradiction.
\end{proof}
\subsection{Multiplicity inequality for $n$-unstable maps}
In order to classify $n$-unstable maps, it will be useful to employ
a basic inequality involving the multiplicities of the bad hole.
\begin{lemma}\label{depth-multiplicity-inequality}
If $f \in\mathcal{U}_n$ and $\mathtt{h}$ is the bad hole of $f$, then
$$2 d_\mathtt{h} (f) + {m_\mathtt{h} (\hat{f})} > d,$$
equivalently
$$2 \overline{d}_\mathtt{h} (f) + {\overline{m}_\mathtt{h} (\hat{f})} > 1.$$
\end{lemma}
\begin{proof}
Assume that $\hat{f} (\mathtt{h}) = \mathtt{h}$. If $\overline{m}_{\mathtt{h}}(\hat f )\not=1$, then
$$\dfrac{1}2 \le \overline{d}_{\mathtt{h}} (f^n)= \overline{d}_{\mathtt{h}} (f) \cdot \sum_{k=0}^{n-1}
\overline{m}_{\mathtt{h}}(\hat f^k) = \overline{d}_{\mathtt{h}} (f) \cdot \dfrac{1- \overline{m}_{\mathtt{h}}(\hat f )^n}{1- \overline{m}_{\mathtt{h}}(\hat f )}<\dfrac{\overline{d}_{\mathtt{h}} (f)}{1- \overline{m}_{\mathtt{h}}(\hat f )}.$$
If $\overline{m}_{\mathtt{h}}(\hat f )=1$, then
$$2 \overline{d}_\mathtt{h} (f) + {\overline{m}_\mathtt{h} (\hat{f})}\ge \frac{2}{d}+1 > 1.$$
Now assume $\hat{f} (\mathtt{h}) \neq \mathtt{h}$. Since
there is a unique bad hole, it follows that $\overline{d}_{\hat f(\mathtt{h})} (f^{n-1}) < 1/2$. Indeed, for otherwise,
$$\overline{d}_{\hat f(\mathtt{h})} (f^{n}) = \overline{d}_{\hat f(\mathtt{h})} (f^{n-1}) +\overline{m}_{\hat f(\mathtt{h})} (\hat f^{n-1}) \,
\overline{d}_{\hat f^{n-1} (\mathtt{h})}(f) \ge 1/2,$$
which implies that $\hat f(\mathtt{h})$ is the bad hole.
Therefore, {if $\hat f$ is nonconstant, we have}
$$\dfrac{1}2 \le \overline{d}_\mathtt{h} (f^n) = \overline{d}_{\mathtt{h}} (f) + \overline{m}_{\mathtt{h}} (\hat{f}) \cdot \overline{d}_{\hat f(\mathtt{h})} (f^{n-1}) < \overline{d}_\mathtt{h} (f) + \dfrac{ \overline{m}_\mathtt{h} (\hat{f})}2.$$
When $\hat{f}$ is constant ($\neq \mathtt{h}$) we have that $d_\mathtt{h}(f) > d/2$. In fact, otherwise we would have
$ {d}_{\mathtt{h}}(f^n) = d_\mathtt{h}(f) \cdot d^{n-1} \le d^n/2$ which would contradict $f \in {\mathcal{U}}_n$. Therefore $d_\mathtt{h}(f)=(d+1)/2$ and $$2 \overline{d}_\mathtt{h} (f) + {\overline{m}_\mathtt{h} (\hat{f})}=\frac{d+1}{d}> 1.$$
\end{proof}
\begin{remark}
The previous lemma suggests that the $n$-unstable set ${\mathcal{U}}_n \subset \P^{2d+1}$ has
codimension $d-1$.
Indeed, parametrize maps $f$ in $\overline{\Rat}_d = \P^{2d+1}$ locally by the location of the zeros $c_1, \dots, c_d$, the poles $p_1, \dots, p_d$ and the value $a$ of $f$ at $z=\infty$.
Then having a hole of depth $D$ corresponds to a union of codimension $2D-1$
linear subvarieties of the local parameters
$(c_1, \dots, c_d, p_1, \dots, p_d, a) \in \mathbb{C}^{2d+1}$.
Having multiplicity $M$ at that hole corresponds to $M-1$ equations on the local parameters.
This suggests that the codimension of having a hole of depth $D$ and multiplicity $M$
is $2D+M -2$. Since the codimension of maps $f$ in ${\mathcal{U}}_n$ with constant induced map is large we may assume that $\hat f$ is non-constant.
In the case that $\hat{f}$ is non-constant and $D < (d+1)/2$, then from Proposition \ref{orbit-hole} we obtain
an extra equation for ${\mathcal{U}}_n$. Thus the codimension should be at least $2D+M-1$.
From the above lemma one would conclude that in $\P^{2d+1}$ the codimension of such $f \in {\mathcal{U}}_n$
is at least $d $.
In the case that $D=(d+1)/2$, then $2D+M -2\ge d$ and the codimension is also at least $d$.
Hence one should expect the dimension of
${\mathcal{U}}_n$ to be at most $d+1$. After projecting ${\mathcal{U}}_n$ to moduli space the dimension of $I(\Phi_n)$ should be at most $d-2$.
For $d \ge 3$ and sufficiently large $n$, in
Corollary~\ref{dimension} we exhibit $(d-2)$-dimensional subsets
of $I(\Phi_n)$ formed by conjugacy classes of $n$-unstable maps.
\end{remark}
\subsection{Bad hole of depth $1$}
If the bad hole $\mathtt{h}$ has depth $1$, then the induced map $\hat f$ is a polynomial or a monomial of degree $d-1$:
\begin{proposition}\label{depth1poly}
Let $d \ge 3$ and consider $f\in{\mathcal{U}}_n$ such that $d_{\mathtt{\infty}}(f)=1$ where $\mathtt{h=\infty}$ is the bad hole of $f$. Then,
\begin{enumerate}
\item
$\hat f$ is a degree $d-1$ polynomial or,
\item $\hat f$ is modulo an affine change of coordinates the
monomial $z^{-(d-1)}$.
\end{enumerate}
\end{proposition}
\begin{proof}
By Lemma~\ref{depth-multiplicity-inequality}, we have that $m_\mathtt{h} (\hat f) = d-1$. Hence $\deg \hat f = d-1$ and $f$ has a unique hole. It follows that there exists a smallest integer $k\ge 1$ such that $\hat f^k(\infty)=\infty$. For otherwise, $d_{\hat f^\ell(\infty)}(f)=0$ for all $\ell\ge 1$ and $d_\infty(f^n)=d^{n-1} < d^n /2$ (Lemma \ref{depth-iteration}) but this contradicts the lower bound $d_\infty(f^n)\ge d^n/2$ due to the fact that
$\infty$ is the bad hole (Lemma \ref{bad hole}).
Thus
$$\frac{d^n}{2}\le d_{\infty}(f^n)\le d^{n-1}+\sum_{i\ge1}d^{n-ik-1}(d-1)^{ik}=\frac{d^{n+k-1}}{d^k-(d-1)^k}.$$
Therefore,
$$d^k-(d-1)^k\le 2d^{k-1}.$$
Hence $k=1$ or $2$. Indeed, if $k\ge 3$, then
$$d^k\left(1-\left(1-\dfrac{1}{d}\right)^k\right)\ge d^k\left(1-\left(1-\dfrac{1}{d}\right)^3\right)=3d^{k-1}-3d^{k-2}+d^{k-3}>2d^{k-1}.$$
If $k=1$, then $\hat f^{-1}(\infty)=\{\infty\}$, since $m_{\infty}(\hat f)=d-1$. Therefore, $\hat f$ is a degree $d-1$ polynomial.
In the case that $k=2$, we claim $m_{\hat f(\infty)}(\hat f)=d-1$, for otherwise $m_{\hat f(\infty)}(\hat f)\le d-2$. Then we would have
$$d_{\infty}(f^n)\le d^{n-1}+\sum_{i\ge 1}d^{n-1-2i}(d-1)^i(d-2)^i=\frac{d^{n+1}}{d^2-(d-1)(d-2)}<\frac{d^n}{2},$$
which is a contradiction. Changing coordinates so that $\hat f (\infty) = 0$ we have that $\hat f (z) = 1/z^{d-1}$ after conjugacy by an appropriate dilation $z \mapsto \lambda z$.
\end{proof}
\section{Non-constant induced map}\label{deep}
As mentioned in the introduction, for $d=2$, Theorem \ref{main} is a consequence of DeMarco's result, see \cite[Theorem 5.1]{DeMarco07}. In this section, we always assume $d\ge 3$.
Our aim is to prove Theorem~\ref{Thm:perturbation} and the following implication in Theorem \ref{main}:
\begin{theorem}
\label{non-constant}
If $f \in {\mathcal{U}_n}$ and $\deg \hat f \ge 1$, then $[f] \in I(\Phi_n)$.
\end{theorem}
\subsection{General strategy}
In this subsection, we state the general strategy to prove Theorem \ref{Thm:perturbation} and Theorem \ref{non-constant}.
From Proposition~\ref{depth1poly} it directly follows that we may organize the proofs of theorems~\ref{Thm:perturbation} and \ref{non-constant} in cases according to the following proposition:
\begin{proposition}
\label{cases}
Given $f \in {\mathcal{U}_n}$ such that $\deg \hat f \ge 1$ denote by $\mathtt{h}$ the bad hole of $f$.
Let ${\mathcal{O}}(\mathtt{h})$ be the forward orbit of $\mathtt{h}$ under $\hat f$ and denote by
$\# {\mathcal{O}}(\mathtt{h})$ its cardinality.
Then one of the following cases hold:
\begin{itemize}
\item[Case 0.] $d_{\mathtt{h}}(f) \ge 2$ and $n \le \# {\mathcal{O}}(\mathtt{h})\le \infty$.
\item[Case 1.] $d_{\mathtt{h}}(f) \ge 2$, $\# {\mathcal{O}}(\mathtt{h}) < n$ and $h$ is strictly preperiodic.
\item[Case 2.] $d_{\mathtt{h}}(f) \ge 2$, $\# {\mathcal{O}}(\mathtt{h}) < n$ and ${\mathcal{O}}(\mathtt{h})$ is a periodic superattracting orbit.
\item[Case 3.] $d_{\mathtt{h}}(f) \ge 2$, $\# {\mathcal{O}}(\mathtt{h}) < n$ and ${\mathcal{O}}(\mathtt{h})$ is a periodic but not superattracting orbit.
\item[Case 4.] $d_{\mathtt{h}}(f) = 1$ and $\# {\mathcal{O}}(\mathtt{h}) =1$.
\item[Case 5.] $d_{\mathtt{h}}(f) = 1$ and $\# {\mathcal{O}}(\mathtt{h}) =2$.
\end{itemize}
\end{proposition}
With the exception of $d=3$ in cases 4 and 5, we produce
for each $\lambda$ in the complement of a finite subset of $\mathbb{C}$, a degenerate holomorphic family $g_{\lambda,t}$ of degree $d$ such that $g_{\lambda,0} = f$. The construction is implemented so that
for a conveniently chosen holomorphic family of M\"obius transformations $M_t$, we have that
$M_t^{-1} \circ g^n_{\lambda, t} \circ M_t \to G_\lambda$ as $t \to 0$, where $G_\lambda$ is some stable map
of degree $d^n$ (i.e. in $\Rat^{s}_{d^n}$).
Thus, $\Phi_n ([g_{\lambda,t}]) \to [G_\lambda]$ as $t \to 0$.
The construction is also implemented so that
the GIT-classes $[G_\lambda]$ vary with $\lambda$, which allows us to conclude that $\Phi_n$ has no continuous extension to $[f]$.
The construction site is the Berkovich projective line ${\mathbf{P}^1}$. For $d \ge 4$, in Berkovich space language our constructions prove Theorem~\ref{Thm:perturbation}.
We start by regarding the induced map $\hat f$ as an element of $\L(z)$ acting on ${\mathbf{P}^1}$ (see Section~\ref{complex-action})
and prescribe a priori the family $M_t$ above to be $$M_t (z) = h_0 + t z.$$
We let $\zeta_0 = M_t ( \xi_g)$ and proceed to construct $g_{\lambda,t}$ so that we have control over the
surplus multiplicities of $g^n_{\lambda,t}$ in all the directions $\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}$.
Starting from $\hat f$ there is plenty of flexibility in order to construct $g_{\lambda,t}$ such that $g_{\lambda,0} = f$.
For simplicity, assume that $\infty$ is not a hole of $f$. For each hole $w$ of $f$ of depth $d_w$, we may choose
$2d_w$ arbitrary points $c_1(w), \dots, c_{d_w}(w), p_1(w), \dots, p_{d_w}(w)$ in $B^-_1(w) = \{z \in \L : |z-w| < 1\}$.
Then we let
$$g (z)= \hat f (z) \cdot \prod_{w \in \operatorname{Hole}(f)} \prod_{i=1}^{d_w} \dfrac{z-c_i(w)}{z-p_i(w)} \in \L(z).$$
It follows that the coefficient reduction of $g$ is exactly $f$.
We exploit the flexibility to choose the zeros and poles in $B^-_1(w)$ to construct $g_{\lambda,t}$ controlling the
surplus multiplicities of $g^n_{\lambda,t}$ in all directions at $\zeta_0$.
Then we apply Lemmas~\ref{depth-surplus} and~\ref{main-lemma} to obtain the depths of the holes of $G_\lambda$, which is the coefficient reduction of $M_t^{-1} \circ g^n_{\lambda, t} \circ M_t$, and use the numerical criteria given by Proposition~\ref{stability-proportional-depth} to certify that $G_\lambda$ is stable. The flexibility of the choices involved will also allow us to verify that $[G_\lambda]$ is not constant (with respect to $\lambda$).
In the exceptional cases 4 and 5 with $d=3$ we were unable to obtain a one parameter family $[G_\lambda]$ as above since the situation turns out to be less flexible, in a certain sense. However, to establish Theorem~\ref{non-constant} we produce two degenerate holomorphic families $f_{t}$ and $g_t$ of degree $d$ rational maps such that $f_0=g_{0} = f$ but $[f_t^n]$ and $[g_t^n]$ converge to distinct elements in $\overline{\mathrm{rat}}_{3^n}$.
The outline of this section is as follows: Section \ref{subsection:notation} contains convenient notations for later use.
Section~\ref{strictly-preperiodic} is devoted to prove that maps which fall into case 1 satisfy the conclusion of Theorem \ref{Thm:perturbation} for any $d\ge 3$ and have GIT-classes in $I(\Phi_n)$. Similarly, in Section~\ref{periodic-superattracting}
we simultaneously address maps that fall into cases 0 or 2,
and in Section~\ref{periodic-simple} maps in case 3. Cases 4 and 5 are dealt with in Sections~\ref{polynomial} and~\ref{monomial}, respectively.
\subsection{Notation}\label{subsection:notation}
When $f \in {\mathcal{U}_n}$ is clear from context, we will freely use the following notation.
The bad hole of $f$ will be denoted by $\mathtt{h}$.
For all $j \ge 0$, set
\begin{eqnarray*}
h_j & =& \hat{f}^j (\mathtt{h}), \\
d_j & =& d_{h_j}(f), \\
m_j &=& m_{h_j}(\hat{f}).
\end{eqnarray*}
Thus the bad hole will be denoted by $\mathtt{h}$ or $h_0$ according to convenience.
Note that $d_j \le (d+1)/2$ and $m_j \le d-1$ for all $j$.
It will be also convenient to work with the proportional depths and multiplicities:
\begin{eqnarray*}
\overline{d}_j &=& \dfrac{d_j}{d}, \\
\overline{m}_j &=& \dfrac{m_j}{d}.
\end{eqnarray*}
Thus, $\overline{d}_j \le (d+1)/2d$ and $\overline{m}_j <1$, for all $j$.
The iterated proportional depths and multiplicities are
\begin{eqnarray*}
\overline{m}_{\mathtt{h}}(\hat{f}^k) & =& \dfrac{m_{\mathtt{h}}(\hat{f}^k)}{d^k} = \prod_{j=0}^{k-1} \overline{m}_j,\\
\overline{d}_{\mathtt{h}}(f^k) & =& \dfrac{d_{\mathtt{h}}(f^k)}{d^k} =
\sum_{j=0}^{k-1} \overline{d}_j \cdot \overline{m}_{\mathtt{h}}(\hat{f}^j).
\end{eqnarray*}
Recall that the threshold proportional depths for (semi)stability are
$$\mu^-(d)=
\begin{cases}
\dfrac{1}{2} & \text{if}\ d\ \text{is even},\\
\\
\dfrac{d-1}{2d} & \text{if}\ d\ \text{is odd},
\end{cases}$$
and
$$\mu^+ (d) = 1 - \mu^- (d) = \begin{cases}
\dfrac{1}{2} & \text{if}\ d\ \text{is even},\\
\\
\dfrac{d+1}{2d} & \text{if}\ d\ \text{is odd}.
\end{cases}$$
Since $f\in{\mathcal{U}}_n$, by Proposition \ref{stability-proportional-depth},
\begin{eqnarray*}
\overline d_{0} (f^n) & > & \mu^+ (d^n), \quad \text{ if } \hat f^n(\mathtt{h})\neq\mathtt{h},\\
\overline d_{0} (f^n) & \ge & \mu^+ (d^n), \quad \text{ if } \hat f^n(\mathtt{h}) = \mathtt{h}.\\
\end{eqnarray*}
\subsection{Strictly preperiodic bad hole}
\label{strictly-preperiodic}
In this section, we deal with the maps in Case 1 of Proposition \ref{cases} and we prove that GIT-classes of such maps lie in
$I(\Phi_n)$. More precisely, we prove
\begin{proposition}
\label{strictly-preperiodic-proposition}
Consider $f \in {\mathcal{U}_n}$ such that $\deg \hat f \ge 1$ and $d_{\mathtt{h}}(f) \ge 2$.
Assume that there exist $1 \le q \le n$ and $0 \le \ell \le n-q$ such that $\hat{f}^{\ell+q}(\mathtt{h}) =\hat f^q (\mathtt{h})$. Then Theorem \ref{Thm:perturbation} holds and $[f] \in I(\Phi_n)$.
\end{proposition}
Thus, throughout this subsection we consider $f$ as in the statement of the above proposition.
\medskip
The proof of the proposition is given after we state and prove three lemmas. The first lemma provides us with a lower bound for the total depth along the orbit of $\mathtt{h}$. The second is the construction of $g_{\lambda,t}$ and the third lemma studies the relevant surplus multiplicities.
\begin{lemma}\label{0implies3}
The depths satisfy
$$\sum_{j=0}^{q+\ell-1} d_j \ge 3.$$
Moreover, $\deg \hat f \ge 2$ and $d\ge 5$.
\end{lemma}
\begin{proof}
By contradiction, suppose that the inequality is false. Then
$d_0=2$ and $d_j=0$ for $1\le j\le q+\ell-1$. The bad hole $h_0$ is strictly preperiodic,
so we would have
$$d_{h_0}(f^n)=d_0d^{n-1} > \frac{d^n+1}{2},$$
which would imply that $d<4$. Taking into account that $\deg\hat f\ge 1$ and $d_0=2$,
we would have that $d=3$ and $\deg\hat f=1$. Therefore, $\hat f$ would have no strictly preperiodic points, giving us the desired contradiction.
Since $\hat f$ has a strictly preperiodic point, $\deg \hat f\ge 2$. Then
$$d=\sum_{j=0}^{q+\ell-1} d_j+\deg\hat f\ge 3+2=5.$$
\end{proof}
As previously mentioned, our construction starts regarding $\hat f$ as a rational map in $\L(z)$ acting on ${\mathbf{P}^1}$. It is convenient to introduce the relevant
geometric situation in ${\mathbf{P}^1}$ before stating the basic properties of our construction, compare with Figure~\ref{fig-123}. Let
\begin{eqnarray*}
\zeta_0 &:= & \xi_{h_0,|t|},\\
\zeta_j &:= & \hat{f}^j (\zeta_0), \quad j \ge 1.
\end{eqnarray*}
Observe that $\zeta_j$ lies in the direction at the Gauss point containing
$h_j$.
Let $X$ be the convex hull of $\zeta_0, \dots, \zeta_{n-1}$, thus
$$X = \bigcup_{0 \le j \le n-1} [\xi_g,\zeta_j].$$
Denote by $\vec{w}_\infty \in T_{\zeta_{0}} {\mathbf{P}^1}$ the direction at $\zeta_0$
containing the Gauss point. Let $\vec{w}_z$ the direction at $\zeta_0$ containing $h_0 + z t$. The directions $\vec{w}_0, \vec{w}_1, \vec{w}_\lambda$ containing
$h_0, h_0+t, h_0 + \lambda t$ will play an important role in our constructions.
For $j=1, \dots, n-1$, let $\vec{v}_j \in T_{\zeta_{j}} {\mathbf{P}^1}$ denote the direction containing $h_j$.
\begin{figure}
\center{
\includegraphics{123}}
\caption{Sketch of geometric situation in ${\mathbf{P}^1}$ for Section~\ref{strictly-preperiodic}}
\label{fig-123}
\end{figure}
In the next lemma, for $\lambda \in \mathbb{C}$ outside a finite set we construct $g_{\lambda, t} (z) \in \mathbb{C}[\lambda, t] (z) \subset \L(z)$.
The rational map $g_{\lambda, t} (z)$ acting on the Berkovich projective line should be regarded as a ``perturbation'' of $\hat{f}$.
In $\overline{\Rat}_d$ we will have that $g_{\lambda, t} \to f$ as $t \to 0$. So regarded as elements of $\overline{\Rat}_d$ the maps $g_{\lambda, t}$ are ``perturbations''
of $f$. The action in ${\mathbf{P}^1}$ will agree with that of $\hat{f}$ ``close'' to the Gauss point. In fact, it agrees with $\hat{f}$ in $X$.
However, the degree of $\hat{f}$ is increased by conveniently
adding zeros and poles to $\hat{f}$. We do so as to ``spread out'' the depth multiplicity of the bad hole $h_0$ in different directions at $\zeta_0$ aiming at
having a stable reduction $g^n_{\lambda,t}$ at the point $\zeta_0$ (equivalently, $M_t^{-1} \circ g^n_{\lambda,t} \circ M_t \to G_\lambda \in \Rat^s_{d^n}$). We ``put'' surplus multiplicity $1$ in several directions at $\zeta_0$ so that under iterations these directions do not fall into directions with positive surplus multiplicities. However, the directions $\vec{w}_0$ and $\vec{w}_\infty$ will increase their surplus multiplicity under iterations but our construction is so that we obtain the necessary upper bounds for the iterated surplus multiplicity.
Given a direction $\vec{v}$ at some point in ${\mathbf{P}^1}$ we denote by
$s_\lambda(\vec{v})$ the surplus multiplicity of $g_{\lambda,t}$ in that direction.
See Figure~\ref{fig-preperiodic-family} for a sketch of the points and directions involved in the construction of $g_{\lambda,t}$ given in the next lemma.
\begin{lemma}
\label{preperiodic-family}
There exists $g_{\lambda, t} (z) \in \mathbb{C}[\lambda, t] (z) \subset \L(z)$
of degree $d$ such that for all $\lambda$ in the complement of a finite subset of $\mathbb{C}$,
the following statements hold:
\begin{enumerate}
\item The coefficient reduction $g_{\lambda,0}$ of $g_{\lambda,t}$ is $f$.
\item For all $\xi \in X$,
\begin{eqnarray*}
g_{\lambda,t}(\xi) &=& \hat{f}(\xi),\\
T_\xi g_{\lambda,t} & = & T_\xi \hat{f}.
\end{eqnarray*}
\item There exist pairwise distinct $c_2, \dots, c_{d_0-1} \in \mathbb{C} \setminus \{ 0, 1 \}$
such that for $\vec{w} \in T_{\zeta_{0}} {\mathbf{P}^1} \setminus \{ \vec{w}_\infty \}$,
$$ s_\lambda(\vec{w}) =
\begin{cases}
1 & \text{ if } \vec{w}\in\{\vec{w}_{1}, \vec{w}_\lambda,
\vec{w}_{c_2}, \dots, \vec{w}_{c_{d_0-1}}\}, \\
0 & \text{ otherwise.}
\end{cases}$$
\item $s_\lambda (\vec{v_j})=d_j$ for $j=1, \dots, n-1$.
\end{enumerate}
\end{lemma}
\begin{proof}
Pick $N$ sufficiently large such that the point $\xi_{h_n,|t|^N}$ lies in the segment $]\zeta_n, h_n[ \subset {\mathbf{P}^1}$. For $1\le j<q+\ell$, set $\xi_{j}=\xi_{h_j,|t|^N}$ and $h_i^+=h_i-t^N$.
Define
$$\gamma(z)=\prod_{j=1}^{q+\ell-1}\left(\frac{z-h_i^+}{z-h_i}\right)^{d_j}=\prod_{j=1}^{q+\ell-1}\left(1+\frac{t^N}{z-h_i}\right)^{d_j}.$$
Let $z_1, \dots, z_m \in \P^1$ be the holes of $f$ outside the set $\{h_0, \dots, h_{q+\ell-1}\}$ with corresponding depths $\delta_1, \dots, \delta_m$. We may assume that $z_j \in \L$ for all $j$. Let
$$\beta(z) = \prod_{j=1}^m \left(1+\dfrac{t^N}{z-z_j} \right)^{\delta_j}.$$
Now choose pairwise distinct $c_2,\cdots,c_{d_0-1}\in \mathbb{C}\setminus\{0, 1\}$ and set
$$\alpha(z)=\prod_{j=2}^{d_0-1}\left(1+\frac{t^N}{z-(h_0+c_it)}\right).$$
Consider
$$g_{\lambda,t}(z)=\left(1+\frac{t^N}{z-(h_0+t)}\right)\left(1+\frac{t^N}{z-(h_0+\lambda t)}\right)\alpha(z)\beta(z)\gamma(z)\hat f(z).$$
For all $\lambda$ distinct from $0,1, c_2, \dots, c_{d_0-1}$ statement (1) follows from the formula of $g_{\lambda,t}(z)$.
Statements (2)--(5) are a direct consequence of Corollary~\ref{perturbation-c}.
\end{proof}
\begin{figure}
\includegraphics{preperiodic}
\caption{Sketch of points and directions involved in Lemma~\ref{preperiodic-family}.}
\label{fig-preperiodic-family}
\end{figure}
Given $\zeta \in {\mathbf{P}^1}$ and a direction $\vec{v} \in T_\zeta {\mathbf{P}^1}$, for a map $g_{\lambda,t}$
as in the previous lemma, we will consistently denote by ${s}^j_\lambda(\vec{v})$ the surplus multiplicity of
$g^j_{\lambda,t}$ in the direction $\vec{v}$ and by ${m}^j_\lambda(\vec{v})$ the corresponding multiplicity.
For $j=1$, we omit the superscript. Similarly, $\overline{s}^j_\lambda(\vec{v})$ and $\overline{m}^j_\lambda(\vec{v})$
denote the corresponding proportional multiplicities.
Our aim is to control $s^n_\lambda(\vec{w})$ for $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$.
\begin{lemma}\label{preperiodic-surplus}
Let $g_{\lambda,t}$ be such that (1)--(5) of Lemma~\ref{preperiodic-family} hold.
Then for all but finitely many $\lambda \in \mathbb{C}$
the following statements
also hold:
\begin{enumerate}
\item If $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ is distinct from $\vec{w}_\lambda, \vec{w}_0, \vec{w}_1, \vec{w}_{c_2} \dots, \vec{w}_{c_{d_0-1}}, \vec{w}_\infty$, then
$${s}^n_\lambda (\vec{w}) =0.$$
\item $$\overline{s}_\lambda^n(\vec{w}_0) \le {\mu^-(d^n)}.$$
Moreover, if $d_{j_0}\ge 1$ for some $1\le j_{0}\le q+\ell-1$, then
$$0<\overline{s}_\lambda^n(\vec{w}_0).$$
\item If $\vec{w} = \vec{w}_1, \vec{w}_\lambda$ or $\vec{w}_{c_j}$ for some $j=2, \dots, d_0-1$, $$ \overline{s}^n_\lambda (\vec{w}) = \dfrac{1}{d.}$$
\item $$\overline{m}_\lambda^n (\vec{w}_\infty) + \overline{s}_\lambda^n(\vec{w}_\infty) < \mu^-(d^n).$$
\end{enumerate}
\end{lemma}
\begin{proof}
By construction, in directions $\vec{w}$ at $\zeta_0$ distinct from $\vec{w}_\lambda, \vec{w}_0, \vec{w}_1, \vec{w}_{c_2}, \dots, \vec{w}_{c_{d_0-1}}, \vec{w}_\infty$ the map $g_{\lambda,t}$ has zero surplus multiplicity.
For $j \ge 1$, at $\zeta_j$ the only directions that may have positive surplus multiplicities are $\vec{v}_j$ and the direction of the Gauss point.
For all $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ we have that $T_{\zeta_0} g^j_{\lambda,t} (\vec{w})$ agrees with $T_{\zeta_0} \hat{f}^j_{\lambda,t} (\vec{w})$.
In view of Section~\ref{complex-action} we know that $T_{\zeta_0} \hat{f}^j_{\lambda,t} (\vec{w})$
is the direction $\vec{v}_j$ or the direction of the Gauss point at $\zeta_j$ if and only if
$\vec{w} = \vec{w}_0$ or $\vec{w}=\vec{w}_\infty$, respectively.
Thus, $s^n_\lambda (\vec{w}) =0$ for all directions $\vec{w}$ at $\zeta_0$ distinct from $\vec{w}_\lambda, \vec{w}_0, \vec{w}_1, \vec{w}_{c_2}, \dots, \vec{w}_{c_{d_0-1}}, \vec{w}_\infty$. Hence we have proven statement (1). Moreover, for $\vec{w} = \vec{w}_1, \vec{w}_\lambda$ or $\vec{w}_{c_j}$ for some $j=2, \dots, d_0-1$, we have that
$s^n_\lambda (\vec{w}) = d^{n-1} s_\lambda (\vec{w}) = d^{n-1}$ and statement (3) also follows.
For statement (2), we apply the formula for $\overline{s}_\lambda^n(\vec{w}_0)$ given by
Lemma \ref{surplus-iterate} taking into account that $s_\lambda(\vec{w}_0)=0$
and that proportional multiplicities are bounded above by $1$ to obtain
$$\overline{s}_\lambda^n(\vec{w}_0) =\overline{m}_\lambda (\vec{w}_0)
\overline{s}_\lambda^{n-1}({\vec{v}_1} ) \le
\overline{s}_\lambda^{n-1}({\vec{v}_1} ),$$
since $\vec{v}_1=T_{\zeta_0}g_{\lambda,t}(\vec{w}_0)$.
Note
$\overline{s}_\lambda^{n-1}({\vec{v}_1} )\le \mu^-(d^{n})$. For otherwise, we would have that $\overline{s}_\lambda^{n}({\vec{v}_1} ) >\overline{s}_\lambda^{n-1}({\vec{v}_1} ))> \mu^-(d^n)$. Since $\overline{s}_\lambda^{n}({\vec{v}_1} )$ may be written as a rational number with denominator $d^n$, it follows that $\overline{s}_\lambda^{n}({\vec{v}_1} ) \ge 1/2$ and hence $h_1$ would be a bad hole. By the uniqueness of the bad hole
(Lemma \ref{bad hole}), we would conclude that
$h_0=h_1$, which is a contradiction with the strict preperiodicity of $h_0$. Thus $\overline{s}_\lambda^n(\vec{w}_0) \le \mu^-(d^n)$.
Moreover, it follows from Lemma \ref{surplus-iterate} that $0<\overline{s}_\lambda^n(\vec{w}_0)$ since by Proposition \ref{preperiodic-family} (2) and (5) we have $\overline s_\lambda (T_{\zeta_0}(g_{\lambda,t}^{j_0}(\vec{w}_0))=d_{j_0}/d>0$. Hence statement (2) holds.
In order to prove statement (4),
recall that
$$\deg_{\zeta_0} g_{\lambda,t}^n + \sum_{\vec{w} \in T_{\zeta_0} {\mathbf{P}^1} }s^n_\lambda(\vec{w})=d^n,$$
(see Equation~(\ref{surplus-sum}) in Section~\ref{berkovich}).
Since $T_{\zeta_j} g_{\lambda,t} = T_{\zeta_j} \hat f$, we have
$\deg_{\zeta_0} g_{\lambda,t}^n = m_0 \cdots m_{n-1} = m^n_\lambda(\vec{w}_\infty)$. Therefore,
$$\overline{m}^n_\lambda(\vec{w}_\infty) + \overline{s}^n_\lambda(\vec{w}_\infty) =
1 - \sum_{\substack{\vec{w}\not=\vec{w}_\infty \\ \vec{w} \in T_{\zeta_0} {\mathbf{P}^1} }}\overline{s}^n_\lambda(\vec{v}).$$
Now
\begin{eqnarray*}
\sum_{\substack{\vec{w}\not=\vec{w}_\infty \\ \vec{w} \in T_{\zeta_0} {\mathbf{P}^1}} }\overline{s}^n_\lambda(\vec{v}) &=&
\frac{d_0}{d} + \overline{s}^n_\lambda(\vec{w}_0)\\
&= &{\overline{d}_0} + \sum_{j=1}^{n-1} \overline{m}_\lambda^j(\vec{w}_0)\overline{s}_\lambda(T_{\zeta_0}g^j_{\lambda,t}(\vec{w}_0))\\
&= &{\overline{d}_0}+ \sum_{j=1}^{n-1} \overline{m}_0^j \overline{d}_j = \overline d_0(f^n).
\end{eqnarray*}
By hypothesis $f \in {\mathcal{U}}_n$ and $\hat f^n (h_0) \neq h_0$, it follows that
$$\mu^{+}(d^n) < \overline d_0(f^n),$$
Thus
$$\overline{m}^n_\lambda(\vec{w}_\infty) + \overline{s}^n_\lambda(w_\infty) < 1-\mu^+(d^n) = \mu^{-} (d^n),$$
and statement (4) follows.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{strictly-preperiodic-proposition}]
For $\lambda$ in the complement of the finite set where the previous lemmas
hold, we let
$$G_\lambda(z) = \lim_{t \to 0} M^{-1}_t \circ g^n_{\lambda,t}\circ M_t (z),$$
where $M_t (z) = h_0 + t z$. Note that if we regard $M_t$ as a degree $1$ rational map
in $\L(z)$, we conclude that the coefficient reduction of $M^{-1}_t \circ g^n_{\lambda,t}\circ M_t$
is $G_\lambda$.
The direction $\vec{w}$ at $\zeta_n$ that contains $\zeta_0$ is the direction containing the Gauss point. Since $\vec{w}_\infty$ is the
unique direction at $\zeta_0$ which maps onto
$\vec{w}$ under $T_{\zeta_o} g_{\lambda,t}^n$, we may apply Corollary~\ref{relative-position} and the previous lemma to conclude that the proportional depths of all the holes of $G_\lambda$ are bounded above
by $\mu^-(d^n)$. Then $G_\lambda \in \Rat^s_{d^n}$ according to Proposition~\ref{stability-proportional-depth}.
It only remains to show $[G_\lambda]$ is not constant in $\lambda$.
If there exists $1\le j_0\le q+\ell-1$ such that $d_{j_0}\ge 1$, by Lemma \ref{preperiodic-surplus} we have $\{0, \infty, 1, \lambda\}\subset\mathrm{Hole}(G_\lambda)$. If $d_j=0$ for all $j\ge 1$, then $d_0\ge 3$ and $\{c_2, \infty, 1, \lambda\}\subset\mathrm{Hole}(G_\lambda)$. In both cases,
$\mathrm{Hole}(G_\lambda)$ has at least $4$ elements including $\infty, 1$ and $ \lambda$ and we claim that there exists $\lambda_0\not=\lambda_1$ such that $[G_{\lambda_0}] \neq [G_{\lambda_1}]$. Indeed, the list of cross ratios of the holes of $G_\lambda$ cannot be independent of $\lambda$. For otherwise, they would be uniformly bounded away from $0$ and $\infty$. However, when $\lambda$ approaches $1$ or $\infty$, at least one cross ratio approaches $0$ or $\infty$.
Hence there are non-conjugate choices for $G_\lambda$. Therefore, the construction is such that Theorem \ref{Thm:perturbation} holds and hence $[f]\in I(\Phi_n)$.
\end{proof}
\subsection{Periodic superattracting or large bad hole orbit}
\label{periodic-superattracting}
In this subsection we show that for $n$-unstable maps $f$ which fall into cases 0 and 2 of Proposition~\ref{cases}, we have that $[f] \in I(\Phi_n)$.
\begin{proposition}
\label{depth>1}
Given $n \ge 2$ assume that $f\in{\mathcal{U}}_n$ with non-constant induced map $\hat{f}$ and the bad hole $\mathtt{h}$ such that $d_{\mathtt{h}}(f) \ge 2$.
Assume that $\# {\mathcal{O}} (\mathtt{h}) \ge n$ or ${\mathcal{O}} (\mathtt{h})$ is a periodic superattracting orbit.
Then Theorem \ref{Thm:perturbation} holds and $[f] \in I(\Phi_n)$.
\end{proposition}
Observe that in this case ${\mathcal{O}} (\mathtt{h})$ can be infinite. The hard case is when the orbit is in fact periodic of period
less than $n$. However, since the same construction applies for $\# {\mathcal{O}} (\mathtt{h}) \ge n$, we deal simultaneously with both situations. In a first reading we suggest to assume that ${\mathcal{O}}(\mathtt{h})$ is periodic of small period compared to $n$.
Throughout this subsection we consider $f$ as in the statement of the above proposition.
As before we regard $\hat{f}$ as an element of $ \L (z)$ which acts on ${\mathbf{P}^1}$
and proceed to construct $g_{\lambda,t}$. The points and directions involved
in the construction are illustrated in Figure~\ref{fig-periodic-super}.
The holomorphic families $g_{\lambda,t}$ are obtained in the first lemma further below.
These holomorphic families $g_{\lambda,t}$ depend on the choice of some integers which will be adjusted in
the proof of the proposition at the end of the subsection.
Let us start labeling some points and directions in our construction site ${\mathbf{P}^1}$, see Figure~\ref{fig-periodic-super}.
Let
$$\ell = \min \{ p , n : \hat{f}^p (\mathtt{h}) = \mathtt{h} \}.$$
That is, $\ell =n$ unless $\mathtt{h}$ is periodic of period $p < n$.
Without loss of generality we assume that $h_j \neq \infty$ for all $j$.
Here we are going to consider $M_t^{-1} \circ g_{\lambda,t} \circ M_t$ for $M_t (z) = h_0 + t^2 z$.
Thus the relevant point $\zeta_0$ in ${\mathbf{P}^1}$ is given by $\zeta_0 = M_t (\xi_g)=\xi_{0,|t|^2}$ and its forward orbit is
$$\zeta_j = \hat{f}^j (\zeta_0 ) = \xi_{h_j, |t|^{2 {m}_{\mathtt{h}}(\hat{f}^j)}}.$$
All the construction depends on an integer $k_\star$ with $0< k_\star <n$. This integer $k_\star$ will be adjusted later
so that the reduction of $g^n_{\lambda,t}$ at $\zeta_0$ is stable for a generic value of $\lambda$ (equivalently, the coefficient reduction of $ M_t^{-1} \circ g_{\lambda,t} \circ M_t$ is stable.)
Given an integer $k_\star$ such that $0< k_\star <n$, apply the division algorithm to write
$$k_\star = q_\star \ell + r_\star,$$
where $q_\star \ge 0$ and $0 \le r_\star < \ell$.
The idea again is to spread the surplus multiplicities along the orbit of $\zeta_0$ to obtain good bounds for the surplus multiplicities in all directions at $\zeta_0$. The directions that are more difficult to
control are $\vec{v}_0, \vec{v}_\infty \in T_{\zeta_0} {\mathbf{P}^1}$ which denote the directions of $h_0$ and the Gauss point, respectively. Intuitively, the integer $k_\star$ is going to be related to the iterate so that the surplus multiplicity in the direction $\vec{v}_0$ will stop to increase. However, one pays the cost that in the direction $\vec{v}_\infty$ the surplus multiplicity will start to increase faster after the corresponding iterate. Achieving the perfect balance is the key of the construction.
To spread the surplus multiplicities appropriately, we focus on the iterates between $\ell q_\star$ and $\ell (q_\star +1) -1$ of $\zeta_0$ and introduce a pair of Berkovich type II points $\xi^\pm_r$ above and below $\zeta_{\ell q_\star + r}$, where $r=0,\dots, \ell-1$, as follows.
Let $$\mu_r = {m}_{\mathtt{h}}(\hat{f}^{\ell q_\star + r}) $$ and
$$
\xi^\pm_r =
\xi_{h_r,|t|^{{2\mu_r} \pm 1}}.
$$
At each $\xi^\pm_r$ choose a direction $\vec{u}^\pm_r$ not containing $h_r$ nor the Gauss point.
\begin{figure}
\includegraphics{FigureDepth2}
\caption{Sketch of points and directions involved in the proof of Proposition~\ref{depth>1}.}
\label{fig-periodic-super}
\end{figure}
Let us intuitively describe some aspects of our construction. For each $r=0, \dots, \ell-1$ we may place $d_r$ zeros and poles in the direction of $h_j$ at the Gauss point.
To achieve the required balance, for $r > 0$, we put all of them in the directions $\vec{u}_r^\pm$. We do so
as to have the highest possible surplus multiplicity in $\vec{u}_r^+$ for $0<r < r_\star$ (here $r_\star$ corresponds to the iterate $k_\star$). That is, we put all the available surplus multiplicity (i.e. $d_r$) in a direction which is below $\zeta_{q_\star \ell + r}$ but above $\zeta_{(q_\star +1) \ell + r}$. In turn for $r > r_\star$ we put the available multiplicity above $\zeta_{q_\star \ell + r}$ in the direction $\vec{u}_r^-$. For $r=r_\star$ (i.e. around $\zeta_{k_\star}$) we put some of surplus multiplicity above and some below in a proportion that will be adjusted later in order to achieve the aforementioned balance. For $r=0$, we also put multiplicity in directions at $\zeta_{q_\star \ell}$.
The precise properties of our construction are stated in the lemma below including how we spread multiplicities for $r=0$.
As in the previous section we denote by $s_\lambda (\vec{v})$ the surplus multiplicity of $g_{\lambda,t}$ in the direction $\vec{v}$. Also it is convenient to let $X$ be the convex hull of the points $\zeta_j$ for $j=0, \dots, \max\{(q_\star+1) \ell -1,n\}$. That is,
$$X = \bigcup_{j=0 }^{n'} [\xi_g,\zeta_j]$$
where $n'=\max\{(q_\star +1)\ell -1,n\}$.
In a first reading we suggest to suppose $r_\star \neq 0$ (i.e. $\ell$ does not divide $k_\star$).
\begin{lemma}\label{superattacting-case}
Let $d_{r_\star}^\pm \ge 0$ be integers such that
$$d_{r_\star}^+ + d_{r_\star}^- =d_{r_\star}$$
and if $r_\star =0$, then $d_{0}^+ \ge 2$.
There exists $g_{\lambda, t} (z) \in \mathbb{C}[\lambda,t] (z) \subset \L(z)$
of degree $d$ such that for $\lambda$ in the complement of finite set in $\mathbb{C}$, the following statements hold:
\begin{enumerate}
\item
The coefficient reduction $g_{\lambda,0}$ of $g_{\lambda,t}$ is $f$.
\item For all $\xi \in X$,
\begin{eqnarray*}
g_{\lambda,t}(\xi) &=& \hat{f}(\xi),\\
T_\xi g_{\lambda,t} & = & T_\xi \hat{f}.
\end{eqnarray*}
\item In $T_{\zeta_{\ell q_\star}} {\mathbf{P}^1}$, let
$\vec{w}_0$ be the direction of $h_0$ and $\vec{w}_\infty$ be the direction
of the Gauss point.
There exists two directions $\vec{w}_1$ and $\vec{w}_\lambda$
with surplus multiplicities $1$ such that the cross ratio $[\vec{w}_0, \vec{w}_1, \vec{w}_\lambda, \vec{w}_\infty] = \lambda$.
\item If $r_\star \neq 0$, then
\begin{eqnarray*}
\label{eq:1}
s_\lambda (\vec{u}^+_0) &=& d_0-2,\\
s_\lambda (\vec{u}^+_r) &=& d_r, \,\, \mbox{ for } \,\,\, r=1, \dots, r_\star-1,\\
s_\lambda (\vec{u}^\pm_{r_\star}) &=&d_{r_\star}^\pm, \\
s_\lambda (\vec{u}^-_r) &=& d_r, \,\, \mbox{ for } \,\,\, r=r_\star +1, \dots, \ell-1.
\end{eqnarray*}
\item If $r_\star =0$, then
\begin{eqnarray*}
\label{eq:2}
s_\lambda (\vec{u}^-_{0}) &=& d_0^-,\\
s_\lambda (\vec{u}^+_{0}) &=& d_0^+-2,\\
s_\lambda (\vec{u}^-_r) & =& d_r, \,\, \mbox{ for } \,\,\, r=1, \dots, \ell-1.
\end{eqnarray*}
\end{enumerate}
\end{lemma}
\begin{proof}
Let $N$ be a sufficiently large integer.
For each $r =0, \dots, \ell-1$, choose $u^{\pm}_r \in \mathbb{C}\setminus \{0\}$ such that
$p_r^\pm =h_r+u_r^\pm t^{2 \mu_r \pm 1}$ is in the direction $\vec{u}_r^\pm$ at $\xi_r^\pm$ and
consider the degree $1$ map with a pole at $p_r^\pm$ and a zero at $p_r^\pm - t^N$:
$$\gamma_r^{\pm} (z) = 1+\dfrac{t^N}{z-p_r^\pm}.$$
Let $z_1, \dots, z_m \in \P^1$ be the holes of $f$ outside the set $\{h_0, \dots, h_{\ell-1}\}$ with corresponding depths $\delta_1, \dots, \delta_m$. We may assume that $z_j \in \mathbb{C} \subset \L$ for all $j$. Let
$$\beta(z) = \prod_{j=1}^m \left(1+\dfrac{t^N}{z-z_j} \right)^{\delta_j}.$$
Now consider
$$\alpha_\lambda (z) = \left(1+\dfrac{t^N}{z- (h_0+ \lambda t^{\mu_0})} \right) \cdot
\left(1+\dfrac{t^N}{z- (h_0+ t^{\mu_0})} \right).$$
If $r_\star = 0$, then let
$$g_{\lambda,t} (z) = \hat{f}(z) \cdot \alpha_\lambda (z) \cdot \beta(z) \cdot
\left(\gamma_0^{+} (z) \right)^{d^+_0-2} \cdot \left(\gamma_0^{-} (z) \right)^{d^-_0}
\prod_{r=1}^{\ell-1} \left(\gamma_r^{-} (z) \right)^{d_r}. $$
If $r_\star \neq 0$, then let
$$g_{\lambda,t} (z) = \hat{f}(z) \cdot \alpha_\lambda (z) \cdot \beta(z) \cdot
\left(\gamma_0^{+} (z) \right)^{d_0-2} \cdot
\left(\gamma_{r_\star}^{+} (z) \right)^{d^+_{r_\star}} \cdot
\left(\gamma_{r_\star}^{-} (z) \right)^{d^-_{r_\star}}
\prod_{r=1}^{r_\star-1}
\left(\gamma_r^{+} (z) \right)^{d_r} \prod_{r=r_\star+1}^{\ell-1} \left(\gamma_r^{-} (z) \right)^{d_r}. $$
Statement (1) follows from the above formulas for $g_{\lambda,t}$. For $N$ sufficiently large,
Lemma \ref{construction} guarantees that statements (2)--(5) hold.
\end{proof}
By construction if $\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}$ is a direction for which $s_\lambda^n (\vec{v}) >0$,
then $\vec{v}$ maps in $q_\star \ell$ iterates onto $\vec{w}_1$ or $\vec{w}_\lambda$, $\vec{v} =\vec{v}_0$, or $\vec{v} = \vec{v}_\infty$. We analyze the surplus multiplicities $s_\lambda^n (\vec{v})$ in our next lemma for $\vec{v} \neq \vec{v}_\infty$. Our control of
$s_\lambda^n (\vec{v}_\infty)$ will follow from equation \eqref{surplus-sum}
\begin{lemma}
Let $g_{\lambda,t}$ be as above. Then the following statements hold:
\begin{enumerate}
\item
$$\overline{s}_\lambda^n(\vec{v}_0) = \left(\sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_{\mathtt{h}}(\hat{f}^j) \right)+
\dfrac{d_{r_\star}^+}{d} \cdot \overline{m}_{\mathtt{h}}(\hat{f}^{k_\star}) -
\dfrac{2}{d} \cdot \overline{m}_{\mathtt{h}}(\hat{f}^{q_\star \ell}).$$
\item There are $2 m_\mathtt{h}(\hat f^{q_\star\ell})$ directions $\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}$ such that $T_{\zeta_0} \hat{f}^{q_\star \ell} (\vec{v}) = \vec{w}_1$ or $\vec{w}_\lambda$,
\item If $\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}$ is such that $T_{\zeta_0} g_{\lambda,t}^{q_\star \ell} (\vec{v}) = \vec{w}_1$ or $\vec{w}_\lambda$, then
$$\overline{s}^n_\lambda (\vec{v}) = \frac{1}{d^n}.$$
\end{enumerate}
\end{lemma}
\begin{proof}
Since $T_{\zeta_j}g_{\lambda,t} = T_{\zeta_j} \hat f$, for $0\le j\le n-1$, the direction $T_{\zeta_0}g_{\lambda,t}^j(\vec{v}_0)$ is the direction containing $h_j$.
Therefore,
$$ m_{\lambda}(T_{\zeta_0}g_{\lambda,t}^j(\vec{v}_0)) = m_j$$
and
$$ s_{\lambda}(T_{\zeta_0}g_{\lambda,t}^j(\vec{v}_0))=
\begin{cases}
d_j & 0\le j\le q_\star\ell-1,\\
d_0-2 & j=q_\star\ell, \\
d_j & q_\star\ell+1\le j\le k_\star-1,\\
d_{k_\star}^+ & j=k_\star,\\
0 & j\ge k_\star.
\end{cases}$$
Statement (1) now follows from Lemma \ref{surplus-iterate}. Note $T_{\zeta_0} \hat{f}^{q_\star \ell} $ has degree $m_\mathtt{h}(\hat f^{q_\star\ell})$ and neither $\vec{w}_1$ nor $\vec{w}_\lambda$ is a critical value of $T_{\zeta_0}g_{\lambda,t}^{q_\star \ell} $. Hence statement (2) holds. Let $\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}$ satisfying $T_{\zeta_0} g_{\lambda,t}^{q_\star \ell} (\vec{v}) = \vec{w}_1$ or $\vec{w}_\lambda$,
$$ s_{\lambda}(T_{\zeta_0}g_{\lambda,t}^j(\vec{v}))=
\begin{cases}
0& 0\le j < q_\star\ell,\\
1& j=q_\star\ell, \\
0 & q_\star\ell+1\le j\le n-1.
\end{cases}$$
Thus statement (3) holds.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{depth>1}]
Now we have to adjust $k_\star$ and $d^\pm_{r_\star}$. We choose $k_\star$ so that the first term in the previous lemma's formula for $\bar{s}_\lambda^n(\vec{v}_0)$ is as large as allowed in order to have stable reduction for $g_{\lambda,t}^n$ at $\zeta_0$.
That is,
for $1\le i\le n$, define
$$\mu_i= \sum_{j=0}^{i-1} \overline{d}_j \cdot \overline{m}_j.$$
Then $\mu_i$ is nondecreasing. Let $k_\star \ge 1$ be the largest integer such that if $\hat f^n(h_0)\neq h_0$, then $\mu_{k_\star}\le \mu^+(d^n)$; if $\hat f^n(h_0)=h_0$, then $\mu_{k_\star}<\mu^+(d^n)$. Note $\overline d_{h_0}(f^n)=\mu_n$. By Proposition \ref{stability-proportional-depth}, such a $k_\star<n$ exists.
Write $k_\star = \ell q_\star + r_\star$ where $q_\star \ge 0$ and $ 0 \le r_\star < \ell$. Now it is time to choose $d_{r_\star}^+$. Again the idea is to choose it as large as stable reduction allows.
More precisely, choose $ d_{r_\star}^+$ with $0 \le d_{r_\star}^+ \le d_{r_\star}$ and such that if $\hat f^n(h_0)\neq h_0$, then
$$
\sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_j +
\dfrac{d_{r_\star}^+-2}{d} \cdot\overline m_{k_\star}\le\mu^-(d^n)\le\mu^+(d^n)< \sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_j + \dfrac{d_{r_\star}^+}{d} \cdot\overline m_{k_\star};$$
if $\hat f^n(h_0)=h_0$, then
$$
\sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_j +
\dfrac{d_{r_\star}^+-2}{d} \cdot\overline m_{k_\star}<\mu^-(d^n)\le\mu^+(d^n)\le \sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_j + \dfrac{d_{r_\star}^+}{d} \cdot\overline m_{k_\star}.
$$
Moreover, when $r_\star=0$, we choose $d_{r_\star}^+ \ge 2$.
Now let $g_{\lambda, t}$ be the family given by the previous lemmas associated to the above choices
of $k_\star$ and $d_{r_\star}^+$. To check that $g_{\lambda,t}^n$ has stable reduction at $\zeta_0$ it is convenient to define
$$\Delta:=\sum_{j=0}^{k_\star-1} \overline{d}_j \cdot \overline{m}_j + \dfrac{d_{r_\star}^+}{d} \cdot\overline m_{k_\star}.$$
Note that
$$\overline{s}^n_\lambda (\vec{v}_0) = \Delta - \dfrac{2}{d} \cdot \overline{m}_{q_\star \ell}.$$
We claim that if $\hat f^n(h_0)\neq h_0$,
$$\Delta - \dfrac{2}{d} \cdot \overline{m}_{q_\star \ell}\le \mu^-(d^n);$$
if $\hat f^n(h_0)= h_0$,
$$\Delta - \dfrac{2}{d} \cdot \overline{m}_{q_\star \ell}< \mu^-(d^n).$$
Indeed, $$\dfrac{1}{d} \cdot \overline{m}_{k_\star} = \dfrac{1}{d} \cdot \overline{m}_{q_\star \ell} \cdot \overline{m}_{r_\star} \le \dfrac{1}{d} \cdot \overline{m}_{q_\star \ell}.$$
It follows that if $\hat f^n(h_0)\neq h_0$, we have $\overline{s}^n_\lambda (\vec{v}_0)\le\mu^-(d^n)$; if $\hat f^n(h_0)= h_0$, we have $\overline{s}^n_\lambda (\vec{v}_0)<\mu^-(d^n)$.
Now we proceed to find an upper bound for $\overline{s}^n_\lambda (\vec{v}_\infty)$.
Since $g_{\lambda,t}^n (\zeta_0) \neq \zeta_0$, we have that
$$ \sum_{\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}} \overline{s}^n_\lambda (\vec{v}) = 1.$$
Moreover,
$$\sum_{\substack{\vec{v} \in T_{\zeta_0} {\mathbf{P}^1}\\ \vec{v} \neq \vec{v}_0, \vec{v}_\infty}}
\overline{s}^n_\lambda (\vec{v}) = \dfrac{2}{d} \cdot \overline{m}_{q_\star \ell}.$$
Thus $$\overline{s}^n_\lambda (\vec{v}_\infty) =1 - \Delta.$$
It follows that if $\hat f^n(h_0)\neq h_0$, we have $\overline{s}^n_\lambda (\vec{v}_\infty)<\mu^-(d^n)$; if $\hat f^n(h_0)=h_0$, we have $\overline{s}^n_\lambda (\vec{v}_\infty)\le\mu^-(d^n).$
After change of coordinates we may assume that $h_0 =0$. For all but finitely many $\lambda \in \mathbb{C}$, we let
$$G_\lambda (z) = \lim_{t \to 0} \dfrac{g_{\lambda,t}^n (t^2z)}{t^2} \in \overline{\Rat}_{d^n}.$$
Then if $\hat f^n(h_0)\neq h_0$, the induced map $\widehat G_\lambda=[1:0]\in\P^1$; if $\hat f^n(h_0)= h_0$, the induced map $\widehat G_\lambda=[0:1]\in\P^1$. By Proposition \ref{stability-proportional-depth}, it follows that $G_\lambda$ is stable. Thus in moduli space,
$$\Phi_n([g_{\lambda,t}]) = [g_{\lambda,t}^n] \to [G_\lambda] \in \overline{\rat}_{d^n}$$
while
$$[g_{\lambda,t}] \to [f] \in \overline{\rat}_d.$$
The holes of $G_\lambda$ are at $0$, $\infty$ and the preimage under $z \mapsto z^{m_\mathtt{h}(\hat f^{q_\star\ell})}$ of $1$ and $\lambda$.
Hence the cross ratios of the holes vary with $\lambda$. For otherwise, these cross ratios would be bounded away from $0$ and $\infty$, which is clearly not the case when $\lambda$
converges to $0, 1,$ or $ \infty$.
Thus, $[G_\lambda]$ is not constant on $\lambda$. The construction is such that Theorem \ref{Thm:perturbation} holds and it follows that $[f] \in I(\Phi_n)$.
\end{proof}
\subsection{Periodic but not superattracting bad hole orbit}
\label{periodic-simple}
Now we deal with the maps in Case 3 of Proposition \ref{cases}. Our goal is to prove
\begin{proposition}
\label{simple}
Given $n \ge 2$ assume that $f\in{\mathcal{U}}_n$ has non-constant induced map $\hat{f}$ and the bad hole $\mathtt{h}$ is such that $d_{\mathtt{h}} (f) \ge 2$. If $\mathtt{h}$
has a critical point free periodic orbit under $\hat{f}$ of period $\ell < n$,
then Theorem \ref{Thm:perturbation} holds and $[f] \in I(\Phi_n)$.
\end{proposition}
First we prove that Proposition \ref{simple} holds under the assumption $$\sum_{j=0}^{\ell-1} d_j \ge 3.$$
Afterwards we consider the exceptional case in which $h_0 = \mathtt{h}$ is the only hole in its orbit and $h_0$ has depth exactly $2$.
Our construction of $g_{\lambda,t}$ will be so that at the point $\zeta_0 = \xi_{h_0,|t|}$ it has stable reduction, for $\lambda$ in the complement of a finite set.
Let $$\zeta_j = \hat{f}^j(\zeta_0).$$
As before, let
$$X = \bigcup_{j=1}^{\ell-1} [\xi_g,\zeta_j].$$
In $T_{\zeta_{0}} {\mathbf{P}^1}$, let
$\vec{w}_0$ be the direction of $h_0$ and let $\vec{w}_\infty$ be the direction
of the Gauss point.
\subsubsection{Proof of Proposition~\ref{simple}: the generic case}
We consider $f \in{\mathcal{U}}_n$ as in the statement of the previous proposition and
assume that $$\sum_{j=0}^{\ell-1} d_j \ge 3.$$
Without loss of generality we also assume that $h_j \neq \infty$ for all $j$. Note that since $m_{h_j} (\hat{f}) =1$ for all $j$, we have that $\zeta_j = \xi_{h_j,|t|}$.
\begin{lemma}
\label{construction4a}
There exists $g_{\lambda, t} (z) \in \mathbb{C}[\lambda, t] (z) \subset \L(z)$
of degree $d$ such that for all $\lambda$ in the complement of a finite subset of $\mathbb{C}$, the following statements hold:
\begin{enumerate}
\item
The coefficient reduction $g_{\lambda,0}$ of $g_{\lambda,t}$ is $f$.
\item For all $\xi \in X$,
\begin{eqnarray*}
g_{\lambda,t}(\xi) &=& \hat{f}(\xi),\\
T_\xi g_{\lambda,t} & = & T_\xi \hat{f}.
\end{eqnarray*}
\item
In $T_{\zeta_{0}} {\mathbf{P}^1}$, there exists two directions $\vec{w}_1$ and $\vec{w}_\lambda$, each
with surplus multiplicity $1$, such that the cross ratio $[\vec{w}_0, \vec{w}_1, \vec{w}_\lambda, \vec{w}_\infty] = \lambda$.
\item For all directions $\vec{v}$ in $T_{\zeta_{j}} {\mathbf{P}^1}$ not containing
the Gauss point,
$s_\lambda(\vec{v}) \le 1$.
\item
There exists $0\le j\le \ell-1$ such that the direction $T_{\zeta_{0}}g_{\lambda,t}^j(\vec{w}_0)$ has nonzero surplus multiplicity.
\end{enumerate}
\end{lemma}
\begin{proof}
We work with subscripts mod $\ell$ so that $\hat f (\zeta_j) = \zeta_{j+1}$ for all $j=0, \dots, \ell-1$.
In coordinates for $T_{\zeta_j} {\mathbf{P}^1} \equiv \mathbb{C} \cup \{ \infty \}$ where $\infty$ corresponds to the direction of the Gauss point, the map $T_{\zeta_j} \hat f$ is affine.
For $0\le j\le \ell-1$ and $1\le i\le d_j$, we choose complex numbers $c_i^{(j)}$ and denote
by $v_i^{(j)}$ the direction in $T_{\zeta_j} {\mathbf{P}^1}$ containing $h_j + c_i^{(j)} t$. Our construction will be so that $v_i^{(j)}$ has surplus multiplicity $1$ for all $i$ and $j$. Our choice is such that the following hold:
\begin{itemize}
\item $c_{1}^{(0)}=1$. This will guarantee that in the direction $\vec{w}_1$ at $\zeta_0$ that contains $h_0 + t$ the surplus multiplicity is $1$.
\item If $d_0\ge 3$, then $c_{2}^{(0)}=0$. The objective of this choice is that if we have a sufficiently deep bad hole $h_0$, then we will have surplus multiplicity $1$ in the direction $\vec{w}_0$ at $\zeta_0$.
\item If $d_0 =2$ and $j_0$ is the smallest $j \ge 1$ such that $d_j \neq 0$, then $c_{1}^{(j_0)}=0$. The idea here is that if the depth of $h_0$ is small, then the direction $\vec{w}_0$ maps in $j_0$ iterates onto the direction of $h_{j_0}$ at $\zeta_{j_0}$ which will have surplus multiplicity $1$.
\item For all $j=0, \dots, \ell-1$ and all $ 1 \le i < k \le d_j$, we have $v_i^{(j)} \neq v_{i'}^{(j)}$.
\end{itemize}
Remark that to apply Lemma \ref{construction} it will be sufficient to just take $N=2$.
Now set
$$\alpha_0(z)=\prod_{i=1}^{d_0-1}\left(1+\frac{t^2}{z-(h_0+c^{(0)}_{i}t)}\right),$$
and for $1\le j\le\ell-1$, set
$$\alpha_j(z)=\prod_{i=1}^{d_j}\left(1+\frac{t^2}{z-(h_j+c^{(j)}_{i}t)}\right).$$
Let $z_1, \dots, z_m \in \P^1$ be the holes of $f$ outside the set $\{h_0, \dots, h_{\ell-1}\}$ with corresponding depths $\delta_1, \dots, \delta_m$. We may assume that $z_j \in \mathbb{C} \subset \L$ for all $j$ and let
$$\beta(z) = \prod_{j=1}^m \left( 1+\dfrac{t^2}{z-z_j} \right)^{\delta_j}.$$
Define
$$g_{\lambda,t}(z)=\left(1+\frac{t^2}{z-(h_0+\lambda t)}\right)\hat f(z)\beta(z)\alpha_0(z)\prod_{j=1}^{\ell-1}\alpha_j(z).$$
Statement (1) follows the formula for $g_{\lambda,t}$. Statement (2) follows Lemma \ref{construction}. Taking $\lambda$ outside the finite set of $\mathbb{C} \setminus \{ 0,1\}$ for which $\vec{w}_\lambda$ agrees with $v_0^{(j)}$ for some $1 \le j \le d_0$ we have that statement (3) holds by construction and since $c_{1}^{(0)}=1$.
By our choice of $c^{(j)}_i$, for any direction $\vec{v}\in T_{\zeta_j}{\mathbf{P}^1}$ not containing the Gauss point, we have that $s_\lambda(\vec{v}) \le 1$. That is, statement (4) holds.
If $d_0\ge 3$, then the direction $\vec{w}_0$ has surplus multiplicity $1$ since $c_{2}^{(0)}=0$.
If $d_0=2$, then $T_{\zeta_{0}}g_{\lambda,t}^{j_0}(\vec{w}_0)$ is $\vec{v}_{1}^{(j_0)}$ which is also a direction with surplus multiplicity $1$ since $c_{1}^{(j_0)}=0$. Therefore, statement (5) holds.
\end{proof}
\begin{lemma}
\label{surplus4a}
Let $g_{\lambda,t} \in \mathbb{C}[\lambda,t](z) \subset \L(z)$ be as in the previous lemma. Then for all $\lambda \in \mathbb{C}$ in the complement of a finite set, the following statements hold:
\begin{enumerate}
\item If $\vec{w}\in T_{\zeta_0}{\mathbf{P}^1}$ is a direction not containing the Gauss point, then
$$\overline s_\lambda^n(\vec{w}) < \mu^-(d^n).$$
\item $$\overline s^n_\lambda(\vec{w}_\infty)=1 - \dfrac{1}{d^n} - \overline{d_0}(f^n).$$
\end{enumerate}
\end{lemma}
\begin{proof}
Since $$1 \le \deg \hat f \le d - \sum_{j=0}^{\ell-1} d_j \le d -3,$$
we have that $d \ge 4$.
For any $\vec{w}\in T_{\zeta_0}{\mathbf{P}^1}$ not containing the Gauss point, we have $s_\lambda(T_{\zeta_0} \hat f^j (\vec{w}))\le 1$ for all $0 \le j \le n-1$.
Therefore,
$$\overline{s}_\lambda^n (\vec{w}) = \sum_{j=0}^{n-1} \dfrac{1}{d^j} \cdot \overline{s}_\lambda(T_{\zeta_0} \hat f^j (\vec{w})) \le \sum_{j=0}^{n-1} \dfrac{1}{d^{j+1}} = \dfrac{1}{d-1} \cdot \left(1 - \dfrac{1}{d^n} \right) < \mu^{-}(d^n),$$
since $d \ge 4$.
Using that $$\overline s_\lambda (T_{\zeta_0} \hat g_{t,\lambda}^j (\vec{w}_\infty)) = 1 - \dfrac{1}{d} - \dfrac{d_j}{d},$$
we have $$\overline s_\lambda^n(\vec{w}_\infty) = \sum_{j=0}^{n-1} \dfrac{1}{d^j} \overline s_\lambda (T_{\zeta_0} \hat g_{t,\lambda}^j (\vec{w}_\infty)) = 1 - \dfrac{1}{d^n} - \sum_{j=0}^{n-1} \dfrac{d_j}{d^{j+1}}
= 1 - \dfrac{1}{d^n} -\overline{d_0}(f^n). $$
\end{proof}
Now we finish the proof of Proposition~\ref{simple}
under the assumption that
$$\sum_{j=0}^{\ell-1} d_j \ge 3.$$
For $\lambda$ in the complement of the finite set where the previous lemmas
hold, we let $$G_\lambda(z) = \lim_{t \to 0} M_t^{-1}\circ g^n_{\lambda,t}\circ M_t (z)$$
where $M_t (z) = h_0 + t z$. As in the previous cases, we conclude that the coefficient reduction of $M_t \circ g_{\lambda,t}^n \circ M^{-1}_t$
is $G_\lambda$.
By Corollary~\ref{relative-position} and the previous lemma, we have that $\overline{d}_z (G_\lambda) < \mu^- (d^n)$ for all $z \neq \infty$. Moreover,
$$\overline d_{\infty}(G_\lambda)=
\begin{cases}
\overline s^n(\vec{w}_\infty)+\dfrac{1}{d^n} &\ \text{if}\ \hat f^n(h_0)\neq h_0,\\
\overline s^n(\vec{w}_\infty) &\ \text{if}\ \hat f^n(h_0)=h_0.
\end{cases}
$$
Taking into account that $f\in{\mathcal{U}}_n$,
we have that if $\hat f^n(h_0)\neq h_0$, then $\overline{d_0}(f^n) > \mu^+ (d^n)$; if $\hat f^n(h_0) = h_0$, then $\overline{d_0}(f^n) \ge \mu^+ (d^n)$. Therefore,
$$\overline{d}_\infty (G_\lambda) < \mu^+ (d^n),$$
and we conclude that $G_\lambda$ is stable.
The maps $g_{\lambda,t}$ were constructed so that for all but finitely many $\lambda$,
$$\{0,1,\infty,\lambda\}\subset\mathrm{Hole}(G_{\lambda}).$$
Therefore, the list of cross ratios
of the holes of $G_\lambda$ is not constant with respect to $\lambda$
and $[G_\lambda] \in \overline{\rat}_{d^n}$ is non-constant. Thus, Theorem \ref{Thm:perturbation} holds and $[f]$ lies in $I(\Phi_n)$. \hfill $\Box$
\subsubsection{Proof of Proposition~\ref{simple}: the exceptional cases}
Here we assume that $f\in{\mathcal{U}}_n$ is a map as in the statement of the proposition such
that $\sum d_j< 3$.
Note that if $d \ge 5$, by Lemma ~\ref{depth-multiplicity-inequality}, we have $d_0 \ge 3$. Hence, $d = 3$ or $4$, $d_0=2$ and $d_1= \cdots=d_{\ell-1}=0$.
Since $f$ is semistable, $h_1 \neq h_0$.
Thus we may assume that $h_0 =0$ and $h_1 = \infty$.
\begin{lemma}
\label{construction4b}
There exists $g_{\lambda, t} (z) \in \mathbb{C}[\lambda, t] (z) \subset \L(z)$
of degree $d$ such that for all $\lambda$ in the complement of a finite subset of $\mathbb{C}$, the following statements hold:
\begin{enumerate}
\item The coefficient reduction $g_{\lambda,0}$ of $g_{\lambda,t}$ is $f$.
\item For all $\xi \in X$,
\begin{eqnarray*}
g_{\lambda,t}(\xi) &=& \hat{f}(\xi).\\
T_\xi g_{\lambda,t} & = & T_\xi \hat{f}, \quad \text{ if } \xi \neq \zeta_j.
\end{eqnarray*}
\item $\deg_{\zeta_0} g_{\lambda,t} = 2$ and $\deg_{\zeta_j} g_{\lambda,t} =1$ for all $j=1, \dots, \ell-1$.
Moreover, $T_{\zeta_k} g_{\lambda, t}$ is independent of $\lambda$, for all $0 \le k \le \ell-1$.
\item If $\vec{w}$ is a direction at $\zeta_0$ such that $m_\lambda (\vec{w}) > 1$, then
$T_{\zeta_0} g_{\lambda, t}^j (\vec{w})$ is not the direction containing the Gauss point at $\zeta_j$ for all $j=1, \dots, n$.
\item There exist directions $\vec{u}_0, \vec{w}_1 \in T_{\zeta_0} {\mathbf{P}^1}$
independent of $\lambda$ such that $\vec{w}_1 \neq \vec{w}_\infty$, $T_{\zeta_0} g_{\lambda,t} (\vec{w}_1)$
is the direction containing the Gauss point at $\zeta_1$, and $T_{\zeta_0} g^\ell_{\lambda,t} (\vec{u}_0) =\vec{w}_1$.
\item There exists a direction $\vec{u}_\lambda \neq \vec{w}_{\infty}$ in $T_{\zeta_0} {\mathbf{P}^1}$ with non-zero surplus multiplicity. Moreover, $s_\lambda ( \vec{u}_\lambda) = 1$. Furthermore, the cross ratio $[ \vec{u}_0, \vec{w}_1, \vec{u}_\lambda, \vec{w}_\infty] = \lambda$.
\end{enumerate}
\end{lemma}
\begin{proof}
Observe that
$\zeta_1 = \hat{f} (\zeta_0) =\xi_{0, |t|^{-1}} $ and for $k = 2, \dots, \ell-1$ we have that $\zeta_k = \hat{f}^k (\zeta_0) = \xi_{h_k, |t|}$. For $k \neq 1$, we consider the
coordinate of $T_{\zeta_k} {\mathbf{P}^1}$ that identifies $w \in \P^1 \equiv \mathbb{C} \cup \{\infty \}$ with
the direction containing $h_k + w t$. In $T_{\zeta_1} {\mathbf{P}^1}$, the coordinate $w \in \P^1$ corresponds to the direction containing $t/w$. For $0 \le k \le \ell-1$, in these coordinates, $T_{\zeta_k} \hat{f} (w) = a_k w$ for some $0 \neq a_k \in \mathbb{C}$.
We construct $g_{\lambda,t}$ in two steps.
Given $\beta \in \mathbb{C} \setminus \{0\}$ to be chosen later,
first we consider
$$\phi_{\beta,t} (z) = \dfrac{z-t+\beta^2 t}{z-t} \cdot \hat{f}(z),$$
and select a convenient $\beta$. Our selection of $\beta$ is so that the assertions corresponding to statements (1)--(4) hold for $\phi_{\beta,t}$. Although statements (1)--(3) are rather straightforward, statement (4) is more subtle. That is, we have to show that critical directions at $\zeta_0$ are not eventually mapped onto the direction of the Gauss point. The existence of an appropriate parameter $\beta$ will be obtained by analyzing the situation when $\beta$ is arbitrarily small.
In fact, a direct computation shows that $\phi_{\beta,t} (\xi) = \hat{f} (\xi)$ for all
$\xi \in [\xi_g,\zeta_j]$ and all $0 \le j \le \ell-1$. Moreover,
\begin{eqnarray*}
T_{\zeta_0} \phi_{\beta, t} (w) & = & a_0 w \cdot \dfrac{w -1 + \beta^2}{w-1}, \\
T_{\zeta_1} \phi_{\beta, t} (w) & = & a_1 w, \\
T_{\zeta_k} \phi_{\beta, t} (w) & = & a_k w + \dfrac{h_{k+1}}{h_k} \beta^2, \quad \text{ for \,\,} 2 \le k \le \ell-1. \\
\end{eqnarray*}
Let $A=\prod\limits_{j=0}^{\ell-1}a_j$. It follows that there exists $C \in \mathbb{C}$ such that
$$q_\beta(w) := T_{\zeta_0} \phi^\ell_{\beta, t} (w) =
A w \cdot \dfrac{w -1 + \beta^2}{w-1} +C \beta^2 = Aw + \beta^2 \left(
\dfrac{A w}{w-1} +C \right).$$
The directions with multiplicity $2$ at $\zeta_0$ under $\phi_{\beta,t}$ correspond to the critical points $w = 1 \pm \beta$ of $T_{\zeta_0} \phi_{\beta, t}$ which are also the critical points of $q_\beta$.
We claim that there exists $\beta$ such that the directions corresponding to $w = 1 \pm \beta$ do not map
to the direction containing the Gauss point (i.e., $w=\infty$) under $T_{\zeta_0} \phi_{\beta,t}^k$ for all $k=1, \dots, n$.
For otherwise, there exists $p_\pm$ such that $p_\pm \ell <n$ and
$$q_\beta^{p_\pm} (1 \pm \beta) = 1$$
for all $\beta$. In particular, this occurs for $\beta$ arbitrarily close to $0$, and therefore $A^{p} =1$ for some $p$ dividing $p_\pm$, since
$q^{p_\pm}_\beta(1 \pm \beta)= A^{p_\pm} + O(\beta)$. Assuming $p$ is the smallest number such that $A^{p} =1$, it follows that
$$q_{\beta}^{p} (1 + w \beta) = 1 + \left( w + \dfrac{1}{w} \right) \beta + o(\beta).$$
Let $$P(w) = w + \dfrac{1}{w}$$
and observe that the critical points $\pm 1$ of $P$ have infinite forward orbit converging to the (double) parabolic point at infinity. Therefore, $P^m (\pm 1) \neq \infty$ for all $m$.
Since $P(0) = \infty$, also $P^m (\pm 1) \neq 0$.
Thus, given $k$,
$$q_{\beta}^{kp} (1 \pm \beta) = 1 + P^k (\pm 1) \beta + o(\beta) \neq 1$$
for $0 \neq \beta$ sufficiently close to $0$.
Hence we may choose $\hat \beta$ such that $q_{\hat \beta}^{kp} (1 \pm \hat\beta) \neq 1$ for all $k$ such that $k\ell < n$.
Once chosen $\hat \beta$ we continue with the second step of the construction of $g_{\lambda,t}$.
Let $z_1, \dots, z_m \in \P^1$ be the holes of $f$ which are not $h_0$ with corresponding depths $\delta_1, \dots, \delta_m$. We may assume that $z_j \in \L$ for all $j$. Let
$$\gamma(z) = \prod_{j=1}^m \left(1+\dfrac{t^N}{z-z_j} \right)^{\delta_j}.$$
Let $u_0 \in \mathbb{C} $ be such that $q_{\hat{\beta}} (u_0) =1$. Note that $T_{\zeta_0} \phi_{\hat{\beta},t}^{\ell+1} (u_0) = \infty$.
Given $\lambda \in \mathbb{C} \setminus \{0,1\}$, let $a(\lambda)$ be the affine function such
that the cross ratio $[ u_0, 1, a(\lambda), \infty] =\lambda$.
Now we can introduce $g_{\lambda,t} \in \mathbb{C}[\lambda,t](z)$ as
$$g_{\lambda,t} (z) = \dfrac{z - (a(\lambda) t - t^N)}{z - (a(\lambda) t+t^N)}\gamma(z)\phi_{\hat\beta,t} (z),$$
where $N \in \mathbb{Z}_{>0}$ is sufficiently large.
Statement (1) follows from the above formula.
Note that for $N$ large, $g_{\lambda,t} (\xi) = \phi_{\hat \beta, t} (\xi) = \hat f (\xi)$ and $T_{\xi} g_{\lambda,t} = T_{\xi} \phi_{\hat \beta, t}$ for all $\xi \in [\xi_g,\zeta_k]$ and all $k$. Therefore, statements (2) and (3) hold.
Statement (4) follows from our careful choice of $\hat \beta$. Taking $\vec{u}_0$ (resp. $\vec{w}_1, \vec{u}_\lambda$) as the direction corresponding to $w=u_0$
(resp. $w=1, w=a(\lambda)$), statements (5) and (6) follow.
\end{proof}
\begin{lemma}
Let $g_{\lambda,t}$ be such that (1)--(6) of Lemma~\ref{construction4b} hold.
Then for all $\lambda$ in the complement of a finite subset of $\mathbb{C}$, the following statements hold:
\begin{enumerate}
\item $$\overline{s}_\lambda (\vec{w}_\infty) = 1 - \dfrac{1}{d^n} - \overline{d}_0 (f^n).$$
\item If $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ is such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{w})$ contains the Gauss point for some $0 \le k < n$, then
$$0 \neq \overline{s}_\lambda^n (\vec{w}) \leq \overline{s}^n_\lambda (\vec{w}_\infty).$$
\item If $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ is such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{w}) =\vec{u}_\lambda$ for some $k$, then
$$\overline{s}_\lambda^n (\vec{w}) = \dfrac{1}{d^{k+1}}.$$
\item If $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ is such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{w})$
is distinct from $\vec{u}_\lambda$ and does not contain the Gauss point, for all $k=0, \dots, n-1$, then $\overline{s}_\lambda^n (\vec{w}) =0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Consider the subset $\Lambda$ of $\mathbb{C} \setminus \{0,1\}$ such that for all $\lambda \in
\Lambda$ the following statements hold:
\begin{itemize}
\item[(i)] $\vec{w}_1 \neq T_{\zeta_0} g_{\lambda,t}^{k\ell}(\vec{u}_\lambda)$ for all $k$ such that $k \ell < n$.
\item[(ii)] $\vec{u}_\lambda \neq T_{\zeta_0} g_{\lambda,t}^{k \ell}(\vec{u}_\lambda)$ and $m_\lambda(T_{\zeta_0} g_{\lambda,t}^{k \ell}(\vec{u}_\lambda))=1$ for all $k$ such that $k \ell < n$.
\item[(iii)] If $w\in T_{\zeta_0} {\mathbf{P}^1}$ is such that $m_\lambda(\vec{w})=2$, then $T_{\zeta_0} g_{\lambda,t}^{k \ell}(\vec{w}) \neq \vec{u}_\lambda $ for all $k$ such that $k \ell < n$.
\end{itemize}
Since $T_{\zeta_0} g_{\lambda,t}^\ell$ is a rational self-map of $T_{\zeta_0} {\mathbf{P}^1} \equiv \P^1$ independent of $\lambda \in \mathbb{C} \setminus \{0,1\}$ and conditions (i)--(iii) are violated for
finitely many $\vec{u}_\lambda \in T_{\zeta_0} {\mathbf{P}^1}$, we conclude that $\Lambda$ is the complement of a finite subset of $\mathbb{C}$.
Let $\lambda \in \Lambda$.
By Formula~(\ref{surplus-sum}), for all $j=0, \dots, n-1$,
$$\overline s_\lambda (T_{\zeta_0} g_{t,\lambda}^j (\vec{w}_\infty)) = 1 - \dfrac{1}{d} - \dfrac{d_j}{d}.$$
As in the proof of Lemma~\ref{surplus4a}, it follows that statement (1) holds.
Statement (4) is a direct consequence of Lemma~\ref{construction4b} (6) together with the fact that for $j=1,\dots,{\ell-1}$, any direction at $\zeta_j$ not containing the Gauss point
has zero surplus multiplicity, since $d_j =0$.
For statement (2), we consider $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{v})$ contains the Gauss point for some $0 \le k < n$. We may assume that $k$ is the minimal
iterate with this property. Therefore, from (i) above and Lemma~\ref{construction4b} (6),
$$\overline{s}_\lambda^n (\vec{w}) = \sum^{n-1}_{j=0}
\overline{m}_\lambda^j(\vec{w}) \overline{s}_\lambda (T_{\zeta_0} g_{\lambda,t}^j (\vec{w})) =
\sum^{n-1}_{j=k}
\overline{m}_\lambda^j(\vec{w}) \overline{s}_\lambda (T_{\zeta_0} g_{\lambda,t}^j (\vec{w}_\infty)).$$
Moreover, from statement (4) of the previous lemma, $\overline m_\lambda^j (\vec{w}) = d^{-j}$ and hence
$$\overline{s}_\lambda^n (\vec{w}) =
\sum^{n-1}_{j=k}
d^{-j} \overline{s}_\lambda (T_{\zeta_0} g_{\lambda,t}^j (\vec{w}_\infty)) \le \sum^{n-1}_{j=0}
d^{-j} \overline{s}_\lambda (T_{\zeta_0} g_{\lambda,t}^j (\vec{w}_\infty))=\overline{s}_\lambda^n (\vec{w}_\infty).$$
That is, statement (2) holds.
Finally, if $\vec{w} \in T_{\zeta_0} {\mathbf{P}^1}$ is such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{w}) =\vec{u}_\lambda$ for some $k$, then by (i), (ii) and Lemma~\ref{construction4b} (6),
$$\overline{s}_\lambda^n (\vec{w}) =
\overline{m}_\lambda^k(\vec{w}) \overline{s}_\lambda (\vec{u}_\lambda),$$
and by (iii) we have $\overline{m}_\lambda^k(\vec{w}) = d^{-k}$. Thus statement (3) holds.
\end{proof}
Now we finish the proof of Proposition~\ref{simple}
under the assumption that
$$\sum_{j=0}^{\ell-1} d_j < 3.$$
We will only consider $\lambda$ for which the previous lemmas hold and
let $$G_\lambda (z) = \lim_{t \to 0} \dfrac{g^n_{\lambda,t}(tz)}{t}.$$
We claim that $G_\lambda$ is stable.
The holes of $G_\lambda$ correspond to directions $\vec{w}$ at $\zeta_0$ such that one of the following holds:
\begin{enumerate}
\item There exists $0 \le k \le n$ such that $T_{\zeta_0} g_{\lambda,t}^k (\vec{w})$ is the direction
at $\zeta_k$ containing the Gauss point.
\item There exists $0 \le k < n$ such that $\ell$ divides $k$ and
$T_{\zeta_0} g_{\lambda,t}^k (\vec{w})$ is the direction containing $a(\lambda) t$ at $\zeta_0$.
\end{enumerate}
Since $f\in{\mathcal{U}}_n$, if $\hat f^n(h_0)\neq h_0$, then
$\overline{d}_0(f^n)>\mu^+(d^n)$; if $\hat f^n(h_0)=h_0$, then $\overline{d}_0(f^n)\ge\mu^+(d^n)$.
Therefore, if $\vec{w}\in T_{\zeta_0}{\mathbf{P}^1}$ and $k$ is minimal so that (1) holds, then
$$\overline{s}_\lambda^n(\vec{w}) \le \overline{s}_\lambda^n (\vec{w}_\infty) = 1 - \dfrac{1}{d^n} - \overline{d}_0 (f^n).$$
If $\hat f^n(h_0)\neq h_0$, then
$$\overline{s}_\lambda^n(\vec{w}) \le \overline{s}_\lambda^n (\vec{w}_\infty)<\mu^-(d^n)- \frac{1}{d^n};$$
if $\hat f^n(h_0)=h_0$, then
$$\overline{s}_\lambda^n(\vec{w}) \le \overline{s}_\lambda^n (\vec{w}_\infty)\le \mu^-(d^n)-\frac{1}{d^n}.$$
Now let $\vec{w}$ be a direction at $\zeta_0$ and $k$ minimal such that (2) holds.
Then
$$\overline{s}_\lambda^n(\vec{w}) = \dfrac{1}{d} \cdot \dfrac{1}{d^k} < \mu^-(d^n).$$
Now consider a hole $z$ of $G_\lambda$. Then the direction $\vec{w}$ at $\zeta_0$ containing $zt$ satisfies either (1) or (2).
In case (1), by Corollary~\ref{relative-position}, since $\overline{m}^n(\vec{w})= d^{-n}$,
$$\overline{d}_z(G_\lambda) =
\begin{cases}
\overline{s}_\lambda^n (\vec{v}) + \dfrac{1}{d^n} & \text{if}\ \hat f^{n}(h_0)\not=h_0,\\
\overline{s}_\lambda^n (\vec{v}) & \text{if}\ \hat f^{n}(h_0)=h_0.
\end{cases}
$$
Thus if $\hat f^n(h_0)\neq h_0$, then $\overline{d}_z(G_\lambda)<\mu^-(d^n)$; if $\hat f^n(h_0)=h_0$, then $\overline{d}_z(G_\lambda)\le\mu^-(d^n)$.
In case (2),
$$\overline{d}_z(G_\lambda) = \overline{s}_\lambda^n(\vec{w}) < \mu^-(d^n).$$
The induced map $\widehat G_\lambda$ is $[1:0]\in\P^1$ if $\hat f^n(h_0)\neq h_0$; the map $\widehat G_\lambda$ has degree $1$ if $\hat f^n(h_0)=h_0$. Hence by Proposition \ref{stability-proportional-depth}, we have that $G_\lambda$ is stable for all $\lambda \in \Lambda$.
Also by considering the list of cross ratios of the holes of $G_\lambda$, we know $[G_\lambda]$ is not a constant. Therefore, Theorem \ref{Thm:perturbation} holds and $[f]\in I(\Phi_n)$. \hfill $\Box$
\subsection{Polynomial induced map}
\label{polynomial}
In this section, we deal with the maps in Case 4 of Proposition \ref{cases} and prove
\begin{proposition}\label{poly-ind}
Let $d \ge 3$ and consider $f\in{\mathcal{U}}_n$ such that $d_{\mathtt{h}}(f)=1$, where $\mathtt{h}$ is the bad hole. If $\hat{f}(\mathtt{h}) = \mathtt{h}$, then $[f] \in I(\Phi_n)$. Additionally assume that $d\ge 4$. Then Theorem \ref{Thm:perturbation} holds.
\end{proposition}
By Proposition~\ref{depth1poly}, the induced map $\hat{f}$ is conjugate to a polynomial of degree $d-1$
where $\mathtt{h}$ corresponds to $\infty$. That is, after a change of coordinates
$$\hat{f}(z) = z^{d-1} + a_{d-2} z^{d-2} + \cdots + a_0,$$
for some $a_j \in \mathbb{C}$.
To prove Proposition \ref{poly-ind}, we study two cases: $d \ge 4$ and $d=3$. In the former case, we construct a degenerate holomorphic family $g_{\lambda, t}$ of rational maps such that $g_{\lambda, t}$ converges to $f$ as $t\to 0$, but the limit of $[g^n_{\lambda,t}]$ varies with $\lambda$. To produce such a family $g_{\lambda, t}$, we first perturb the polynomial $\hat f$ by adding two finite poles to $\hat f$ (decreasing by two the multiplicity of infinity) and obtain a rational map $\phi_{\lambda,t}$ of the same degree as $\hat f$. Then we construct the family $g_{\lambda, t}$ by adding a zero and a pole to $\phi_{\lambda, t}$. The added zero and pole are suitably chosen so that also $\phi_{\lambda,t}$ and $g_{\lambda,t}$ agree in an appropriate subset of ${\mathbf{P}^1}$. We will show that there exists $\zeta_0\in(\xi_g,\infty)$ such that the coefficient reduction of $g^n_{\lambda, t}$ is stable and varies with $\lambda$. The later case ($d=3$) is more subtle. Since $\deg\hat f=2$, we were unable to construct appropriate families depending on $\lambda$. In this case we construct two holomorphic families of rational maps with distinct (semi)stable limits under the iteration map. One family is the analog of the one from the case $d \ge 4$. The other family is constructed by adding a hole and a pole directly to the map $\hat f$.
To ease notation, in this and subsequent sections, we will set $\chi_\alpha = \xi_{0, |t|^{-\alpha}}$ for $\alpha\in\mathbb{R}$. It parametrizes the arc from $0$ to $\infty$ inside ${\mathbf{P}^1}$ by $\alpha$.
\subsubsection{Proof of Proposition \ref{poly-ind}: $d\ge 4$ case.}
Consider for the moment $N >1$ to be adjusted in the sequel. For $\lambda\in\mathbb{C}\setminus\{0,1\}$, consider the degree $d-1$ rational map
$$\phi_{\lambda,t}(z)=\frac{1}{(1-t^Nz)(1-\lambda t^Nz)}\hat f(z) \in \L(z).$$
Given $\alpha >0$, we have that
$$\phi_{\lambda,t} (\chi_\alpha) =
\begin{cases}
\chi_{ (d-1) \alpha} & 0 < \alpha \le N,\\
\chi_{ (d-3) \alpha + 2N} & \alpha > N.
\end{cases}
$$
In standard coordinates where we identify $z \in \P^1$ with the
direction at $\chi_\alpha$ containing $z t^{-\alpha}$, we have
$$T_{\chi_\alpha} \phi_{\lambda,t} (z) =
\begin{cases}
z^{d-1} & 0 < \alpha <N,\\
\dfrac{1}{(1-z)(1-\lambda z)} z^{d-1} & \alpha =N,\\
\lambda^{-1} z^{d-3} & \alpha > N.
\end{cases}
$$
For an integer $k$ also to be chosen later such that $0\le k\le n-1$, we let $$\alpha_0 = \dfrac{N}{(d-1)^k}.$$
Set
\begin{eqnarray*}
\zeta_0 &=& \chi_{\alpha_0},\\
\zeta_j &=& \phi^j_{\lambda,t} (\zeta_0) \, \text{ for } j \ge 1.
\end{eqnarray*}
Observe that $\zeta_k = \chi_{N}$.
Now let $\vec{v}_\infty\in T_{\zeta_n}{\mathbf{P}^1}$ be the direction containing $\infty$. Pick $c_1\not=c_2\in\mathbb{L}$ in the direction $\vec{v}_\infty$. We put an extra zero at $c_2$ and
an extra pole at $c_1$. That is, we consider
$$g_{\lambda,t}(z)=\left(1+\frac{c_1-c_2}{z-c_1}\right)\phi_{\lambda,t}(z).$$
Then $g_{\lambda,t}\to f$ as $t\to 0$. Choose $N$ large such that Lemma \ref{construction} implies that $\zeta_j = g^j_{\lambda,t} (\zeta_0)$ and $T_{\zeta}g_{\lambda,t}=T_\zeta\phi_{\lambda,t}$ for all $\zeta\in [\zeta_0,\zeta_{n-1}]$. Also we conclude that the surplus multiplicity of $g_{\lambda,t}$ in the direction at $\chi_\alpha$ containing the Gauss point $\xi_g$ is $0$ for $\alpha \le N$ and
$2$ otherwise. Moreover, the surplus multiplicity in the direction at $\chi_\alpha$ containing $\infty$ is $1$ for all $\chi_\alpha \in ]\xi_g, \zeta_n]$.
To ease notation, set
\begin{eqnarray*}
\mu&=&(d-1)/d,\\
\theta&=&(d-3)/d.
\end{eqnarray*}
Denote by $\vec{w}_0$ and $\vec{w}_\infty$ the directions at $\zeta_0$ containing $0$ and $\infty$, respectively. For the direction $\vec{w}_0$, from above we have
$$\overline{s}_\lambda(T_{\zeta_0}g_{\lambda,t}^j(\vec{w}_0))=
\begin{cases}
0 & 0\le j\le k,\\
\dfrac{2}{d} & k+1 \le j,
\end{cases}$$
and
$$\overline{m}_\lambda^j(\vec{w}_0)=
\begin{cases}
\mu^j & 1\le j\le k+1,\\
\mu^{k+1}\theta^{j-k-1} & k+2 \le j.
\end{cases}$$
Also, for the direction $\vec{w}_\infty$, we have
$$\overline{s}_\lambda(T_{\zeta_0}g_{\lambda,t}^j(\vec{w}_\infty))=\dfrac{1}{d}\ \ \ \ j\ge 0,$$
and
$$\overline{m}_\lambda^j(\vec{w}_\infty)=
\begin{cases}
\mu^j & 1\le j\le k,\\
\mu^k\theta^{j-k} & k+1 \le j.
\end{cases}$$
Then by Lemma \ref{surplus-iterate}, if $n>k+1$,
$$\overline{s}_\lambda^n (\vec{w}_0) =\frac{2}{d}\mu^{k+1}\sum_{j=0}^{n-k-2}\theta^j=\frac{2}{3}\mu^{k+1}(1-\theta^{n-k-1}),$$
and
$$\overline{s}_\lambda^n (\vec{w}_\infty) =\frac{1}{d}\sum_{j=0}^{k}\mu^j+\frac{1}{d}\mu^k\theta\sum_{j=0}^{n-k-2}\theta^j=1-\frac{2}{3}\mu^k-\frac{1}{3}\mu^k\theta^{n-k}.$$
If $n=k+1$, then
$$\overline{s}_\lambda^n (\vec{w}_0) =0,$$
and
$$\overline{s}_\lambda^n (\vec{w}_\infty) =\dfrac{1}{d}\sum_{j=0}^{n-1}\mu^j=1-\mu^n>\mu^+(d^n)$$
since $\hat f^n$ fixes $\infty$. Moreover, for a direction $\vec{w}\in T_{\zeta_0}{\mathbf{P}^1}$, if $T_{\zeta_0}g^k_{\lambda}(\vec{w})$ contains $t^N$ or $\lambda t^N$, we have
$$\overline{s}_\lambda^n (\vec{w})=\dfrac{1}{3d^{k+1}}(1-\theta^{n-k-1}).$$
Note that as $k$ decreases, $\overline{s}_\lambda^n (\vec{w}_\infty)$ decreases, and for $k=0$,
$$\overline{s}_\lambda^n (\vec{w}_\infty)=\frac{1}{3}(1-\theta^n)<\mu^+(d^n).$$
Hence there exists $0\le k\le n-2$ such that
$$\frac{2}{3}\mu^{k+1}+\frac{1}{3}\mu^{k+1}\theta^{n-k-1}\le\mu^-(d^n)\le\mu^+(d^n)<\frac{2}{3}\mu^k+\frac{1}{3}\mu^k\theta^{n-k}.$$
For such $k$, let $M_t(z) = t^{-\alpha_0} z$, and consider
$$G_{\lambda} = \lim_{t \to 0} M_t^{-1} \circ g^n_{\lambda,t} \circ M_t.$$
By Corollary \ref{relative-position},
$$\overline d_0(G_{\lambda})=\overline{s}^n (\vec{w}_0)+\mu^{k+1}\theta^{n-k-1}=\frac{2}{3}\mu^{k+1}+\frac{1}{3}\mu^{k+1}\theta^{n-k-1}\le\mu^-(d^n),$$
and
$$\overline d_\infty(G_{\lambda})=\overline{s}_\lambda^n (\vec{w}_\infty)<\mu^-(d^n).$$
Moreover, if $z$ is a $(d-1)^k$-th root of unity or a $(d-1)^k$-th root of $\lambda$, then
$$\overline d_z(G_{\lambda})=\frac{1}{3d^{k+1}}(1-\theta^{n-k-1})<\mu^-(d^n).$$
Note the induced map $\widehat G_\lambda=[1:0]$. Hence by Proposition \ref{stability-proportional-depth}, we have that $G_{\lambda}$ is stable. Moreover, $[G_\lambda] \in \overline{\rat}_{d^n}$ varies with $\lambda$ since $G_\lambda$ has holes at $0, \infty$, the $(d-1)^k$-th roots of $1$ and the $(d-1)^k$-th roots of $\lambda$. Note that the action of $\phi_{\lambda, t}$ and hence of $g_{\lambda, t}$ on the segment $[\xi_g,\infty]$ is independent of $\lambda$. Therefore, the construction is such that Theorem \ref{Thm:perturbation} holds and hence $[f]\in I(\Phi_n)$. \hfill $\Box$
\subsubsection{Proof of Proposition \ref{poly-ind}: cubic case.}\label{cubic-poly} In this case, $\deg\hat f=2$ and $n \ge 2$. As before, we consider
$$\phi_{t}(z)=\frac{1}{1 - t^Nz}\hat f(z),$$
where $N >1$ is to be adjusted in the sequel.
Given $\alpha >0$, note that
$$\phi_{t} (\chi_\alpha) =
\begin{cases}
\chi_{2\alpha} & 0 < \alpha \le N,\\
\chi_{\alpha +N} & \alpha > N.
\end{cases}
$$
In standard coordinates where we identify $z \in \P^1$ with the
direction at $\chi_\alpha$ containing $z t^{-\alpha}$, we have
$$T_{\chi_\alpha} \phi_{t} (z) =
\begin{cases}
z^2 & \alpha <N,\\
\dfrac{z^2}{1 - z} & \alpha =N,\\
- z & \alpha > N.
\end{cases}
$$
Now consider $\zeta_j$ for $j \ge 0$ and $g_t(z)$ as in the proof of previous case. It follows that
$$\overline{s}(T_{\zeta_0}g_{t}^j(\vec{w}_0))=
\begin{cases}
0 & 0\le j\le k,\\
\dfrac{1}{3} & k+1 \le j \le n.
\end{cases}$$
Choose the largest $k$ such that $0\le k\le n-2$ and
$$2^{k-1} \left(\frac{1}{3^{k}} + \frac{1}{3^n}\right) > \mu^-(3^n).$$
For such $k$, let $M_t (z) = t^{-N/2^k} z$ and consider
$$G = \lim_{t \to 0} M_t^{-1} \circ g^n_{t} \circ M_t.$$
If $z$ is a $2^k$-th root of unity, then
$$\overline d_z(G)= \dfrac{1}{2} \left(\frac{1}{3^{k+1}} - \frac{1}{3^n}\right).$$
A direct computation shows that $G$ is semistable since
$$\overline{d}_0(G)=2^{k} \left(\frac{1}{3^{k+1}} + \frac{1}{3^n}\right).$$
Now we consider
$$\psi_{t} (z) =\dfrac{t^Sz -1}{t^S z -(1+t)}\hat{f}(z),$$
where $S >1$ is to be adjusted in the sequel.
Observe that for all $\alpha>0$, we have
$$\psi_t (\chi_\alpha) = \chi_{2 \alpha}$$ and
$$T_{\chi_\alpha}\psi_{t} (z) = z^{2}.$$
For $\tilde{k}$ also to be chosen later such that
$1\le \tilde k < n$, let $$\tilde{\alpha}_0 = \dfrac{S}{2^{\tilde k}}$$
and $$\tilde\zeta_0 = \chi_{\tilde{\alpha}_0}.$$
For $j \ge 1$, let $$\tilde\zeta_j = \psi^j_{t} (\tilde\zeta_0)$$
and observe that $\tilde\zeta_{\tilde k} =\chi_S$.
Denote by $\vec{v}_0$ and $\vec{v}_\infty$ the directions at $T_{\tilde\zeta_0} {\mathbf{P}^1}$ containing $0$ and $\infty$, respectively. Then
\begin{eqnarray*}
\overline{s}^n (\vec{v}_0) &=& \left( \dfrac{2}{3} \right)^{\tilde k+1} - \left( \dfrac{2}{3} \right)^{n},\\
\overline{s}^n (\vec{v}_\infty) &=&1-\left( \dfrac{2}{3} \right)^{\tilde k}.
\end{eqnarray*}
Consider one of the $2^{\tilde k}$ directions $\vec{v}$ at $\tilde\zeta_0$ which under
$T_{\tilde\zeta_0} \psi_{t}^{\tilde k}$ maps onto
$\vec{v}_1$ (the direction containing $t^{-S}$).
Then
$$\overline{s}^n (\vec{v}) = \frac{1}{3^{\tilde k+1}}.$$
Let $L_t (z) = t^{-\tilde\alpha_0} z$ and consider
$$F= \lim_{t \to 0} L_t^{-1} \circ\psi^n_{t} \circ L_t.$$
Then the holes of $F$ are at $0, \infty$ and the $2^{\tilde k}$th-roots of unity. Moreover,
the proportional depths are
$$\overline{d}_0(F) = \overline{s}^n (\vec{v}_0) + \left( \dfrac{2}{3} \right)^n,$$
$$\overline{d}_\infty(F) = \overline{s}^n (\vec{v}_\infty).$$
If $z$ is a $2^{\tilde k}$-root of unity, then
$$\overline{d}_z(F)= \frac{1}{3^{\tilde k+1}}.$$
Since $\mu^+(3^n)\le\overline{d}_\infty(f^n)=1-(2/3)^n$ there exists $\tilde k <n$ such that $(2/3)^{\tilde k+1} \le\mu^-(3^n) <\mu^+(3^n)<(2/3)^{\tilde k}$.
The induced map is $\widehat F=[1:0]$ and it follows that $F$ is stable for this value of $\tilde k$.
Now we claim that $G$ and $F$ are not in the same GIT-class. For otherwise, $G$ is stable with $k=\tilde k$ and $d_{z}(G)=d_z(F)$ if $z$ is a $2^{k}$th-root of unity. However, if $k=\tilde k$, we have
$$d_{z}(G)=\frac{1}{2\cdot 3^{k+1}}\left(1-\frac{1}{3^{n-k-1}}\right)< \frac{1}{3^{k+1}}=d_z(F),$$
which is a contradiction. \hfill $\Box$
\begin{corollary}
\label{dimension}
The indeterminacy locus
$I(\Phi_n)$ contains a complex subvariety of dimension $d-2$ for $n$ sufficiently large.
\end{corollary}
\begin{proof}
The map $\mathbb{C}^{d-2} \to \overline{\rat}_d$ given by
$$(a_0, \dots, a_{d-3}) \mapsto [a_0 + \cdots + a_{d-3} z^{d-3} + z^{d-1}]$$
is finite--to--one. For $n$ such that $(1 - 1/d)^n < 1/2$, the image of
this map is contained in ${\mathcal{U}}_n$.
By the previous
proposition, this image lies in $I(\Phi_n)$.
\end{proof}
\subsection{Monomial induced map}
\label{monomial}
In this section, we deal with the maps in Case 5 of Proposition \ref{cases} and prove
\begin{proposition}\label{monomial-ind}
Let $d \ge 3$ and consider $f\in{\mathcal{U}}_n$ such that $d_{\mathtt{h}}(f)=1$, where $\mathtt{h}$ is the bad hole of $f$. If $\hat{f}(\mathtt{h}) \neq \mathtt{h}$, then $[f] \in I(\Phi_n)$. Additionally assume that $d\ge 4$. Then Theorem \ref{Thm:perturbation} holds.
\end{proposition}
In view of Proposition~\ref{depth1poly}, modulo conjugacy, $\hat{f} (z) =z^{-(d-1)}$ and $\mathtt{h} = \infty$.
We write $n=2m$ or $2m - 1$ for some $m \ge 1$.
Then
$$\overline{d}_\infty (f^n) = \dfrac{1}{d} \sum_{j = 0}^{m-1} \left(\dfrac{d-1}{d} \right)^{2j}.$$
Since $f \in{\mathcal{U}}_n$, we have that $\overline{d}_\infty (f^n ) \ge 1/2$ with strict inequality if $n$ is odd. Hence $n\ge 3$.
The strategy to prove the proposition is similar to the previous case (polynomial induced map) with the extra complication that the bad hole now has period $2$.
\subsubsection{Proof of Proposition \ref{monomial-ind}: $d\ge 4$ case.}
Consider for the moment $N >1$ to be adjusted in the sequel. For $\lambda\in\mathbb{C}\setminus\{0, 1\}$, let
$$\phi_{\lambda,t}(z)=(1-t^Nz)(1-\lambda t^Nz)\hat f(z).$$
Since $\hat{f} (z) =z^{-(d-1)}$, the map $\phi_{\lambda,t}$ also has degree $d-1$.
For $\alpha \in \mathbb{R}$, we have
$$ \phi_{\lambda,t} (\chi_\alpha) =
\begin{cases}
\chi_{-(d-1)\alpha} & \alpha \le N,\\
\chi_{-(d-3)\alpha -2N} & \alpha > N.
\end{cases}
$$
In standard coordinates where we identify $z \in \P^1$ with the
direction at $\zeta$ containing $z t^{-\alpha}$, we have
$$T_{\chi_\alpha} \phi_{\lambda,t} (z) =
\begin{cases}
z^{-(d-1)} & \alpha <N,\\
(1-z)(1-\lambda z) z^{-(d-1)}& \alpha =N,\\
\lambda z^{-(d-3)} & \alpha > N.
\end{cases}
$$
As in the proof of the previous proposition,
for an even integer $k$ to be chosen later such that $0\le k\le n-1$, we let $$\alpha_0 = \dfrac{N}{(d-1)^k}$$
and set
\begin{eqnarray*}
\zeta_0 &=& \chi_{\alpha_0},\\
\zeta_j &=& \phi^j_{\lambda,t} (\zeta_0), \, \text{ for } j \ge 1.
\end{eqnarray*}
Observe that $\zeta_k = \chi_N$.
Now let $\vec{v}_\infty\in T_{\zeta_{2m}}{\mathbf{P}^1}$ be the direction containing $\infty$. Pick $c_1\not=c_2\in\mathbb{L}$ in the direction $\vec{v}_\infty$. Consider
$$g_{\lambda,t}(z)=\left(1+\frac{c_1-c_2}{z-c_1}\right)\phi_{\lambda,t}(z).$$
Then $g_{\lambda,t}\to f$ as $t\to 0$. Choose $N$ large such that Lemma \ref{construction} implies that $\zeta_j = g^j_{\lambda,t} (\zeta_0)$ and $T_{\zeta}g_{\lambda,t}=T_\zeta\phi_{\lambda,t}$
for $\zeta\in [\zeta_{2m-1},\zeta_{2m-2}]$. Moreover, a direction at such $\zeta$ has positive surplus multiplicity if and only if it contains $\infty$. In this case, the surplus multiplicity is $1$. \par
Again we set $\mu=(d-1)/d$ and $\theta=(d-3)/d$.
Denote by $\vec{w}_0$ and $\vec{w}_\infty$ the directions at $T_{\zeta_0} {\mathbf{P}^1}$ containing $0$ and $\infty$, respectively.
For the direction $\vec{w}_\infty$, we have
$$\overline{s}_\lambda(T_{\zeta_0}g_{\lambda,t}^j(\vec{w}_\infty))=
\begin{cases}
\dfrac{1}{d} &\text{even}\ j\ge 0,\\
0 &\text{odd}\ j\ge 1,
\end{cases}$$
and
$$\overline{m}_\lambda^j(\vec{w}_\infty)=
\begin{cases}
\mu^j & 1\le j\le k,\\
\mu^k\theta^{(j-k+1)/2}\mu^{(j-k-1)/2} & \text{odd}\ j\ge k+1,\\
\mu^k\theta^{(j-k)/2}\mu^{(j-k)/2} & \text{even}\ j\ge k+2.
\end{cases}$$
Then by Lemma \ref{surplus-iterate},
$$\overline{s}_\lambda^n (\vec{w}_\infty) =\frac{1}{d}\sum_{j=0}^{k/2}\mu^{2j}+\frac{1}{d}\mu^k\sum_{j=1}^{m-k/2-1}\mu^j\theta^{j}.$$
Let $\vec{w}\in T_{\zeta_0} {\mathbf{P}^1}$ be a direction such that $T_{\zeta_0}g_{\lambda,t}^k(\vec{w})$ is a direction containing $t^N$ or $\lambda t^N$. Then there are $2(d-1)^k$ many such $\vec{w}$, and under $g_{\lambda,t}^n$ each of them has proportional surplus multiplicity
$$\overline s^n_\lambda(\vec{w})=\frac{1}{d}\cdot\frac{1}{d^{k+1}}\sum_{j=1}^{m-k/2-1}\mu^j\theta^{j-1}.$$
Thus
$$\overline s^n_\lambda(\vec{w}_\infty)+2(d-1)^k\overline s^n_\lambda(\vec{v})=\frac{1}{d}\sum_{j=0}^{k/2}\mu^{2j}+\frac{1}{d}\mu^{k+2}\sum_{j=0}^{m-k/2-2}\mu^j\theta^{j}=\frac{1}{d}\sum_{j=0}^{k/2+1}\mu^{2j}+\frac{1}{d}\mu^{k+2}\sum_{j=1}^{m-k/2-2}\mu^j\theta^{j}.$$
Set
$$a_k=\frac{1}{d}\sum_{j=0}^{k/2}\mu^{2j}+\frac{1}{d}\mu^k\sum_{j=1}^{m-k/2-1}\mu^j\theta^{j}.$$
Then $a_k$ decreases as even $k$ decreases. Note that
$a_{2m-2}=\overline{d}_\infty (f^n)$ and $a_{0}<\mu^-(d^n)\le \mu^+(d^n)$. Thus there exists even integer $0\le k\le 2m-2$ such that
$$\begin{cases}
a_k<\mu^-(d^n)\le\mu^+(d^n)\le a_{k+2} &\text{if}\ n \text{ is even},\\
a_k\le\mu^-(d^n)\le\mu^+(d^n)<a_{k+2} &\text{if}\ n \text{ is odd}.
\end{cases}$$
For this $k$, we have
$$\begin{cases}
\overline s^n_\lambda(\vec{w}_\infty)<\mu^-(d^n)\ \text{and}\ \sum\limits_{\vec{v}\not=\vec{w}_0}\overline s^n_\lambda(\vec{v})\ge \mu^+(d^n) &\text{if}\ n \text{ is even},\\
\overline s^n_\lambda(\vec{w}_\infty)\le\mu^-(d^n)\ \text{and}\ \sum\limits_{\vec{v}\not=\vec{w}_0}\overline s^n_\lambda(\vec{v})> \mu^+(d^n) &\text{if}\ n \text{ is odd}.
\end{cases}$$
For such $k$, let $M_t(z) = t^{-\alpha_0} z$ and consider
$$G_{\lambda} = \lim_{t \to 0} M_t^{-1} \circ g^n_{\lambda,t} \circ M_t.$$
Since $g^n_{\lambda,t}(\zeta_0)\not=\zeta_0$, the induced map $\widehat G_{\lambda}$ is a constant. Hence
$$\sum_{z\in\P^1}\overline d_z(G_{\lambda})=1.$$
By Lemma \ref{relative-position}, if $\vec{v}_z\not=\vec{w}_0\in T_{\zeta_0}{\mathbf{P}^1}$, we have $\overline d_z(G_{\lambda})=\overline s_\lambda^n(\vec{v}_z)$.
Thus
$$\overline d_\infty(G_{\lambda})=\overline{s}_\lambda^n (\vec{w}_\infty)$$
and
$$\overline d_0(G_{\lambda})=1-\sum_{\vec{v}\not=\vec{w}_0}\overline s_\lambda^n(\vec{v}).$$
It follows that
$$\begin{cases}
\overline d_\infty(G_{\lambda})<\mu^-(d^n)\ \text{and}\ \overline d_0(G_{\lambda})\le \mu^-(d^n) &\text{if}\ n \text{ is even},\\
\overline d_\infty(G_{\lambda})\le\mu^-(d^n)\ \text{and}\ \overline d_0(G_{\lambda})<\mu^-(d^n) &\text{if}\ n \text{ is odd}.
\end{cases}$$
Moreover, if $z$ is a $(d-1)^k$-th root of unity or a $(d-1)^k$-th root of $\lambda$, then
$$\overline d_z(G_{\lambda})=\frac{1}{d}\cdot\frac{1}{d^{k+1}}\sum_{j=1}^{m-k/2-1}\mu^j\theta^{j-1}<\mu^-(d^n).$$
Note that if $n$ is even, the induced map $\widehat G_{\lambda}$ is $[1:0]\in\P^1$; if $n$ is odd, $\widehat G_{\lambda}$ is $[0:1]\in\P^1$. Hence by Proposition~\ref{stability-proportional-depth}, $G_{\lambda}$ is stable. Moreover, $[G_\lambda]$ varies with $\lambda$. Note that the action of $\phi_{\lambda, t}$ and hence of $g_{\lambda, t}$ on the segment $[0,\infty]$ is independent of $\lambda$. Therefore, the construction is such that Theorem \ref{Thm:perturbation} holds and hence $[f]\in I(\Phi_n)$. \hfill $\Box$
\subsubsection{Proof of Proposition \ref{monomial-ind}: cubic case.}\label{cubic-monomial}
In this case, $\deg\hat f=2$. First as in the previous case, consider
$$\phi_{t}(z)=(1-t^Nz)\hat f(z),$$
where $N >1$ is sufficiently large.
Given $\alpha\in\mathbb{R}$, note that
$$\phi_{t} (\chi_\alpha) =
\begin{cases}
\chi_{-2\alpha} & \alpha \le N,\\
\chi_{-\alpha -N} & \alpha > N.
\end{cases}
$$
In standard coordinates where we identify $z \in \P^1$ with the
direction at $\zeta$ containing $z t^{-\alpha}$, we have
$$T_{\chi_\alpha} \phi_{t} (z) =
\begin{cases}
z^{-2} & \alpha <N,\\
(1-z)z^{-2}& \alpha =N,\\
- z^{-1} & \alpha > N.
\end{cases}
$$
Now consider $\zeta_j$ for $j \ge 0$ and $g_t(z)$ as in the proof of previous case.
Set
$$a_k=\frac{1}{3}\sum_{j=0}^{k/2}\left(\frac{2}{3}\right)^{2j}+\frac{1}{3}\left(\frac{2}{3}\right)^k\sum_{j=1}^{m-k/2-1}\left(\frac{2}{3}\right)^j\left(\frac{1}{3}\right)^{j}.$$
Choose an even integer $0\le k\le 2m-2$ such that
$$\begin{cases}
a_k<\mu^-(3^n)\le\mu^+(3^n)\le a_{k+2} &\text{if}\ n \text{ is even},\\
a_k\le\mu^-(3^n)\le\mu^+(3^n)<a_{k+2} &\text{if}\ n \text{ is odd}.
\end{cases}$$
For such $k$, let $M_t(z) = t^{-N/2^k} z$
and consider
$$G = \lim_{t \to 0} M_t^{-1} \circ g^n_{t} \circ M_t.$$
If $z$ is a $2^k$-th root of unity, then
$$\overline d_z(G)=\frac{1}{3^{k+2}}\sum_{j=1}^{m-k/2-1}\left(\frac{2}{3}\right)^{j}\left(\frac{1}{3}\right)^{j-1},$$
A direct computation shows that $G$ is stable since
$$\overline d_\infty(G)=a_k,$$
$$\overline d_0(G)=1-\sum_{z\not=0}\overline d_z(G)=1-a_{k+2},$$
and
if $n$ is even, $\widehat G=[1:0]\in\P^1$; if $n$ is odd, $\widehat G=[0:1]\in\P^1$.
Now we consider
$$\psi_{t} (z) =\dfrac{t^Sz -1}{t^S z -(1+t)}\hat{f}(z),$$
where $S >1$ is sufficiently large.
Observe that given $\alpha \in \mathbb{R}$, we have $\psi_t (\chi_\alpha) = \chi_{2 \alpha}$ and
$$T_{\chi_\alpha}\psi_{t} (z) = z^{-2}.$$
For an even integer $\tilde k$ to be chosen later with $1 \le \tilde k < n$, let $$\tilde{\alpha}_0 = \dfrac{S}{2^{\tilde k}}$$
and $$\tilde\zeta_0 = \chi_{\tilde{\alpha}_0}.$$
For $j \ge 1$, let $$\tilde\zeta_j = \psi^j_{t} (\tilde\zeta_0)$$
and observe that $\tilde\zeta_{\tilde k} = \chi_{S}$.
Let $\vec{v}_0$ (resp. $\vec{v}_\infty$) be the direction in $T_{\tilde\zeta_0}{\mathbf{P}^1}$ containing $0$ (resp. $\infty$). Observe that $\tilde \zeta_{\tilde k}$ is contained in the direction $T_{\tilde\zeta_0}\psi^j_{t} (\vec{v}_0)$ or in $T_{\tilde\zeta_0}\psi^j_{t} (\vec{v}_\infty)$ for all $j \neq\tilde k$.
Thus,
$$\overline{s}^n (\vec{v}_0) + \overline{s}^n (\vec{v}_\infty) + \dfrac{1}{3} \left(\dfrac{2}{3} \right)^{\tilde k} = \sum_{j=0}^{n-1} \dfrac{1}{3}\left(\dfrac{2}{3} \right)^{j}= 1 - \left(\dfrac{2}{3} \right)^{n}.$$
Since $T_{\tilde\zeta_0}\psi^j_{t} (\vec{v}_\infty)$ contains $\tilde\zeta_{\tilde k}$ only for $j$ even with $j <\tilde k$, we have
$$\overline{s}^n (\vec{v}_\infty) = \dfrac{1}{3} \sum_{j=0}^{\tilde k/2 -1} \left(\dfrac{2}{3} \right)^{2j}.$$
Choose $\tilde k$ even such that if $n$ is even,
$$\dfrac{1}{3} \sum_{j=0}^{\tilde k/2 -1} \left(\dfrac{2}{3} \right)^{2j} <\mu^-(3^n)<\mu^+(3^n)\le
\dfrac{1}{3} \sum_{j=0}^{\tilde k/2} \left(\dfrac{2}{3} \right)^{2j},$$
and if $n$ is odd,
$$\dfrac{1}{3} \sum_{j=0}^{\tilde k/2 -1} \left(\dfrac{2}{3} \right)^{2j}\le\mu^-(3^n)<\mu^+(3^n)<
\dfrac{1}{3} \sum_{j=0}^{\tilde k/2} \left(\dfrac{2}{3} \right)^{2j}.$$
Thus, if $n$ is even,
$$ \overline{s}^n (\vec{v}_\infty) < \mu^-(3^n)\ \text{and}\ \overline{s}^n (\vec{v}_0) \le \mu^-(3^n) - \left(\dfrac{2}{3} \right)^{n},$$
and
if $n$ is odd,
$$ \overline{s}^n (\vec{v}_\infty) \le \mu^-(3^n)\ \text{and}\ \overline{s}^n (\vec{v}_0) < \mu^-(3^n) - \left(\dfrac{2}{3} \right)^{n}.$$
If $\vec{v}$ is one of the $2^{\tilde k}$ directions at $\tilde \zeta_0$ for which $T_{\tilde \zeta_0}\psi^{\tilde k}_{t} (\vec{v})$
contains $t^{-S}$, then
$$\overline{s}^n(\vec{v}) =\frac{1}{3^{\tilde k+1}}.$$
Now let $L_t(z) = t^{-\tilde\alpha_0 z}$ and
$$F= \lim_{t \to 0} L_t^{-1} \circ\psi^n_{t} \circ L_t \in \overline{\Rat}_{d^n}.$$
The holes of $F$ are $0$, $\infty$, and all $z$ such that $T_{\tilde \zeta_0}\psi^{\tilde k}_{t} (z)=1$. Moreover, we have
$$\overline{d}_0(F)=\overline{s}^n (\vec{v}_0)+\left(\dfrac{2}{3} \right)^{n}.$$
If $\vec{v}_z$ is one of the $2^{\tilde k}$ directions at $\tilde \zeta_0$ for which $T_{\tilde \zeta_0}\psi^{\tilde k}_{t} (\vec{v}_z)$
contains $t^{-S}$, then
$$\overline{d}_z(F)=\overline{s}^n (\vec{v}_z) =\frac{1}{3^{\tilde k+1}}<\mu^-(3^n).$$
Note that if $n$ is even, $\widehat F=[1:0]$, and if $n$ is odd, $\widehat F=[0:1]$. It follows that $F$ is stable.
Now we claim that $G$ and $F$ are not in the same GIT-class. For otherwise, $k=\tilde k$ and $d_{z}(G)=d_z(F)$ if $z$ is a $2^{k}$th-root of unity. However, if $k=\tilde k$, we have
$$d_{z}(G)=\frac{1}{3^{k+2}}\sum_{j=1}^{m-k/2-1}\frac{1}{3^{2j-1}}< \frac{1}{3^{k+1}}=d_z(F),$$
which is a contradiction. \hfill $\Box$
\section{Constant induced map}
\label{constant}
In this section, we deal with the constant induced map case and complete the proof of Theorem \ref{main}. More precisely, we prove
\begin{proposition}
\label{constant-p}
If $f\in I(d)\cup{\mathcal{U}}_n$ is semistable and has constant induced map $\hat f$. Then $[f]\in I(\Phi_n)$.
\end{proposition}
First we establish some general properties of degenerate rational maps
with constant induced maps. Then in Section~\ref{proof-constant}
we prove the above proposition.
\subsection{Stable and $n$-unstable maps with constant induced map}
Recall that
$$I(d)=\{f=H_f\hat f\in\P^{2d+1}: \hat f=c\in\P^1, H_f(c)=0\}. $$
By Proposition \ref{stability-depth}, we immediately have
\begin{lemma}\label{deg4}
For $d\ge 2$, then
$\mathrm{Rat}_d^s\cap I(d)=\emptyset$ if and only if $d=2$ or $3$.
\end{lemma}
For degrees $d\ge 4$, we have a
lower bound on the number of holes of the maps in $\mathrm{Rat}_d^s\cap I(d)$:
\begin{lemma}\label{num-hole}
For $d\ge 4$, if $f\in\mathrm{Rat}_d^s\cap I(d)$, then $f$ has at least $3$ holes.
\end{lemma}
\begin{proof}
We proceed by contradiction. Suppose $f$ has two holes, $h_1, h_2$ with depths $d_1,d_2$. Write $f=H_f\hat f$. Then $\deg H_f=d$ and $d_1+d_2=d$. Since $f\in\mathrm{Rat}_d^s$, by
Proposition~\ref{stability-depth}, $d_1\le d/2$ and $d_2\le d/2$. Hence $d_1=d_2=d/2$. Since $f\in I(d)$, the induced map $\hat f$ is a constant which is a hole, say $h_1$. Again by Proposition~\ref{stability-depth}, we have $d_1< d/2$, since $f\in \mathrm{Rat}_d^s$, which is a contradiction.
\end{proof}
For the $n$-unstable set we have
\begin{lemma}\label{deg0}
Consider $f \in{\mathcal{U}}_n$ with bad hole $\mathtt{h}$.
If $\deg\hat f=0$, then $d_{\mathtt{h}}(f)=(d+1)/2$.
\end{lemma}
\begin{proof}
It is a direct consequence of Lemma~\ref{depth-multiplicity-inequality}
since $\deg\hat f=0$ and therefore $m_{\mathtt{h}} (\hat f ) =0$.
\end{proof}
If $f \in I(d)$ is strictly semistable or $f$ is $n$-unstable with constant induced map, then $f$ is GIT conjugate to a degenerate map with non-constant induced map:
\begin{lemma}\label{const-to-nonconst}
For odd $d\ge 3$, set
$$F([X:Y])=X^{\frac{d+1}{2}}Y^{\frac{d-1}{2}}[1:0]\in\P^{2d+1}$$
and
$$G([X:Y])=X^{\frac{d-3}{2}}Y^{\frac{d-1}{2}}[X^2:Y^2]\in\P^{2d+1}.$$
Then
\begin{enumerate}
\item $[F]=[G]$.
\item If $f\in(\mathrm{Rat}_d^{ss}\setminus\mathrm{Rat}_d^s)\cap I(d)$ or $f\in{\mathcal{U}}_n$ with constant induced map $\hat f$, then $[f]=[F]$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $M_t(z)=tz$. Then as $t\to 0$,
$M_t\circ G\circ M_t^{-1}\to F$.
Since $F$ and $G$ are in $\mathrm{Rat}_d^{ss}\setminus\mathrm{Rat}_d^s$, we have $[F]=[G]$.
When $f \in {\mathcal{U}}_n$ has constant induced map, by Lemma ~\ref{deg0}, it follows that $f$ is M\"obius conjugate to
$$f_{-1}([X:Y])=X^{\frac{d+1}{2}}H_{-1}(X,Y)[1:0],$$
where $H_{-1}(X,Y)$ is a degree $(d-1)/2$ homogenous polynomial and $H_{-1}([0:1])\not=0.$
When $f\in(\mathrm{Rat}_d^{ss}\setminus\mathrm{Rat}_d^s)\cap I(d)$, by Proposition ~\ref{stability-depth} we have that $f$ is M\"obius conjugate to
$$f_1([X:Y])=X^{\frac{d-1}{2}}H_1(X,Y)[0:1], $$
where $H_1(X,Y)$ is a degree $(d+1)/2$ homogenous polynomial and $H_1([0:1])\not=0$.
For $i\in\{-1, 1\}$, set $M_{t,i}(z)=t^iz^{-i}$. Then as $t\to 0$,
$$M_{t,i}\circ f_i\circ M_{t,i}^{-1}\to F.$$
Note $f_{-1},f_1$ and $F$ are semistable. We have $[f]=[F]$.
\end{proof}
\subsection{Proof of Proposition~\ref{constant-p}}
\label{proof-constant}
If $f$ satisfies the assumptions in Lemma \ref{const-to-nonconst} (2), then $[f]=[G]$, where $G$ is defined as in Lemma \ref{const-to-nonconst}. We can apply either Proposition \ref{depth>1} or \ref{poly-ind} to $G$ depending whether $d\ge 4$ or $d=3$ to conclude that $[f] \in I(\Phi_n)$. So it is sufficient to assume $f\in\mathrm{Rat}_d^s\cap I(d)$.
By Lemmas \ref{deg4} and \ref{num-hole}, we may assume $d\ge 4$ and normalize such that $f$ has holes at $0,1$ and $\infty$. Thus $f$ has the form:
$$f([X:Y])=H_f(X,Y)\hat f([X:Y])=X^{d_0}Y^{d_\infty}(X-Y)^{d_1}\prod_{i=2}^k(X-c_iY)^{d_i}[0:1],$$
where $1\le d_1\le d/2$, $1\le d_i<d/2$ for $i\in\{0,2,\cdots,k,\infty\}$ and $c_2,\cdots, c_k$ are distinct points in $\mathbb{C}\setminus\{0,1\}$. \par
For $t\in\mathbb{C}\setminus\{0\}$, set
$$g_t([X:Y])=H_{f}(X,Y)[t:1]$$
and
$$h_t([X:Y])=\frac{H_{f}(X,Y)}{Y}[tX:Y].$$
Then $g_t$ and $h_t$ are stable but not in $I(d)$ for sufficiently small $t\not=0$. Moreover, $g_t$ and $h_t$ converges to $f$ as $t\to 0$. Hence $[g_t]$ and $[h_t]$ converge to $[f]$ as $t\to 0$.
Note
$$g_t^n([X:Y])=(H_f(X,Y))^{d^{n-1}}[t:1]$$
and
$$h_t^n([X:Y])=\prod_{m=0}^{n-1}\left(\frac{H_f(t^mX,Y)}{Y}\right)^{d^{n-1-m}}[t^nX:Y].$$
Then for sufficiently small $t\not=0$, by Lemma \ref{stability-depth}, $g_t^n$ and $h_t^n$ are stable.
Set $g_n=\lim\limits_{t\to 0}g_t^n$ and $h_n=\lim\limits_{t\to 0}h_t^n$.
Then we have
$$g_n([X:Y])=(H_f(X,Y))^{d^{n-1}}[0:1].$$
If $m \neq 0$,
then $$H_f (t^m X, Y) = t^{m d_0} X^{d_0}Y^{d_\infty}(t^m X-Y)^{d_1}\prod_{i=2}^k(t^m X-c_iY)^{d_i} \to X^{d_0} Y^{d-d_0}$$
in the projective space of the vector space of homogenous polynomials of degree $d$.
Taking into account that
$$\prod_{m=1}^{n-1} \left( X^{ d_0} Y^{(d-d_0-1)} \right)^{d^{n-1-m}} = \left( X^{ d_0} Y^{(d-d_0-1)} \right)^{\frac{d^{n-1}-1}{d-1}}, $$
it follows that
$$h_n([X:Y]) = \left(\frac{H_f(X,Y)}{Y}\right)^{d^{n-1}} \left( X^{ d_0} Y^{(d-d_0-1)} \right)^{\frac{d^{n-1}-1}{d-1}} Y [0:1].$$
After collecting all powers of $X$ and $Y$ we obtain
$$h_n([X:Y])=\left(\frac{H_f(X,Y)}{X^{d_0}Y^{d_\infty}}\right)^{d^{n-1}} X^{\frac{d^n-1}{d-1}d_0}Y^{d^{n-1}d_\infty-\frac{d^{n-1}-1}{d-1}d_0}[0:1].$$
Note $d_0(g_n)<d^n/2$, $d_0(h_n)<d^n/2$ and the depths of all other holes of $g_n$ and $h_n$ are $\le d^n/2$. Therefore $g_n$ and $h_n$ are stable. Thus, as $t\to 0$, $\Phi_n([g_t])$ converges to $[g_n]$ and $\Phi_n([h_t])$ converges to $[h_n]$. However, $[g_n]\neq [h_n]$ since $d_0(g_n)\not=d_0(h_n)$ and the induced map for both $g_n$ and $h_n$ is the constant map $[0:1]$. Thus $[f]\in I(\Phi_n)$ for all $n\ge 2$.
\hfill $\Box$
\bibliographystyle{siam}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,891 |
{"url":"https:\/\/divisbyzero.com\/2018\/02\/13\/finite-differences-of-polynomials\/","text":"# Finite differences of polynomials\n\nIt is interesting watching my kids go through the school math curriculum. Since I\u2019m a math professor, one would think that I would know all of the school-aged math. While that is mostly true, sometimes the teachers and textbooks use unfamiliar terminology for familiar mathematical ideas. (\u201cOh, ____ is just ___,\u201d I\u2019ve said multiple times.)\n\nMy son is now in Algebra 2, and for the first time, he showed me something that I\u2019ve never seen before\u2014the relationship between polynomials and finite differences.\n\nTake any polynomial, such as $f(x)=7x^3+4x^2-2x+15,$ and any arithmetic sequence, such as 0, 2, 4, 6,\u2026 Plug these values into the polynomial. Take the neighboring pairwise differences. So, for instance, $f(2)-f(0)=83-15=68.$ Then take the neighboring pairwise differences of those values, and so on. It turns out that the\u00a0nth level will consist entirely of the same nonzero value if, and only if, the polynomial has degree\u00a0n. Wow! That\u2019s so cool!\n\nHere\u2019s my worked-out degree-3 example. Notice that after three levels, we yield the constant value 336:\n\nMy son\u2019s textbook (Algebra 2, by Larson, Boswell, Kanold, and Stiff)\u00a0shared this fact but provided no explanation for why it is true (which, as a mathematician, I find very disappointing). So, I had to work it out myself.\n\nIt turns out that more is true\u2014if the polynomial has degree\u00a0n with\u00a0leading coefficient\u00a0c,\u00a0and is the difference between terms in\u00a0the arithmetic sequence, then the final value is $a^n c\\cdot n!.$ In particular, if a=1 and c=1, then the pairwise difference process will terminate with n!. Notice that in my example, the final value is $2^3\\cdot 7\\cdot 3!=336.$\n\nWhy is this true?\n\nHere\u2019s a proof by induction on the degree of the polynomial. As the base case, consider a degree-1 polynomial: p(x)=cx+b. Then,\n\n$p(x+a)-p(x)=(c(x+a)+b)-(cx+b)=ac=a^1c\\cdot 1!,$\n\nso the base case holds.\n\nNow, assume that the result is true for any polynomial of degree n-1, for some n\u22652. We will prove that it is true for a polynomial of degree\u00a0n.\u00a0Let $p(x)=cx^n+dx^{n-1}+\\text{l.o.t.},$ where c\u22600 and \u201cl.o.t.\u201d means \u201clower order terms.\u201d We see that\n\n$p(x+a)-p(x)=(c(x+a)^n+d(x+a)^{n-1}+\\text{l.o.t.})-(cx^n+dx^{n-1}+\\text{l.o.t.})=cx^n+acnx^{n-1}+dx^{n-1}+\\text{l.o.t.}-cx^n-dx^{n-1}+\\text{l.o.t.}=acnx^{n-1}+\\text{l.o.t.}.$\n\nLet us call this polynomial q(x).\n\nNotice that because the leading coefficient\u00a0acn\u00a0is nonzero,\u00a0\u00a0q(x) has degree n-1. By our inductive hypothesis, after n-1 pairwise differences, the polynomial\u00a0q(x)\u00a0will yield a constant value $acn\\cdot a^{n-1}(n-1)!=a^ncn!.$ Thus, for p,\u00a0the process terminates after\u00a0steps with the constant value $a^n c n!.$ This proves the theorem.\n\nAfter playing around with this, I googled it, and\u2014no surprise\u2014the mathematics of finite differences has a long history. Also, it is not difficult to see the resemblance of these calculations to the calculation of the derivative (using the definition of the derivative).","date":"2018-04-22 02:37:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 9, \"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\": 0.8306077718734741, \"perplexity\": 478.61267821356427}, \"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-2018-17\/segments\/1524125945484.58\/warc\/CC-MAIN-20180422022057-20180422042057-00553.warc.gz\"}"} | null | null |
#import "MULogViewController.h"
#import "MULogEntry.h"
@interface MULogViewController () {
NSMutableArray *_logEntries;
MKServerModel *_serverModel;
NSDateFormatter *_dateFormatter;
}
@end
@implementation MULogViewController
- (id) initWithServerModel:(MKServerModel *)serverModel {
self = [super init];
if (self == nil)
return nil;
_serverModel = serverModel;
[_serverModel addDelegate:self];
_logEntries = [[NSMutableArray alloc] init];
_dateFormatter = [[NSDateFormatter alloc] init];
[_dateFormatter setTimeStyle:NSDateFormatterLongStyle];
return self;
}
- (void) dealloc {
[_serverModel removeDelegate:self];
[_logEntries release];
[super dealloc];
}
- (void) didReceiveMemoryWarning {
[super didReceiveMemoryWarning];
}
#pragma mark -
- (void) viewWillAppear:(BOOL)animated {
self.navigationItem.title = @"Log";
}
#pragma mark -
//
// A user joined the server.
//
- (void) serverModel:(MKServerModel *)server userJoined:(MKUser *)user {
MULogEntry *entry = [[MULogEntry alloc] initWithText:@"User Joined"];
[_logEntries addObject:entry];
[entry release];
[self.tableView insertRowsAtIndexPaths:[NSArray arrayWithObject:[NSIndexPath indexPathForRow:0 inSection:0]] withRowAnimation:UITableViewRowAnimationTop];
}
//
// A user left the server.
//
- (void) serverModel:(MKServerModel *)server userLeft:(MKUser *)user {
MULogEntry *entry = [[MULogEntry alloc] initWithText:@"User Left"];
[_logEntries addObject:entry];
[entry release];
[self.tableView insertRowsAtIndexPaths:[NSArray arrayWithObject:[NSIndexPath indexPathForRow:0 inSection:0]] withRowAnimation:UITableViewRowAnimationTop];
}
#pragma mark Table view data source
- (NSInteger) numberOfSectionsInTableView:(UITableView *)tableView {
return 1;
}
- (NSInteger) tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section {
return [_logEntries count];
}
- (UITableViewCell *)tableView:(UITableView *)tableView cellForRowAtIndexPath:(NSIndexPath *)indexPath {
static NSString *CellIdentifier = @"logCell";
UITableViewCell *cell = [tableView dequeueReusableCellWithIdentifier:CellIdentifier];
if (cell == nil) {
cell = [[[UITableViewCell alloc] initWithStyle:UITableViewCellStyleSubtitle reuseIdentifier:CellIdentifier] autorelease];
}
// Ascending
NSUInteger idx = 0;
if (_logEntries.count > 0) {
idx = (_logEntries.count - 1) - indexPath.row;
}
MULogEntry *entry = [_logEntries objectAtIndex:idx];
cell.textLabel.text = [entry text];
cell.detailTextLabel.text = [_dateFormatter stringFromDate:[entry date]];
return cell;
}
- (void)tableView:(UITableView *)tableView didSelectRowAtIndexPath:(NSIndexPath *)indexPath {
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,929 |
Home » Accumulator » Bladder Accumulators »
Bladder Accumulators
Three Symptoms Of A Bad Accumulator
Bladder accumulators or Gas-charged accumulators are widely used nowadays as this phenomenon is used thoroughly for the hydraulic system. The multipurpose functionality of the accumulator has retained a sense of importance. The accumulator charges every part of the hydraulic system.
A few of its significance surpasses the role of preservation as well as storing the energy. The energy is stored to maintain the leakage of the fluid. Other than that, it maintains the thermal conductivity as the accumulator seeks to concentrate on the compensation of energy.
The hydraulic accumulator absorbs the shock waves generated in the system and tries to compensate for the energy. It has been said that a hydraulic system can work without maintenance for twelve years. According to the law, which states that the integrity of the accumulator can be reserved for a long time.
However, the strain of working multiple tasks to maintain the efficiency of the result it is producing indicates the need to conduct a maintenance service now and then. In this article, we will tell you the symptoms of a bad bladder accumulator so that you can conduct an analysis easily.
What Are The Symptoms Of A Bad Hydraulic Accumulator?
Getting acquainted with the law and regulation regarding the supervision of the accumulator is a very important task that you need to accomplish. Only by knowing what they are, you can execute every step. But if you see any of the symptoms of a bad accumulator occurring before twelve years, you need to authorize a supervision session. Let us discuss what those symptoms are.
Noise: The occurrence of noise in the hydraulic system indicates a malfunction in the system. The noises occur due to aeration or cavitation. The cavitation or rotation is the result of the contamination in the hydraulic fluid.
The air present in the hydraulic system makes a banging noise when it is being compressed and decompressed whenever it is circulated through the system. The noise can even occur due to the creation of foaming in the hydraulic fluid.
The aeration then strengthens the process of composition which causes damage to piston-type hydraulic accumulators and others due to lack of lubrication, loss of seals, and overheating.
The air gets into the fluid through the pump's inlet. So ensuring that the inlet is in a good condition is important. The facets that you have to keep in mind are written below.
The integrity of the pump's inlet
The condition of the clamps should be tightly fitted as flexible lines could cause the air to be injected.
The fluid level should be maintained.
You have to check the condition of the pump shaft seal. If the seal.is leaking then you might have to replace it.
The cavitation appears mainly when the fluid crosses the excessive limit of the system the ever excessive fluid creates pressure in the system that falls below the vapor pressure of the hydraulic fluid. It causes the vapor cavity to appear within the fluid. That is how noise occurs.
High fluid temperature: The temperature of the fluid should not exceed above 180°F. Going above the set limit means the degradation of the Parker accumulator. If the viscosity falls low below the optimum level of the components of the system then the temperature of the fluid is bound to shoot up.
The hydraulic system emits heat through the reservoir. You have to ensure that the amount of fluid in the reservoir does not exceed.
No obstructions should be spotted in the reservoir. If any airflow is found in the reservoir then it would lead to the inclusion of first and debris.
Ensure that the inspection of the heat exchanger has been carried out.
The core should not be blocked.
Slow response: If the outcome of the machine has slowed down compared to what it has already been, then it is a sign that the lack of speed in the flow actuator is causing the slow operation.
Most Asked Questions On The Internet:
What is a bladder accumulator?
The bladder accumulator is generally a chamber that stores a compressed gas to provide the whole hydraulic system with power. It has a flexible bladder which makes the operation run smoothly. bladder accumulators contain a forged pressure vessel with numerous fittings to seal the openings so that air doesn't slither.
How does a bladder accumulator work?
The hydraulic pump boosts the pressure of the system which forces the fluid into the accumulator. The directional control valve controls the flow of the fluid to go in and out. A piston-type hydraulic accumulator then compresses the volume of the gas to maintain the precharge pressure and pressure.
What is the purpose of the bladder accumulator?
The Parker accumulator is an energy storage device that enables the hydraulic system to withstand the extremities of the pressure put through the whole mechanism. It responds quickly to the operation as it smoothens the pulsation, reduces noise.
The piston-type hydraulic accumulator is widely used because of its application. All you have to do is keep in mind the three points to conserve the integrity of the accumulator.
Bladder Accumulators, hydraulic accumulator, parker accumulator, piston type hydraulic accumulator
Older Major Three Functions Of A Directional Control Valve
Accumulator, Piston Type Hydraulic Accumulator
The work of Piston Type Accumulators In Hydraulic Systems
Accumulators are commonly used for improving efficiency in pumps. They deliver steadier, sustainable operation, and largely preserve a massive amount of power in case of electrical...
Piston Pumps: An Overview Of Type & Work
Everybody commonly uses piston pumps, but most of us are unaware of its function and the variety. Although it is a part of everyday life, we are in lack of knowledge about piston &...
A Thorough Study Of Hydraulic Accumulators
Hydraulic accumulators are a type of accumulator that has the power to strengthen the working capacity by delivering a smoother and more durable operation ser... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,631 |
José Elias Moedim Júnior (25 de septiembre de 1976), más conocido como Zé Elias, es un exfutbolista brasileño y se desempeñaba como centrocampista. Cabe destacar que ha jugado en 12 clubes de 7 países distintos.
Clubes
Palmarés
SC Corinthians
Copa de Brasil: 1995
FC Internazionale
Copa de la UEFA: 1998
Olympiacos FC
Super Liga de Grecia: 2000-01, 2001-02, 2002-03
Santos FC
Campeonato Brasileño de Serie A: 2003-04
Personas del estado de São Paulo
Futbolistas de la selección de fútbol de Brasil en los años 1990
Futbolistas de Brasil en la Copa de Oro de la Concacaf de 1996
Futbolistas del Sport Club Corinthians Paulista
Futbolistas del Bayer 04 Leverkusen en los años 1990
Futbolistas del Inter de Milán
Futbolistas del Bologna Football Club 1909
Futbolistas del Olympiacos de El Pireo
Futbolistas del Genoa Cricket & Football Club en los años 2000
Futbolistas del Santos Futebol Clube
Futbolistas del Football Club Metalurh Donetsk
Futbolistas del Guarani Futebol Clube
Futbolistas del AC Omonia Nicosia
Futbolistas del Rheindorf Altach
Hermanos futbolistas
Nacidos en São Paulo | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,371 |
using Quilt4.Service.Entity;
using Quilt4Net.Core.DataTransfer;
namespace Quilt4.Service.Converters
{
internal static class IssueResponseConverter
{
public static IssueResponse ToIssueResponse(this RegisterIssueResponseEntity response, string webUrl)
{
return new IssueResponse
{
Ticket = response.Ticket.ToString(),
IssueKey = response.IssueKey,
ServerTime = response.ServerTime,
IssueTypeUrl = webUrl + "#/issueType/" + response.IssueTypeKey,
IssueUrl = webUrl + "#/issue/" + response.IssueKey,
};
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,418 |
\section{Introduction}
\label{intro}
In accreting neutron stars with an extended magnetosphere matter proceeds along the fields towards the magnetic poles. The infalling matter slows down and settles at the base to form an accretion mound confined by the magnetic field. Local magnetic field will be distorted by the pressure of the accreted matter. As mounds of larger mass are built, leading to larger field distortions, MHD instabilities may be triggered \cite{litwin01} which will cause matter to escape from the confined mound. The magnetic structure inside the accretion column and its local dynamics will have important concequences on the observed X-ray emission from such system. For neutron stars with high surface magnetic field ($\sim 10^{12}$), the X-ray emission show cyclotron scattering features which depend on the local magnetic field, and hence can be used as a tool to probe the structure of the accretion column.
For the work in this presentation we consider a neutron star (mass $\sim 1.4 M_{\bigodot}$ and radius $\sim 10$ km) with surface field ($\sim 10^{12}$ G) with an accretion of radius $\sim 1$km. We have studied the structure of the axisymmetric accretion mound by recasting the time-independent Euler equation as the Grad-Shafranov (hereafter GS) equation. We solve the GS equation for accretion mounds of different shapes and sizes and discuss the effect of mass loading of the field on the shape of the mound. We show from MHD simulations that stable mounds can be built only up to a threshold height, beyond which MHD instabilities are triggered. We discuss the effect of the field distortion and local dynamics on the emitted CRSF spectra.
\section{Grad-Shafranov solutions for accretion mounds}\label{sec.GS}
Following the formalism of Mukherjee et.~al \cite{mukherjee12}, \cite{mukherjee13a}, \cite{mukherjee13b}, \cite{mukherjee13c} we model the accretion mound by solving for the static Euler equation. Assuming axisymmetry one can express the poloidal magnetic field in terms of the flux function ($\psi$) to write the Grad-Shafranov (GS) equation for the static accretion mound in force equilibria:
\begin{eqnarray}
r\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial \psi}{\partial r}\right) + \frac{\partial ^2 \psi}{\partial z^2} &=& -4 \pi r^2 \rho g \frac{dh_0(\psi)}{d\psi} \quad \quad \quad \, \mbox{ (Cylindrical coordinates)} \label{eq.gs.cyl} \\
\frac{\partial ^2 \psi}{\partial r^2}+\frac{\sin \theta}{r^2} \frac{\partial }{\partial \theta}\left(\frac{1}{\sin \theta}\frac{\partial \psi}{\partial \theta}\right) &=& -4 \pi r^2 \sin ^2 \theta \rho g \frac{d h_0(\psi)}{d \psi} \quad \mbox{(Spherical coordinates)} \label{eq.gs.sph}
\end{eqnarray}
where $\rho$ is the density and $g=1.86\times10^{14} \mbox{ g cm}^{-2}$ is the acceleration due to gravity. The mound height function $h_0(\psi)$ is an arbitrary function of the magnetic flux ($\psi$) which defines the shape of the mound. We have used the Paczynski equation of state \cite{pacz83} \cite{mukherjee13b} $p= (8 \pi/15)m_ec^2\left(\frac{m_ec}{h}\right)^3 x_F^5/\left(1+16/25x_F^2\right)^{1/2}$, which closely approximates (to within 1.5\%) the $T=0$ degenerate Fermi equation of state for electrons. The density is obtained from the Fermi momentum ($x_F=\frac{1}{m_e c} \left(\frac{3 h^3}{8 \pi \mu _e m_p}\right)^{1/3} \rho ^{1/3}$) as:
\begin{equation}\label{fermiP}
x_F = \frac{5}{4}\left( \frac{\xi ^2 - 8/3 + \xi \sqrt{16/9 + \xi ^2}}{32/9} \right) ^{1/2} \; ; \; \xi=\frac{16}{15} \frac{\mu _e m_p}{m_e c^2} \left(h_0(\psi) - h\right) + 1 \end{equation}
The variable $h$ denotes the vertical height, which for cylindrical system is the coordinate $z$ and $r$ for spherical polar coordinates, such that $h_0-h$ is the height above the neutron star surface. The structure of accretion mounds on high field pulsars have been previously studied by solving the GS equation in cylindrical coordinates (eq.~\ref{eq.gs.cyl}) \cite{mukherjee12}. Although realistic mound profiles can be of complex shapes depending on the mass-loading of the field lines by the accretion disc (or mass capture from winds), in this work we consider smoothly varying mound height profiles (e.g. $h(\psi)=h_c\left(1-\left(\psi/\psi_a\right)^2\right)$) for ease in numerical computations. The shape of the mound will however depend on the mass distribution in different flux-tubes. For a cylindrical coordinate system, the mass per flux tube is approximately related to the accretion rate per unit area as (following eq.~14, eq.~15 and eq.~17 of Hameury (1983) \cite{hameury83}):
\begin{equation}\label{eq.mdot}
\dot{m}t\simeq B_0 \int ^{h_0(\psi)}_{0} \rho(\psi _0,z) r(\psi_0) \frac{\partial r(\psi _0)}{\partial \psi} dz
\end{equation}
where $B_0=10^{12}$ G is the scale magnetic field. Integrating eq.~(\ref{eq.mdot}) using the GS solutions obtained previously we get the mass loading profile of a given mound (see Fig.~\ref{GSsol}). For the mound with exponential profile, the maximum mass is concentrated in flux tubes near the axis. For filled parabolic and hollow mounds, the maximum mass is in the flux tubes near the middle of the mound. In real systems a similar behaviour is expected as the mass-loading will be effective over a finite range of radii at the accretion disc with a maxima near the middle. Realistic models of accretion mounds should consider mass loading profiles from numerical studies of accretion funnel flows \cite{ghosh78}, \cite{romanova02} to determine the mound height function.
\begin{figure}
\centering
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {GSexp.pdf}
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {GSpara.pdf}
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {GSholo.pdf}
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {dmdsi_exp.pdf}
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {dmdsi_para.pdf}
\includegraphics[width = 4.5cm, height = 4.5cm,keepaspectratio] {dmdsi_holo.pdf}
\caption{\small Top: GS solutions for exponential $\left(h_0=h_c\exp{\left(-2\psi/\psi _a\right)}\right .$, $\psi _a$ being the flux function at $r=1$km), parabolic $\left( h_0=h_c\left(1-\left(\psi/\psi _a\right)^2 \right)\right)$ and hollow $\left(h_0=h_c/0.25\left(0.25-\left(\psi/\psi _a - 0.5\right)^2\right)\right)$ mound profiles. The red dotted line represents the top of the mound. Bottom: Mass per flux tube vs radius for respective mound profiles.}
\label{GSsol}
\end{figure}
The solutions discussed so far were for neutron stars with surface fields $\sim 10^{12}$G with polar cap radius 1 km. To study the effect of low field pulsars with larger polar cap, we have solved the GS equation in spherical coordinates (eq.~\ref{eq.gs.sph}). A typical solution for a neutron star with surface field $10^{12}$G and $\sim 1.5$ km polar cap is presented in Fig.~\ref{GSsph}. We have repeated the analysis of Mukherjee et.~al 2012 \cite{mukherjee12} to evaluate the GS equation for mounds with different base magnetic fields, and now also for different polar cap radii. As reported earlier in Mukherjee et.~al 2012 \cite{mukherjee12} the GS solver does not converge beyond a threshold height for a given surface field. Additionally, we find that the threshold height also depends on polar cap size. From numerical fits we find that threshold heights depends on the surface field and the polar cap radius (for a parabolic profile) as
\begin{equation}\label{eq.threshold}
h_T=54\mbox{ m}\left(\frac{B}{10^{12}{\rm G}}\right)^{0.41} \left(\frac{R_p}{1{\rm km}}\right)^{0.42}
\end{equation}
The above scaling can be understood by considering the magnetostatic equilibrium between the pressure gradients and the tension from the curved field line: $ \frac{\partial p}{\partial r} \simeq B_z \frac{\partial B_r}{\partial z}$. Approximating the spatial derivatives with their length scales in either direction, and assuming that the radial component of the field is approximately related to the vertical component as $B_r=B_zR_p/h_T$ ($h_T$ being the threshold mound height), we get $p \simeq B_z^2 R_p^2/h_T^2$. The solutions do not converge when the local pressure gradients are larger than the tension from field curvature, resulting in the formation of closed magnetic loops. As the regions of maximum magnetic tension are near the middle of the mound where $x_F \sim 1$, the pressure can be approximated as $p\propto \rho ^{5/3} \propto h_T^{5/2}$, where the last relation follows from the approximation $x_F << 1$ in eq.~\ref{fermiP}. Using the above relations we get the threshold height as $h_T \propto B_z^{4/9}R_p^{4/9}$. The exponents thus obtained are very close to the values obtained from numerical fits (in eq.~\ref{eq.threshold})\footnote{The approximate analysis in Mukherjee et.~al (2012) \cite{mukherjee12} is similar in nature. However, effects due to bending of the field lines were not accounted for properly, as has been done in Titos et.~al \cite{titos13}, following which we get a better agreement between approximate exponents and the values obtained numerically}.
\begin{figure}
\centering
\includegraphics[width = 12cm, height = 12cm,keepaspectratio] {sphGS.pdf}
\caption{\small GS solution for a mound of height $h_c=21$ m with a polar cap radius $R_p\sim 1.5$ km with surface field $B_p=10^{11}$G. The white lines depict the magnetic field. The density is represented in coloured contours, with the blue-dotted line marking the top of the mound.}
\label{GSsph}
\end{figure}
\section{MHD instabilities in the mounds}
\begin{figure}
\centering
\includegraphics[width = 12cm, height = 12cm,keepaspectratio] {65m_5_106.pdf}
\caption{\small Closed magnetic loops in 2D simulations for a $h_c=65$m mound with surface field $10^{12}$G. As the added mass descends due to gravity, it drags the field lines resulting in the formation of closed magnetic bubbles.}
\label{instability2D}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 8cm, height = 8cm,keepaspectratio] {visit0003.pdf}
\caption{\small 3D density contour of a perturbed mound of height $h_c=70$m at $t\sim 1 t_A$. The radial finger-like structures along the toroidal direction are due to pressure driven interchange instabilities.}
\label{instability3D}
\end{figure}
The high $\beta$ plasma inside the mound along with highly curved magnetic field structures are prone to interchange and ballooning type pressure driven instabilities. MHD simulations of perturbed GS solutions confirm the presence of MHD instabilities beyond a threshold mound \cite{mukherjee13a} \cite{mukherjee13c}. 2D axisymmetric simulations \cite{mukherjee13a} show that gravity driven modes cause formation of closed loops inside mounds beyond a threshold size as added mass descends (as in Fig.~\ref{instability2D}). The threshold height from 2D simulations corresponds to the threshold of the GS solutions (as discussed in Sec.~\ref{sec.GS}) due to the non-convergence of the solutions. In 3D simulations, interchange type pressure driven instabilities take precedence \cite{mukherjee13c}. Multiple finger like channels are formed along the toroidal direction at the outer radial edges (see Fig.~\ref{instability3D}). Matter streams settles in magnetic valleys and flows outwards radially. We find a new threshold mound size beyond which MHD instabilities operate, lower than that of 2D simulations. For system with surface polar field $\sim 10^{12}$G, the instabilities are triggered for mounds of size larger than $5 \times 10^{-13} M_{\bigodot}$, which is in rough agreement to the instability threshold previously predicted by Litwin et.~al \cite{litwin01}. The onset of instabilities will have important concequences for the local dynamics. Leakage of matter on addition of mass beyond the instability threshold will limit further deformation of field lines. Also, radial outflows from the base of the `leaky column' will result in the spread of hot plasma on the surface which may manifest itself as an increase in the area of black body radiation from the surface of the star as is seen in RX J0440.9+4431 \cite{carlo13}.
\section{Effects on CRSF}
Scattering in the presence of strong magnetic field results in the formation of cyclotron resonance scattering features (CRSF), which appear as absorption like features in the X-ray spectra from high field neutron stars. The line energy of the CRSF depends on the local magnetic field:
\begin{equation}\label{cyclotronE}
E_n = m c^2 \frac{\left(1+2n(B/B_{\rm crit})\sin ^2 \theta\right)^{1/2} -1}{\sin ^2 \theta} \frac{1}{1+z}
\end{equation}
where $B_{\rm crit}=(m^2 c^3)/(e \hbar)$, $\theta$ is the angle between the scattering particle and the local magnetic field and $z=1/(\sqrt{1-(2GM)/(Rc^2)})-1$ is the gravitational red-shift for a neutron star of mass $M$ and radius $R$. Accretion induced distortions in the local field will be imprinted on to the emergent spectra. Mukherjee et.~al \cite{mukherjee12} \cite{mukherjee13a} have shown that field distortion can give rise to complex asymmetric profiles with multiple dip-like features by approximately evaluating the line profiles integrated over the mound. Overlying accretion column in accreting system will mask the effect of the field distortion, but if the emission region comes closer to the mound surface effects of field distortion will become apparent. Indeed, larger equivalent widths of the CRSF in V\,0332+53 for spectra closer to the surface (at lower luminosities) \cite{tsygan10} along with the reported asymmetry of the line profile \cite{katja05} \cite{nakajima10} hints to the possibility that the region of emission may have complex non-dipolar field structure.
From the current GS solutions we find that for heights below 200-300 m, there is significant distortion in the local field (see Fig.~\ref{Bcompare}), which can affect the CRSF spectra. As magnetic field is pushed outwards by pressure of accreted matter in filled mounds, magnetic field is larger than dipole value for $r \gtrsim 700$ m. For hollow mounds, field is enhanced on both sides as it is pushed outwards on both the inner and outer edges , lowering in value in the central region near the apex of the mound. If the CRSF is originating from regions of distorted fields, the inferred magnetic field will not reflect the dipole moment of the neutron star, as is commonly associated in the current studies so far. Comparison of magnetic field obtained from CRSF and that from tracking the Alfv\'en radius through disc-magnetosphere interaction or QPO studies etc. will show the extent of local non-dipolar nature of the magnetic field.
\begin{figure}
\centering
\includegraphics[width = 7cm, height = 7cm,keepaspectratio] {bratio_para.pdf}
\includegraphics[width = 7cm, height = 7cm,keepaspectratio] {bratio_holo.pdf}
\caption{\small Left: Ratio of field from GS solution to dipole value ($B_{\rm GS}/B_{\rm dip}$) for a filled parabolic mound of central height $h_c = 55$m. Even for heights $\sim 500$m, the GS solution shows $\sim 5\%$ deviation from dipole value. Right: $B_{\rm GS}/B_{\rm dip}$ for a hollow mound of apex height 38m.}
\label{Bcompare}
\end{figure}
\def\apj{ApJ}%
\def\mnras{MNRAS}%
\def\aap{A\&A}%
\defApJ{ApJ}
\defPhR{PhR}
\defApJS{ApJS}
\defPASA{PASA}
\defPASJ{PASJ}
\defNature{Nature}
\defMmSAI{MmSAI}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,716 |
//! @param xmlVersion is the xml version for the processing instruction, eg, L"1.0"
//! @param indent is indentation string used, eg L"\t"
//! @param str is the string to be written.
//! Traverse the tree recursively and write them to file.
//! Write start element (eg, <element>) to unicode file without indentation.
//! Write end element (eg, </element>) to unicode file without indentation.
//! Traverse the tree recursively and write them to file without indentation.
//! Escape the XML string(eg, L"<>" to L"<>" | {
"redpajama_set_name": "RedPajamaC4"
} | 7,263 |
MELISSA WYNDHAM
P R O J E C T S
Based in the UK and with projects worldwide, Melissa Wyndham Ltd features as one of House and Garden's leading interior design practices.
Since 1985 they've developed a portfolio of understated elegance, and a body of work spanning classic country houses, modern flats, and ski chalets. They're admired for their flexible approach, artful pairing of the contemporary and antique, and ability to work collaboratively with their clients to meet both practical needs and aesthetic tastes.
Vanessa Macdonald, who now leads the company, trained under and then worked alongside the late Melissa Wyndham for 14 years. Prior to that, she studied Art History at the University of British Columbia, completed a graduate degree in Fine Arts in New York City, and served in Private Client Services at Sotheby's London.
With both Canadian and Venezuelan roots, Vanessa has made the UK her home. She's married with two teenage sons and splits her time between Oxfordshire and London. Melissa Wyndham's design studio also includes Alice McFarlane and Vincent Dane.
Left: Vanessa Macdonald
"Expect sophisticated and elegant classic style with a modern twist."
Our design portfolio includes modern flats in New York and London, flawless English country houses, private US residences, and ski chalets in the Alps.
"Their style is a marriage of the best of English decoration with immaculate modern detail."
Our work has been featured in Architectural Digest, Country Life, The English Home, The Telegraph Magazine, and The World of Interiors, and has appeared on the cover of House & Garden.
MELISSA WYNDHAM LTD. | 0207 352 2874 | INFO@MELISSAWYNDHAM.COM | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,398 |
Enjoy this simple craft just in time for Easter Holidays. Making some Mini Popsicle Stick Easter Animals together with your children is loads of fun.
Start by coloring the craft stick with white and yellow paint. Let them dry.
Cut the head shapes, round one from yellow construction paper for chicken and white circle with ears for the bunny.
Draw (or glue) the beak, nose and ears. Stick on a pair of googly eyes (or draw them).
Once the paint on craft sticks is dry, place it at the center of the pipe cleaner. Start wrapping the pipe cleaners around them, twisting them on the back side of the craft stick.
Continue wrapping the pipe cleaner until you are done.
Tuck the loose end under the wrapped pipe cleaner.
Stick on the head with glue and you have your Easter craft stick puppets ready. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,935 |
Флаг муниципального образования город Ковро́в Владимирской области Российской Федерации — опознавательно-правовой знак, служащий официальным символом муниципального образования.
В настоящее время муниципальное образование город Ковров не имеет официально утверждённого флага.
Описание
«Флаг городского округа представляет собой прямоугольное полотнище с отношением ширины к длине 2:3, багряного цвета с односторонним изображением в центре флага герба городского округа. Габаритная ширина изображения герба городского округа должна составлять 2/5 части длины полотнища флага».
Обоснование символики
Геральдическое описание герба гласило: «В зелёном щите два сидящих зайца, смотрящих в разные друг от друга стороны. В верхней части — герб Владимирской области. Щит увенчан золотой башенной короной, о трёх зубцах, мурованной чёрным. За щитом два накрест положенных золотых молотка, соединённых Александровской лентой».
История
Флаг города Коврова был установлен уставом города Коврова Владимирской области, по крайней мере, в 1997 году. В приложении № 2 к уставу приводилось «Положение о флаге города Коврова». В ходе муниципальной реформы 2006 года был принят устав муниципального образования город Ковров Владимирской области, которым в названии «Положения» и в описании флага было изменено название муниципального образования.
25 января 2012 года, решением Совета народных депутатов города Коврова № 24, был утверждён новый герб муниципального образования со следующим описанием: «В зелёном поле на золотой земле два серебряных сидящих, обращённых друг от друга зайца. В вольной части (в верхнем правом углу) — герб Владимирской области. Щит увенчан золотой башенной короной о пяти видимых зубцах с двумя обнаженными, скрещёнными за короной мечами с серебряными клинками и золотыми рукоятями, и обрамлен лентой ордена Трудового Красного Знамени».
В этот же день рассматривался новый вариант флага, разработанный на основе герба города Коврова. Специалистами геральдической палаты было предложено следующее описание флага: «Прямоугольное зелёное полотнище с отношением ширины к длине 2:3, с жёлтой полосой вдоль нижнего края (в 3/10 полотнища) и в зелёной части двумя белыми, обращёнными друг от друга зайцами, сидящими вплотную к полосе». Все присутствующие депутаты возмутились цветовым решением флага и вынесли данный вопрос на публичное обсуждение, после которого и будет принято соответствующее решение.
25 апреля 2012 года, решением Совета народных депутатов города Коврова № 125, приложение № 2 к уставу города Коврова, которым устанавливался флаг городского округа, признано утратившим силу.
Примечания
Символы Коврова
Флаги городов России
Флаги с изображением львов
Флаги с изображением корон
Флаги с изображением молота
Заяц в геральдике | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,704 |
Q: helpicon functionality on form element using Jquery I am having below form with helpIcon for every form element with description.
How we can show help when we are hover on help icon and mouseout using jquery?
I am quit new for jquery. so please can someone give simple implementation?
<table>
<tbody>
<tr><td>
Name:
</td>
<td>
<input type="text" name="name" class="dipika">
</td>
<td>
<img src="help.png" id="imgNamehelp">
</td>
<td><div id="divNamehelp" style="display: none;"> This is Full Name</div></td>
</tr>
<tr><td>
Color:
</td>
<td>
<input type="radio" value="red" name="Color" class="dipika"> Red
<input type="radio" value="yellow" name="Color" class="dipika"> Yellow
</td>
<td>
<img src="help.png" id="imgColorhelp">
</td>
<td><div id="divColorhelp" style="display: none;"> This is color choice</div></td>
</tr>
<tr><td>
Hobbies:
</td>
<td>
<input type="checkbox" value="cricket" name="cricket" class="dipika"> Cricket
<input type="checkbox" value="kabdi" name="kabdi" class="dipika"> Kabadi
</td>
<td>
<img src="help.png" id="imgHobbieshelp">
</td>
<td><div id="divHobbieshelp" style="display: none;"> This is Hobbies choice</div></td>
</tr>
<tr><td>
Address:
</td>
<td>
<textarea class="dipika"></textarea>
</td>
<td>
<img src="help.png" id="imgAddresshelp">
</td>
<td><div id="divAddresshelp" style="display: none;"> This is Hobbies choice</div></td>
</tr>
</tbody>
</table>
Can anybody help with this issue? Thanks in advance.
A: $(document).ready(function(){
$('img[src="help.png"]').mouseover(function(){
$('#'+getDivId(this)).fadeIn(500);
}).mouseout(function(){
$('#'+getDivId(this)).fadeOut(500);
});
function getDivId(helpId){
var helpId = $(helpId).attr('id');
return helpId.replace('img','div');
}
});
A: $(document).ready(function(){
$('img[src="help.png"]').hover(function(){
$('#'+getDivId(this)).fadeIn(500);
},function(){
$('#'+getDivId(this)).fadeOut(500);
});
function getDivId(helpId){
var helpId = $(helpId).attr('id');
return helpId.replace('img','div');
}
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,113 |
What Will Art Be in the Future?
Posted on February 18, 2022 February 18, 2022 AuthorApril
Technology has certainly affected all art mediums, and even commercial printers get to play a part in this constantly changing arena. Throughout history, art changed with the times, and today is no different. Artists now have the ability to create and share works in ways never before possible.
Art as we know it is changing shape, literally. New space-age materials allow for the creation of objects which were never previously thought possible, including everything from printable rocket engines to architecturally sound carbon fiber objects. 3D printers are one example of this kind of technology.
Can a 3D printer be used for art?
3D printing helps artists transform ideas into tangible works of art. Artists from creative and entertainment domains can truly unleash their imagination to create new and exciting objects. 3D printed art models aim to expand design horizons and foster a culture of aesthetic innovation.
The question is, what will art be in the future? Will advances in technology drive it, or will humanity see a return to more traditional mediums? This is an important question, especially when considering how this affects professional printing companies.
3D printers change the way things are made. As stated above, objects which were previously impossible to make can now be realized with this technology. For example, for commercial printers who work in product design, visualizations for meetings and presentations will no longer need to be hand-made scale models; they can simply be printed.
3D printers change the way things are sold. For example, a company creates a product that it wants to sell, but its salespeople have a difficult time bringing samples with them while out on sales calls. Now they can simply print out physical samples that clients can hold in their hands and better see the final product.
What kind of art is trending now?
The most popular art today is commonly referred to as Contemporary Art. Contemporary art encompasses many styles, including Modern, Abstract, Impressionism, Pop Art, Cubism, Surrealism, Fantasy, Graffiti, and Photorealism. In addition, today's popular mediums include painting, sculpture, mixed media, photography, and digital art.
More Virtual Art Exhibitions
An Increase in 3D Motion Graphic Design
Renewed Appreciation for Nature
Increasing Popularity of Contemporary African Art
Continued Increase in Street Art
Contemporary art is associated with expression and experimentation. The roots of contemporary art began in the early 1960's, particularly abstract expressionism. It continues to be popular today and has many different styles, such as Pop Art (Jasper Johns & Andy Warhol) and Graffiti (Keith Haring). Contemporary art can be seen in galleries, films, music videos, advertisements, and broadways shows.
The Internet has had a monumental effect on the distribution of contemporary art. With nearly everybody using some form of social media (Facebook, Instagram, Twitter), sharing art is as easy as a click of a button. Artists can post their work online and it can be seen by millions at any given moment. Art is more accessible than it has ever been, allowing for art to be seen on a global level.
Hopefully, as the future progresses, there will be an increase in artistic education and art appreciation worldwide. Contemporary art will continue to advance, evolve, and inspire millions of people around the world every day.
What art is trending in 2021?
The growing popularity of the virtual world is one of the artistic themes we've noticed. Furthermore, with this has come an increase in digital 3D motion graphic design. This is a cutting-edge and exciting art form that has caught the attention of budding artists all around the world.
Future of art and design
Art has a future that will continue to branch into new forms, including continuing to integrate new technology. Both Virtual Reality and Augmented Reality offer excellent ways to create immersive work, where the viewer can experience the artwork utilizing a headset or phone.
Technologies are revolutionizing the artistic process at each stage: ideation, production, and launch. Technology is empowering creativity at the highest level, resulting in ground-breaking new art forms, from architects employing digital whiteboards to design structures to artists videoconferencing with marine biologists to preserve ecosystems to musicians collaborating with coders to create immersive VR.
Future of digital art
The future of digital art can be used as an instrument to communicate with people, express ideas and emotions, and make changes in society. The future of AI art is one that can make machines more human. The use of AI technology is still limited, but it can be used to create AI creations in the future.
Traditional mediums such as canvas and lithography may get pushed to the side in favor of these new technologies, but that's not necessarily a bad thing. Painting has been around for centuries and will remain relevant even when technology is capable of producing realistic 3D replicas of people or objects. It won't die out anytime soon, but it may certainly evolve.
Games such as World of Warcraft and Minecraft may become even more popular in the future thanks to these new printing technologies. Artists will be able to design their own objects like swords or pickaxes and print them out with ease, giving kids a brand-new way to play outside (or inside). It's only a matter of time before we see 3D printers in schools.
In the past, it was possible to determine what art would look like because there were restrictions imposed on artists by society and their tools. An artist who wanted to paint a picture couldn't exactly go out and buy a set of oils if all they had access to were acrylics. The same idea applied to sculptors. The point is that limitations force creativity, which leads to innovation. What happens when the limitations are removed? That remains to be seen.
In conclusion, art will have a strong focus on community building. Technology is merging the creation and production of art with storytelling, making them very popular among millennials. In addition, this technology will help to continue building connectivity, which is important in our time since it has been a very relevant topic for artists today.
Then again, who knows, but there's no doubt in my mind that it will still exist in some form or another!
Tagged 3d printing, art and design, future of art, marketing, marketing design, printer detroit michigan, royal oak printer
How Can People Keep Active and Busy After They Retire?
Why Are Dogs Good for Your Mental Health? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,453 |
\section{\label{sec:level1}Introduction}
Noncollinear spins are generic to the systems with spin spirals, spin helicoids, canted spins, and ferromagnetic (antiferromagnetic) domain walls. Such spin systems have attracted signif\hspace*{.5pt}icant interest in recent research activities since they are potential candidates for ultra-dense magnetic storage devices \textcolor{blue}{\cite{Fert152,Sampaio839,Beach611}}. Noncollinear spin helix mainly evolve in the presence of inversion symmetry breaking in such systems and yields various unusual physical phenomena like electric excitation of magnon \textcolor{blue}{\cite{Ramirez406,Tokura510}} and magnetic skyrmions \textcolor{blue}{\cite{Muhlbauer,Rossler}}. Various microscopic driving mechanisms would produce noncollinear spin ordering, and among them, Dzyaloshinskii--Moriya interaction (DMI) is the most common in many of these systems \textcolor{blue}{\cite{Dzyaloshinskii658,Moriya760}}.
Noncollinear spin arrangements in helimagnets also result from the presence of competing ferromagnetic and antiferromagnetic Heisenberg interactions between spins \textcolor{blue}{\cite{Mostovoy817,Masuda,Zinke,Villain977}}. Considering exchange interactions model in a magnetic f\hspace*{.5pt}ield of intensity H, the Hamiltonian \textbf{H} for a triangular lattice can be expressed through the following Eq. (1)
\begin{equation}
\textbf{H}=-J_{1}\Sigma_{<i,j>}S_{i}.S_{j}-J_{2,3}\Sigma_{<<i,j>>}S_{i}.S_{j}-H\Sigma_{i}S_{i}
\end{equation}
where, $\Sigma_{<i,j>}$ and $\Sigma_{<<i,j>>}$represent the sum over the nearest neighbor (\textit{nn}) and third nearest neighbor (\textit{nnn}), respectively \textcolor{blue}{\cite{Mostovoy817}}. For ferromagnetic state ($J_{1}>0$) with only \textit{nn} interaction, the ground state of such system would be commensurate to the underlying lattice. However, on the other hand, for competing interactions such as $J_{1}>0$ and the AF third-neighbor interaction $J_{3}<0$, an incommensurate ground state is more favorable \textcolor{blue}{\cite{Mostovoy817,Reatto979}}. In other words, when \textit{nnn} interaction become dominant, the ground state is often an incommensurate spiral structure \textcolor{blue}{\cite{Mostovoy817,Reatto979,Kimura1007,Fishman1109,Fishman_prb}}. In such systems, the magnitude of \textit{nnn} interaction is larger than the \textit{nn} interaction. Although the ordered magnetic structures are normally robust, exciting phase diagrams as a function of temperature and applied magnetic f\hspace*{.5pt}ield are often encountered \textcolor{blue}{\cite{Mostovoy817,Reatto979,Kimura1007,Fishman1109,Fishman_prb}}.
Transition metal dihalides have long been studied as prototypes of antiferromagnets with a triangular lattice \textcolor{blue}{\cite{Ward1282,Tokura1311,Tokura1411}}. Most of these investigations were focused on the coupling between magnetism and ferroelectricity in MnI$_{2}$ and NiBr$_{2}$ \textcolor{blue}{\cite{Tokura1311,Tokura1411}}. The latter system has been considered as a model system for spin triangular lattice \textcolor{blue}{\cite{Robbins1588,Meriel1680,Terrier1982}}. When the temperature reduces, the NiBr$_{2}$ displays a paramagnetic-antiferromagnetic phase transition at $T_{{\rm N}}$ = 44.0(1) K and then antiferromagnetic-incommensurate spiral transition at $T_{{\rm m}}$ = 22.8(1) K \textcolor{blue}{\cite{Robbins1588,Meriel1680,Terrier1982}}.
The ground state of the incommensurate spiral exhibits a threefold degeneracy with respect to the three equivalent wave vectors of the lattice. This threefold degeneracy is predicted to be a source of exotic ordered states, namely, multiple-\textit{q} state \textcolor{blue}{\cite{Okubo2012}}. This multiple-\textit{q} state may not be favorable at lowest temperature, but they may be stabilized at moderate temperature by thermal fluctuations and finite external magnetic fields. Such multiple-\textit{q} state corresponds to the Skyrmion lattice state which may occur in such a system, as considered theoretically by Okubo et al. Phys. Rev. Lett. \textbf{108}, 017206 (2012) \textcolor{blue}{\cite{Okubo2012}}.
In this work, we address the formation of a multiple-\textit{q} state in the NiBr$_{2}$ triangular frustrated Heisenberg system. We have investigated various properties of the incommensurate spiral phase in the single crystal NiBr$_{2}$ using dc(ac) magnetization and magnetic susceptibilities, neutron diffraction, and small angle neutron scattering. The experimental f\hspace*{.5pt}indings suggest that NiBr$_{2}$ has a helimagnetic ground state in good agreement with the previous observations \textcolor{blue}{\cite{Tokura1311,Tokura1411,Robbins1588,Meriel1680}}. The ground state is reached upon reducing the temperature through a commensurate phase. In an applied magnetic field and at finite temperature, we could not find evidence for multiple-\textit{q} state occurrence. Instead, we found that the system responds to the structural symmetry of 120$^\circ$ by populating the three equivalent wave vectors in the incommensurate state. Moreover, we emphasize on a new transition from the ground state incommensurate spiral phase to commensurate one under magnetic f\hspace*{.5pt}ield (H$>$ 4 T) applied perpendicular to the $c$-axis. The f\hspace*{.5pt}ield-induced phase appears to be the same as the high temperature phase, i.e., NiBr$_{2}$ exhibits a spin order transition, where helical spins can regain high temperature antiferromagnetic phase even the system temperature is at well below $T_{\mathrm{m}}$. The unique response of spin-helix phase under external f\hspace*{.5pt}ield (H) indicates that even a slight difference in the magnetic interactions is critically reflected in the respective response in frustrated magnets.
\section{\label{sec:level2}METHODS AND EXPERIMENTAL DETAILS}
High-quality single crystals of NiBr$_{2}$ were grown at the University of Amsterdam, using a self-flux growth method starting from a stoichiometric mixture of Ni, and Br, as reported in Tokunaga et al. and Kurumaji et al. \textcolor{blue}{\cite{Tokura1311,Tokura1411}}. The samples have been studied with the average weight of 10 mg. The single crystals had typical dimensions of approximately $4 \times 4 \times 2\, \mathrm{mm}^{3}$. NiBr$_{2}$ crystals are sensitive to air/moisture; therefore, they are kept in the vacuum to prevent any degradation of the sample's quality. NiBr$_{2}$ crystal forms naturally layered shape with the \textit{c}-axis perpendicular to planes that eff\hspace*{.5pt}iciently cleave. The nickel ions occupy a single site in the Bravais lattice as shown in Fig.~\href{Color online}{\textcolor{blue}{1(a)}}. Magnetic measurements at various temperatures in the range 2~K to 300~K were carried out using the MPMS 7T device and PPMS 14T (Quantum Design) systems, in f\hspace*{.5pt}ields up to 14 T, applied within and out of the plane directions. The measurements and analyses of the dc(ac) susceptibility have been conducted on samples that were f\hspace*{.5pt}ixed to quartz sample holders using a diamagnetic glue.
NiBr$_{2}$ crystal originating from the same batch used for both magnetic measurements as well as for neutron diffraction experiments, which were carried out on the diffractometers E2 and E4 and the small-angle neutron scattering instrument V4 at the BER II reactor of the Helmholtz-Zentrum Berlin (HZB) \textcolor{blue}{\cite{Yokoachiya1717}}. In the part, magnetic f\hspace*{.5pt}ields up to 5 T were applied along and perpendicular to the trigonal axis of the single-crystalline NiBr$_{2}$ using a horizontal-f\hspace*{.5pt}ield cryomagnet that restricts signif\hspace*{.5pt}icantly the scattering geometry. In particular, this makes impossible to survey a large portion of the reciprocal space due to dark angles imposed by the construction of the magnet coils. E2 and E4 instruments use a pyrolytic graphite (002) monochromator selecting the neutron wavelength $\lambda = 2.4$ \AA. The data were collected with two-dimensional position sensitive $^{3}$He detectors that were of different sizes at respective instruments. The f\hspace*{.5pt}ield- and temperature-dependent scans were performed by measuring the peak intensity while ramping the f\hspace*{.5pt}ield/temperature. The integrated peak intensity was found by summing the counts in a small area of the detector surrounding the reflection. On E2 and E4, in order to ref\hspace*{.5pt}ine the magnetic structure of NiBr$_{2}$, we have collected data mainly in the (\textit{hhl}) plane. However, a signif\hspace*{.5pt}icant portion below and above the scattering plane has been covered as well. In addition, on E4, we have collected data also in the (\textit{hk0}) plane.
The crystal and magnetic structure ref\hspace*{.5pt}inements were performed with the program Fullprof \textcolor{blue}{\cite{Roisnel1801,Sears1992}}. The nuclear scattering lengths b(Ni) = 1.03 fm and b(Br) = 0.6795 fm were considered. For the absorption correction (Gaussian integration), we used the absorption coeff\hspace*{.5pt}icient $\mu = 0.12\, \mathrm{cm}^{-1}$. No secondary extinction correction has been applied. $\frac{\lambda}{2}$ f\hspace*{.5pt}ilters were used at both E2 and E4 instruments imply residual higher-order wavelength contamination at a level of less than $10^{-4}$.
In addition, the small-angle neutron scattering measurements were carried out with instrument V4 at HZB that possess a large position-sensitive detector movable at the distance between 1 and 16 m from the sample. The V4, where we have also applied the magnetic f\hspace*{.5pt}ield along the basal plane of the crystal, has been used in a conf\hspace*{.5pt}iguration that covered $0.04\, \mathrm{nm}^{-1} < Q < 1.3\, \mathrm{nm}^{-1}$.
\section{\label{sec:level3}RESULTS AND DISCUSSIONS}
\subsection{\label{sec:results}Magnetization and Susceptibility}
Figure \href{Color online}{\textcolor{blue}{2(a)}} illustrates the temperature dependence of the magnetic susceptibility $\chi= M/H$ of NiBr$_{2}$, where, the $M$ is magnetization, and the $H$ is magnetic f\hspace*{.5pt}ield ranging between 1.0 and 14.0 T, which was applied perpendicular to the \textit{c}-axis. In the case of NiBr$_{2}$ at high temperatures where $\chi= M/H$ measured the magnetization is directly proportional to the applied magnetic field (i.e. linear in the field), the evaluated M/H values are identical to the magnetic susceptibility at different fields are insensitive to the magnetic field strengths.
It is evident, that $\chi(T)$ is f\hspace*{.5pt}ield independent for temperatures above \AC80~K. At lower temperatures, two well-distinguished anomalies can be discerned marking the magnetic phase transition temperature that were determined from maxima of $\frac{\partial T.\chi(T)}{\partial T}$.
At 1.0 T magnetic f\hspace*{.5pt}ield, the transition between the paramagnetic and an antiferromagnetic state is detected at $T_{{\rm N}}$ = 44.6(1)~K. A closer inspection reveals that for increasing the magnetic f\hspace*{.5pt}ield, this transition shifts toward lower temperatures. The second transition between AF and helical spin order takes place at $T_{{\rm m}}$ = 21.8 (1) K. As it is evident from Fig.~\href{Color online}{\textcolor{blue}{2(a)}}, this anomaly also shifts toward lower temperatures as a function of the magnetic f\hspace*{.5pt}ield.
The temperature dependence of $1/\chi(T)$ and their f\hspace*{.5pt}its to a modif\hspace*{.5pt}ied Curie--Weiss law are shown in inset panels of Fig.~\href{Color online}{\textcolor{blue}{2(a and b)}}. The modif\hspace*{.5pt}ied Curie--Weiss law can be expressed by Eq.(2):
\begin{equation}
\chi=C/(T-\theta_{\mathrm{p}}) + \chi_{0}
\end{equation}
where, C represents the Curie constant and $\theta_{\mathrm{p}}$ the paramagnetic Curie temperature. $\chi_{0}$ denotes the temperature-independent terms. The excellent quality of f\hspace*{.5pt}its conf\hspace*{.5pt}irms the modif\hspace*{.5pt}ied Curie--Weiss behavior for the system above $T > 70$ K. The best f\hspace*{.5pt}it of data acquired with 1.0 T ($T> 150$ K) yields the effective moment of $\mu_{{\rm eff}}$ = 2.76 (1) $\mu_{\mathrm{B}}$. This value is approximately the same as the effective moment expected for the S = 1, 3d$^8$ electronic conf\hspace*{.5pt}iguration of Ni ions $(\sqrt{8} = 2.83)$. The effective moment determined from data taken at 14.0 T amounts to 2.66 (1)$\mu_{\mathrm{B}}$. The fitted values of $\theta_{\mathrm{p}}$ amount to 22.2(1) K and 28.1 (1) K for the data recorded 1.0 T and 14.0 T, respectively.
Figure \href{Color online}{\textcolor{blue}{2(b)}} shows the temperature dependence of magnetic susceptibility measured for the various f\hspace*{.5pt}ield applied along the \textit{c}-axis. Also for this orientation, the magnetic susceptibility at higher temperatures is f\hspace*{.5pt}ield independent with anomalies visible only at the lower temperatures. The effective magnetic moment and the paramagnetic Curie temperature are very similar to the other orientation ($\mu_{\mathrm{eff}}$ =2.74 (1)$\mu_{\mathrm{B}}$ and $\theta_{\mathrm{p}}$ = 21.3 (1) K, respectively). The best fit to data taken at 14 T applied along the c-axis is shown in the inset of Fig. \href{Color online}{\textcolor{blue}{2(b)}}.
For $T > 70$ K, the similarity between magnetic susceptibilities with f\hspace*{.5pt}ield applied along and perpendicular to the \textit{c}-axis suggests that the anisotropy in paramagnetic state of NiBr$_{2}$ is negligible. However, this does not hold true for the state below $T_{{\rm N}}$ as magnetization (\hbox{M--H}) curves taken for f\hspace*{.5pt}ield applied perpendicular and along the \textit{c}-axis had shown entirely different magnetization behaviors, as depicted in Fig.~\href{Color online}{\textcolor{blue}{3(a and b)}}. Selectively, for the case when applied magnetic f\hspace*{.5pt}ield within the (\hbox{\hbox{$a$--$b$}}) basal plane, a clear f\hspace*{.5pt}ield-induced transition that shifts with increasing temperature to lower f\hspace*{.5pt}ields is evident. At 3~K, the magnetization step associated with the transition amounts to about 0.036 $\mu_{\mathrm{B}}$/Ni. This is in good agreement with the magnetic susceptibility data. For the second geometry, when f\hspace*{.5pt}ield was applied along the \textit{c}-axis, no such trends were observed.
Now we discuss the insights on phase transitions through measurements of the temperature and magnetic f\hspace*{.5pt}ield dependence of ac magnetic susceptibility, which is a complex value and expressed as $\chi_{ac}$ = $\chi'-i\chi^{\prime\prime}$. The real component, $\chi'$, is related to the reversible magnetization process and always in-phase with the oscillating f\hspace*{.5pt}ield. However, imaginary component, $\chi^{\prime\prime}$, is related to losses during the irreversible magnetization process.
The measurements of ac susceptibilities were carried out for multiple values of applied f\hspace*{.5pt}ields as a function of temperature (measured with H = 1 mT and f = 99 Hz). To minimize the demagnetization effects, the dc magnetic f\hspace*{.5pt}ields were applied within the basal plane (\hbox{\hbox{$a$--$b$}}). Fig.~\href{Color online}{\textcolor{blue}{4(a)}} illustrates the temperature dependence of the ac susceptibilities measured in the vicinity of both transitions. Experimental results over the f\hspace*{.5pt}ield range from 0.1 to 2.0 T show the emergence of maxima across both phase transitions. Over the $T_{{\rm m}}$, ac susceptibility displays a maxima with respect to the temperature; the position of which trending towards higher side (while the amplitude of ac susceptibility decreases) with reducing the strength of applied magnetic f\hspace*{.5pt}ields, similar as in Ref. \textcolor{blue}{\cite{Tokura1411}}. The variation in peak heights and temperatures for various dc magnetic f\hspace*{.5pt}ields shows a monotonic variation with strength of the applied dc f\hspace*{.5pt}ields.
Figure \href{Color online}{\textcolor{blue}{4(b)}} shows the ac susceptibilities as a function of applied magnetic f\hspace*{.5pt}ields for various constant temperatures that range from 5~K to 80~K. At $T=80$~K, the ac susceptibility exhibits a line-shaped curve with low amplitude. At $T = 25$~K; a peak can be seen close to the upper phase boundary of the antiferromagnet state as also seen in Fig. \href{Color online}{\textcolor{blue}{2(a)}}. At even lower temperature near T=20~K, an increase in ac susceptibility peak amplitude is clearly noticed for applied low f\hspace*{.5pt}ields 0.5 T $<$ H $<$ 2.5 T. This modif\hspace*{.5pt}ied ac susceptibility magnitude is also accompanied by a systematic \hbox{enhancement} in the peak f\hspace*{.5pt}ield, which would correspond to the boundary of incommensurate phase in this region.
Reapeating the f\hspace*{.5pt}ield dependence of ac susceptibilities at various temepratures across incommensurate phase result a $\chi$(H,T) curve as illustarted in Figure \href{Color online}{\textcolor{blue}{4(b)}}. Below $T_{{\rm m}}$, a hysteresis nature of $\chi$(H,T) curves clearly evident, which acquired for different temepratures, conf\hspace*{.5pt}irming the helix nature of noncollinear nature of spins.
The shift in peak positions as a function magnetic f\hspace*{.5pt}ield represents the torque required to reverse the helical spin upon flipping the polarization of applied magnetic f\hspace*{.5pt}ield. In this way, it might be possible to
identify the spin reorientation processes, which are responsible for
the unusual field dependence in the ac susceptibility and subsequently investigate the magnetic field effects on the physical interactions driving these phase transition processes. The results reported above indicate the emergence of field-induced incommensurate to commensurate phase transition (as demonstrated further below by neutron scattering),
a generic response that occurs only acros the low temperature boundary of these phases. Note that in Ref. \textcolor{blue}{\cite{Tokura1411}}, this is ascribed to a transition from cycloidal to collinear AF order. The ac susceptibility peak positions in (H,T) dimensions over
the typical ranges map directly onto the field-induced
modulation in incommensurate phase. It should be noted
that at zero field, ac susceptibility diminishes monotonically over the range of temperatures between 5 K
and 20 K.
Figure \href{Color online}{\textcolor{blue}{4(c)}} shows the comparison between $\chi_{{\rm dc}}$ and $\chi'$$_{{\rm ac}}$, both measured at $H_{{\rm dc}}$ = 2 kOe. Their respective temperature derivatives are depicted in the inset of Fig.~\href{Color online}{\textcolor{blue}{4(c)}}. The $\chi'$$_{{\rm ac}}$ provides a precise determination of the long-range magnetic phase ordering temperatures, which can be determined from the $\chi'$$_{{\rm ac}}$ or $d\chi'$$_{{\rm ac}}$$/dT$ maxima. The f\hspace*{.5pt}ield and temperature dependence of the ac susceptibility at 20~K is more complicated. In this \hbox{region}, the ac susceptibility peak amplitudes f\hspace*{.5pt}irst increase before dropping again with dc f\hspace*{.5pt}ields. In contrast, the susceptibility peak temperatures increase monotonically with f\hspace*{.5pt}ield, tracking closely along the phase boundaries across $T_{{\rm m}}$ regimes determined from the experimental values of \vspace*{1.5pt} $\frac{\partial M(T)}{\partial T}$ and $\frac{\partial M(H)}{\partial H}$. This behavior of $\chi_{{\rm ac,dc}} (H,T)$ is a def\hspace*{.5pt}ining characteristic of such transitions, specif\hspace*{.5pt}ically reflecting the combination of ferromagnetic and antiferromagnetic interactions in the triangular NiBr$_{2}$ system, which leads to an intricated influence of applied f\hspace*{.5pt}ields as a function of the changing spin conf\hspace*{.5pt}igurations.
\subsection{\label{sec:N Diffraction} Neutron diffraction on NiBr$_{2}$ crystal}
NiBr$_{2}$ crystallizes in the CdCl$_{2}$ structure, which has the rhombohedral space group \textit{D$^{5}_{3d}$} \textcolor{blue}{\cite{Ward1282,Tokura1311,Tokura1411,Robbins1588,Meriel1680}}. In this space group, the Ni and Br atoms are situated at the Wyckoff positions 3\textit{a}(0,0,0) and 6\textit{c}(0,0,\textit{z}) with 3~\textit{m} and $-$3~\textit{m} local symmetries, respectively \textcolor{blue}{\cite{Robbins1588,Meriel1680}}. The lattice constants are a = 3.723 \AA \hspace*{.5pt} and c = 18.340 \AA.
The space group leads to specif\hspace*{.5pt}ic extinction conditions, for instance, the reflections \textit{hkil}: \textit{$-$h$+$k$+$l} = 3\textit{n} and the reflections \textit{hki0}: \textit{$-$h$+$k} = 3\textit{n}. Above $T_{{\rm N}}$, in the course of neutron diffraction experiments, we could observe only nuclear Bragg reflections.
Figure \href{Color online}{\textcolor{blue}{5(a)}} illustrates the (\textit{hhl}) diffraction pattern recorded at 50~K. As it is evident that only (110), (003), (006) and (009) Bragg reflections are observed (along with signal originating from the aluminum sample holder/cryostat)\textcolor{blue}{\cite{Robbins1588,Meriel1680}}. There were no unexpected Bragg reflections observed. The ref\hspace*{.5pt}inement of observed Bragg reflections leads to a reasonable agreement with literature data with the only structural free parameter z$_{Br}$ = 0.268. As the temperature is decreased below $T_{{\rm N}}$ (but above $T_{{\rm m}}$), new Bragg reflections associated with AF order are observed \textcolor{blue}{\cite{Robbins1588,Meriel1680}}. These can be indexed with a propagation vector $q_{{\rm c}}$ = (0 0 3/2) as shown in Fig.~\href{Color online}{\textcolor{blue}{5(b)}} and which is in well agreement with the literature \textcolor{blue}{\cite{Robbins1588,Meriel1680}}. The diffraction pattern recorded at 26~K conf\hspace*{.5pt}irms that (0 0 3/2), (0 0 9/2), and (0 0 15/2) magnetic reflections show non-zero intensity.
These magnetic satellite structures around the (003) m, (006) m, and (009) m corroborate the antiferromagnetic ordering of Ni spins in (\hbox{\hbox{$a$--$b$}}) basal plane \textcolor{blue}{\cite{Robbins1588,Meriel1680,Yokoachiya1717, Stock}}.
As the magnitude of incommensurate component at lower temperature is expected to be very small, special care has been paid to improve the resolution of the diffractometer. We have utilized high wavelength neutrons with $\lambda=2.4$ \AA produced by a pyrolytic graphite monochromator and graphite filters to remove the $\lambda/2$ contamination. A reduced focus lead on one side to lower intensities but also to decreased vertical divergence needed to separate the magnetic reflections in the vertical direction. To improve the signal to noise ratio we have closed both the horizontal and vertical slits as much as possible while still irradiating the sample completely \textcolor{blue}{\cite{Yokoachiya1717,Stock}}. These scans show that the component of propagation vector along the \textit{c}-axis remains commensurate with a value k$_{{\rm z}}$ = 3/2. However an incommensurate component exists in the (\hbox{\hbox{$a$--$b$}}) basal plane, which is oriented along $<110>$ direction. From 4.2~K to 21.8~K the propagation vector has been found to be temperature dependent. The k$_{{\rm z}}$ component remains commensurate to a value 3/2. The in-plane component is always along $<110>$ direction. The absence of any second, third, or higher harmonics (Fig.~\href{Color online}{\textcolor{blue}{5(c)}}) proves that NiBr$_{2}$ orders with a helical structure within the (\hbox{\hbox{$a$--$b$}}) basal plane in zero f\hspace*{.5pt}ield at $T$ = 4.2~K. The planes remain coupled antiferromagnetically along the c-axis direction \textcolor{blue}{\cite{Robbins1588,Meriel1680,Stock}}. The magnetic satellites regain its shape similar to commensurate antiferromagnetic phase upon applying an in-plane magnetic f\hspace*{.5pt}ield (H $\leq$ 4~T) as shown in Fig.~\href{Color online}{\textcolor{blue}{5(d)}}. This spin order transition of magnetic satellite peak is reported in the incommensurate phase for a triangular spin systems. A signif\hspace*{.5pt}icant change in the diffraction patterns reported at T=21.1K, as the $q_{{\rm c}}$ Bragg reflections split (see Fig.~\href{Color online}{\textcolor{blue}{5(c)}}). The estimated propagation is of the $q_{{\rm i}}$ = (q$_{{\rm h}}$ q$_{{\rm h}}$ 3/2), with q$_{{\rm h}}$ $\approx$ 0.03. There are no other magnetic reflections that described by q$_{2}$ are visible at 2~K, zero f\hspace*{.5pt}ield. These scans show that the component of the propagation vector along the $c$-axis remains commensurate with a value k$_{{\rm z}}$ value 3/2 \textcolor{blue}{\cite{Robbins1588,Meriel1680,Yokoachiya1717,Stock}}.
Magnetic measurements have indicated for the f\hspace*{.5pt}ield applied within the (\hbox{\hbox{$a$--$b$}}) plane, a magnetic phase transition around 3 T. Figure \href{Color online}{\textcolor{blue}{5(d)}} shows the pattern recorded at 2~K and 4 T. As can be seen, it is identical to the pattern recorded at 26~K (see Fig.~\href{Color online}{\textcolor{blue}{5(b)}}). This can be at best seen from the insets of the respective panels. In the case of data taken at 50~K, one can also notice a detectable short-range ordered signal. Such f\hspace*{.5pt}ield-induced spin reorientation in single crystal of incommensurate NiBr$_{2}$ is observed with a magnetic f\hspace*{.5pt}ield applied perpendicular to \textit{c}-axis. Spin reorientation occurred continuously with increasing f\hspace*{.5pt}ield. At sample temperatures of 4.2~K and 22~K, a complete spin reorientations were observed with critical f\hspace*{.5pt}ields (H $\geq$ 4~T). These f\hspace*{.5pt}indings will be further examined within the framework of spin-flop coupling in the next section.
Further, we have recorded the quality of crystalline lattice, rocking curves of NiBr$_{2}$ crystal, concentrating on a small region of reciprocal space at a constant \textit{q}-value. Figure \href{Color online}{\textcolor{blue}{6(a)}} shows the rocking curve measured through the (0 0 3) nuclear Bragg reflection at 55~K in the \hbox{absence} of any applied magnetic f\hspace*{.5pt}ields. The full width of the reflection at half maximum (FWHM) amounts to 1.9\textdegree, which is about four times more than the resolution of the used instrument. A comparative rocking curve through the magnetic (0 0 3/2) Bragg reflection were recorded at 23~K in zero f\hspace*{.5pt}ield, as depicted in Fig.~\href{Color online}{\textcolor{blue}{6(b)}}. The FWHM for (0 0 3/2) reflection is larger than the corresponding nuclear reflection (0 0 3). The shape is similar to the nuclear reflection suggesting that magnetic ordering takes place in the whole volume of the sample, without any influence of crystal mosaicity.
Figure \href{Color online}{\textcolor{blue}{6(c)}} shows the temperature dependence of the magnetic (0 0 3/2) Bragg reflection recorded with zero f\hspace*{.5pt}ield upon cooling at the top of the reflection. The solid red line represents the best f\hspace*{.5pt}it of the experimental data using Eq. (3):
\begin{equation}
I(T)=b+I_{0}(1-\frac{T}{T_{{\rm N}}})^{2\beta}
\end{equation}
where, b denotes the background intensity, $I_{0}$ represents the intensity at 0~K, and $\beta$ is the critical parameter related to dimensionality of the system. The best f\hspace*{.5pt}it to this empirical formula, which is valid in the critical region near the magnetic phase transition, leads to a good description of the data above 40~K. The magnetic phase transition occurs at $T_{{\rm N}}= 44.5$ (6) K with $\beta$ = 0.30 (1). However, the observation of scattered intensity (see the non-negligible intensity above the background level in the Fig.~\href{Color online}{\textcolor{blue}{6(c)}} highlighted by the dashed line) above the $T_{{\rm N}}$ points to the presence of critical scattering in this material. The second-order transition at $T_{{\rm m}}$ takes place at 21.6 (5) K. Below this temperature, the intensity is signif\hspace*{.5pt}icantly above its high temperature background level. However, in this case, as it is shown in Fig.~\href{Color online}{\textcolor{blue}{6(c)}}, the intensity originates from new Bragg reflections that contribute to the original position of the (0 0 3/2) Bragg reflection due to a f\hspace*{.5pt}inite resolution of the E4 instrument and spread of the intensity originating from different magnetic Bragg reflections.
Figure \href{Color online}{\textcolor{blue}{7(a)}} shows diffraction pattern recorded on E4 at 2~K with zero magnetic f\hspace*{.5pt}ield around the (0 0 3/2) reciprocal space position, using projection on the $\omega{-}\upsilon$ plane, where $\omega$ is the rotational angle of the sample and $\upsilon$ is the deviation angle from the scattering plane. As it is evident, the original magnetic Bragg reflection splits into six reflections indexed by three propagation vectors $q_{1}$ = ($q_{h} \hspace*{1.5pt} q_{h} \hspace*{1.5pt} 3/2$), $q_{2}$ = (-$q_{h}\hspace*{1.5pt} 2q_{h} \hspace*{1.5pt} 3/2$), and $q_{3}$ = (-2$q_{h} \hspace*{1.5pt} q_{h} \hspace*{1.5pt} 3/2$) (and associated opposite vectors). Using the UB matrix ref\hspace*{.5pt}ined from very few nuclear Bragg reflections and positions of the maxima, it follows that $q_{h}$ = 0.027 (1) is in agreement with the literature \textcolor{blue}{\cite{Svergun1987,Cameron2016}}. The six magnetic reflections can be indexed as ($-$0.027 $-$0.027 3/2), ($-$0.054 0.027 3/2), ($-$0.027 0.054 3/2), (0.027 0.027 3/2), (0.054 $-$0.027 3/2) and (0.027 $-$0.054 3/2). These propagation vectors are incommensurate with the crystal structure. The projection of Fig.~\href{Color online}{\textcolor{blue}{7(a)}} on the rotational axis, $\omega$ is shown in Fig.~\href{Color online}{\textcolor{blue}{7(b)}}, suggesting that these six reflections have inhomogeneous intensity distributions.
The existence of six magnetic propagation vectors at low temperatures raises question whether one deals with a homogeneous state where the Ni magnetic moments are modulated in the whole volume of the sample by all the propagation vectors or whether one deals with spatially disjoint domains, each having one propagation vector. However, from the fact that, for instance, the reflection described by the propagation vector $q_{3}$ is by about 50 $\% $ more intense than the $-q_{2}$ reflection (see Fig.~\href{Color online}{\textcolor{blue}{7(a)}}), the former scenario can be ruled out. This conf\hspace*{.5pt}irms that the magnetic structure consists of volume-separated domains.
From a symmetry point-view, the most distinctive properties of the incommensurate spiral state are from the 120\textdegree\ structures. The ground state displays threefold degeneracy, \textit{i.e.}, there are three equivalent directions of wave vectors on the lattice. A f\hspace*{.5pt}irst-order transition is associated with a breaking of the threefold C$_{3}$ lattice symmetry. The ordered state is a single-\textit{q} state with three equivalent wave vector directions. This threefold degeneracy could also be in principle a source of exotic ordered states, e.g., various types of multiple-\textit{q} state where multiple wave vectors coexist. The helix structure is realized at the magnetic ground state below $T_{{\rm N}}$, where spins rotate within the plane perpendicular to the magnetic modulation vector.
Figure \href{Color online}{\textcolor{blue}{8(a)}} shows the temperature dependence of the diffracted intensities projected on the $\omega$-axis along with projections on the $\omega{-}\upsilon$ plane (Fig.~\href{Color online}{\textcolor{blue}{8(b)}}) recorded at various temperatures. As can be seen, the six incommensurate reflections collapse at $T_{{\rm m}}$ to a single (0 0 3/2) reflection. Figure \href{Color online}{\textcolor{blue}{8(c)}} illustrates the temperature dependence of the intensity around (0 0 3/2). Total diffracted intensity is included, irrespective of whether originating from a commensurate or incommensurate magnetic state. The absence of any anomaly around $T_{{\rm N}}$ suggests that the (0 0 3/2) reflection just splits into six reflections at lower temperature without any signif\hspace*{.5pt}icant change in the magnetic structure/magnetic moment magnitude.
Symmetry analysis for both types of propagation vectors (commensurate and incommensurate) leads to conclusion that Ni magnetic moments are oriented either along the \textit{c}-axis or perpendicular to it. The fact that we do observe intensities at (0 0 3/2) position, which suggests that the moments are not along the trigonal axis and are oriented within the plane. However, due to high symmetry of the crystal structure, it is not possible to determine the direction of these moments within the (\hbox{\hbox{$a$--$b$}}) basal plane.
The best f\hspace*{.5pt}it to data shown in Fig.~\href{Color online}{\textcolor{blue}{5(b)}}, i.e., data taken in zero f\hspace*{.5pt}ields at 25~K, leads to Ni moment magnitude of 2.8 (2) $\mu_{\mathrm{B}}$. This value is somewhat larger with respect to the literature data \textcolor{blue}{\cite{Robbins1588,Meriel1680}}. The feasible magnetic structures are shown in Fig.~\href{Color online}{\textcolor{blue}{9}}. It is noteworthy here that this is one of the several possible magnetic structures that agree equally with the data. The difference between solution shown in Fig.~\href{Color online}{\textcolor{blue}{9(a and b)}} is that Ni moments are collinear in Fig.~\href{Color online}{\textcolor{blue}{9(a)}} and smaller than in Fig.~\href{Color online}{\textcolor{blue}{9(b)}}. The best f\hspace*{.5pt}it to data taken at 2~K, 0 T as shown in Fig.~\href{Color online}{\textcolor{blue}{5(c)}} using six propagation vectors $q_{1}$ = ($-$0.027 $-$0.027 3/2), $q_{2}$ = ($-$0.054 0.027 3/2), $q_{3}$ = ($-$0.027 0.054 3/2), $q_{4}$ = (0.027 0.027 3/2), $q_{5}$ = (0.054 $-$0.027 3/2), and $q_{6}$ = (0.027 $-$0.054 3/2) leads to Ni magnetic moment of 3.0 (3) $\mu_{\mathrm{B}}$. The direction of Ni moments are changing within the basal plane in an incommensurate manner, making with respect to each other about $\alpha$ = 9.5$^\circ$ as one moves within the plane along \textit{a}-axis as illustrated in Fig.~\href{Color online}{\textcolor{blue}{9(c)}}. This solution is in good agreement with the literature data \textcolor{blue}{\cite{Robbins1588,Meriel1680,Terrier1982}}. The population of six magnetic domains is about $18\%$:$10\%$:$22\%$:$13\%$:$19\%$:$18\%$.
In this section, we would elaborate in more detail about the spin-flop mechanism across the incommensurate phase that is induced in the presence of an external magnetic f\hspace*{.5pt}ield (H $\geq$ 4~T) applied within the (\hbox{\hbox{$a$--$b$}}) plane. The high temperature magnetic phase is retained, which suggests that only the magnetic reflections indexable with $q_{c}$ = (0 0 3/2) are present (see Fig.~ \href{Color online}{\textcolor{blue}{5(b) $\&$ 5(d)}}). This is also shown in Fig.~ \href{Color online}{\textcolor{blue}{10(c)}}, which shows the f\hspace*{.5pt}ield dependence of the intensity projected on $\omega$-axis with increasing f\hspace*{.5pt}ield. Merely the intensities of AF reflections are reduced with respect to the zero f\hspace*{.5pt}ield value, pointing to a smaller staggered magnetic moment of 2.5 (2) $\mu_{\mathrm{B}}$. No signif\hspace*{.5pt}icant ferromagnetic component could be resolved.
In Fig.~\href{Color online}{\textcolor{blue}{10(b)}}, the temperature dependence of the intensity around (0 0 3/2) position is measured with increasing temperature in a f\hspace*{.5pt}ield of 2 T applied within the (\hbox{\hbox{$a$--$b$}}) plane. As can be seen, the lower transition shifts with respect to zero f\hspace*{.5pt}ields toward lower temperatures while the upper transition stays approximately unchanged that in well agreement with magnetization data. As the f\hspace*{.5pt}ield increases at 2~K above the critical f\hspace*{.5pt}ield of about 3 T, only the magnetic reflections indexable with $q_{c}$ = (0 0 3/2) are present. In Fig.~\href{Color online}{\textcolor{blue}{10(c)}}, the temperature dependence of the intensity around (0 0 3/2) position measured with increasing temperature in a f\hspace*{.5pt}ield of 5 T applied within the (\hbox{\hbox{$a$--$b$}}) plane is shown. Also, in this case, the $T_{{\rm N}}$ remains equal to approximately 44~K. This transition can be interpreted as in the presence of weak anisotropic energy, at a critical magnetic f\hspace*{.5pt}ield the two sub-lattice magnetization rotates suddenly to a direction perpendicular to the easy magnetization direction, i.e., (\hbox{\hbox{$a$--$b$}}) basal plane.
\subsection{\label{sec:phase}Phase diagram}
The magnetic phase diagram of NiBr$_{2}$ near critical temperature $T_{{\rm m}} (T_{{\rm N}})$ has been drawn from the measurements of bulk magnetization, susceptibility, and neutron diffraction. A decrease of the magnetic phase boundary at $T_{{\rm N}}$ results in a substantial increase of the satellite \textit{q}-vector. As shown in the Fig.~\href{Color online}{\textcolor{blue}{11}}, for an intermediate temperature range in the (\hbox{$a$--$b$}) plane, a commensurate magnetic structure between 45~K and 22~K was found by a least-squares ref\hspace*{.5pt}inement to be modulated with moments lying in the (\hbox{$a$--$b$}) plane. At lower temperatures and moderate magnetic f\hspace*{.5pt}ield in the (\hbox{$a$--$b$}) plane, an incommensurate magnetic structure between 22~K and 4~K was found by a least-squares ref\hspace*{.5pt}inement to be modulated with moments lying in the (\hbox{$a$--$b$}) plane with angle of $\gamma$ = 9.6\textdegree\ . The magnetic phase diagram for an external f\hspace*{.5pt}ield applied in the \textit{c}-axis direction was also determined that conf\hspace*{.5pt}irms that upon reducing temperature, one commensurate magnetic structure phase can evolve. The inset shows the extended f\hspace*{.5pt}ield range that suggests for the f\hspace*{.5pt}ield applied within the (\hbox{$a$--$b$}) plane, yet another f\hspace*{.5pt}ield-induced transition around 60 T.
This behavior in $\chi(H, T)$ may be a def\hspace*{.5pt}ining characteristic of such transitions, mainly reflecting the unusual combination of ferromagnetic and antiferromagnetic interactions in the NiBr$_{2}$ system that leads to the complicated influence of applied f\hspace*{.5pt}ields on the changing spin conf\hspace*{.5pt}igurations along the commensurate--incommensurate boundary. Indeed, it is not a universal characteristic of systems with competing ferromagnetic and antiferromagnetic interactions, even when these lead to metamagnetic transitions. The present results thus suggest further study to determine whether such a response is a unique along spin reorientation boundaries-or their analogue occurring in other systems. As Rastelli {et al.} \textcolor{blue}{\cite{Reatto979}} predicted, when $J_{nn}$ and $J_{nnn}$ are both positive, the lattice is ferromagnetic, but when either or both are negative, helical or antiferromagnetic. The helical magnetic structure found in NiBr$_{2}$ at low temperature results from a very delicate and fortuitous balance of the various $J_{i}$. Elevated temperature forces the system across boundary from incommensurate to commensurate phase. The interplay between competing interactions lead to amplitude-modulated magnetic structures with a periodicities that do not match the periodicity of crystal lattice \textcolor{blue}{\cite{Shirane2070,Noda2105,Vekua2206,Randy}}.
Here, it is noteworthy to mention that one of our motivations to undertake the study was to identify whether NiBr$_{2}$ is hosting skyrmions. We have used the SANS method with field applied within the basal plane, similar to studies performed on MnSi \textcolor{blue}{\cite{Muhlbauer}}. We have surveyed a large range of fields up to 4 T and temperatures, focusing mainly around magnetic phase transition lines as shown in Fig.~\href{Color online}{\textcolor{blue}{11}}. Within the given coverage range we did not find any signatures of skyrmions.
\subsection{\label{sec:summmary}SUMMARY AND CONCLUSIONS}
In conclusion, by measuring the magnetic properties and neutron diffraction of model spin triangular system of NiBr$_{2}$, we demonstrated that the variation of uniform exchange, saturated magnetic moment, and helical wave vector systematically as a function of the temperature and applied magnetic f\hspace*{.5pt}ield. The changes we measure in the magnetic properties are dominated by the variation of anisotropy and exchange interactions. These observations are supported by the models as predicted by Rastelli {et al.} \textcolor{blue}{\cite{Reatto979}}, after simplif\hspace*{.5pt}ied assumption of mutual competition in neighboring exchange interactions. Despite this, we demonstrated a magnetic f\hspace*{.5pt}ield-induced phase transition across the incommensurate phase, which entirely transform the low-temperature incommensurate phase into the high temperature commensurate spin structure. This behavior of incommensurate phase is purely governed by spin-flop transition. On the methodological side, our work demonstrates that combining state-of-the-art neutron scattering experiments with magnetization measurements, we were enable to extract def\hspace*{.5pt}initive microscopic information from the spin triangular system of NiBr$_{2}$. These f\hspace*{.5pt}indings accelerate the search for exotic quantum states in helimagnetic systems through the screening of many related materials having direct technological implications. As a conclusion, one may foresee that the accurate understanding the role of neighboring exchange interactions in the triangular spin systems paves the way to exploratory research on f\hspace*{.5pt}ield-induced phase transitions where the choice of spin helix and topological magnetic texture will be a tool to improve the performances.
\begin{acknowledgments}
We would like to thank M. Reehuis, F. Yokaichiya and U. Keiderling for technical help with the neutron data collection. Authors would like to acknowledge Dr. Abhisekh Singh, CIFC, IIT (BHU) for assisting the magnetic measurments. SKM acknowledges the funding from the IIT(BHU) with grant no. IIT(BHU)/R\&D/SM/2016-17/4453.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,275 |
10.4236/pp.2017.812028
Stage-Specific Changes on Plasmodium yoelii yoelii Following Treatment with Hintonia latiflora Stem Bark Extract and Phytochemical-Antioxidant Evaluation
Elba Carrasco-Ramírez1, Perla Y. López-Camacho2, Armando Zepeda-Rodríguez3, Patricia Bizarro-Nevares3, Filiberto Malagón-Gutiérrez1, Gustavo Basurto-Islas4, Norma Rivera-Fernández1*
1Laboratorio de Malariología, Departamento de Microbiología y Parasitología, Facultad de Medicina, Universidad Nacional Autónoma de México, CDMX, México
2Departamento de Ciencias Naturales e Ingeniería, Universidad Autónoma Metropolitana, CDMX, México
3Deparatmento de Biología Celular y Tisular, Facultad de Medicina, Universidad Nacional Autónoma de México, CDMX, México
4Departamento de Ingenierías Química Electrónica y Biomédica, División de Ciencias e Ingeniería, Universidad de Guanajuato, Campus León, Guanajuato, México
Received: October 30, 2017; Accepted: December 5, 2017; Published: December 8, 2017
Malaria endemic zones are mostly located on third world countries, where antimalarials are not easily found or patients cannot afford them, and in consequence, they must turn toward natural products or phytomedicines. In the present study, the effect of Hinotnia latiflora (Hl) methanolic stem bark extract (HlMeOHe) on the ultrastructure of the asexual intraerythrocytic stages of Plasmodium yoelii yoelii (Pyy) after a Peters' four-day oral treatment was assessed by transmission electron microscopy (TEM), as well as the parasite development on blood smears, analyzed by light microscopy. Likewise, extract was subjected to qualitative tests adopting standard procedures for identification of phytoconstituents; its antioxidant activity was evaluated according to the method of Brand-Williams and by the radical cation decolorization assay. Results showed higher percentage of rings and lower percentage of trophozoites and schizonts in the treated animals, in comparison with those of the control groups, which demonstrated lower percentage of rings and trophozoites, and schizonts in higher number. Images of TEM showed in some treated parasites, mild parasite membranes, organelle swelling and ribosomal depletion. The phytochemical profile demonstrated that the extract contains alkaloids, tannis, steroids, terpenoids, flavonoids, phenolics and saponins. The obtained values of the half maximal inhibitory concentration (IC50) in µg/mL, for both antioxidant assays were of 423.83 and 202.95 respectively. It is concluded that HlMeOHe altered the development of the intraerythrocytic asexual stages and the ultrastructure of Pyy, and due to its phytochemical constituents, showed an in vitro antioxidant activity.
Hintonia latiflora, Plasmodium yoelii yoelii, Malaria, Ultrastructural Changes, Phytochemicals, Antioxidant, Natural Products
Malaria is still a public health problem in third world countries where people cannot afford or have access to the conventional antimalarial drugs. It is estimated that 80% of malaria patients living in poor endemic areas treat themselves with plants and never assist to any formal health facility [1] [2] . Artemisinin combination therapies in these countries are often used to treat children and pregnant women, consequently, patients at lower risk, attempt to find less expensive and more accessible alternatives such as natural products. Artemisia annua, Cinchona bark, Cryptolepis sanguinolenta and Cochlospermum planchonii, are some of the plants officially approved to be used as phytomedicines to treat malaria in Africa [1] [3] . In Mexico, the stem bark of Hintonia latiflora (Sessé & Moc. Ex. DC.) Bullok Rubiaceae, commonly known as copalchi, is frequently consumed to treat gastrointestinal disorders, diabetes and malaria [4] ; in Europe, capsules of copalchi micronized cortex are consumed as an antioxidant supplement [5] . In previous studies, the stem bark of Hl has demonstrated in vitro and in vivo antimalarial efficacy in a good to moderate range, respectively, and an excellent antipyretic effect on murine malaria model [6] [7] , however, more studies need to be done concerning its toxicity, pharmacokinetics and biological activity in order to obtain valuable clinical information to sustain their use as an adjuvant treatment to control malaria, as, even though the World and Health Organization (WHO) promote the use of natural products, a synthetic or natural medicine must achieve good scientific evidence of safety and efficacy before being commercialized [2] . There are no published data showing the effect of HlMeOHe on the development of the asexual intraerythrocytic stages and morphology of the parasite, therefore, in the present study, the development of the asexual stages and the ultrastructural changes of Pyy induced by the treatment with HlMeOHe were evaluated, as well as its phytochemical profile and in vitro antioxidant activity.
2.1. Animals
15 CD-1 male mice weighing 28 g were used for the study and were obtained from the Faculty of Medicine, UNAM vivarium. Mice were divided into 3 groups of five mice each. Animal management was performed according to the Mexican Official Norm NOM-062-ZOO-1999 for the production, care, and use of laboratory animals in accordance to international guidelines and approved by the Ethical and Research committee at School of Medicine, UNAM (project 095/2016).
2.2. Parasites
P. yoelii yoelii lethal strain was obtained from the London School of Hygiene and Tropical Medicine and maintained by serial passages in CD1 mice.
2.3. Plant Material
Stem bark of H. latiflora was collected at the UNAM Chamela's Biological Station, Jalisco México, latitude 19˚29'52.90375 N, longitude 105˚02'41.33116W, H: (Ell. Height) 77.679 m, identified and prepared as described by Rivera et al. (2014) [7] . A voucher sample (collect number 7772) of the plant was deposited at Dr. Salvador Nava y Esparza Herbarium (UAMIZ) collect number 83,519 by Jhony Anacleto. The stem bark of Hl was prepared as described by Rivera et al., (2014) [7] . Permission to collect stem bark samples was obtained from Chamela UNAM Research Institute of Biology.
2.4. Extraction
Sixty grams of dried powdered stem bark of Hl was extracted with methanol in a solid-liquid system for 72 h. The solvent was evaporated in vacuo to afford 10 g of extract [7] . Methanol extract was used because it was reported previously that this extractant showed the higher biological activity and lower toxicity to the mice [7] .
2.5. Biological Experiment
Mice were infected according Rivera et al. (2013) [8] . A four-day suppression test [9] was used to evaluate the effect of the extract on the parasites. By oral gavage, five mice received 1000 mg/kg of HlMeOHe and five others received the vehicle (tween 80%) at a concentration of 0.04%; five mice remained as Pyy-untreated control group. Selection of the extract dose, was made regarding the LD10 work dose reported previously [7] . On the fifth day post-treatment, a blood smear was made to all mice and the percent individual parasitemia was estimated [6] . The development of the intraerythrocytic asexual stages was evaluated by light microscopy, classifying the parasites into three groups: rings, trophozoites, and schizonts [10] . At 5th day post infection (PI), 1000 cells on blood smears were counted per mouse to obtain the percentage of rings, trophozoites and schizonts. The results obtained from both HlMeOHe and tween 80 group, were compared with those obtained on the untreated Pyy mice.
At fifth day post-treatment, 25 µL of peripheral blood from a cut of mice tail vein from the treated and untreated animals was collected in 200 µL Eppendorf® tubes and immediately mixed with citrate 3.8% in a ratio of 9 parts blood to 1-part anticoagulant. The samples were mixed by gentle inversion. Whole blood samples were centrifuged at 1500 × g for 10 minutes at room temperature to obtain infected erythrocytes for TEM [11] .
2.6. Transmission Electron Microscopy
Blood samples were fixed for 1 h in 2.5% glutaraldehyde in sucrose-cacodylate buffer, then, centrifuged at 1500 rpm, washed three times in cacodylate buffer and postfixed for 1 h with 2% osmium tetroxide in sucrose-cacodylate buffer. The samples were dehydrated in a graded 30% to 100% ethanol series at 4˚C for 10 min each, and embedded in Araldita 6005 epoxy resin (Electron Microscopy Sciences). Thin sections were stained with lead citrate and uranyl acetate and observed using a JEM 1010 electron microscope [11] . All studies were made by duplicate.
2.7. Phytochemical Evaluation
The analysis was conducted as described by Yadav and Agarwala (2011) [12] . 10 mg of extract was mixed with a few drops of HCl for the detection of alkaloids, turbidity in the suspension indicated the presence of alkaloids. Tannins were detected by dissolving 0.01 g of extract in 0.2 mL of water and a few drops of FeCl3 0.1%, the change to dark blue indicated the presence of tannins. Steroids were detected by mixing 10 mg of extract with 0.4 mL of acetic anhydride and 0.4 ml of sulfuric acid; the change to red violet indicated the presence of steroids. For the detection of terpenoids, 10 mg of extract was dissolved in 0.4 mL of chloroform and 0.6 mL of concentrated sulfuric acid without shaking, formation of brown ring indicates the presence of terpenoids. Flavonoids were detected by mixing 5 mL of plant extract with a few drops of ethanolic FeCl3, the formation of red color indicated the presence of flavonoids. For total phenols, 5 mL of alcoholic solution of the plant extract was mixed with FeCl3 solution. The change in color to dark blue indicated the presence of phenols. Saponins were detected by dissolving 10 mg of the extract in 1 mL of distilled water and few drops of olive oil; the formation of emulsion indicated the presence of saponins.
2.8. In Vitro Antioxidant Capacity Determinations
The 1,1-diphenyl-2-picrylhydrazyl (DPPH) radical scavenging capacity of the extract was determined according to the method of Brand-Williams [13] . DPPH radicals have an absorption maximum at 518 nm, which disappears with reduction by an antioxidant compound. The DPPH• solution in methanol (0.2 mM) was prepared daily. 150 μL of this solution was mixed with 50 µL of methanolic plant extract solution at concentrations of 25, 50, 125, 250 and 500 μg/mL. Absorbance reading was taken at 518 nm 5 min after initial mixing. Butylated hydroxytoluene (BHT) and ascorbic acid were used as positive controls in the same concentrations (25, 50, 125, 250 and 500 μg/mL). All determinations were carried out in triplicate. The percentage of inhibition of ABTS+• was calculated using the following formula:
% inhibition = [(AB − AE)/AB] × 100, where AB = absorbance of the blank sample, and AE = absorbance of the plant extract.
The free radical scavenging capacity of the plant extract was performed using the 2,2'-azino-bis (3-ethylbenzothiazoline-6-sulphonic acid) (ABTS) radical cation decolorization assay [14] , which is based on the reduction of ABTS+• radicals. ABTS was dissolved in deionized water to a 7 mM concentration. ABTS radical cation (ABTS+•) was produced by reacting ABTS solution with 2.45 mM potassium persulfate and allowing the mixture to stand in the dark at room temperature for 12 - 16 h before use. The ABTS+• solution was then diluted in ethanol to an absorbance of 0.7 (±0.02) at 734 nm. Plant extract (50 μL) at concentrations of 25, 50, 125, 250 and 500 μg/mL were mixed with 150 μL of the ABTS+• solution. Absorbance reading was taken at 734 nm 10 min after initial mixing. BHT and ascorbic acid were used as positive controls in the same concentrations (25, 50, 125, 250 and 500 μg/mL). All determinations were carried out in triplicate. The percentage of inhibition of ABTS+• was calculated by using the same formula as given above.
Obtained data were analyzed with one-way analysis of variance (ANOVA) and Tukey's test using GraphPad Prism® software, version 7. All analyzed data had a normal distribution, and statistical significance was set at P < 0.05.
3.1. Antimalarial Efficacy
All mice showed parasites in their blood on the fifth day of sampling. At 5th day, the mean parasitemia for the Pyy and Tween 80 control group was of 41% and 38.30% respectively, while the HlMeOHe treated mice showed a parasitemia of 5.60%. Mice from Pyy and Tween group died at 6 days postinfection (PI) with parsitemias over 69%, while HlMeOHe treated mice died at 14 days PI with a parasitemia of 22.6% (Table 1). At 5th day PI, on the infected untreated group, the percentages of Pyy intraerythrocytic stages were observed as follows: rings 7.4, trophozoites 26.4 and schizonts 7.2 and in the Tween control group, the observed percentages were: 7.2 rings, 24.2 trophozoites and 6.9 schizonts. In the HlMeOHe treated group, rings, trophozoites and schizonts were seen in percentages of 15.2, 7.06 and 1.4, respectively (Table 2).
3.2. Light Microscopy Observations
Under light microscopy, Pyy and Pyy-Tween 80 mice, showed a high number of infected red blood cells in peripheral blood at 5 days PI; rings, trophozoites, schizonts, and merozoites with visible pigment, were observed on the blood smears. Smears of HlMeOHe treated mice showed a small number of infected
Table 1. Effect of 1000 mg/kg−1 of methanolic stem bark extract of Hintonia latiflora on CD-1 Pyy infected mice.
Results are reported in mean ± standard error of two independent studies. *P < 0.05 versus Pyy control group. PO = per os, SID = every 24 h.
Table 2. Activity of 1000 mg/kg−1 of methanolic stem bark extract of Hintonia latiflora on Plasmodium yoelii yoelii intraerythrocytic stages.
Results are reported in mean ± standard error of two independent studies. *P < 0.05 versus Pyy control group.
red blood cells at 5 days PI; only a few rings and trophozoites were observed (Figure 1), and in some samples between 1 - 2 schizonts per smear were detected. It was difficult to assess parasite structural changes at a light microscope level.
On the TEM images obtained from the parasites from the untreated Pyy and Tween 80 groups, well preserved intraerythrocytic parasites were observed. Almost all the evaluated samples showed stages of rings, trophozoites and schizonts; merozoites stages were present only in a few samples and in a reduced number; rhoptries, micronemes and dense granules were the organelles most frequently observed in merozoites. Rings were located peripherally in the erythrocytes. Almost all the evaluated ring stages presented a thin and flat discoidal appearance with cytoplasm housing their nucleus and free ribosomes. A digestive vacuole as large as 40% of the parasite volume was observed in almost all the rings. In some samples, crystallized malarial pigment was found. Trophozoites showed a thick border of cytoplasm housing organelles like nucleus, mitochondrion, and ribosomes; in some samples clusters of vesicles on the cytoplasm were seen. Schizogony budding was observed in some specimens, free merozoites were difficult to find. Different parasite stages can be observed on Figure 2.
On the treated parasites, the most common observed stages were rings, followed by few trophozoites. Schizonts were difficult to find and in some samples a very few number of free damage merozoites were observed. Parasites treated
Figure 1. Blood smears of Pyy and HlMOHe-Pyy mice. (a) 5th day sampling Pyy mice; rings, trophozoites, schizonts and merozoites are observed; (b) HlMOHe treated Pyy mice at 5th day PI, only ring stages were observed; (c) Pyy mice at the day of the death (6 days postinfection), very low number of infected red blood cells with all the asexual parasite stages are observed; (d) HlMOHe treated Pyy mice at 14 days PI, infected red blood cells are observed containing mostly rings and trophozoites stages; m (merozoite), r (ring), s (schizont), t (trophozoite).
with HlMeOHe, showed in general, an amorphous shape and swollen plasma and cellular membranes. In some cases, cell membranes seemed to be disintegrated in specific zones and occasionally membranous debris were observed; however, a complete disintegration of the cell membranes was never seen. Most of the parasites depicted vacuoles in their cytoplasm and ribosomal depletion. In almost all the samples, endoplasmic reticulum was difficult to observe. All parasites presented a normal food vacuole, except in some cases (around 10% of the observed parasites), where a minimally swollen membrane was detected. The nucleus membrane appeared to be a little bit swollen in some parasites. In the more severe cases, very few parasites showed a complete destruction of their organelles (approximately less than the 3% of the evaluated parasites). Images of treated parasites are depicted in Figure 3.
3.4. Phytochemical Profile and Antioxidant Activity
HlMeOHe stem bark extract showed an effective free radical scavenging activity against radicals DPPH and ABTS (Table 3 and Table 4). Extract radical scavenging capacity was compared with that of the ascorbic acid and BHT. In the DPPH assay, the extract showed an IC50 value of 423.83 µg/mL, while BHT and ascorbic acid depicted IC50 values of 805.50 and 139.84 µg/mL respectively, and
Figure 2. Transmission electron micrographs of the asexual intraerythrocytic stages of Pyy, recovered at 5th day sampling. In all the images, the density of the parasites and the red blood cells are the same. Intraerythrocytic parasites are surrounded by the parasitophorous vacuole membrane. (a) Ring stage surrounded by ribosomes with a sausage like nucleus and a large digestive vacuole; (b) Immature schizont with developing merozoites; (c) Schizont with mature merozoites showing the apical prominence and different organelles as nucleus, ropthries and dense granules; ap (apical prominence), arrows (rhoptries), dv (digestive vacuole), n (nucleus). Bars 1 µm.
under same conditions; therefore, the extract demonstrated an effective antioxidant capacity. In the ABTS assay, the concentration of the extract required to quench 50% of the ABTS radical cation was higher compared to that of both controls, with values of 202.95, 2.45 and 32.59 µg/mL for the extract, BHT and ascorbic acid, respectively.
Research on anti-malarial extracts, must include toxicity and pharmacokinetics studies as well as the evaluation of their efficacy on killing the parasites by modifying its ultrastructure, which can be very useful to identify a possible mechanism
Figure 3. Transmission electron micrographs of the asexual intraerythrocytic stages of Pyy-HlMeOHe, recovered at 5th day sampling. (a) Ring stage with a slightly swollen digestive vacuole membrane and swollen mitochondria; (b) Ring stage with a flattened like appearance, showing depletion of ribosomes (*) and undistinguishable organelles; (c)-(d) Trophozoite with extensive loss ribosomal areas, swollen mitochondria and small digestive vacuoles; (e) Trophozoite exhibiting severe damage and total disorganization; (f) Free merozoites with vacuoles in their cytoplasm, swollen organelles and small digestive vacuoles; m (mitochondria), n (nucleus), dv (digestive vacuoles), arrow malaria pigment. Bars (a)-(b) 2 µm, (c)-(d) 1µm, (e)-(f) 2 µm.
Table 3. Phytochemical analysis of Methanolic extract of Hintonia latiflora stem bark.
+ = detected.
of action [10] . In the present study, the effect of 1000 mg/kg of HlMeOHe on the development of the asexual intraerythrocytic stages and on the ultrastructure of Pyy, was evaluated. Likewise, the phytochemical profile and the in vitro
Table 4. Antioxidant capacity of Hintonia latiflora methanolic stem bark extract IC50 (µg/mL).
Results are the mean ± standard error of three independent experiments, *P < 0.05 versus BHT and ascorbic acid.
antioxidant activity of the extract were performed. It is important to carry out phytochemical and antioxidant profiles, even if some constituents of the plant are partially known, because the variations in the chemical composition and consequently, on the biological activity of herb compounds, are associated mainly with the geographic origin where the plant was grown and collected [15] .
The antimalarial efficacy obtained with the 4-day test, regarding the parasitemia and the mice survival time after 5th day, showed that all treated mice maintained lower parasitemias and presented a survival time of 14 days with respect to the control groups; these results agree with those reported by Rivera et al. (2014) [7] . The percentage of rings observed on the blood smears of the treated animals at 5th day sampling, was almost double the percentage observed on the control groups, and in consequence, the percentage of trophozoites and schizonts decreased in the treated mice. These results showed that the extract may have a parasitostatic effect, retarding the development from rings to schizonts stages. Nevertheless, the development of the parasite continued and the treated mice finally died despite the low parasitemia reported at the day of the death. Cell-cycle delays and recrudescence in malaria, have been reported after treatment with atovaquone, atovaquone plus proguanil or mefloquine [16] [17] , and several studies have identified ring-stage quiescence mechanism of survival during exposure to monotherapies with artemisinin drugs [18] [19] [20] [21] . In other studies, it was observed that 50% of nonimmune patients, experience treatment failure if artemisinins are given as monotherapy [22] . Natural products tested in vivo and in vitro against malaria, like hydroethanolic crude extract of Ajuga remota and a natural triterpene obtained from olive pomace, also reported a parsitostatic effect, delaying the development of mature rings or trophozoites [23] [24] .
It is not known how Hl stem bark extracts are metabolized in vivo and in vitro, nonetheless, the results obtained in this work suggest that its constituents could be rapidly metabolized and eliminated within hours in the treated mice, and in consequence, constituents could not remain in plasma time enough to kill the parasites. It is important to consider that the schizogonic rhythm of rodent malarias in the blood of mice varies from one species to the other [25] . Murine antimalarial drugs studies are performed mostly with asynchronous strains that are less sensitive than synchronous strains because of the delayed penetration of merozoites into red blood cells [25] .
HlMeOHe damaged the ultrastructure of the asexual intraerythrocytic stages of Pyy, hence, morphological alterations were observed on different parasite stages. As cited by Sachanonta et al. (2011) [10] , it is important to guarantee that the evaluated changes in the specimens treated with drugs or other substances, are the consequence of the exposure to these molecules than a result of a wrong TEM process. In the present study, proper TEM conditions were used to ensure that the ultrastructural changes on Pyy, were a direct effect of Hl extract.
The most frequently observed lesions on the treated parasites were ribosomal depletion and cytoplasmic vacuolization, and in some cases, disruption and swelling of cell and organelles membranes; equivalent ultrastructural changes have been reported on plasmodium parasites after exposure to quinine, piperaquine and artesunate [10] . Disruptions in cell and organelle membranes were observed in Plasmodium falciparum parasites exposed in vitro to artemisisnin. [26] Even in the more severe cases of parasite destruction, the digestive vacuoles appeared to preserve their integrity, therefore, it seems that HlMeOHe does not interfere with the physiological function of this organelle, as most antimalarials do. Same results have been obtained by other authors; Ellis et al. (1985) [27] , observed ultrastructural changes as membrane and ribosomal organization disruption on Plasmodium berghei after 30 minutes treatment with artemisinin but no changes were noted in the digestive vacuoles. The effect of Qinghao extract and artemisinin on the ultrastructure of Plasmodium berghei by a Peters' four-day oral treatment, was evaluated by You You TU research team; their results showed that treated parasites depicted cell and organelle membrane swelling, and in extreme cases, the structure of some parasites disappeared but residual food vacuoles were still contained inside the infected erythrocytes [28] . The absence of free ribosomes observed in some treated specimens, may suggest, that the mode of action of the HlMeOHe could be related with the protein synthesis of the parasite. Moreover, the disruption of some membrane-bound organelles suggested that some extract constituents could be involved in both oxidative phosphorylation and anaerobic glycolysis effects which can act as a sink for excess intracellular calcium [29] [30] , thus, more studies need to be done to identify the mechanism by which Hl affects malaria parasites.
In our CD1 mice, Pyy produces pathology findings and clinical features of complicated malaria; infected mice die with high parasitemia (>80%), severe anemia, hemoglobinuria and hypoglycemia six to seven days PI and always exhibit multi-organic involvement [31] . Pathogenesis in complicate malaria is originated by three main mechanisms: cytoadherence, oxidative stress and reactive oxygen species (ROS) production and exacerbated release of proinflammatory citokines, predominantly tumor necrosis factor [32] . ROS are produced during hemoglobin degradation by the intracellular parasite, during the adherence of infected red blood cells to the endothelium and during the production of proinflammatory citokines [33] [34] [35] . Additionally, malaria infection decreases the antioxidant defense system [36] . Antioxidants limit the ROS oxidative damage to biological systems, therefore, because of their antioxidant constituents, the use of natural products is increasingly growing.
The phytochemical profile revealed that HlMeOHe, contains among other metabolites, flavonoids, phenols, tannins and terpenoids. In general, polyphenols have demonstrated benefits as antioxidants. Flavonoids have a planar structure with hydroxyl groups and double bond in position C2-C3, which give them capacity as a chelators, free radical scavengers and inhibitors of enzymes that produce free radicals [37] . It is known that secondary metabolites, such as alkaloids, flavonoids, tannins and other phenolic compounds are responsible of antioxidant and antimicrobial activities in most plants [38] . Oxidative stress is one of the main pathological mechanisms by which the parasite produces severe damage to the host, therefore, the antioxidant capacity of the HlMeOHe, could explain, at least in part, the beneficial effects showed in the treated mice. The survival time of these animals, could be due to a boosting of the immune system or maybe to an inhibition of the exacerbated production of proinflammatory cytokines produced by the antioxidants metabolites of the extract; phenolic constituents and flavonoids exhibit anti-inflammatory properties and reduce the levels of prostaglandin, thus reduce fever. This action was observed in a previous study done by our research group, in which a marked decrease of body temperature was observed in mice following the first few minutes after HlMeOHe treatment [7] .
The methanolic extract of the stem bark of Hintonia latiflora, delayed the development of the asexual intraerythrocytic stages of Pyy, specifically from rings to trophozoites and schizonts, moderately altered the parasite ultrastructure and depicted an in vitro antioxidant activity that could help the host immune system during a complicated malarial infection. Nevertheless, the use of the stem bark of Hintonia latiflora as an antimalarial treatment must be taken with caution as it seems to just delay the parasite development. Our results leave an open door to continue the research with Hl, regarding primarily on its possible action into the host immune system.
This study was supported by Projects DGAPA-PAPIIT UNAM IA203015, IA206217 and CONACYT CB 182003. The authors thank the technical assistance of Francisco Pasos Nájera for the art work process and Professor Josefina Bolado, Head of the Scientific Paper Translation, División de Investigación, Facultad de Medicina, UNAM, for editing the English-language version of this paper.
Carrasco, R.E., López, C.P., Zepeda, R.A., Bizarro, N.P., Malagón, G.F., Basurto, I.G. and Rivera, F.N. (2017) Stage-Specific Changes on Plasmodium yoelii yoelii Following Treatment with Hintonia latiflora Stem Bark Extract and Phytochemical-Antioxidant Evaluation. Pharmacology & Pharmacy, 8, 381-395. https://doi.org/10.4236/pp.2017.812028
1. Willcox, M. (2011) Improved Traditional Phytomedicines in Current Use for the Clinical Treatment of Malaria. Planta Medica, 77, 662-671. https://doi.org/10.1055/s-0030-1250548
2. WHO (2015) Guidelines for the Treatment of Malaria, 2nd Edition. World Health Organization, Geneva. [Internet] [cited 2017 June 26]; Available from http://www.who.int/malaria/publications/atoz/9789241549127/en/
3. WHO Monographs (2009) Medicinal Plants Commonly Used in the Newly Independent States (NIS). World Health Organization [Internet]. [Cited 2017 June 26]; Available from http://apps.who.int/medicinedocs/es/m/abstract/Js17534en/
4. Biblioteca Digital de la Medicina Tradicional Mexicana? [Internet]. Mexico [update 2009; cited 2017 June 2017]. Avilable from http://www.medicinatradicionalmexicana.unam.mx/index.php
5. Bruguera, M., Herrera, S., Lázaro, E., Madurga, M., Navarro, M. and de, Abajo, F. (2017) Hepatitis aguda asociada al consumo de copalchi. A propósito de 5 casos. Gastroenterología y Hepatología, 30, 66-68. https://doi.org/10.1157/13099265
6. Argotte, R.R., Ramírez, A.G., Rodríguez, G.M., Ovilla, M.M., Lanz, M.H., Rodríguez, M.H., González, C.M. and Alvarez, L. (2006) Antimalarial 4-Phenylcoumarins from the Stem Bark of Hintonia latiflora. Journal of Natural Products, 69, 1442-1444. https://doi.org/10.1021/np060233p
7. Rivera, N., López, Y., Rojas, M., Fortoul, T., Reynada, D., Reyes, J., et al. (2014) Antimalarial Efficacy, Cytotoxicity, and Genotoxicity of Methanolic Stem Bark Extract from Hintonia latiflora in a Plasmodium yoelii yoelii Lethal Murine Malaria Model. Parasitology Research, 113, 1529-1536. https://doi.org/10.1007/s00436-014-3797-9
8. Rivera, N., Marrero, P.Y., Arán, V.J., Martínez, C. and Malagón, F. (2013) Biological Assay of a Novel Quinoxalinone with Antimalarial Efficacy on Plasmodium yoelii yoelii. Parasitology Research, 112, 1523-1527. https://doi.org/10.1007/s00436-013-3298-2
9. Peters, W. and Robinson, B.L. (1992) The Chemotherapy of Rodent Malaria. XLVII. Studies on Pyronaridine and Other Mannich Base Antimalarials. Annals of Tropical Medicine & Parasitology, 86, 455-465. https://doi.org/10.1080/00034983.1992.11812694
10. Sachanonta, N., Chotivanich, K., Chaisri, U., Turner, G., Ferguson, D., Day, N., et al. (2011) Ultrastructural and Real-time Microscopic Changes in P. falciparum-Infected Red Blood Cells Following Treatment with Antimalarial Drugs. Ultrastructural Pathology, 35, 214-225. https://doi.org/10.3109/01913123.2011.601405
11. Bizarro, P., Acevedo, S., Nino-Cabrera, G., Mussali-Galante, P., Pasos, F., Avila-Costa, M.R., et al. (2003) Ultrastructural Modifications in the Mitochondrion of Mouse Sertoli Cells after Inhalation of Lead, Cadmium or Lead-Cadmium Mixture. Reproductive Toxicology, 17, 561-566. https://doi.org/10.1016/S0890-6238(03)00096-0
12. Yadav, R.N.S. and Agarwala, M. (2011) Phytochemical Analysis of Some Medicinal Plants. Journal of Phytology, 3, 10-14.
13. Brand, W.W., Cuvelier, M.E. and Berset, C. (1995) Use of a Free Radical Method to Evaluate Antioxidant Activity. LWT-Food Science and Technology, 28, 25-30. https://doi.org/10.1016/S0023-6438(95)80008-5
14. Re, R., Pellegrini, N., Proteggente, A., Pannala, A., Yang, M. and Rice-Evans, C. (1999) Antioxidant activity Applying an Improved ABTS Radical Cation Decolorization Assay. Free Radical Biology and Medicine, 26, 1231-1237. https://doi.org/10.1016/S0891-5849(98)00315-3
15. Ying, T.S., Liu, W., Liu, J., Yin, D. and Zhao, X. (2015) Influence of Ecological Factors on the Production of Active Substances in the Anti-Cancer Plant Sinopodophyllum hexandrum (Royle). PLos One, 10, e0122981. https://doi.org/10.1371/journal.pone.0122981
16. Thapar, M.M., Gil, J.P. and Bjorkman, A. (2005) In Vitro Recrudescence of Plasmodium falciparum Parasites Suppressed to Dormant State by Atovaquone Alone and in Combination with Proguanil. Transactions of The Royal Society of Tropical Medicine and Hygiene, 99, 62-70. https://doi.org/10.1016/j.trstmh.2004.01.016
17. Veiga, M.I., Ferreira, P.E., Schmidt, B.A., Ribacke, U., BjÖrkman, A., Tichopad, A., et al. (2010) Antimalarial Exposure Delays Plasmodium falciparum Intra-Erythrocytic Cycle and Drives Drug Transporter Genes Expression. PLoS ONE, 5, e12408. https://doi.org/10.1371/journal.pone.0012408
18. Codd, A., Teuscher, F., Kyle, D.E., Cheng, Q. and Gatton, M.L. (2011) Artemisinin-Induced Parasite Dormancy: A Plausible Mechanism for Treatment Failure. Malaria Journal, 8, 56. https://doi.org/10.1186/1475-2875-10-56
19. Teuscher, F., Gatton, M.L., Chen, N., Peters, J., Kyle, D.E. and Cheng, Q. (2010) Artemisinin-Induced Dormancy in Plasmodium falciparum: Duration, Recovery Rates, and Implications in Treatment Failure. The Journal of Infectious Diseases, 202, 1362-1368. https://doi.org/10.1086/656476
20. Tucker, M.S., Mutka, T., Sparks, K., Patel, J. and Kyle, D.E. (2012) Phenotypic and Genotypic Analysis of In Vitro-Selected Artemisinin-Resistant Progeny of Plasmodium falciparum. Antimicrobial Agents and Chemotherapy, 56, 302-314. https://doi: 10.1128/AAC.05540-11
21. Witkowski, B., Lelièvre, J., Barragán, M.J., Laurent, V., Su, X.Z., Berry, A., et al. (2010) Increased Tolerance to Artemisinin in Plasmodium falciparum Is Mediated by a Quiescence Mechanism. Antimicrobial Agents and Chemotherapy, 54, 1872-1877. https://doi.org/10.1128/AAC.01636-09
22. Meshnick, S.R., Taylor, T.E. and Kamchonwongpaisan, S. (1996) Artemisinin and the Antimalarial Endoperoxides: From Herbal Remedy to Targeted Chemotherapy. Microbiology Reviews, 60, 301-315.
23. Moneriz, C., García, M.P., Granados, G.A., Bautista, M.J., Diez, A. and Puyet, A. (2011) Parasitostatic Effect of Maslinic Acid. I. Growth Arrest of Plasmodium falciparum Intraerythrocytic Stages. Malaria Journal, 10, 82. https://doi.org/10.1186/1475-2875-10-82
24. Nardos, A. and Makonnen, E. (2017) In Vivo Antiplasmodial Activity and Toxicological Assessment of Hydroethanolic Crude Extract of Ajuga remota. Malaria Journal, 16, 25. https://doi.org/10.1186/s12936-017-1677-3
25. Landau, I., Caillard, V., Beaute-Lafitte, A. and Chabaud, A. (1993) Chronobiology and Chronotherapy of Malaria: Investigations with Murine Malaria Models. Parassitologia, 35, 55-57.
26. Maeno, Y., Toyoshima, T., Fujioka, H., Ito, Y., Meshnick, S.R., Benakis, A., et al. (1993) Morphologic Effects of Artemisinin in Plasmodium falciparum. The American Journal of Tropical Medicine and Hygiene, 49, 485-491. https://doi.org/10.4269/ajtmh.1993.49.485
27. Ellis, D.S., Li, Z.L., Gu, H.M., Peters, W., Robinson, B.L., Tovey, G., et al. (1985) The Chemotherapy of Rodent Malaria, XXXIX. Ultrastructural Changes Following Treatment with Artemisinine of Plasmodium berghei Infection in Mice, with Observations of the Localization of [3H]-Dihydroartemisinine in P. falciparum in Vitro. Annals of Tropical Medicine & Parasitology, 79, 367-374. https://doi.org/10.1080/00034983.1985.11811933
28. Tian, G., Li, Y.C., Wang, J.Y., Ji, X.G., Yang, L. and Tu, Y.Y. (2008) Effects of Qinghao Extract on the Ultrastructure of Plasmodium berghei. Acta Parasitology et Medica Entomologica Sinica, 4, 13-15.
29. Beller, M., Thiel, K., Thul, P.J. and Jäckle, H. (2010) Lipid Droplets: A Dynamic Organelle Moves into Focus. FEBS Letters, 3584, 2176-2182.
30. Haynes, R.K. and Krishna, S. (2004) Artemisinins: Activities and Actions. Microbes and Infection, 6, 1339-1346. https://doi.org/10.1016/j.micinf.2004.09.002
31. Rivera, N., Samanta, E.R., Menchaca, A., Zepeda, A., García, L.E., Salas, G., et al. (2013) Blackwater Fever like in Murine Malaria. Parasitology Research, 112, 1021-1029. https://doi.org/10.1007/s00436-012-3224-z
32. Milner, D.A. (2017) Malaria Pathogenesis. Cold Spring Harb Perspect Med. https://doi.org/10.1101/cshperspect.a025569
33. Ginsburg, H. and Atamna, H. (1994) The Redox Status of Malaria-Infected Erythrocytes: An Overview with an Emphasis on Unresolved Problems. Parasite, 1, 5-13. https://doi.org/10.1051/parasite/1994011005
34. Makarenko, V.V., Usatyuk, P.V., Yuan, G., Lee, M.M., Nanduri, J., Natarajan, V., et al. (2014) Intermittent Hypoxia-Induced Endothelial Barrier Dysfunction Requires ROS-Dependent MAP Kinase Activation. American Journal of Physiology. Cell Physiology, 306, 745-752. https://doi.org/10.1152/ajpcell.00313.2013
35. Bullock, G.C., Richardson, C.L., Schrott, V., Gunawardena, N.D., Cole, T.N., Corey, C.G., et al. (2015) The Role of Mitochondrial Metabolism and Redox Signaling in Iron Deficiency Anemia. Blood, 126, 2145-2145.
36. Reis, D., Comim, P.A., Hermani, C.M., Silva, F., Barichello, P.T., et al. (2010) Cognitive Dysfunction Is Sustained after Rescue Therapy in Experimental Cerebral Malaria, and Is Reduced by Additive Antioxidant Therapy. PLOS Pathogens, 6, e1000963. https://doi.org/10.1371/journal.ppat.1000963
37. Atmani, D., Chaher, N., Atman, D., Berboucha, M., Debbache, N. and Boudaoud, H. (2009) Flavonoids in Human Health: From Structure to Biological Activity. Current Nutrition and Food Science, 5, 225-235. https://doi.org/10.2174/157340109790218049
38. Xu, Y., Burton, S., Kim, Ch. and Sismour, E. (2016) Phenolic Compounds, Antioxidant, and Antibacterial Properties of Pomace Extracts from Four Virginia-Grown Grape Varieties. Food Science & Nutrition, 4, 125-133. https://doi.org/10.1002/fsn3.264
● PP Subscription
●Most popular papers in PP
●About PP News | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 576 |
\section{Introduction}
Modern societies critically rely on distribution networks of various kinds.
Typically, a distribution network is depicted as a graph where resources can enter the network via supply vertices and leave the network via demand vertices, together with edges that connect the supply, demand and additional internal vertices. Often, flow capacity constraints and cost functions are assigned to the edges.
Distribution networks can be divided into two classes, depending on whether the vertices can store resources or not. If the vertices can only distribute resources without storage, we refer to this type of distribution networks as static. The study of static distribution networks is a broad research topic which has a long history and a large number of applications \cite{Aronson1989}. One celebrated result is the max-flow min-cut theorem \cite{Ford1956}. The static distribution problem is closely related to monotropic programming problems which enjoy a complete and symmetric duality theory \cite{rockafellar1984network}.
Differently from static distribution networks, in dynamical distribution networks vertice can have storage of resources. This type of models has many applications in, e.g., communication networks \cite{Ephremides1989,Segall1977}, transportation networks \cite{Moreno1995,Como2015,Lovisari2014,Moss1982}, hydraulic networks \cite{SCHOLTEN2017}, flow networks \cite{DANIELSON2013,wei2013}, and inventory and production systems \cite{Bertsimas2006,Blanchini00}.
In this paper, we analyze the robustness of a basic dynamical distribution networks where we assign a set of single integrators to the vertices (with state variables corresponding to storage). All the integrators are controlled by the flows on the edges. On each edge, the flow is the weighted storage difference of the adjacent vertices. Furthermore, unknown in/outflows may enter or leave the network through some of the vertices. The aim here is to minimize the induced $\ensuremath{\mathcal{L}}_2$ gain from the in/outflows to the output of the network by allocating the weights on the edges, which will be called optimal weight allocation problem in this paper. The results of this problem can be relevant when designing robust multi-agent systems. Especially, our setup is similar to the setting in \cite{rai2012}, when one considers the in/outflows as malicious attacks whose goal is to maximize the differences of the storages of the vertices. Then by solving the optimal weight allocation problem, the effect of the worst attack will be minimized. The distribution networks considered in this paper can be seen as linear time-invariant port-Hamiltonian systems \cite{schaftSIAM}, but also resides in the category of state-space symmetric systems \cite{Nagashio2005,Qiu1996,WILLEMS1976,Yang2001}. One useful property of the state-space symmetric system is that its $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ is attained at the zero frequency \cite{TAN2001}.
One closely related problem to the optimal weight allocation, where the connection will be clear in the primary part of the paper, is the positive semi-definiteness of weighted Laplacian with both negative and positive weights. This problem is of salient importance in distributed algorithms \cite{Altafini2013,Xia2016,Xiao2003}. This problem was considered by many authors. In \cite{Zelazo2014}, the authors provided one sufficient and necessary condition, using effective resistance, for a special weighted graph, namely those where the negatively weighted edges are isolated in different cycles in the graphs spanned by the positive edges. Under the same assumption, the authors of \cite{ChenY2016} re-derived the result in \cite{Zelazo2014} by using geometrical and passivity-based approaches. For general weighted graphs, one sufficient and necessary condition was proposed in \cite{ChenW2016,ChenW2017} using pseudo-inverse of weighted (with negative ones) Laplacian. Here we propose a sufficient and necessary condition using the effective resistance matrix of the positive subgraph from $\ensuremath{\mathcal{H}}_\infty$ approach.
The contributions of this paper are listed as follows. First, we derive a bounded real lemma type of result for state-space symmetric systems. Second, the problem of minimizing the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of the dynamical distribution networks subject to the allocation of the flow capacities is formulated as a semi-definite program. Third, we present a necessary and sufficient condition of positive semi-definiteness of weighted Laplacians, with negative and positive weights, i.e., signed Laplacians.
The structure of the paper is as follows. Some preliminaries will be given in Section \ref{s:preli}. The considered class of dynamic distribution networks and the corresponding weights allocation problem, and the problem of positive semi-definiteness of weighted Laplacian are formulated in Section \ref{ProbForm}. The main results are presented in Section \ref{distnet} and \ref{s:Positive-Laplacian}. Conclusions and future work are given in Section \ref{Concl}.
The notations used in the current paper are collected as follows.
\textbf{Notation.} A positive definite (positive semi-definite) matrix $M$ is denoted as $M\succcurlyeq 0$ ($M\succ 0$).
The element on the $i^{\text{th}}$ row and $j^{\text{th}}$ column of a matrix $M$ is denoted $M_{ij}$. The pseudo-inverse of $M$ is $M^\dagger$. Recall that, for any finite dimensional square matrix $M$, the induced $\ell_2$ norm, denoted by $\|M\|_2$, is the largest singular value which is denoted by $\bar{\sigma}(M)$. The image of a matrix $M$ is $\im M$. The identity matrix is denoted as $I$. The vectors $\one_n$ represents a $n$-dimensional column vector with each entry being $1$.
We will omit the subscript $n$ when no confusion arises. The Euclidean norm of a vector $x$ is denoted as $\|x \|_2$. Given a set $\ensuremath{\mathcal{S}}$, $\inte\{S\}$ denotes its interior.
\section{Preliminaries}\label{s:preli}
In this section, we briefly review some essentials about graph theory \cite{Bollobas98} and robust analysis \cite{zhou1998essentials}.
\subsection{Graph Theory}\label{GraphT}
An undirected graph $\mathcal{G}=(\mathcal{W},\mathcal{V},\mathcal{E})$ consists of a finite set of vertices $\mathcal{V}=\{v_1,...,v_n\}$, a set of edges $\mathcal{E}=\{ \mathcal{E}_1,...,\mathcal{E}_m\}$ that contains unordered pairs of elements of $\ensuremath{\mathcal{V}}$, and a set of corresponding edge weights $\mathcal{W}=\{w_1,...,w_m\}$.
The set of neighbours to vertice $i$ is
\begin{equation*}
N_i=\{v_j|(v_i,v_j)\in \mathcal{E} \}.
\end{equation*}
The graph Laplacian $L\in \mathbb{R}^{n \times n}$ is defined component-wise as
\begin{equation*}\label{Laplacian}
L_{w,ij} =
\begin{cases}
\sum_{ j\in N_i} w_{ij} & \quad \text{if } i=j, \\
-w_{ij} & \quad \text{if } j\in N_i \setminus \{i\}\\
0 & \quad \text{if } j\notin N_i,
\end{cases},
\end{equation*}
where both positive and negative weights are allowed. Given an arbitrary orientation for each edge, the incidence matrix $B\in~\ensuremath{\mathbb R}^{n \times m}$ is defined as
\begin{equation*}\label{incid}
B_{ij}=\begin{cases}
1 & \quad \text{if } \mathcal{E}_j \ \text{starts in vertice} \ v_i, \\
-1 & \quad \text{if } \mathcal{E}_j \ \text{ends in vertice} \ v_i, \\
0 & \quad \text{else}.
\end{cases}
\end{equation*}
These two matrices are related by $L_w=BWB^T$, where $W=~\diag(w_1,...,w_m)$. If $W \succcurlyeq 0$, i.e., there are only positive edges, then it is well-known that the eigenvalues of $L_w$ can be structured as
$
0=\lambda_1\leq\lambda_2\leq...\leq \lambda_n,
$
where the eigenvector corresponding to $\lambda_1=~0$ is $\mathds{1}$. If $W = I$, the Laplacian is denoted without subscript as $L$.
If a graph $\ensuremath{\mathcal{G}} = (\ensuremath{\mathcal{V}},\ensuremath{\mathcal{E}},\ensuremath{\mathcal{W}})$ has both positive and negative weights, we separate the edges set $\ensuremath{\mathcal{E}}$ into $\ensuremath{\mathcal{E}}_+$ and $\ensuremath{\mathcal{E}}_-$, which contains the positive and negative edges, respectively. Thus $\ensuremath{\mathcal{E}}=\ensuremath{\mathcal{E}}_-\cup \ensuremath{\mathcal{E}}_+$. Accordingly, the weight matrix $W = W_+ - W_-$, where $W_+$ ($W_-$) is the absolute value of the weights corresponding the positive (negative) edges. The Laplacian matrix is referred to as signed Laplacian which can be decomposed as
\begin{equation*}
L_w =L_{w+} - L_{w-} : = B_{+}W_{+}B_{+}^T - B_{-}W_{-}B_{-}^T ,
\end{equation*}
where $B_{+}$ and $B_{-}$ are incidence matrices corresponding to the positive and negative sub-graphs, respectively.
The undirected and connected graph without self-loops and with only positive weights can be associated with electrical networks \cite{Klein93}. One important concept is the \emph{effective resistance matrix}, see e.g., \cite{bullo2014,Klein93}, which is defined as
\begin{equation*}
\Gamma = B^TL_w^{\dagger}B,
\end{equation*}
where $L_w^{\dagger}$ is the Moore-Penrose pseudo-inverse of $L_w$ and $B$ is the incidence matrix.
\subsection{$\mathcal{L}_2$-Norm and induced $\mathcal{L}_2$-Gain}
In this subsection, we recall some definitions from robust control. The notations used in this paper are fairly standard and are consistent with \cite{zhou1998essentials}, \cite{RANTZER2015}.
The space of square-integrable signals $f:[0,\infty)\rightarrow \mathbb{R}^n$ is denoted by $L_2[0,\infty)$.
For the linear time-invariant system
\begin{align}\label{e:linear-sys}
\dot{x} & = Ax+Bu, \\ \nonumber
y & = Cx+Du,
\end{align}
the transfer matrix is $\mathbb{G}(s)=C(sI-A)^{-1}B+D$, which has the impulse response
\begin{equation*}
g(t)~=\ensuremath{\mathcal{L}}^{-1}\{\mathbb{G}(s)\}~=~Ce^{At}B\mathbf{1}_+(t)+D\delta(t),
\end{equation*}
where $\delta(t)$ is the unit impulse and $\mathbf{1}_+(t)$ is the unit step defined as
\begin{align}\nonumber
\mathbf{1}_+(t) = \begin{cases}
1, t\geq 0,\\
0, t<0.
\end{cases}
\end{align}
If $x(0)=0$, then we have
$
y(t) = \int_{0}^t g(t-\tau) u(\tau) d\tau.
$
Then the induced $\mathcal{L}_2 \text{- gain}$ is defined as
\begin{align}\nonumber
\|g\|_{2-ind} = \sup_{u\in L_2[0, \infty)}\frac{\|y\|_2}{\|u\|_2} = \sup_{u\in L_2[0, \infty)}\frac{\|g*u\|_2}{\|u\|_2},
\end{align}
where
$\|u(t)\|_2 =\Big(\int_{0}^{\infty}|u(t)|_2^2 dt\Big)^{\frac{1}{2} }.$
This induced $\mathcal{L}_2 \text{- gain}$, i.e., $\|g\|_{2-ind}$ or $\|\mathbb{G}\|_{2-ind}$, is often called the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$, denoted as $\|\mathbb{G}\|_\infty$. It is well-know that for stable systems we have that $\|\mathbb{G}\|_\infty= \sup_{\omega\in\mathbb{R}}\bar{\sigma}\{\mathbb{G}(j\omega)\}$,
where $\bar{\sigma}(A)$ denotes the largest singular value of the matrix $A$.
If the matrices in \eqref{e:linear-sys} satisfy $A=A^\top$, $D=D^\top$ and $C=B^\top$, the system is referred to as state-space symmetric system \cite{TAN2001}. Here, we present a bounded real lemma type of result with respect to state-space symmetric system. Notice that for internally positive LTI system, the bounded real lemma was established in \cite{Tanaka2011}.
\begin{lemma}\label{lm:bounded-real-ssss}
Consider any state-space symmetric system \eqref{e:linear-sys} with ${A \prec 0}$ and $D=0$. Then the following conditions are equivalent,
\begin{enumerate}
\item $\|\mathbb{G}\|_\infty \leq \gamma$,
\item the inequality
\begin{align}\label{e:Hinf-Riccati}
PA + AP + BB^\top+ \frac{1}{\gamma^2}PBB^\top P \preccurlyeq 0
\end{align}
has a solution $P=\gamma I$.
\end{enumerate}
\end{lemma}
The proof is given in Appendix.
\begin{remark}
In the previous lemma, we gave an explicit solution for the Riccati inequality \eqref{e:Hinf-Riccati} where the bounded real lemma can only guarantee the existence of the solutions.
\end{remark}
\section{Problems Formulation}\label{ProbForm}
In this paper, we first consider the weight allocation problem in the scenario of dynamical distribution networks, which is defined on a graph $\ensuremath{\mathcal{G}}=(\ensuremath{\mathcal{V}},\ensuremath{\mathcal{E}})$ with $|\ensuremath{\mathcal{V}}|=n$ and $|\ensuremath{\mathcal{E}}|=m$. Consider the dynamic model
\begin{align}\label{e:plant}
\dot{x}(t) & = B u(t) + E d(t),
\end{align}
where $B$ is the incidence matrix of the graph $\ensuremath{\mathcal{G}}$, $x\in\ensuremath{\mathbb R}^n$ is the system state whose components represent the storage levels in the vertices, $u\in\ensuremath{\mathbb R}^m$ is the controlled flows on the edges, $E\in\ensuremath{\mathbb R}^{n\times k}$ is an assigned matrix and $d(t)\in\ensuremath{\mathbb R}^k$ is an unknown external in/outflows. Here we assume that the image of $E$ is a subset of the image of $B$, i.e., the inflow is equal to the outflow. To simplify the composition, we further assume that, for all $i = 1,\ldots,k$, the $i$th column of $E$ consists of one element which is $\alpha_i>0$ (inflow) and one element $-\alpha_i$ (outflow), while the rest of the elements are zero. Without specification, we set $\alpha_i=1$. A \emph{port} is a set of vertices (terminals) to where the external flows which enter and leave the network sum to zero. Thus, $E$ defines $k$ ports. One example of system \eqref{e:plant} is depicted as Fig. \ref{figure main ex1}.
The condition $\im E\subset \im B$ is a standard assumption, in order to have a stable distribution network, for example in \cite{Blanchini00} which is recalled in the following remark.
\begin{remark}
In \cite{Blanchini00}, the authors considered a distribution network with constraints on the storages, flows and external in/outflows as
\begin{align*}
x(t) & \in \ensuremath{\mathcal{X}} := \{ x\in\ensuremath{\mathbb R}^n \mid x^-\leq x \leq x^+ \} \\
u(t) & \in \ensuremath{\mathcal{U}} := \{ u\in\ensuremath{\mathbb R}^m \mid u^-\leq u \leq u^+ \} \\
d(t) & \in \ensuremath{\mathcal{D}} := \{ d\in\ensuremath{\mathbb R}^k \mid d^-\leq d \leq d^+ \}
\end{align*}
where $x^+,x^-,u^+,u^-,d^+,d^-$ are assigned vectors and the inequalities hold component-wisely. First, it was proved that
the existences of a state-feedback control $u(t)\in\ensuremath{\mathcal{U}}$ and a set of initial conditions $\ensuremath{\mathcal{X}}_0\subset \ensuremath{\mathcal{X}}$, such that for every $x(0)\in\ensuremath{\mathcal{X}}_0$, the solutions of \eqref{e:plant} satisfy
\begin{align*}
x(t)\in\ensuremath{\mathcal{X}}, \quad \forall d(t)\in\ensuremath{\mathcal{D}}, \quad t\geq 0
\end{align*}
if and only if $E\ensuremath{\mathcal{D}}\subset -B\ensuremath{\mathcal{U}}$.
Then it was proved that for any $\bar{x}\in\ensuremath{\mathcal{X}}$, the existence of a state feedback control law $u$, for system \eqref{e:plant}, such that $x(t)\in\ensuremath{\mathcal{X}}$, $u(t)\in\ensuremath{\mathcal{U}}$, and
\begin{align*}
\lim_{t\rightarrow \infty} x(t) = \bar{x}, \quad \forall d(t)\in\ensuremath{\mathcal{D}}
\end{align*}
if and only if
\begin{align*}
E\ensuremath{\mathcal{D}} \subset -\inte\{B\ensuremath{\mathcal{U}}\}.
\end{align*}
It can be seen that one necessary condition to have $E\ensuremath{\mathcal{D}}\subset -B\ensuremath{\mathcal{U}}$ and $E\ensuremath{\mathcal{D}} \subset -\inte\{B\ensuremath{\mathcal{U}}\}$ is $\mathds{1}^\top E = 0$ which implies that the image of $E$ is a subset of the image of $B$ for connected graphs.
\end{remark}
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}
\tikzstyle{EdgeStyle} = [thin,double= black,
double distance = 0.5pt]
\useasboundingbox (0,0) rectangle (4cm,4.5cm);
\tikzstyle{VertexStyle} = [shading = ball,
ball color = white!100!white,
minimum size = 20pt
inner sep = 1pt,]
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_1$},x=1.5cm,y=3.5cm]{v1}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_2$},x=0cm,y=2cm]{v2}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_3$},x=1.5cm,y=0.5cm]{v3}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_4$},x=3cm,y=2cm]{v4}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_5$},x=0cm,y=3.5cm]{v5}
\draw
(v1) edge[->,>=angle 90,thick]
node[right]{$\ensuremath{\mathcal{E}}_1$} (v2)
(v2) edge[->,>=angle 90,thick]
node[left]{$\ensuremath{\mathcal{E}}_2$} (v3)
(v3) edge[->,>=angle 90,thick]
node[left]{$\ensuremath{\mathcal{E}}_3$} (v1)
(v1) edge[->,>=angle 90,thick]
node[right]{$\ensuremath{\mathcal{E}}_4$} (v4)
(v4) edge[->,>=angle 90,thick]
node[left]{$\ensuremath{\mathcal{E}}_5$} (v3)
(v1) edge[->,>=angle 90,thick]
node[above]{$\ensuremath{\mathcal{E}}_6$} (v5)
(v5) edge[->,>=angle 90,thick]
node[left]{$\ensuremath{\mathcal{E}}_7$} (v2)
(v4) edge[dashed,->,>=angle 90,thick]
node[near end, above]{$d_1$} (4cm, 2cm)
(0cm,4.5cm) edge[dashed,->,>=angle 90,thick]
node[near start,left]{$d_1$} (v5)
(-1cm,3.5cm) edge[dashed,->,>=angle 90,thick]
node[near start,below]{$d_2$} (v5)
(v3) edge[dashed,->,>=angle 90,thick]
node[near end,above]{$d_2$} (2.5cm,0.5cm);
\end{tikzpicture}
\caption{Distribution network \eqref{e:plant} on the graph. The state $x_i$ is the storage at the vertex $v_i$. The flows on the corresponding edges are denoted as $u_i$. The orientations on the edges are consistent with the incidence matrix. The vertices $v_5$ has inflow $d_1$ and $d_2$, the vertices $v_4$ and $v_3$ have outflow $d_1$ and $d_2$, respectively. In this case, $E\in \ensuremath{\mathbb R}^{5\times 2}$ whose first and second column are $[0,0,0,-1,1]^\top$ and $[0,0,-1,0,1]^\top$, respectively.}\label{figure main ex1}
\end{center}
\end{figure}
In this paper, we consider the flows on the edges are proportional to the state differences of the adjacent vertices. More precisely, the flows are given as
\begin{align}\label{e:controller}
u = W B^\top x,
\end{align}
where the diagonal matrix $W\in\ensuremath{\mathbb R}^{m\times m}$ is the control gain. The output $y\in\ensuremath{\mathbb R}^k$, which measures the state difference at each port, is given as
\begin{align}\label{e:output}
y= & E^Tx.
\end{align}
This form of the output can be due to the physical constraints of the distribution network, i.e., only the state differences at the ports can be measured. Furthermore, for SISO dynamical distribution networks defined on some special graphs, it can be shown that the induced $\ensuremath{\mathcal{L}}_2$ gain from $d$ to $y$ in \eqref{e:output} is the largest among all $y =C x$ with $C\in\ensuremath{\mathbb R}^{1\times n}$ and $C\mathds{1}=0$. See Corollary \ref{cor:complete} in appendix for details.
Now the closed-loop is,
\begin{equation}\label{flowsys}
\begin{split}
\dot{x}= & -L_wx+Ed, \\
y= & E^Tx,
\end{split}
\end{equation}
where $L_w$ is the graph Laplacian of $\ensuremath{\mathcal{G}}=(\ensuremath{\mathcal{W}},\ensuremath{\mathcal{V}},\ensuremath{\mathcal{E}})$ and $\ensuremath{\mathcal{W}}$ is the set of weights specifying by control gain $W$ in \eqref{e:controller}.
We are ready to introduce two problems which we shall tackle in this paper.
\textbf{Optimal Weight Allocation:} For a given graph and a positive constant $c$,
\begin{align}\label{oriopt}
\min_{W} &\|\mathbb{G}\|_\infty \\ \nonumber
s.t., & \sum w_i =c, \ w_i \geq 0,
\end{align}
where $\mathbb{G}$ is the transfer function (from $d$ to $y$) of the system (\ref{flowsys}), $W=~\diag(w_1,...,w_m)$ and $w_i$, for $i=1,...,m$, are the weights on the edges, and $c$ is a positive constant.
\textbf{Positive semi-definite Laplacian:} Given a weighted graph $\ensuremath{\mathcal{G}}$ with both positive and negative edge weights, what are the upper bounds on the magnitudes
of the negative weights in order to have the Laplacian to be positive semi-definite?
The following two sections are devoted to these two problems, respectively. Before proceeding, this section is closed with following physical interpretation of distribution networks.
\begin{example}
One physical interpretation of the system (\ref{flowsys}) is a basic model of a dynamic flow network, where there are water reservoirs on the vertices and pipes on the edges. The reservoirs are identical cylinders and the pipes are horizontal. The state $x$ is constituted by the water levels in the reservoirs and the pressures are proportional to the water levels. The flow in the pipes are passively driven by pressure difference between the reservoirs. The weights $ \mathcal{W}$ are representing the capacities of the pipes, in terms of diameter and friction. The passive flow from reservoir $i$ to reservoir $j$ is then $q_{ij}=~w_{ij}(x_i-x_j)$. The external input $d$ can e.g. be interpreted as flow in pumps which are distributing water inside the network. The output $y$ is then the difference between water levels of the reservoirs which the pumps are pumping to and the reservoirs which the pumps are pumping from.
\end{example}
\medskip
\section{$\mathcal{H}_\infty$-norm of the distribution network}\label{distnet}
In this section, we shall solve the optimal weight allocation problem by reformulating problem \eqref{oriopt} as an equivalent optimization problem with LMIs as constraints, which can then be efficiently solved numerically using, e.g., CVX \cite{cvx}. The main result of this section is presented as follows.
\begin{theorem}\label{thm:main}
Consider the system (\ref{flowsys}), where $\mathcal{G}$ is an undirected graph and each port belongs to exactly one connected component of $\mathcal{G}$. Suppose $L_w \succcurlyeq 0$. Then
\begin{enumerate}
\item the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of (\ref{flowsys}) is finite, and
\item the following statements are eqvivalent:
\begin{itemize}
\item the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ is less than or equal to $\gamma$.
\item the following LMI is satisfied,
\begin{equation}\label{newLMI}
\begin{bmatrix}
L_w & E \\
E^\top & \gamma I_k
\end{bmatrix}\succcurlyeq 0.
\end{equation}
\end{itemize}
\end{enumerate}
\end{theorem}
\begin{proof}
We show that the theorem is true for the case where $\mathcal{G}$ has exactly one connected component. This is without loss of generality since if $\mathcal{G}$ has more than one connected component, the same procedure can be done for each component and the LMIs can be merged with a common $\gamma$. Denote
\begin{align}\label{e:U}
U^\top=~[\frac{1}{\sqrt{n}}\mathds{1}_n, u^\top_2, \ldots, u^\top_n] \ \text{and} \ U_2^\top~=~[u^\top_2, \ldots, u^\top_n],
\end{align}
for which
$U L_w U^\top =~\diag(0, \lambda_2, \ldots,\lambda_n)=:~\Lambda$. Denote $\hat{\Lambda}=~\diag(\lambda_2,\ldots,\lambda_n)$. Then the system (\ref{flowsys}) has equal $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ as the system
\begin{align*}
\dot{\tilde{x}} & = - \Lambda \tilde{x} + UE d, \\
z & = E^\top U^\top \tilde{x}.
\end{align*}
Notice that the first row of $UE$ is zero, thus the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of the system (\ref{flowsys}) equals the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of the system
\begin{equation}\label{e:flow_P_mini}
\begin{aligned}
\dot{\hat{x}} & = - \hat{\Lambda} \hat{x} + U_2E d, \\
z & = E^\top U_2^\top \hat{x}.
\end{aligned}
\end{equation}
Due to the symmetry of the system and by Theorem 6 in \cite{TAN2001}, the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of the system (\ref{e:flow_P_mini}) is $\|E^\top U_2^\top \hat{\Lambda}^{-1} U_2E \|_{2}$, which is finite. The $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of the system \eqref{flowsys} is then less than or equal to $\gamma$ if and only if
\begin{equation} \nonumber
\|E^\top U_2^\top \hat{\Lambda}^{-1} U_2E \|_{2} \leq \gamma.
\end{equation}
By the property of real symmetric matrix, we can further rewrite the previous constrain as $E^\top U_2^\top \hat{\Lambda}^{-1} U_2E \preccurlyeq \gamma I_{k}$. By Schur complement, we have
\begin{align} \nonumber
\begin{bmatrix}
\hat{\Lambda} & U_2 E \\
E^\top U_2^\top & \gamma I_{k}
\end{bmatrix} \succcurlyeq 0,
\end{align}
which is equivalent to
\begin{equation} \nonumber
\begin{bmatrix}
\Lambda & UE \\
E^\top U^\top & \gamma I_{k}
\end{bmatrix} \succcurlyeq 0.
\end{equation}
By pre and post multiplication of matrix $\diag(U^\top, I_k)$ and $\diag(U, I_k)$, respectively, the previous inequality is transformed to
\begin{equation} \nonumber
\begin{bmatrix}
L_w & E \\
E^\top & \gamma I_k
\end{bmatrix} \succcurlyeq 0.
\end{equation}
Then the conclusion follows.
\end{proof}
\begin{remark}
Notice that in Theorem \ref{thm:main}, the weighted Laplacians $L_w$ can have both positive and negative weights. The result still holds as long as $L_w$ is positive semi-definite.
\end{remark}
\begin{remark}
By Theorem \ref{thm:main}, the problem (\ref{oriopt}) is equivalent to the following semi-definite programming (SDP) problem
\begin{equation}\label{optnew}
\begin{aligned}
\min_W & \quad \gamma \\
s.t., & \ \
\begin{bmatrix}
L_w & E \\
E^\top & \gamma I_k
\end{bmatrix} \succcurlyeq 0, \\
& \ \ \sum w_i = c, \ w_i \geq 0,
\end{aligned}
\end{equation}
which can be efficiently solved by e.g., CVX.
As one numerical example, we consider problem \eqref{optnew} defined on the graph in Fig. \ref{figure main ex1} with $c=8$. Then the optimal weights are $w_1=0, w_2 =1.0427, w_3 = 2, w_4 = 3.0427, w_5 = 0.9573, w_6 = 0.9573,$ and $w_7 = 0$. Here the minimum is $\gamma = 1$. It can be seen that the flows on the first and seventh edge do not contribute to the minimization of $\ensuremath{\mathcal{H}}_{\infty} \text{- norm}$ of this network. The mechanism of this weight allocation is under investigation.
\end{remark}
It is worth mentioning that in a recent work \cite{Pirani2017}, the authors considered a $\ensuremath{\mathcal{H}}_\infty$ design problem for system \eqref{flowsys} with grounded Laplacian with respect to the topology, instead of weight allocation.
In Theorem \ref{thm:main}, we proved that the inequality \eqref{newLMI} is satisfied if and only if the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ is less than or equal to $\gamma$. Moreover, by the bounded real lemma for state-space symmetric systems, i.e., Lemma \ref{lm:bounded-real-ssss}, we have that the following two statements are equivalent
\begin{itemize}
\item $\|\mathbb{G}\|_\infty \leq \gamma$,
\item the Riccati inequality
\begin{equation}\label{flow2feb}
-PL_w-L_w^TP+EE^T+ \frac{1}{\gamma^2}PEE^TP \preccurlyeq 0.
\end{equation}
is satisfied with the solution $P=\gamma I$.
\end{itemize}
In this case, \eqref{flow2feb} is simplified as
\begin{equation}\label{LMI94}
-L_w+\frac{EE^T}{\gamma}\preccurlyeq 0.
\end{equation}
In next section, we shall focus on the positive semi-definiteness of weighted Laplacian, which is a key assumption in Theorem \ref{thm:main}. It turns out that the inequality \eqref{LMI94} plays a crucial role.
\medskip
\section{Positive Semidefiniteness of Signed Laplacians}\label{s:Positive-Laplacian}
In this section, we consider the positive semidefiniteness of signed Laplacian matrices. The main result of this section is formulated in the following theorem, we establish the relation between the magnitude of the negative weights and the effect resistance matrix of subgraph $\ensuremath{\mathcal{G}}_+$. In \cite{Zelazo2014}, the authors assumed that for any $(i,j)\in\ensuremath{\mathcal{E}}_-$ and $(i',j')\in\ensuremath{\mathcal{E}}_-$ being two distinct pairs of vertices, there is no cycle in $\ensuremath{\mathcal{G}}_+$ containing $i,j,i'$ and $j'$. Here we relax the condition to general graphs.
\begin{theorem}\label{th:iff-semip}
The Laplacian matrix $L$ is positive semidefinite if and only if
\begin{enumerate}
\item for any $e_- = (i,j)\in\ensuremath{\mathcal{E}}_-$, $i,j$ belong to one connected component of $\ensuremath{\mathcal{G}}_+$, and
\item the magnitude of the negative weights satisfies
\begin{align}\label{e:iff-semipositive}
\ensuremath{\mathcal{W}}_-^{-1} \succcurlyeq B_-^\top L_{w+}^\dagger B_-.
\end{align}
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{Sufficiency:}
Since for any $e_- = (v_i,v_j)\in\ensuremath{\mathcal{E}}_-$, $i,j$ belong to one connected component of $\ensuremath{\mathcal{G}}_+$, by Theorem \ref{thm:main}, we have the system
\begin{equation}\label{e:sys-semipositive-Laplacian}
\begin{aligned}
\dot{x} & = -L_{w+} x + B_-\sqrt{\ensuremath{\mathcal{W}}_-} d \\
y & = \sqrt{\ensuremath{\mathcal{W}}_-} B^\top_- x
\end{aligned}
\end{equation}
has finite $\ensuremath{\mathcal{H}}_\infty$ norm. Furthermore, by Lemma \ref{lm:bounded-real-ssss}, we have that $L_w = L_{w+} -B_-\sqrt{\ensuremath{\mathcal{W}}_-}\sqrt{\ensuremath{\mathcal{W}}_-} B^\top_- \succcurlyeq 0$ is equivalent to the induced $\ensuremath{\mathcal{L}}_2$ gain of system \eqref{e:sys-semipositive-Laplacian} is less than or equal to $1$.
To prove that $\ensuremath{\mathcal{W}}_-^{-1} \succcurlyeq B_-^\top L_{w+}^\dagger B_-$ implies $L_w\succcurlyeq 0$, we only focus on the case that $\ensuremath{\mathcal{G}}_+$ is connected, i.e., there is only one connected component, without loss of generality. Since
\begin{align*}
\sqrt{\ensuremath{\mathcal{W}}_-}B_-^\top L_{w+}^\dagger B_-\sqrt{\ensuremath{\mathcal{W}}_-}
= \sqrt{\ensuremath{\mathcal{W}}_-}B_-^\top U^\top_2 \Lambda_+^{\dagger} U_2 B_-\sqrt{\ensuremath{\mathcal{W}}_-},
\end{align*}
where $U_2$ is given as in \eqref{e:U} such that $U_2 L_{w+} U^\top_2 = \Lambda_+$, and the induced $\ensuremath{\mathcal{L}}_2$ gain from $d$ to $y$ of system \eqref{e:sys-semipositive-Laplacian} is $\|\sqrt{\ensuremath{\mathcal{W}}_-}B_-^\top U^\top_2 \hat{\Lambda}_+^{-1} U_2 B_-\sqrt{\ensuremath{\mathcal{W}}_-}\|_2$, we have
\begin{align*}
& \ensuremath{\mathcal{W}}_-^{-1} \succcurlyeq B_-^\top L_{w+}^\dagger B_- \\
\Longleftrightarrow \quad & \|\sqrt{\ensuremath{\mathcal{W}}_-}B_-^\top L_{w+}^\dagger B_-\sqrt{\ensuremath{\mathcal{W}}_-} \|_2\leq 1 \\
\Longleftrightarrow \quad & \|\sqrt{\ensuremath{\mathcal{W}}_-}B_-^\top U^\top_2 \hat{\Lambda}_+^{-1} U_2 B_-\sqrt{\ensuremath{\mathcal{W}}_-}\|_2 \leq 1.
\end{align*}
Then the conclusion follows.
\emph{Necessity:} First, it can be verified that if there exists an edge $e_- = (v_i,v_j)\in\ensuremath{\mathcal{E}}_-$ such that $v_i,v_j$ belong to different connected components of $\ensuremath{\mathcal{G}}_+$, $L$ can not be positive semidefinite. More precisely, suppose $\ensuremath{\mathcal{G}}_+$ has $N$ connected components, and the vertices set $\ensuremath{\mathcal{V}}$ can be divided as $\ensuremath{\mathcal{V}}=\ensuremath{\mathcal{V}}_1\cup \cdots \cup \ensuremath{\mathcal{V}}_N$ with $|\ensuremath{\mathcal{V}}_i| = n_i$ and $\sum_{i=1}^{N}n_i = n$. Furthermore, w.l.o.g., suppose $e_-=(v_i,v_j)\in\ensuremath{\mathcal{E}}_-$ such that $v_i$ and $v_j$ belongs to the first and second component, respectively. Denote the Laplacian of the graph $(\ensuremath{\mathcal{V}},\ensuremath{\mathcal{E}}\setminus\{(i,j)\})$ as $\tilde{L}_{w-}$. Then by choosing $v^\top = (\mathds{1}_{n_1}, \frac{1}{2}\mathds{1}_{n_2}, 0,\ldots,0)^\top$, we have
\begin{align*}
v^\top L_w v \leq -\frac{1}{4} \ensuremath{\mathcal{W}}_{-,ij} <0
\end{align*}
where $\ensuremath{\mathcal{W}}_{-,ij}>0$ is the magnitude of the negative weights of $(v_i,v_j)$, which is contradict to the positive semi-definiteness of $L_w$.
With the item 1) holding, the necessity of \eqref{e:iff-semipositive} follows directly from the sufficiency part of the proof.
\end{proof}
\begin{remark}
Notice that the matrix $B_-^\top L_{w+}^\dagger B_-$ is a submatrix of the effective resistance matrix of $\ensuremath{\mathcal{G}}_+$.
When there is only one negative edge, the condition \eqref{e:iff-semipositive} is equivalent to Theorem III.3 in \cite{Zelazo2014}. However, for the multiple negative edges, the result in Theorem \ref{th:iff-semip} is more general than Theorem III.4 in \cite{Zelazo2014} in the sense that there are no constraints on the positions of negative edges.
\end{remark}
The intuition of Theorem \ref{th:iff-semip} is illustrated in the following example.
\begin{example}\label{ex:equivalence-posi-distri}
Consider a weighted graph with two negative edges given as in Fig. \ref{figure main2}. Suppose the negative weights of $(v_5,v_3)$ and $(v_5,v_4)$ are $-w_8$ and $-w_9$, respectively. Recall that the network in Fig.\ref{figure main ex1} represents a dynamical distribution network with two ports and in/outflows $d_1$ and $d_2$, respectively, and only positive edge weights. By setting
\begin{align}\label{e:E}
E^\top = \begin{bmatrix}
0 & 0 & 0 & -\sqrt{w_9} & \sqrt{w_9}\\
0 & 0 & -\sqrt{w_8} & 0 & \sqrt{w_8}
\end{bmatrix}
\end{align}
and by \eqref{LMI94}, we have the Laplacian of the graph in Fig. \ref{figure main2} is positive semi-definite, if and only if the dynamical distribution network in Fig. \ref{figure main ex1} with $E$ defined as \eqref{e:E} has $\ensuremath{\mathcal{H}}_\infty$-norm no larger than $1$.
As an numerical example, we consider the positive weights of the graph in Fig. \ref{figure main2} are identical to one. In this case the submatrix of the effective resistance matrix
\begin{equation*}
B_-^\top L_{w+}^\dagger B_-=\begin{bmatrix}
1.1429 & 0.7143 \\
0.7143 & 0.9048
\end{bmatrix}.
\end{equation*}
It can be verified that by choosing $w_8 = w_9 = 0.5$, we have that \eqref{e:iff-semipositive} holds. In this case, the eigenvalues of $L_w$ are $0, 0.2, 2.6, 4.2, 5$. However, by choosing $w_8 = 0.7, w_9 = 0.5$, which violates \eqref{e:iff-semipositive}, the eigenvalues of $L_w$ are $-0.04, 0, 2.4, 4.2, 5$.
\end{example}
\begin{figure}[ht]
\centering
\begin{tikzpicture}
\tikzstyle{EdgeStyle} = [thin,double= red,
double distance = 0.0pt]
\useasboundingbox (0,0) rectangle (4cm,4cm);
\tikzstyle{VertexStyle} = [shading = ball,
ball color = white!100!white,
minimum size = 20pt
inner sep = 1pt,]
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_1$},x=1.5cm,y=3.5cm]{v1}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_2$},x=0cm,y=2cm]{v2}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_3$},x=1.5cm,y=0.5cm]{v3}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_4$},x=3cm,y=2cm]{v4}
\Vertex[style={minimum
size=0.2cm,shape=circle},LabelOut=false,L=\hbox{$v_5$},x=0cm,y=3.5cm]{v5}
\draw
(v5) edge[dashed, -,>=angle 90,thick, red]
node[right]{} (v4)
(v2) edge[-,>=angle 90,thick]
node[left]{} (v3)
(v3) edge[dashed,-,>=angle 90,thick, red]
node[left]{} (v5)
(v3) edge[-,>=angle 90,thick]
node[left]{} (v1)
(v2) edge[-,>=angle 90,thick]
node[left]{} (v1)
(v1) edge[-,>=angle 90,thick]
node[right]{} (v4)
(v4) edge[-,>=angle 90,thick]
node[left]{} (v3)
(v1) edge[-,>=angle 90,thick]
node[above]{} (v5)
(v5) edge[-,>=angle 90,thick]
node[left]{} (v2);
\end{tikzpicture}
\caption{ The network used in Example \ref{ex:equivalence-posi-distri} where the graph has two negative edges $(v_1,v_2)$ and $(v_1,v_3)$ (red colored).}\label{figure main2}
\end{figure}
\section{Conclusions}\label{Concl}
For a basic dynamic distribution networks, we have derived an optimization set up with LMIs as constraints, which minimizes the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ with respect to the allocation of the weights on the edges. Furthermore, by using a bounded real lemma for state-space symmetric systems, we have interpreted the Riccati inequality for distribution networks as a definiteness criterion of a Laplacian to a graph containing both positive and negative weights on the edges. Moreover, we have provided a sufficient and necessary condition, using effective resistance matrix of the subgraph spanned the positive edges, for the positive semi-definiteness of the Laplacian with both positive and negative edges.
A related future topic is the problem of minimizing the $\ensuremath{\mathcal{H}}_\infty \text{- norm}$ of dynamic flow networks with respect to topology, more precisely, a limited amount of edges is to be allocated in a graph with fixed vertices. Another future topic is to consider a fixed graph (both topology and weights), but consider saturation of the flow on the edges. The problem is then to minimize the induced $\ensuremath{\mathcal{L}}_2$-gain with respect to allocation of the saturation limits.
\section{Acknowledgment}
The first author would like to acknowledge Dr. Mohammad Pirani for the valuable discussions.
\section{Appendix}
\begin{proof}[Proof of Lemma \ref{lm:bounded-real-ssss}]
Notice that $\textit{(2)}$ implies $\textit{(1)}$ is guaranteed by the bounded real lemma. Hence we only show that $\textit{(1)}$ implies $\textit{(2)}$.
By using Theorem 6 in \cite{TAN2001}, we have that $\|\mathbb{G}\|_\infty = \|-BA^{-1}B\|_2$. Hence, by Schur complement, $\|\mathbb{G}\|_\infty \leq \gamma$ implies
\begin{align*}
\begin{bmatrix}
-A & B \\
B^\top & \gamma I
\end{bmatrix} \succcurlyeq 0,
\end{align*}
which is equivalent to $A+\frac{1}{\gamma}BB^\top \preccurlyeq 0$ and $\gamma\geq 0$. Then it is straightforward to see that $P=\gamma I$ is a solution to \eqref{e:Hinf-Riccati}.
\end{proof}
\begin{corollary}\label{cor:complete}
Consider a SISO dynamical distribution network
\begin{equation}\label{e:system-general-output}
\begin{aligned}
\dot{x} & = - L_w x + Ed, \\
y & = C x,
\end{aligned}
\end{equation}
defined on a connected graph, with $E\in\ensuremath{\mathbb R}^{n\times 1}$ and $C\in\ensuremath{\mathbb R}^{1\times n}$ satisfying $\mathds{1}^\top E = C\mathds{1}=0$, then the $H_\infty-$norm is upper bounded by
\begin{align}
\frac{\|C\|_2\|B\|_2}{\lambda_2}.
\end{align}
Furthermore, suppose that the eigenvalues of $L_w$ satisfy $0=\lambda_1<\lambda_2=\cdots=\lambda_n$, then we have
\begin{align*}
E^\top = \arg \max_{C} & \quad \|\mathbb{G}\|_\infty, \\
s.t. & \quad \|C\|_2 = \|E\|_2.
\end{align*}
\end{corollary}
\begin{proof}
The notations in this proof are consistent with the ones in the proof of Theorem \ref{thm:main}.
The $H_\infty$ norm of system \eqref{e:system-general-output} is
\begin{align*}
& \sup_{\omega\in\ensuremath{\mathbb R}} |C U_2^\top (j\omega I + \hat{\Lambda})^{-1} U_2E| \\
= & \sup_{\omega\in\ensuremath{\mathbb R}} | \sum_{i=1}^{n-1} \frac{\hat{C}_i \hat{E}_i}{j\omega+\lambda_{i+1}} | \\
\leq & \sup_{\omega\in\ensuremath{\mathbb R}} \sum_{i=1}^{n} \frac{|\hat{C}_i \hat{E}_i|}{\sqrt{\omega^2+\lambda^2_{i+1}}} \\
\leq & \frac{\|C\|_2 \|E\|_2}{\lambda_2}
\end{align*}
where $\hat{C}_i$ and $\hat{E}_i$ are the $i$th components of the vectors $CU_2^\top $ and $U_2E$, respectively, and the last inequality is based on the fact that $\|CU_2^\top\|_2 = \|C\|_2$ and $\|U_2 E\|_2 = \|E\|_2$.
If we further have $\lambda_2=\cdots=\lambda_n$, i.e., the previous upper bound can be achieved if and only if $U_2 C^\top = U_2 E$. Then since $U_2^\top U_2 = I - \frac{1}{n}\mathds{1}\mathds{1}^\top$ and $\mathds{1}^\top E = C\mathds{1}=0$, we have $C= E^\top$. Thus the conclusion follows.
\end{proof}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,516 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/434221\/can-e-fracq4-pi-epsilon-0-r2-be-directly-derived-from-differential-form\/434228","text":"# Can $E=\\frac{q}{4\\pi\\epsilon_0 r^2}$ be directly derived from differential form of Maxwell equations?\n\nThe electric field of a point charge $$q$$ is well known to be $$\\mathbf E=\\frac{q}{4\\pi\\epsilon_0 |\\mathbf r|^3}\\hat{\\mathbf r}$$\n\nThis can be derived easily from integral form of Gauss\u2019s law. Taking $$V$$ as a sphere of radius $$r$$ centered at the point charge, $$\\int_{\\partial V}\\mathbf E\\cdot d\\mathbf A=\\frac{q}{\\epsilon_0}$$ $$4\\pi r^2 E=\\frac{q}{\\epsilon_0}\\implies E=\\frac{q}{4\\pi\\epsilon_0 r^2}$$\n\nHowever, I can\u2019t see how this can be derived directly from the differential form of Maxwell\u2019s equations. Maxwell\u2019s equations only says the divergence and curl of electric field is zero everywhere away from the point charge, which isn\u2019t quite useful at all.\n\nHow can the formula for electric field of a point charge derived directly from differential form of Maxwell\u2019s equations?\n\n(Note: it is of course possible to convert the differential form into integral form and the proceed as I\u2019ve shown. However, that\u2019s somewhat trivial and I do not consider that as a direct derivation.)\n\n\u2022 @AaronStevens My emphasis is on \u2018differential form\u2019 and \u2018derived directly\u2019. \u2013\u00a0Szeto Oct 13 '18 at 4:24\n\u2022 The charge distribution is not continuous and the differential form of Maxwell's equations would involve a kind of Dirac delta function for the charge density, since it is infinite at the origin and zero everywhere else. \u2013\u00a0Tob Ernack Oct 13 '18 at 4:31\n\u2022 To add to what @TobErnack it's saying, the fact that the divergence is not $0$ at the point charge should be useful. \u2013\u00a0BioPhysicist Oct 13 '18 at 4:34\n\u2022 @TobErnack But how can this be useful for the derivation of the formula of electric field? \u2013\u00a0Szeto Oct 13 '18 at 4:52\n\nWhen you have a point charge, the charge density is not really defined, as it is basically infinite at the origin, and zero everywhere else. You would need a distribution rather than a function to describe it. The derivation here is not completely rigorous as I am treating $$\\delta_{\\mathbb{R}^3}(\\vec{r})$$ as a normal function when doing the integrations...\n\nWe can use the Dirac delta function over $$\\mathbb{R}^3$$ to write $$\\rho(\\vec{r}) = q\\delta_{\\mathbb{R}^3}(\\vec{r})$$\n\nThen the Gauss equation gives $$\\nabla \\cdot \\vec{E} = \\frac{q}{\\varepsilon_0}\\delta_{\\mathbb{R}^3}(\\vec{r})$$\n\nSince the problem is spherically symmetric, we compute the divergence in spherical coordinates and use $$\\vec{E} = E_r \\hat{r}$$ (where $$E_r$$ is the radial component of $$\\vec{E}$$ and $$\\hat{r}$$ is a unit radial vector): $$\\nabla \\cdot \\vec{E} = \\frac{1}{r^2}\\frac{d}{dr}(r^2E_r)$$\n\nSo we get Gauss's equation as: $$\\frac{d}{dr}(r^2E_r) = \\frac{q}{\\varepsilon_0}r^2\\delta_{\\mathbb{R}^3}(r\\hat{r}) \\tag{*}$$\n\nFor $$r \\gt 0$$ the equation is just $$\\frac{d}{dr}(r^2E_r) = 0$$, so that $$r^2E_r = C$$ for some constant $$C$$.\n\nTherefore $$E_r = \\frac{C}{r^2}$$ for $$r \\gt 0$$.\n\nTo find the actual value of $$C$$, we integrate (*) from $$0$$ to $$\\infty$$ and use $$r^2E_r = C$$ to find $$C = \\frac{q}{\\varepsilon_0}\\int_{0}^{\\infty}\\delta_{\\mathbb{R}^3}(r\\hat{r})r^2dr$$ (note that this is not fully rigorous since the usual Fundamental theorem of calculus applies to functions, not distributions).\n\nNow the main property of $$\\delta_{\\mathbb{R}^3}(\\vec{r})$$ is that $$\\iiint\\limits_{\\mathbb{R}^3} \\delta_{\\mathbb{R}^3}(\\vec{r}) dV = 1$$\n\nUsing spherical coordinates, this is $$1 = \\int\\limits_{0}^{\\infty}\\int\\limits_{-\\pi}^{\\pi}\\int\\limits_{0}^{2\\pi}\\delta_{\\mathbb{R}^3}(r\\hat{r})r^2\\sin\\varphi \\text{ d}\\theta\\text{ d}\\varphi \\text{ d}r = \\int\\limits_{0}^{\\infty}\\delta_{\\mathbb{R}^3}(r\\hat{r})r^2\\text{ d}r \\int\\limits_{-\\pi}^{\\pi}\\sin\\varphi\\text{ d}\\varphi\\int\\limits_{0}^{2\\pi}d\\theta = 4\\pi\\int\\limits_{0}^{\\infty}\\delta_{\\mathbb{R}^3}(r\\hat{r})r^2\\text{ d}r$$\n\nTherefore $$C = \\frac{q}{4\\pi\\varepsilon_0}$$ and we obtain $$\\vec{E} = \\frac{q}{4\\pi\\varepsilon_0r^2}\\hat{r}$$\n\nOn the other hand, this derivation is not actually too different from the one using the integral version of Gauss's theorem, since we performed an integration and used the 1-d fundamental theorem of calculus to find $$C$$, which is equivalent to using the divergence theorem in 3-d.\n\n\u2022 For physicists using Dirac delta functions under integrals like this is rigourous :) \u2013\u00a0BioPhysicist Oct 13 '18 at 5:33\n\u2022 Is it possible not to utilize spherical symmetry? \u2013\u00a0Szeto Oct 13 '18 at 7:37\n\u2022 @Szeto Sorry for the late reply. If you dont assume spherical symmetry I think it is still possible to solve it, but you will have to use expansions of $E$ and $\\delta_{\\mathbb{R}^3}$ in spherical harmonics. Then you can compute the spherical expansion of $\\nabla\\cdot E$. At this point you will be comparing coefficients in the left and right hand sides, and only the coefficient corresponding to a spherically symmetric term will be nonzero. (We are using the completeness and orthogonality property of spherical harmonics in this argument). \u2013\u00a0Tob Ernack Jun 30 '19 at 5:36\n\u2022 @TobErnack If you have time and if you don\u2019t mind, please demonstrate the maths behind. You might want to add it to your answer. Lots of thanks! \u2013\u00a0Szeto Jun 30 '19 at 5:38\n\nIt'd be hard to get the multiplicative factor, I think, but the basic principle is not hard:\n\nThe divergence in spherical coordinates is handily provided by Wikipedia as$$\\nabla\\cdot A = {1 \\over r^2}{\\partial \\left( r^2 A_r \\right) \\over \\partial r} + {1 \\over r\\sin\\theta}{\\partial \\over \\partial \\theta} \\left( A_\\theta\\sin\\theta \\right) + {1 \\over r\\sin\\theta}{\\partial A_\\varphi \\over \\partial \\varphi}.$$Looking for a spherically symmetric field gives just $${d \\left( r^2 E_r \\right) \\over d r} = r^2~ {\\rho(r)\\over\\epsilon_0}.$$ To get the proportionality right one would want to solve for $$\\rho(r)=\\{3q\/(4\\pi R^3)\\text{ if } r< R\\text{ else }0\\}$$ with boundary condition $$E_r(r=0)=0.$$","date":"2020-08-05 01:26:35","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\": 37, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9427691698074341, \"perplexity\": 154.53099605400317}, \"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-2020-34\/segments\/1596439735906.77\/warc\/CC-MAIN-20200805010001-20200805040001-00347.warc.gz\"}"} | null | null |
**The MRCP PACES Handbook**
Second Edition
**The MRCP PACES Handbook**
Second Edition
Saira Ghafur
CCT in Respiratory and General Internal Medicine and Harkness
Fellow in Health Care Policy and Practice
New York City, New York, USA
Parminder K Judge
ST6 in Nephrology
Oxford, England, UK
Richard Kitchen
Consultant in Palliative Medicine
Glasgow, Scotland, UK
and
Samuel Blows
Consultant Geriatrician and Acute Physician
Ipswich, England, UK
Foreword by
Dr Fiona Moss
Dean of the Royal Society of Medicine
London, England, UK
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2017 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-4987-8632-4 (paperback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (<http://www.copyright.com/>) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
<http://www.taylorandfrancis.com>
and the CRC Press Web site at
<http://www.crcpress.com>
Contents
Foreword
Preface
Authors
Acknowledgements
List of Abbreviations
The PACES Examination
Top Tips for PACES Success: An Examiner Speaks
1Station 1: Respiratory
Hints for the Respiratory Station
Bronchiectasis
Chronic Obstructive Pulmonary Disease
Consolidation
Cystic Fibrosis
Fibrotic Lung Disease
Lung Cancer
Old Tuberculosis
Pleural Effusion
Patient with Previous Lung Surgery
Respiratory Station Summary
2Station 1: Abdominal
Hints for the Abdominal Station
Abdominal Case with a Normal Abdomen
Chronic Liver Disease
Generalised Lymphadenopathy
Hepatosplenomegaly
Multiple Abdominal Scars
Palpable Kidneys
Renal Replacement Therapy
Splenomegaly
Abdominal Station Summary
3Station 2: History Taking
Hints for the History-Taking Station
Patient with an Abnormal Blood Result
Patient with Back Pain
Patient Who Has Collapsed
Patient with a Cough
Patient with Diarrhoea
Patient with Jaundice
Patient with Joint Pains
Patient with Left Arm Weakness
Patient with Visual Disturbance
Patient with Weight Loss
History-Taking Station Summary
4Station 3: Neurology
Hints for the Neurology Station
Cerebellar Syndrome
Hemiparesis
Myotonic Dystrophy
Ocular Palsies
Parkinson' s Disease
Peripheral Neuropathy
Mononeuropathies
Motor Neurone Disease
Multiple Sclerosis
Spastic Paraparesis
Visual Field Defects
Neurology Station Summary
5Station 3: Cardiology
Hints for the Cardiology Station
Aortic Stenosis
Aortic Regurgitation
Eisenmenger' s Syndrome
Hypertrophic Cardiomyopathy
Mitral Stenosis
Mitral Regurgitation
Mitral Valve Prolapse
Mixed Aortic Valve Disease
Mixed Mitral Valve Disease
Prosthetic Heart Valves
Tricuspid Regurgitation
Ventricular Septal Defect
Cardiology Station Summary
6Station 4: Ethics and Communication Skills
Hints for the Ethics and Communication Skills Station
Breaking Bad News
Care in the Deteriorating Patient
Driving Regulations
Initiating a New Therapy
Long-Term Condition
Medical Error
Mental Capacity
New Diagnosis
Organ Transplantation
Ethics and Communication Skills Station Summary
7Station 5: Brief Clinical Encounters
Hints for the Brief Clinical Encounters Station
Ankylosing Spondylitis
Anticoagulation
Diabetic Retinopathy
Facial Nerve Palsy
Falls
Headache (Idiopathic Intracranial Hypertension)
Hyperthyroidism
Neck Lump
Osteoporosis
Proximal Myopathy
Psoriasis
Rheumatoid Arthritis with Carpal Tunnel Syndrome
Systemic Sclerosis
Third Nerve Palsy or Patient Presenting with Diplopia
Brief Clinical Encounters Station Summary
Afterword: Hawks and Doves
Index
Foreword
Postgraduate medical exams are significant both as career-defining milestones for the candidate and as one of the ways of ensuring that doctors who progress have all that it takes to provide excellent patient care. Exams focus the minds of candidates for whom exam content and exam approach define what examiners consider important and so perhaps mould approaches to practise. The Practical Assessment of Clinical Examination Skills (PACES), the final part of the Membership of the Royal Colleges of Physicians (MRCP) examination, is a pivotal step for all aspiring physicians, and this book will be an enormous support for them. But, the _MRCP PACES Handbook_ is so much more than a book for passing exams. A comprehensive book, it succinctly covers much of the necessary content, and for that it will be useful to candidates. Responses and analyses to the questions posed are logical and practical and cleverly proceed from the likely and frequent to the rarefied. It is also very well organised, demonstrating not just how to pass an exam, but also the importance of an ordered approach to clinical reasoning. Lists of key points at the ends of chapters keep the reader on track, and the well-chosen lists of references really do encourage further reading. The real strength of this handbook is that it gets to the heart of the purpose of the PACES examination.
The PACES exam reflects the work done through MRCP UK to develop assessments of all the skills necessary for excellent clinical practice. Of course, this includes physical examination, recognition of clinical signs and differential diagnosis, the stuff of medical examinations for many years. Now, assessments have become more patient centred and more holistic and are designed to demonstrate explicitly the expectation of modern physicians and good medical practice. Clinical judgement, communication, managing patient concerns and maintaining patient welfare are four of the seven skills assessed through the PACES exam, and these skills are very well covered in this handbook.
Not only are technical points covered, but by including, for example, outlines of themes explored, suggestions for candidates and expectations for both candidates and patients, this handbook provides a succinct but in-depth outline of some of the difficult but crucial aspects of medical practice.
The simplicity of approach and the clear language throughout make it clear that this is not a book for the ' superficial learner' wanting a ' quick fix' for an impending exam, but one that encourages understanding and a mature approach to developing all the skills necessary for becoming a good physician.
PACES, the final part of the MRCP exam, is what is says: practical. And reflecting the essence of the PACES exam, this handbook is a supremely practical ' vade mecum' . It is a handbook that not only will be essential for those approaching their PACES exam or those considering becoming a physician, but also will be a useful and continuing guide to ' practise after PACES' for those entering higher training after PACES, and it will be useful for anyone who teaches postgraduate or undergraduate medicine.
This is a book that will be bought and used to help aspiring physicians through their PACES, will encourage development of a patient-centred approach to care and then, for many years after, will stay in easy reach.
**Dr Fiona Moss, CBE, MD, FRC
Dean of the Royal Society of Medicine **
Preface
Observe, record, tabulate, and communicate. Use your five senses. Learn to see, learn to hear, learn to feel, learn to smell, and know that by practice alone you can become an expert.
**Sir William Osler, FRS, FRCP**
The Practical Assessment of Clinical Examination Skills (PACES) should be seen as the defining moment of your medical training when you have been accepted into the fold – the final hurdle you need to jump over before gaining the elusive title of ' Medical Registrar' . The wise words from Sir William Osler sum up the skills needed to achieve this goal.
This book is designed to be a comprehensive revision aid that will be invaluable to candidates studying for their PACES examination. The book is designed to be used at the patient' s bedside to guide clinical examination, as well as having enough detail to cover pertinent points in the case in question.
Each of the chapters contains hints and tips on how to tackle each of the examinations and a summary at the end to consolidate key learning points. The cases represent a collection of those which are most frequently encountered in the PACES examination and are set out in a standardised format. Each chapter has been reviewed by senior clinicians within the specialty. The information within the handbook has been updated and is supported by evidence-based literature and supporting guidelines.
Passing PACES requires not only an in-depth knowledge of the subject of the case but also the fine art of being able to present in a concise and coherent manner. Each case contains a section on presentation which will aid the candidate in perfecting this skill.
Excellent communication skills are the cornerstone of every doctor' s interaction with patients and colleagues and are essential in passing the PACES examination. Uphold the legacy of Dr Kate Granger, who initiated the #hellomynameis campaign, and treat every patient with the utmost respect, kindness and compassion.
PACES is feared as the unachievable milestone; the key to passing the exam is hard work and sheer dedication. The blood, sweat and tears will all be worth it in the end when you have attained the highly sought after title of Membership of the Royal Colleges of Physicians of the United Kingdom (MRCP UK).
If you follow in the footsteps of those who have gone before you, you are guaranteed to succeed.
Good luck – go forth and conquer!
**SG, PKJ, RK and SB**
Authors
**Saira Ghafur** graduated from the University of Dundee Medical School. She has a dual CCT in respiratory and general internal medicine and has a particular interest in community respiratory medicine. She is currently a post-CCT Harkness Fellow in Health Care and Policy at the Commonwealth Fund in New York City, New York.
**Parminder K Judge** graduated from the University of Birmingham Medical School. She is a nephrology registrar on the Oxford rotation. She recently took time out of clinical training to undertake research in clinical trials and epidemiology at the University of Oxford, Oxford, United Kingdom.
**Richard Kitchen** graduated from the University of Leicester Medical School. He has a CCT in palliative medicine (obtained on the West Midlands rotation) and has a particular interest in medical education, having completed a master' s degree in this area. He has recently been appointed as a consultant in palliative medicine at Ardgowan Hospice and Inverclyde Royal Hospital, Greenock, United Kingdom.
**Samuel Blows** graduated from the University of Leicester Medical School. He trained in geriatrics and general internal medicine in London with a sabbatical at Bankstown Hospital, Sydney, Australia. He has an interest in medical education and has recently been appointed as a consultant in complex care/interface geriatrics at Ipswich Hospital, Ipswich, United Kingdom.
The authors all worked together at the University Hospitals of Leicester as junior doctors, during which time they all passed the PACES examination and developed their interest in medical education and evidenced-based medicine.
Acknowledgements
We would like to thank the following people for all their support, knowledge and input in the chapters and cases relating to their area of expertise and their guidance in making this book possible:
**Dr Mark Ardron**
Consultant Geriatrician, University Hospitals of Leicester NHS Foundation Trust
**Dr Parry Blaxill**
Consultant in Respiratory Medicine, Lung Cancer Lead, Mid Yorkshire NHS Foundation Trust
**Dr Binith Cheeran**
Honorary Consultant Neurologist, Oxford University Hospitals NHS Foundation Trust Senior Clinical Fellow, University of Oxford
**Dr Rachael Curley**
Consultant in Respiratory Medicine (Cystic Fibrosis), Sheffield Teaching Hospitals NHS Foundation Trust
**Dr Marni Greig**
Consultant in Diabetes and Endocrine, Sheffield Teaching Hospitals NHS Foundation Trust
**Dr Kar-Ping Kuet**
Consultant Rheumatologist, Sheffield Teaching Hospitals NHS Foundation Trust
**Dr Sapna Ladani**
Consultant Haematologist, University Hospitals of Leicester NHS Foundation Trust Honorary Senior Lecturer, University of Leicester
**Mark Latimour**
LLB (Hons), BA, GradDip (Legal Practice)
**Dr Ajay M Verma**
Consultant Gastroenterologist, Kettering General Hospital
**mDr David Warriner**
SpR in Cardiology, Yorkshire, and the Humber Deanery
List of Abbreviations
A2 | aortic component of heart sound 2
---|---
AA | amyloid A
ABG | arterial blood gas
ABPA | allergic bronchopulmonary aspergillosis
ACE | angiotensin-converting enzyme
ACF | antecubital fossa
ACPA | anticitrullinated protein antibodies
ACTH | adrenocorticotrophic hormone
ADH | antidiuretic hormone
ADLs | activities of daily living
ADPKD | autosomal dominant polycystic kidney disease
AF | atrial fibrillation
AFB | acid-fast bacilli
AFP | alpha-fetoprotein
AKI | acute kidney injury
AL | amyloid L
ALD | adrenoleukodystrophy
ALD | alcoholic liver disease
ALP | alkaline phosphatase
ALS | amyotrophic lateral sclerosis
AMA | antimitochondrial antibody
AMTS | abbreviated mental test score
ANA | antinuclear antibody
APKD | adult polycystic kidney disease
AR | aortic regurgitation
ARB | angiotensin receptor blocker
ARDS | acute respiratory distress syndrome
AS | aortic stenosis
ASA | aminosalicylic acid
ASD | atrial septal defect
AV | arteriovenous
AVA | aortic valve area
AVR | aortic valve replacement
AXR | abdominal X-ray
AVSD | atrioventricular septal defect
BCC | basal cell carcinoma
BP | blood pressure
BTS | British Thoracic Society
CABG | coronary artery bypass graft
CCF | congestive cardiac failure
CDT | _Clostridium difficile_ toxin
CF | cystic fibrosis
CFTR | cystic fibrosis transmembrane conductance regulator Ch chromosome
CK | creatine kinase
CKD | chronic kidney disease
CLL | chronic lymphocytic leukaemia
CML | chronic myeloid leukaemia
CMV | cytomegalovirus
CN | cranial nerve
CO | carbon monoxide
COMT | catechol-o-methyltransferase
COPD | chronic obstructive pulmonary disease
COX | cyclo-oxygenase
CRF | chronic renal failure
CRP | C-reactive protein
CSF | cerebrospinal fluid
CT | computerised tomography
CVA | cerebrovascular accident
CXR | chest X-ray
DCCT | Diabetes Control and Complications Trial
DEXA | dual-energy X-ray absorptiometry
DMARD | disease-modifying antirheumatoid drug
DNase | deoxyribonuclease
DVLA | Driver and Vehicle Licensing Agency
EBUS | endobronchial ultrasound
EBV | Epstein– Barr virus
ECG | electrocardiogram
EDM | end-diastolic murmur
EMA | endomysial antibody
EMG | electromyogram
ENT | ear, nose and throat
ESR | erythrocyte sedimentation rate
ESRD | end-stage renal disease
FBC | full blood count
FEV1 | forced expiratory volume in first second
FiO2 | fraction of inspired oxygen
FSH | follicle-stimulating hormone
FVC | forced vital capacity
GBM | glomerular basement membrane
GBS | Guillain– Barré syndrome
GGT | gamma-glutamyl transpeptidase
GH | growth hormone
GI | gastrointestinal
GIM | general internal medicine
GNRH | gonadotrophin-releasing hormone
GP | general practitioner
HCM | hypertrophic cardiomyopathy
HIV | human immunodeficiency virus
HLA | human leucocyte antigen
HPOA | hypertrophic pulmonary osteoarthropathy
IBD | inflammatory bowel disease
IBS | irritable bowel syndrome
ICD | implantable cardioverter defibrillator
ICU | intensive care unit
IE | infective endocarditis
IGF-1 | insulin-like growth factor 1
IIH | idiopathic intracranial hypertension
IL | interleukin
IMCA | independent mental capacity advocate
INO | internuclear ophthalmoplegia
INR | international normalised ratio
ITP | idiopathic thrombocytopenic purpura
IV | intravenous
IVDU | intravenous drug user
JAK2 | Janus kinase 2
JVP | jugular venous pressure
KCO | carbon monoxide transfer coefficient
LA | left atrium
LABA | long-acting beta-agonist
LAMA | long-acting muscarinic antagonist
LBBB | left bundle branch block
LDH | lactate dehydrogenase
LEMS | Lambert– Eaton myaesthenic syndrome
LFT | liver function test
LH | luteinising hormone
LKM | liver kidney microsomal
LMN | lower motor neurone
LMWH | low-molecular-weight heparin
LP | lumbar puncture
LTOT | long-term oxygen therapy
LUQ | left upper quadrant
LV | left ventricle
LVF | left ventricular failure
LVH | left ventricular hypertrophy
LVOTO | left ventricular outflow tract obstruction
MAOB | monoamine oxidase B
MCPJ | metacarpophalangeal joint
MCV | mean cell volume
MDT | multidisciplinary team
MHC | major histocompatibility complex
MMF | mycophenolate mofetil
MND | motor neurone disease
MR | mitral regurgitation
MRC | Medical Research Council
MRI | magnetic resonance imaging
MS | multiple sclerosis
MVP | mitral valve prolapse
NCS | nerve conduction study
NICE | National Institute for Health and Care Excellence
NIH | National Institutes of Health
NIPPV | noninvasive positive-pressure ventilation
NOAC | novel oral anticoagulant
NSAID | nonsteroidal anti-inflammatory drug
NSCLC | non-small-cell lung cancer
NSTEMI | non-ST elevation myocardial infarction
NYHA | New York Heart Association
OCP | oral contraceptive pill
OD | once daily
OGD | oesophagogastroduodenoscopy
ON | once nightly
OSA | obstructive sleep apnoea
OT | occupational therapist
P2 | pulmonary component of heart sound 2
PACES | Practical Assessment of Clinical Examination Skills
PBC | primary biliary cirrhosis
PD | peritoneal dialysis
PDA | patent ductus arteriosus
PEG | percutaneous endoscopic gastrostomy
PET | positron emission tomography
PFT | pulmonary function test
PLS | primary lateral sclerosis
PMA | progressive muscular atrophy
PPI | proton pump inhibitor
PPM | permanent pacemaker
PR | pulmonary regurgitation
PRN | as needed (pro re nata)
PRV | polycythaemia rubra vera
PTH | parathyroid hormone
PUVA | psoralen and ultraviolet A
QOL | quality of life
RA | rheumatoid arthritis
RF | rheumatoid factor
RIF | right iliac fossa
RRT | renal replacement therapy
S1 | heart sound 1
S2 | heart sound 2
S3 | heart sound 3
S4 | heart sound 4
SA-AG | serum ascites-albumin gradient
SAA | serum amyloid A
SABA | short-acting beta-agonist
SAH | subarachnoid haemorrhage
SALT | speech and language therapist
SAP | serum amyloid P
SBP | spontaneous bacterial peritonitis
SCC | squamous cell carcinoma
SCD | sudden cardiac death
SCL-70 | antitopoisomerase antibodies
SCLC | small-cell lung cancer
SIADH | syndrome of inappropriate antidiuretic hormone secretion
SIGN | Scottish Intercollegiate Guidelines Network
SLE | systemic lupus erythematosus
SOL | space-occupying lesion
SSc | systemic sclerosis
SVCO | superior vena caval obstruction
SVD | structural valve deterioration
TAVI | transcutaneous aortic valve implantation
TB | tuberculosis
TBLB | transbronchial lung biopsy
TFT | thyroid function test
TIA | transient ischaemic attack
TLC | total lung capacity
TLCO | transfer factor of the lung for carbon monoxide
TNF | tumour necrosis factor
TOE | transoesophageal echocardiogram
TPO | thyroid peroxidase
TR | tricuspid regurgitation
TRH | thyroid-releasing hormone
TSH | thyroid-stimulating hormone
tTG | tissue transglutaminase
U1-RNP U1 | ribonucleoprotein
U&Es | urea and electrolytes
UC | ulcerative colitis
UDCA | ursodeoxycholic acid
UIP | usual interstitial pneumonia
UKPDS | United Kingdom Prospective Diabetes Study
UMN | upper motor neurone
US | ultrasound
USS | ultrasound scan
VATS | video-assisted thoracoscopic surgery
VF | ventricular fibrillation
VP | ventriculoperitoneal
VSD | ventricular septal defect
VT | ventricular tachycardia
VTE | venous thromboembolism
WHO | World Health Organization
The PACES Examination
The Practical Assessment of Clinical Examination Skills (PACES) consists of five 20-minute stations that make up a carousel. The examination lasts for a total of 125 minutes, which includes a 5-minute break between each of the stations.
Stations 1 (respiratory/abdominal) and 3 (neurology/cardiovascular) are clinical scenarios lasting 10 minutes each. A maximum of 6 minutes is allowed for examination, followed by 4 minutes for discussion. Candidates will be given written instructions prior to examining the patient.
Stations 2 (history-taking skills) and 4 (communication skills and ethics) last 20 minutes each. In these stations, 14 minutes will be allowed for history taking or communicating with the patient. One minute for reflection is provided to consolidate your thoughts prior to a 5-minute discussion with the examiners.
Station 5 (integrated clinical assessment) involves two brief clinical scenarios of 10 minutes each. Eight minutes are allocated for focused history taking, examination and responding to the patient' s concerns. The remaining 2 minutes are for discussing examination findings, diagnosis and patient' s concerns with the examiners. Candidates can be assessed on a wide range of clinical problems.
There will be two examiners present at each station. Candidates will be assessed in seven key areas. They are not all assessed at every station. The key areas include
1.Physical examination
2.Identifying physical signs
3.Clinical communication
4.Differential diagnosis
5.Clinical judgement
6.Managing patient concerns
7.Managing patient welfare
Examiners will mark candidates as satisfactory, borderline or unsatisfactory in each area. It is necessary to reach a minimum standard in each of these areas, as well as meet the overall pass mark.
REFERENCE
Membership of the Royal Colleges of Physicians of the United Kingdom. (n.d.) PACES examination format. Available from https://www.mrcpuk.org/mrcpuk-examinations/paces/paces-examination-format. Accessed 25 July 2016.
Top Tips for PACES Success: An Examiner Speaks
I have been an examiner for the Practical Assessment of Clinical Examination Skills (PACES) since dinosaurs walked the earth (or at least since the exam started). The following views are entirely my own and based on my observations of hundreds of candidates I have seen during that time.
• **The cases are usually straightforward with clear-cut physical signs.**
Centres are actively discouraged from selecting rare or esoteric conditions. Patients with physical signs sometimes come to the exam on an annual basis, and occasionally their physical signs may become less obvious over time. If this is the case, an allowance will be made during ' calibration' ; a process prior to the exam where the two examiners at that station examine the patient independently, compare findings and make an assessment of the abnormalities a candidate might be reasonably expected to find.
• **I am often asked, ' What is the standard required to pass PACES?'**
There are of course strict marking criteria for each station, but the majority of examiners are ' jobbing' physicians; we don' t expect super specialist levels of knowledge. For most of the time, we are examining cases outside our own speciality. For me, the overall benchmark is, ' Would I feel comfortable with this doctor running the general medical take in my hospital?'
• **Practise, practise, practise!**
I am afraid it' s all too easy to spot a candidate who isn' t wholly comfortable doing a physical examination. Get a senior colleague to critique your exam technique: it is important that you have a structured approach that you could do almost automatically.
• **You will not pass unless you can identify physical signs.**
Part of this is pattern recognition: once you have felt polycystic kidneys, you are likely to correctly identify them in the future. The longer and more diverse your clinical experience is, the more likely you will be to detect physical signs because you have seen them before. When you think about it, the types of physical sign you are likely to come across in each station are fairly limited, so when preparing for PACES, sit down and make a list of your deficiencies and make sure you have seen all the common abnormalities at least once, even if that means visiting other hospitals. I am not generally a fan of commercial courses, but this may be an area where they can help if your exposure to abnormal physical signs is limited. The other common problem is candidates who ' look but do not see' ; in other words, they go through the motions of the examination almost by rote but do not find the abnormalities they are ostensibly looking for.
• **Not every patient in the exam will have something to find.**
So don' t make things up! Abdominal and chest cases particularly may only have a scar to find. In general, if you can' t find anything wrong, then you are probably correct.
• **Station 5 is problematic.**
Station 5 carries a lot of marks, and the time available is limited. You have to be able to talk and examine at the same time; the good news is that the tasks you are given and any examination findings are usually pretty straightforward. Don' t try to examine too many systems at once; read the introductory statement and allow that to focus your history/discussion and examination. The examiners realise you will not be able to do everything, so concentrate on what you think is important.
• **Timing is everything.**
When practising, make sure that you take 6 minutes to complete your abdominal, respiratory, cardiology and focused neurological examinations. Finish too early and you are likely to have missed something and exposed yourself to extra questioning. Too late and you will have failed to pick up the appropriate signs. In general, in the talking stations you should have plenty of time to complete the required task, so go through the history or the communication issue in a steady, structured way, clarifying things where necessary and avoiding repetition. Try not to finish early; if you do, you are very likely to have missed something. When you are sitting outside the station in the 5 minutes after reading the clinical information, list your structured approach to problem, questions to cover in the history-taking station and important issues that need to be addressed (and questions to be answered) in communications skills.
• **Don' t waste time.**
Especially on unnecessary ' peripheral' examinations; spending 3 minutes examining for a collapsing pulse will not give you time to palpate the precordium and listen to the heart properly. If you discover a displaced apex and a diastolic murmur, you can go back and do it later.
• **Honesty is the best policy.**
Don' t make up physical signs because you think they should be there or you think that' s what the examiner wants. In the talking stations, if you don' t know the answer to the surrogate' s question, say so, but explain why and how you will obtain the information.
• **Don' t beat about the bush.**
Candidates who are not confident about things will sometimes be deliberately vague when describing their findings; this is unlikely to help. The examiners will press you to clarify things, so it' s best to come clean early on, describing what you have found, what it could be and why.
• **Always answer the question.**
In the talking station, the surrogate will have been primed with questions or concerns. A good candidate may answer most of these during the consultation. But if asked a direct question by the surrogate, you should usually try to answer it there and then. Occasionally, you may feel you have not gathered enough information to do so, in which case you can say so and return to the question later, but do not forget that failure to address the ' patient' concerns is a common cause of poor marks in Stations 2 and 4.
• **Create an impression.**
Wear comfortable but smart clothing; it' s not a fashion show, but you should look professional – so no jeans or short skirts. Don' t weigh yourself down with equipment; all you need is a stethoscope. Everything else will be provided, and bringing more stuff just means a greater chance of dropping it or leaving it behind at one of the stations.
• **Read the introductory statement carefully.**
Particularly in the neurological station, it should tell you which area to focus on; it is impossible to do a complete neurological examination in 6 minutes. In the other stations, it is likely to guide questioning, particularly when assembling the list of possible causes of the physical findings.
• **Be kind to the patient and try to look as though you are enjoying yourself.**
Not an easy thing to do, but handling the patient with courtesy and good humour will set you in good stead with the examiners and show that you are an experienced clinician.
• **Finally, the examiners genuinely want you to pass.**
You may not believe it or feel like it at the time, but it' s true. We have been in your position ourselves at some stage and realise how nerve racking it is. You will need to meet the required standard, but we will give you every opportunity to demonstrate that you can, given the confines of the examination structure. During questioning, the examiner may try to lead you towards a diagnosis; this is not a plot to try to trick you into saying the wrong thing but is usually a genuine attempt to make you clarify your views on a clinical problem.
**Anonymous reviewer**
Station 1: Respiratory
Hints for the Respiratory Station
Bronchiectasis
Chronic Obstructive Pulmonary Disease
Consolidation
Cystic Fibrosis
Fibrotic Lung Disease
Lung Cancer
Old Tuberculosis
Pleural Effusion
Patient with Previous Lung Surgery
Respiratory Station Summary
HINTS FOR THE RESPIRATORY STATION
•Observe the patient from the end of the bed and comment on chest expansion and respiratory rate.
•Look for clues around the bedside, e.g. inhalers, spacer device, peak flow metre, nebuliser.
•Be aware of the different types of inhalers and the colour design and contents of each inhaler type.
•Always examine the contents of the sputum pot if present.
•If the patient is on oxygen, comment on delivery method and the percentage inspired if appropriate.
•Examine the patient starting from the back, as findings are more likely to be picked up from the back (important, if time is running out).
•Try to look for underlying causes of the pathology identified.
•Look for signs of cor pulmonale where appropriate.
•Complete the examination by telling the examiner you would like to check oxygen saturations, peak flow and spirometry results where appropriate.
•Use ward rounds and clinics as opportunities to practise your clinical examination skills and with presenting cases.
BRONCHIECTASIS
Please examine this patient who has a cough productive of thick sputum.
**FINDINGS**
• _General:_ Age (may help with underlying diagnosis), cachexia, dyspnoea, inhalers, sputum pot (check for thick sputum ± haemoptysis), nebulisers
• _Peripheral:_ Clubbing, signs of cor pulmonale, long line, tunnelled central venous catheter
• _Chest:_ Coarse inspiratory crackles (which may clear with coughing), wheeze, situs inversus, scars from previous operations
**PRESENTATION**
On examination, this patient has evidence of bilateral coarse inspiratory crackles to the midzones and a wheeze. She also has evidence of finger clubbing, and the sputum pot by her bedside has thick dark-green-coloured sputum. The diagnosis in this patient is bronchiectasis. The patient has no evidence of cachexia, situs inversus or previous operations. In this patient, the most likely cause may be as a result of previous infection, and I would like to investigate this further.
•Comment on the patient' s productive cough.
•Look for a sputum pot and be sure to look at the contents.
•Look for an obvious underlying cause, including: possible signs of cystic fibrosis (CF) and chronic obstructive pulmonary disease (COPD).
•If there is no obvious underlying cause, mention that the top three causes are postinfective, COPD and Cystic Fibrosis.
**INVESTIGATIONS**
•Diagnosis
•Chest X-ray (CXR): May be normal, dilated bronchi with thickened walls (' tramline' shadowing), ring shadowing.
•High-resolution CT (HRCT): Bronchial wall dilatation with possible thickening, airway size is greater than accompanying artery size (' signet ring' appearance).
•Sputum: Send for culture and sensitivities ( _Pseudomonas aeruginosa_ is important), atypical screen, acid-fast bacilli (AFB) and fungal cultures.
•Spirometry: May show an obstructive picture (usually forced expiratory volume in first second [FEV1]/forced vital capacity [FVC] ratio < 0.7).
•Bloods: Check serum immunoglobulin levels (immunoglobulin [Ig] A, IgG, IgM) to rule out immunodeficiency as a cause, _Aspergillus_ serology (total IgE, IgE to _Aspergillus_ and _Aspergillus_ precipitins).
•Investigations for the underlying cause of bronchiectasis
•Cystic fibrosis: Two measurements of sweat chloride and cystic fibrosis transmembrane conductance regulator (CFTR) genetic mutation analysis.
•Immunodeficiency: Immunoglobulin screen, functional antibody levels.
•Allergic bronchopulmonary aspergillosis (ABPA) screen.
•Ciliary dysfunction (e.g. primary ciliary dyskinesia): Saccharin test or cilial electron microscopy to assess ciliary function.
•Mechanical obstruction: Computerised tomography (CT) of the chest will locate any obstructive lesions, e.g. foreign body, lymph node compression or tumour.
**MANAGEMENT**
•Specialist physiotherapy with a daily routine for patients
•Antibiotics
•Treatment of acute exacerbations often requires intravenous antibiotic therapy with antipseudomonal agents (piperacillin, ceftazidime, carbapenems, aminoglycosides).
•Oral ciprofloxacin may be used.
•Long-term antibiotics may include nebulised colistin and oral macrolides.
•Bronchodilators
•In those with airflow limitation
•Mucolytics
•Carbocisteine
•Surgery
•Bronchiectasis is rarely sufficiently localised to be amenable to surgery but may be an option in a very small minority of cases.
**QUESTIONS**
1.What is bronchiectasis?
•Abnormal and permanently dilated airways with bronchial wall thickening.
•This is manifest as a cough with the production of thick sputum.
2.What are the causes of bronchiectasis?
•Postinfective bronchial damage: Severe bacterial and viral pneumonias, including measles and pertussis, tuberculosis (TB) and nontuberculous mycobacterial infections
•Mucociliary clearance defects: Cystic fibrosis, primary ciliary dyskinesia, Kartagener' s syndrome, Young' s syndrome
•Immunodeficiency: Primary (immunoglobulin deficiency) and secondary (human immunodeficiency virus [HIV] infection)
•Mechanical: Obstruction (tumour, foreign body)
•Immunological response: Allergic bronchopulmonary aspergillosis
3.What is the differential diagnosis of bilateral lower-zone crackles?
•Bronchiectasis: Coarse crackles heard in early– mid inspiration
•Lung fibrosis: Fine, late (end-inspiratory) crackles; look for clubbing, dry cough and cyanosis, as well as a cause, such as connective tissue disease
•Pulmonary oedema: Fine/coarse bibasal crackles; look for evidence of fluid overload
•Bilateral pneumonia: Coarse crackles; look for pyrexia and bronchial breathing
**KEY POINTS**
•On examination, coarse inspiratory crackles (that clear with coughing), along with finger clubbing, are the hallmark clinical features of bronchiectasis.
•Typical radiological signs seen on HRCT include bronchial wall thickening and the classic ' signet ring' appearance of dilated bronchi.
•Ensure that you are familiar with the common causes of bronchiectasis and can list them in an ordered fashion.
REFERENCE
Pasteur MC, Bilton D, Hill AT. (2010) British Thoracic Society guideline for non-CF bronchiectasis. _Thorax_ 65: 577.
CHRONIC OBSTRUCTIVE PULMONARY DISEASE
Please examine this patient with breathlessness.
**FINDINGS**
• _General:_ Dyspnoea, pursed lip breathing, prolonged expiratory time, use of accessory muscles, wheeze, presence of inhalers and oxygen
• _Peripheral:_ Cyanosis, tar staining on finger nails, bronchodilator (fine) tremor, flap, signs of cor pulmonale
• _Chest:_ Hyperexpanded chest, hyperresonance and reduced breath sounds (over bullae), wheeze, Hoover' s sign (inward movement of the lower rib cage during inspiration associated with COPD)
**PRESENTATION**
This gentleman is dyspnoeic at rest with pursed lip breathing and is on oxygen therapy at 2 L/min. He has nicotine-stained fingernails. There is a widespread wheeze audible throughout both lung fields, and the patient has evidence of cor pulmonale. I note the presence of inhalers at the bedside, including salbutamol and a combination inhaler. The diagnosis in this patient is COPD.
•Comment on external clues, including inhalers, oxygen, nebulisers and a sputum pot.
•Comment on any features of respiratory distress and ensure the respiratory rate is counted.
•Look for features of cor pulmonale (raised jugular venous pressure [JVP], loud pulmonary component of loud second heart sound [P2], peripheral oedema), polycythaemia (plethora), infection (consolidation) and Cushing' s syndrome (corticosteroid use).
**INVESTIGATIONS**
•Diagnosis
•Spirometry with bronchodilator response (to differentiate from asthma). FEV1/ FVC ratio < 0.7 and not fully reversible
•Arterial blood gas (ABG) (type 1 or 2 respiratory failure is possible)
•CXR (hyperinflation, flat hemidiaphragms, bullae, large prominent pulmonary arteries)
•Complications
•Echocardiogram (pulmonary hypertension), CXR (infection/pneumothorax/bullae)
**MANAGEMENT**
•Acute
•Oxygen therapy (with caution and monitoring of arterial blood gases), nebulised bronchodilators, corticosteroids, antibiotics for infection, theophylline, physiotherapy, consideration of noninvasive positive-pressure ventilation (NIPPV) and intubation for more severe cases
•Chronic: Depends on the severity of airway obstruction
•Mild
•PRN inhalers: Short-acting bronchodilator, e.g. salbutamol
•Moderate
•If still breathless and FEV1 > 50%, the patient can have a long-acting beta-agonist (LABA), e.g. salmeterol, or a long-acting muscarinic antagonist (LAMA), e.g. tiotropium.
•If still symptomatic and FEV1 < 50%, the patient can have a combination inhaler (LABA/steroid) or a LAMA.
•For patients who are still symptomatic, regardless of the FEV1, they can be prescribed a combination inhaler and a LAMA. There are also newer inhalers which are a combination of a LABA/LAMA.
•Severe
•Consider home nebulisers, theophyllines and anti-mucolytics.
•Extras
•Smoking cessation advice is key and should be discussed with all patients, as well as offering nicotine replacement therapy and drugs such as varenicline and bupropion to help stop. Smoking cessation therapy is monitored using carbon monoxide (CO) readings (< 10 ppm at 4 weeks).
•Other considerations include nutritional management, pulmonary rehabilitation, vaccinations (pneumococcal and influenza), long-term oxygen therapy (LTOT), surgery, social support, psychological support and palliative care input.
**QUESTIONS**
1.What is cor pulmonale? What is its significance?
•Right-sided cardiac dysfunction secondary to pulmonary hypertension.
The pulmonary hypertension must be of a respiratory cause (chronic lung disease, pulmonary vasculature disorders, neuromuscular disease affecting the respiratory system).
•Untreated cor pulmonale causes right-sided heart failure and death.
2.What are the indications for LTOT?
•LTOT describes oxygen given for > 16 hours/day, with the aim of achieving a PaO2 > 8 kPa.
•It is indicated for those patients with a PaO2 < 7.3 kPa on two consecutive readings at least 3 weeks apart (in a stable patient), or for a PaO2 7.3– 8 kPa in a patient with cor pulmonale.
•All patients with an FEV1 < 30% predicted, signs of right-sided heart failure and oxygen saturations of < 92% should be considered for LTOT.
•Smoking is not a contraindication to LTOT; however, patients need to be advised of the risks associated.
3.How would you classify the severity of COPD?
•Use the National Institute for Health and Care Excellence (NICE) guidelines on classifying the severity of COPD based on the presence of symptoms and the percentage predicted of their FEV1:
•Mild: FEV1 > 80%
•Moderate: FEV1 50%– 80%
•Severe: FEV1 30%– 50%
•Very severe: FEV1 < 30%
4.What surgical interventions can be offered to patients with COPD?
•Bullectomy: In symptomatic patients who have a large bulla and FEV1 < 50% predicted.
•Lung volume reduction surgery (LVRS): This can be considered in patients who have upper-zone dominant emphysema, FEV1 > 20% predicted, PaCO2 < 7.3 kPa and transfer factor of the lung for carbon monoxide (TLCO) > 20%.
•Lung transplantation.
**KEY POINTS**
•Be able to identify what medications inhalers contain, from their colour/design.
•Ensure that you know the criteria for LTOT.
REFERENCES
Hardinge M, et al. (2015) BTS guidelines for home oxygen use in adults. Thorax 70: i1– 43.
National Institute for Health and Care Excellence. (2010) Guidance CG101. Chronic obstructive pulmonary disease in over 16s: Diagnosis and management. Available from https://www.nice.org.uk/guidance/cg101. Accessed 17 August 2016.
Troosters T, et al. (2005) Pulmonary rehabilitation in chronic obstructive pulmonary disease. _Am J Respir Crit Care Med_ 172: 19– 38.
CONSOLIDATION
Please examine this patient' s respiratory system.
**FINDINGS**
• _General:_ Oxygen, nebulisers, sputum pot, haemoptysis, herpes labialis, cough
• _Peripheral:_ Tachypnoea, tachycardia, pyrexia, respiratory distress
• _Chest:_ Decreased chest expansion, dullness to percussion over affected lobe, bronchial breathing, crackles, increased vocal resonance
**PRESENTATION**
This patient has evidence of consolidation at his right base. He is receiving oxygen therapy and is tachypnoeic with a respiratory rate of 28 breaths/minute. There is a sputum pot containing thick green sputum. There is dullness to percussion at the right lung base, and on auscultation, there is evidence of bronchial breathing and coarse crackles at the right base.
•Remember that in the exam situation if the information about the case is handwritten, it is likely that a patient has withdrawn from the exam at short notice and a last-minute replacement may have been recruited from the inpatient wards.
•Such a patient is likely to be more unwell than a stable patient; therefore, comment on any features of respiratory distress, note any infusions or nebulisers and check the FiO2 being delivered.
•Patients at highest risk of pneumonia include immunocompromised, COPD, elderly and alcoholic patients.
**INVESTIGATIONS**
•Diagnosis
•CXR, ABG, sputum culture, routine bloods, blood cultures, atypical pneumonia screen (urine for legionella and pneumococcal antigens, mycoplasma serology) if indicated
**MANAGEMENT**
•Use the CURB-65 score to assess the severity of the pneumonia.
•Confusion (abbreviated mental test score [AMTS] ≤ 8)
•Urea > 7 mmol/L
•Respiratory rate ≥ 30 breaths/minute
•Blood pressure (systolic < 90 mmHg or diastolic < 60 mmHg)
•Age > 65 years
•A score of 0 or 1: Treat as an outpatient.
•A score of 2: Possible short stay in hospital.
•A score of 3– 5: Requires hospitalisation and may require critical care intervention.
•Treatment is with antimicrobials and oxygen therapy where required. In more severe cases, critical care intervention may be required.
•Antimicrobial therapy is guided by local trust guidelines, but generally use amoxicillin ± a macrolide (to cover atypical organisms in severe cases).
•Patients should be followed up after a 6-week interval with a repeat CXR to ensure that the consolidation has resolved and there is nothing sinister underlying it.
•Complications of pneumonia can include pleural effusion, empyema or a lung abscess.
**QUESTIONS**
1.What are the common organisms causing community-acquired pneumonia?
•Common: _Streptococcus pneumoniae_ , _Haemophilus influenzae_ , _Staphylococcus aureus_
•Atypical: _Mycoplasma pneumoniae_ , _Legionella pneumophila_ , _Chlamydophila pneumoniae_ , _Chlamydophila psittaci_ , _Coxiella burnetii_
•Viruses: Influenza, cytomegalovirus (CMV) and varicella-zoster
2.List possible complications of pneumonia.
•Parapneumonic effusion, empyema, cavitation, lung abscess, septic shock, respiratory failure/acute respiratory distress syndrome (ARDS), hepatitis, haemolytic anaemia, erythema multiforme
3.What is the difference between an empyema and a complicated parapneumonic effusion?
•An empyema is pus in the pleural cavity with a pH < 7.2.
•A complicated parapneumonic effusion has a pH < 7.2 but is clear.
•A parapneumonic effusion is clear with a pH > 7.2.
4.How would consolidation be differentiated from an effusion on clinical examination?
•Tactile vocal fremitus: Sound transmission is increased through tissue (consolidation) and decreased through fluid (pleural effusion).
•Whispering pectoriloquy: Is indicative of consolidation when whispered sounds are heard clearly through affected lung tissue.
5.Other than guiding the clinical management of a patient, what other information does the CURB-65 score give?
•The CURB-65 score also indicates the mortality associated with the severity of the pneumonia. A higher score is associated with a higher mortality rate:
•Score 0– 1 = < 5% mortality rate
•Score 2 = 9% mortality rate
•Score 3– 5 = 15%– 40% mortality rate
**KEY POINTS**
•Look closely at the information presented to the candidate, i.e. is it handwritten.
•Patients from an inpatient ward may be more unwell; comment on any respiratory distress and any infusions.
•Be aware of complications associated with pneumonia.
REFERENCE
National Institute for Health and Care Excellence. (2014) Guidance CG191. Pneumonia in adults: Diagnosis and management. Available from https://www.nice.org.uk/guidance/cg191?unlid=2520898732016221192356. Accessed 17 August 2016.
CYSTIC FIBROSIS
This patient has presented with repeated chest infections. Please examine their respiratory system.
**FINDINGS**
• _General:_ Young patient, short stature, cachexia, pallor, dyspnoea, tunnelled central venous catheter (port-a-cath) or long line, inhalers, sputum pot, pinpricks from blood glucose measurement, nebulisers, Creon tablets, percutaneous endoscopic gastrostomy (PEG) tube
• _Peripheral:_ Finger clubbing, signs of cor pulmonale (in patients with severe lung disease)
• _Chest:_ Coarse inspiratory crackles (which may clear with coughing), wheeze
**PRESENTATION**
This young patient has cystic fibrosis. He appears cachectic and also has a port-a-cath _in situ_ . The patient has finger clubbing and also is likely to have CF-related diabetes due to the pinpricks on his fingertips from blood glucose monitoring. On auscultation of his chest, there is evidence of coarse crackles in both lower zones in keeping with bronchiectasis.
•Comment on the patient' s general appearance and look for any extrapulmonary manifestations. Some patients may have had a liver or renal transplant as well.
**INVESTIGATIONS**
•Diagnosis
•Guthrie test.
•Further genetic testing: Δ F508 is the most common mutation.
•Sweat test: Sweat sodium and chloride.
•Complications
•Chest
•Chest X-ray: Bronchial wall thickening, mucous plugging, pneumothorax
•HRCT: Bronchial wall thickening, mucous plugging, bronchiectasis (signet ring sign)
•Sputum culture: Looking for common CF pathogens, atypical infections, AFB and fungal infections
•Spirometry: Obstructive pattern
•CF-related diabetes: Oral glucose tolerance test, Hba1c, blood sugar series and continuous glucose monitoring system
•Pancreatic insufficiency: Faecal elastase levels
•CF-related liver disease: Liver function tests (LFTs), coagulation screen (assess synthetic function of liver), ultrasound scan (USS) of liver.
•Ear, nose and throat (ENT) complications (nasal polyps, sinusitis): CT scan of sinuses
•Osteoporosis/osteopenia: Dual-energy X-ray absorptiometry (DEXA) scan, parathyroid hormone (PTH), calcium and vitamin D
**MANAGEMENT**
•Multidisciplinary team (MDT) approach
•Respiratory physician (CF specialist), CF specialist nurse, physiotherapist, dietitian, clinical psychologist, GP, other medical teams (including endocrine and gastroenterology)
•Specialist physiotherapy
•Antibiotics (preventative nebulised antibiotics, as well as rescue courses during an acute exacerbation)
•See bronchiectasis section.
•Mucolytics
•DNase (nebulised), hypertonic saline
•Nutrition
•Special diet: High calorie/high fat, vitamins and Creon.
•Enteral feeding may be needed.
•Management of extrapulmonary complications, including CF-related diabetes and liver disease
•Psychological support
**QUESTIONS**
1.How common is cystic fibrosis in the UK population?
•Cystic fibrosis occurs in approximately 1 in 2500 live births.
•Cystic fibrosis is an autosomal recessive condition.
•The chance of being a carrier in the United Kingdom is 1 in 25.
2.What organisms are commonly found in the sputum of patients with cystic fibrosis, and which are most important for prognosis?
• _Haemophilus influenzae_ , _Staphylococcus aureus_ (commonly found in CF), _Moraxella catarrhalis_ , _Streptococcus pneumoniae_ , atypical mycobacteria, _Aspergillus fumigatus_ .
• _Burkholderia cepacia_ (in particular cenocepacia), _Pseudomonas aeruginosa_ and _Mycobacterium abcessus_ infection are poor prognostic indicators.
3.List the extrapulmonary manifestations of cystic fibrosis.
•Pancreatic: Malabsorption
•Endocrine: CF-related diabetes – often associated with pancreatic insufficiency
•Hepatobiliary: Gallstones, cirrhosis, portal hypertension
•Intestinal: Distal intestinal obstruction syndrome (meconium ileus equivalent) which is treated with laxatives, including gastrograffin
•Musculoskeletal: Osteoporosis, arthritis, osteopenia
•Sinusitis and nasal polyps
•Male infertility, delayed puberty
4.What are the respiratory complications caused by CF?
•Infective exacerbations
•Pneumothorax
•Haemoptysis
• _Aspergillus_ lung disease
•Respiratory failure
5.Are you aware of any new treatments available for CF?
•Genetic modulators such as ivacaftor are now available for some CF patients with particular genetic defects, e.g. G551D.
•Ivacaftor works on chromosome 7, CFTR gene, G551D defect (7% of CF patients) by making the chloride channel functional.
**KEY POINTS**
•Be able to recognise these patients, taking their age and general appearance into account.
•Recognise and comment on any extrapulmonary manifestations.
•Be aware of common respiratory tract pathogens in cystic ?brosis.
REFERENCE
Ramsey B, et al. (2011) A CFTR potentiator in patients with cystic fibrosis and the G551D mutation. _N Engl J Med_ 365: 1663– 72.
FIBROTIC LUNG DISEASE
Please examine this patient who is breathless.
**FINDINGS**
• _General:_ Dyspnoea, oxygen, cachexia, tachypnoea
• _Peripheral:_ Cyanosis (peripheral and central), clubbing, signs of cor pulmonale
• _Chest:_ Scar (from biopsy), reduced chest expansion, fine end-inspiratory crackles
**PRESENTATION**
This patient has pulmonary fibrosis as evidenced by finger clubbing and fine inspiratory bibasal crackles. The likely underlying cause is systemic sclerosis, as the patient also has evidence of sclerodactyly, telangiectasiae and microstomia.
•This is a standard case which may not be included on its own in the respiratory station and may also appear in Station 5, so look for a possible underlying cause:
•Features of rheumatoid arthritis: Swan neck and boutonniè re deformities, Z-thumb, ulnar deviation at the wrist, nodules at the elbow
•Ankylosing spondylitis: ' Question mark' posture
•Pacemaker and amiodarone facies
•Radiation burns and tattoos: Radiation therapy
•Features of systemic sclerosis: Sclerodactyly, telangiectasiae, beaked nose, furrowing of the mouth
•Also look for features of Cushing' s syndrome (corticosteroid use)
**INVESTIGATIONS**
•Diagnosis
•ABG: Type 1 respiratory failure
•Pulmonary function tests – restrictive picture, reduced transfer factor
•CXR: Peripheral and basal reticular shadowing
•HRCT: Ground-glass appearance
•Cause
•Full history and examination, autoimmune screen, serum angiotensin-converting enzyme (ACE), bronchoalveolar lavage, transbronchial lung biopsy (TBLB) or surgical lung biopsy (if diagnostic doubt)
**MANAGEMENT**
•Discussion at interstitial lung disease MDT
•Smoking cessation
•Pulmonary rehabilitation
•Gastric protection (e.g. proton pump inhibitors) for any patients with symptoms of reflux disease
•LTOT
•Nutritional assessment
•Psychological support
•Palliative care
**QUESTIONS**
1.Please describe the causes of lung fibrosis and divide them into upper- and lower-zone aetiologies.
•Upper zone
•Berylliosis, radiation, extrinsic allergic alveolitis, ankylosing spondylitis, sarcoidosis, TB (mnemonic: ' breast' )
•Lower zone
•Rheumatoid arthritis and other connective tissue diseases, idiopathic, drugs (methotrexate and amiodarone), asbestosis
2.What new treatments are available for the treatment of pulmonary fibrosis (usual interstitial pneumonia [UIP] pattern)?
•Pirfenidone and nintedanib are both antifibrotics which have recently been introduced to slow down the progression of mild– moderate disease. Any patient being considered for treatment needs to be discussed at an interstitial lung disease MDT and, if they fit the required criteria, can be prescribed treatment from a specialist centre.
3.What are the pulmonary manifestations of rheumatoid arthritis?
•Lung fibrosis (which may also be secondary to methotrexate treatment)
•Pleural effusions
•Intrapulmonary nodules (including Caplan' s syndrome)
•Obliterative bronchiolitis
**KEY POINTS**
•Fine end-inspiratory crackles are suggestive of fibrotic lung disease; a key differential is pulmonary oedema.
•Know the causes of pulmonary fibrosis and the lung zones they affect.
•Ensure that any management plan for a patient with pulmonary fibrosis is based around a MDT approach.
•Comment on any evidence of cor pulmonale which signifies advanced disease.
REFERENCE
National Institute for Health and Care Excellence. (2013) Guidance CG163. Idiopathic pulmonary fibrosis in adults: Diagnosis and management. Available from https://www.nice.org.uk/guidance/CG163. Accessed 8 August 2016.
LUNG CANCER
Please examine this patient who has presented with a chronic cough.
**FINDINGS**
• _General:_ Cachexia (may or may not be present), lymphadenopathy, hoarse voice
• _Peripheral:_ Finger clubbing, nicotine staining
• _Chest:_ Scars from previous lobectomy/pneumonectomy, radiation tattoos/burns, reduced chest expansion, tracheal deviation, reduced breath sounds, dullness to percussion, possible pleural effusion
**PRESENTATION**
This patient appears cachectic and has finger clubbing and a hoarse voice. There are palpable cervical lymph nodes. The patient has a small blue tattoo at the front of his chest which is indicative of a radiotherapy tattoo and also has a lateral thoracotomy scar. The diagnosis in this patient would be previous lung cancer (possible recurrence with chronic cough), with the patient having had surgery and radiotherapy in the past for treatment.
Look for the following features:
•Superior vena caval obstruction (SVCO): Raised JVP, oedematous face, distended veins over chest wall and neck, respiratory distress
•Metastases
•Hepatomegaly, bony tenderness, skin lesions, cervical lymph nodes
•Paraneoplastic disorders (see below)
•Pancoast' s syndrome (see below)
**INVESTIGATIONS**
•Diagnosis
•CXR: Mass, pleural effusion, bulky hilum
•CT scan: Look for evidence of masses, lymphadenopathy and metastases
•Positron emission tomography (PET) scan: For evaluation and staging of lung cancers (especially pulmonary nodules, lymph nodes and distant metastases)
•Pleural fluid cytology
•Bronchoscopy and biopsy of mass lesions
•Endobronchial ultrasound (EBUS)– guided biopsy: to assess mediastinal lymph nodes
•CT-guided lung biopsy: For peripheral lung lesions
•Biopsy of peripheral lesions to assess for metastases
•Thoracoscopy: To assess an exudative pleural effusion of unknown aetiology
**MANAGEMENT**
•Referral to lung MDT team.
•Dependent on staging and histology; surgical resection/chemotherapy/ radiotherapy/palliation.
•Surgical resection (pneumonectomy or lobectomy) is suitable for patients with adequate lung function and no medical contraindications.
•Refer to patient' s functional status using the World Health Organization (WHO) performance status.
**QUESTIONS**
1.What are the four main histological types of lung cancer and how common are each of these?
•Lung cancer classification can be divided into small-cell lung cancer (SCLC) and non-small-cell lung cancer (NSCLC).
•SCLC comprises 15% of all cases, whereas NSCLC (adenocarcinoma [35%– 40%], squamous cell carcinoma [25%– 30%] and large-cell carcinoma [10%– 15%]) accounts for the remainder of lung malignancies.
•Small-cell tumours have the worst prognosis, as they have a rapid growth rate and metastasise early.
2.Which paraneoplastic syndromes are associated with lung cancer?
• _Small-cell lung cancer_ is the most common form of cancer associated with paraneoplastic syndromes, including
•Ectopic hormone secretion: Adrenocorticotrophic hormone (ACTH) (Cushing' s syndrome) and antidiuretic hormone (ADH) (causing syndrome of inappropriate antidiuretic hormone secretion [SIADH] – low sodium)
•Lambert– Eaton myasthenic syndrome (LEMS)
• _Squamous cell carcinoma_ : Parathyroid hormone– related peptide release, resulting in hypercalcaemia (note that hypercalcaemia is more frequently a result of bony metastases).
• _Adenocarcinoma_ : Hypertrophic pulmonary osteoarthropathy (HPOA); it results in gross finger clubbing and arthritis with radiological evidence of subperiosteal new bone formation.
3.What are the signs of Pancoast' s syndrome?
•Pancoast' s syndrome is characterised by an apical lung tumour with involvement of the brachial plexus and cervical sympathetic nerves; patients may complain of pain in the shoulder/anterior chest wall and arm weakness, and have wasting of the intrinsic muscles of the hand and an ipsilateral Horner' s syndrome (ptosis, miosis, anhidrosis and enophthalmos).
4.What are the clinical signs and symptoms of SVCO?
•Oedema of the face, neck and upper body
•Prominent neck and chest wall vessels
•Facial plethora
•Stridor
•Headache (often worse on bending forward)
•Dizziness
**KEY POINTS**
•Comment on any past treatment, including surgery and radiotherapy.
•Comment on any associated complications as detailed above.
•Be able to differentiate between a pneumonectomy and lobectomy on clinical examination.
REFERENCE
Lim E, et al. (2010) Guidelines on the radical management of patients with lung cancer. _Thorax_ 65(Suppl III): iii1– 27.
OLD TUBERCULOSIS
Please examine this gentleman' s respiratory system.
**FINDINGS**
• _General:_ May appear well or cachectic
• _Chest:_ Tracheal deviation, thoracotomy scar, rib resection, decreased breath sounds with reduced chest expansion, signs of fibrosis/bronchiectasis, evidence of respiratory failure secondary to thoracoplasty
•Look for a supraclavicular scar from a phrenic nerve crush procedure, scarring from induced pneumothoraces or a lateral thoracotomy scar
**PRESENTATION**
This gentleman appears well with no evidence of respiratory distress. He has a scar in the left supraclavicular fossa with fine inspiratory crepitations in the left upper zone. The patient also has a left-sided chest wall deformity and a lateral thoracotomy scar. The examination findings in this patient would be consistent with old TB with a phrenic nerve crush and thoracoplasty.
**INVESTIGATIONS**
•CXR
•Raised hemidiaphragm on the side of phrenic nerve crush procedure, upper-lobe fibrosis, areas of cavitation
•CT scan
•Areas of fibrosis, loss of lung volume, thickened or calcified pleura, areas of bronchiectatic change
•Spirometry
•Postthoracoplasty may show an obstructive or restrictive defect.
**MANAGEMENT**
•The patient may be completely well and not need treatment.
•They may be investigated for possible recurrence of TB (bronchoscopy for a bronchoalveolar lavage) or manifestations of old TB, such as pulmonary fibrosis.
**QUESTIONS**
1.Outline the current drug therapies available for treatment of TB, and their side effects.
•The standard treatment for TB involves a combination of the following drugs: rifampicin, isoniazid, pyrazinamide and ethambutol. Patients are initially treated with all four drugs for the initial 2 months of therapy, followed by rifampicin and isoniazid for the subsequent 4 months; treatment is complete at 6 months.
•Multi-drug-resistant TB is that which is resistant to rifampicin and isoniazid.
•Drug-induced hepatitis
•Rifampicin, isoniazid and pyrazinamide (check LFTs prior to commencement)
•Optic neuritis
•Ethambutol (visual acuity should be documented before starting treatment)
•Peripheral neuropathy
•Isoniazid (coprescribe pyridoxine)
2.Discuss past surgical treatments of TB.
•Induced pneumothoraces
•To collapse the affected lung; procedure repeated every few weeks
•Phrenic nerve crush
•To paralyse the diaphragm and cause collapse of the underlying lung
•Plombage
•Insertion of polystyrene balls into the chest cavity to cause collapse of underlying lung
•Thoracoplasty
•Ribs around the infected cavity broken and pushed inwards to collapse underlying lung
3.What tests are used in the diagnosis of TB?
•CXR: There may be evidence of an enlarged hilum, consolidation, cavitation, pleural effusion or granulomata.
•Sputum staining: Ziehl– Neelsen staining for acid-fast bacilli.
•Bronchoscopy and washings.
•Biopsies of extrapulmonary manifestations, e.g. lymph nodes.
•Whole-blood interferon or skin tests (Mantoux test).
•Always consider HIV testing in patients with suspected TB.
**KEY POINTS**
•Be aware of the past surgical treatments for TB and the associated clinical signs.
•Be aware of the current treatment recommendations for TB.
•Be aware of the main side effects of anti-TB therapy.
PLEURAL EFFUSION
Please examine this patient' s respiratory system. They have been complaining of worsening shortness of breath.
**FINDINGS**
• _General:_ Dyspnoea, oxygen and walking aids
• _Peripheral:_ Dependent on the underlying cause (see below)
• _Chest:_ Decreased chest expansion, trachea deviated away from side of effusion, stony-dull percussion note on affected side, decreased breath sounds on affected side
**PRESENTATION**
On examination, this patient has evidence of a right-sided pleural effusion. He has reduced chest expansion and a stony-dull percussion note to the right midzone with reduced breath sounds. The likely cause in this patient would be lung cancer, as he has evidence of scars that are likely from a chest drain at the right side of his chest, has finger clubbing and is cachectic.
Search for a cause for the effusion:
•Connective tissue disease: Features of rheumatoid arthritis butterfly facial rash indicative of systemic lupus erythematosus (SLE)
•Cardiac disease: Raised JVP and ankle swelling
•Lung cancer: Clubbing, wasting, Horner' s syndrome, radiation scars, tattoos, lymphadenopathy, previous drain site scars
•Liver disease: Leuconychia, palmar erythema, spider naevi, gynaecomastia, parotid swelling, ascites
•Renal disease: Arteriovenous fistulae, scars from neck lines, peritoneal dialysis catheters, renal transplant
Also look for signs of treatment:
•Scars from pleural taps and chest drains
**INVESTIGATIONS**
•CXR, pleural tap (should be US guided) sent for lactate dehydrogenase (LDH), protein, pH, amylase, glucose, cytology, microscopy and culture
**MANAGEMENT**
•Dependent on underlying cause and the treatment for that (e.g. infection, lung cancer), but can include the following options: therapeutic aspiration, chest drain insertion, thoracoscopy, long-term drain insertion and possible surgical interventions for an empyema
**QUESTIONS**
1.How would you differentiate an exudative effusion from a transudate?
•An exudative effusion is defined by
•An effusion albumin/plasma albumin ratio > 0.5
•An effusion LDH/plasma LDH ratio > 0.6
•A pleural fluid LDH > 2/3 the upper limit of normal serum LDH (Light' s criteria)
2.What are the causes of an exudative and transudative effusion?
•Exudative
•The 4 I' s: Infiltration (neoplasm), infection, infarction (pulmonary embolus), inflammation (rheumatoid arthritis and SLE)
•Transudative
•Cardiac failure, chronic renal disease, chronic liver disease, other rarer causes include Meigs' syndrome
3.List some drugs that may cause a pleural effusion.
•Amiodarone
•Phenytoin
•Methotrexate
•Nitrofurantoin
•Beta-blockers
4.What is the main clinical indication for a thoracoscopy?
•The main indication for a thoracoscopy is to investigate an exudative effusion of uncertain cause, as it yields a better diagnostic rate than a pleural tap and pleural biopsies can also be taken at the same time. A thoracoscopy can also be used for therapeutic purposes with drainage of the effusion and pleurodesis during the procedure.
**KEY POINTS**
•Be able to differentiate between effusion and collapse (stony-dull percussion in an effusion)
•Be aware of the causes of a pleural effusion and be familiar with Light' s criteria for classifying pleural effusions.
•Look for an underlying cause for the effusion.
PATIENT WITH PREVIOUS LUNG SURGERY
This patient is complaining of a cough; please examine their respiratory system.
**FINDINGS**
• _General:_ Dyspnoea, reduced chest expansion, cachexia
• _Peripheral:_ Finger clubbing, peripheral and central cyanosis, reduced muscle bulk under scar site, tracheal deviation towards the side of surgery (more obvious in pneumonectomy)
• _Chest:_ Thoracotomy scar, scars from drain sites, reduced expansion on affected side, dull percussion note on the affected side, decreased breath sounds on affected side
**PRESENTATION**
This patient has evidence of a right-sided pneumonectomy as evidenced by the lateral thoracotomy scar, reduced chest expansion and absence of breath sounds. The absence of breath sounds suggests that the procedure has been a pneumonectomy as opposed to a lobectomy.
•Comment on the possible reason for surgery.
•Lung cancer: Clubbing, cachexia, Horner' s syndrome, radiation scars and tattoos, lymphadenopathy
•Evidence of old TB: COPD (could have been treated with surgery for excision of large bullae and lung reduction surgery)
•Keyhole scars from video-assisted thoracoscopic surgery (VATS)
•Note: The other scenario in this station is that you have a patient who has had a lung transplant; if this is the case, you may find that the only abnormality is a ' clamshell' scar.
**QUESTIONS**
1.How would you differentiate between a lobectomy and a pneumonectomy?
•Pneumonectomy
•The trachea is deviated away towards the side of surgery.
•Decreased breaths sounds (or no sounds) over the whole lung field.
•Reduced chest expansion.
•Lobectomy
•Trachea may be shifted away from the side of surgery.
•Audible breath sounds from the lobes that have not been operated on.
•Chest expansion may be reduced.
2.What are the criteria for lung surgery in lung cancer?
•Patients must have an FEV1 > 1.51, a transfer factor > 50%, no evidence of severe pulmonary hypertension and no evidence of metastatic disease (surgery is beneficial only in peripheral non-small-cell disease).
3.What are the indications for a lung transplant?
•Patients with emphysema (usually with alpha-1 antitrypsin deficiency), idiopathic pulmonary fibrosis, idiopathic pulmonary hypertension, bronchiectasis and cystic fibrosis may be considered for surgery.
4.What are the absolute contraindications for lung transplantation?
•Recent malignancy in the past 2 years
•Substance abuse (alcohol, smoking)
•Chest wall deformity
•Poor social support
•Psychiatric illness
•Advanced extrapulmonary organ dysfunction
•Noncurative infections: HIV
**KEY POINTS**
•Be able to recognise the range of thoracic scars and look for clues to guide to the underlying cause.
•Be aware of indications for lung transplantation and the major contraindications.
REFERENCE
British Thoracic Society. (2001) Guidelines on the selection of patients with lung cancer for surgery. _Thorax_ 56: 89– 108.
RESPIRATORY STATION SUMMARY
•Observe the patient from the end of the bed and count the respiratory rate.
•Take a good look around the bedside and pick up on any clues that may help aid your diagnosis, e.g. sputum pot, inhalers, oxygen mask.
•Look at the inhalers to elicit their contents.
•Ensure that you have time to examine both the front and the back of the chest in your examination – concentrate on the back, as you are more likely to elicit findings and especially if you are running out of time (although this should not be the case for a well-practiced candidate).
•Be able to differentiate between consolidation and an effusion on examination.
•Look for any scars which may aid you with the diagnosis.
•If you think the diagnosis is bronchiectasis and the patient has bibasal coarse crackles, ask them to cough to see if the crackles clear.
•Be able to differentiate the causes of upper- and lower-zone fibrosis.
•In cases such as pulmonary fibrosis and pleural effusion, look for an underlying cause for the pathology.
•If you see a surgical scar, look for clues such as a radiotherapy tattoo to help guide your diagnosis.
•Familiarise yourself with pulmonary function test results for common respiratory conditions such as COPD.
•Be aware of British Thoracic Society (BTS), NICE and Scottish Intercollegiate Guidelines Network (SIGN) guidelines to help with the diagnosis and treatment of patients.
Station 1: Abdominal
Hints for the Abdominal Station
Abdominal Case with a Normal Abdomen
Chronic Liver Disease
Generalised Lymphadenopathy
Hepatosplenomegaly
Multiple Abdominal Scars
Palpable Kidneys
Renal Replacement Therapy
Splenomegaly
Abdominal Station Summary
HINTS FOR THE ABDOMINAL STATION
•Read the candidate information carefully and look closely from the end of the bed for any signs that may help guide you to the specific examination; renal/ haematology/liver/general abdominal.
•For a renal examination, look for evidence of renal replacement therapy (RRT). Always check for arteriovenous (AV) fistulae along the arms and look for fresh venepuncture marks at a fistula site. Also look for evidence of previous internal jugular vein lines or scars from peritoneal dialysis (PD) catheter insertions. Don' t miss nephrectomy and pelvic transplant scars.
•For a liver exam, ensure that you comment on the presence of finger clubbing, leuconychia, palmar erythema, Dupuytren' s contractures and tattoos if present.
•Look in the oral cavity; gum hypertrophy in ciclosporin toxicity, poor dentition in chronic liver disease and glossitis in anaemia.
•If you find evidence of any cervical lymphadenopathy, always go on to check for axillary and inguinal lymphadenopathy.
•Ensure that the patient is adequately exposed and that they are lying as flat as possible prior to palpating the abdomen.
•Always make constant eye contact with the patient when palpating their abdomen to ensure that they are not in any discomfort. Apologise if you do cause any discomfort to the patient.
•Do not panic if you cannot elicit any abnormal findings; the patient may have a normal abdomen. In this case, look for peripheral signs which may help guide you towards a diagnosis.
•If you elicit any hepatomegaly or splenomegaly, always try to comment on the size of the organomegaly. In a case of splenomegaly, this may aid you with your differential diagnosis.
•Look for evidence of peripheral oedema.
ABDOMINAL CASE WITH A NORMAL ABDOMEN
Please examine this patient' s abdomen.
FINDINGS
• _General:_ Cachexia
• _Peripheral:_ Oral/perioral telangiectasia, buccal pigmentation, hypertension, facial flushing, lymphadenopathy, cushingoid features, signs of RRT
• _Abdomen:_ No abnormality
If you finish your examination and have no positive findings, it is likely that you are in one of the three following scenarios:
1.You have been given a patient with a normal abdomen.
2.You may have missed a peripheral sign that points to a diagnosis such as hereditary haemorrhagic telangiectasia (HHT), Peutz– Jeghers syndrome, generalised lymphadenopathy, Cushing' s syndrome, carcinoid syndrome or a patient with end-stage renal disease (ESRD) on RRT
3.You may have missed organomegaly.
**PRESENTATION**
This patient has oral telangiectasia and looks clinically anaemic. There is nothing abnormal to find in the abdomen. The diagnosis may be hereditary haemorrhagic telangiectasia. I would like to know this patient' s full blood count (FBC) and ask if there is a family history of this disorder.
QUESTIONS
1.What is the mode of inheritance of Peutz– Jeghers syndrome?
•Autosomal dominant, caused by mutations in the STK11 (also known as LKB1), which is a tumour suppressor gene
2.What are the manifestations of Peutz– Jeghers syndrome?
•This condition is characterised by dark-coloured freckling from mucocutaneous lesions predominantly on the face (around lips and mouth), oral mucosa and peripheries. It also causes polyps within the gastrointestinal (GI) tract which may be complicated by bleeding, intestinal obstruction and chronic pain and have a high risk of malignancy during their lifetime (particularly breast, colorectal, pancreatic, stomach ovarian, lung and small bowel).
3.What is the mode of presentation of hereditary haemorrhagic telangiectasia?
•Recurrent epistaxis in childhood with red spots on lips, tongue, fingertips ± family history.
•It is also known as Osler– Weber– Rendu syndrome.
•It is characterised by multiple telangiectasia.
•The patient is at risk of haemorrhage from AV malformations, particularly pulmonary and cerebral.
4.What is the mode of inheritance of hereditary haemorrhagic telangiectasia?
•It has an autosomal dominant mode of inheritance. More than 80% of cases are due to mutations in ENG or ACVRL1 genes. MADH4 gene mutations have also been reported.
KEY POINTS
•If you are unable to elicit any positive ?ndings, you may have been given a ' normal abdomen' as your case.
•Ensure that you look carefully for the presence of any peripheral signs to help guide your diagnosis.
•Do not make up any ?ndings.
CHRONIC LIVER DISEASE
Please examine this patient who has presented with abdominal swelling.
FINDINGS
• _General:_ Cachexia, poor dentition, muscle wasting, loss of body hair, gynaecomastia, testicular atrophy, tattoos, pedal oedema
• _Skin:_ Excoriation marks, purpura, spider naevi (most likely to be seen on the chest/ back)
• _Hands:_ Clubbing, palmar erythema, leuconychia, Dupuytren' s contractures
• _Face:_ Icterus, parotid swelling, pallor
• _Abdomen:_ Caput medusae, ascites, hepatomegaly, hepatic bruit, splenomegaly, evidence of previous ascitic taps/drains
PRESENTATION
On examination, this patient has evidence of chronic liver disease. There is leuconychia and palmar erythema. The patient is icteric and there are excoriation marks. There is marked ascites with prominent superficial veins over the abdominal wall.
This is a common presentation, so be sure to look for an underlying cause. The most common cause is alcoholic liver disease (ALD).
CAUSES
•ALD
•Parotid swelling, cachexia, Dupuytren' s contracture
•Viral hepatitis
•Tattoos, injection sites
•Wilson' s disease
•Kayser– Fleischer rings, Parkinsonism, cognitive impairment, features of heart failure
•Haemochromatosis
•Slate-grey pigmentation, evidence of diabetes mellitus, features of heart failure, arthritis
•Alpha-1 antitrypsin deficiency
•Shortness of breath, hyperinflated chest, evidence of cor pulmonale, clubbing (bronchiectasis)
INVESTIGATIONS
•Bloods
•FBC, urea and electrolytes (U&Es), liver function tests (LFTs) including albumin, gamma-glutamyl transpeptidase (GGT), coagulation screen, hepatitis serology, caeruloplasmin, ferritin, autoantibodies (antimitochondrial antibody [AMA], anti– smooth muscle, anti– liver kidney microsomal [LKM], antinuclear antibody [ANA]), alpha-fetoprotein (AFP), thyroid function tests (TFTs), coeliac screen, alpha-1 antitrypsin levels, glucose
•Ascitic fluid
•Gram stain and cell count (> 250 white cells/mm3 is indicative of spontaneous bacterial peritonitis [SBP]), protein concentration, culture
•Imaging
•Ultrasound scan (USS): Check for hepato-/splenomegaly, confirm ascites
•Doppler flow studies of the hepatic/portal vein: Rule out thrombosis
•Computerised tomography (CT) abdomen (adds little if USS is normal)
MANAGEMENT
Dependent on the underlying cause
•Alcohol
•Alcohol avoidance, diuretics, vitamin B compound, thiamine, oesophagogastroduodenoscopy (OGD) to look for oesophageal varices
•Hepatitis
•Hepatitis C: Antiviral agents
•Wilson' s disease
•Chelation: Penicillamine, trientine
•Haemochromatosis
•Venesection (to a ferritin level of below 50 µg/L)
•Alpha-1 antitrypsin deficiency
•Generally supportive; advise not to smoke. Alpha-1 antitrypsin may be an option.
All these patients should be considered for liver transplantation on an individual basis if severity dictates.
**QUESTIONS**
1.What are the signs of decompensated chronic liver disease?
•Encephalopathy: Grade 1 (altered mood/behaviour) to grade 4 (coma)
•Asterixis
•Jaundice
•Ascites
•Hepatic fetor
•Constructional apraxia: Unable to draw a five-pointed star
2.What are the reversible causes of hepatic encephalopathy?
•Alcohol
•Drugs
•GI haemorrhage
•Infection
•Constipation
3.How is ascites investigated?
•The protein level of the ascites is used to split the aetiology by transudative and exudative causes. However, the serum ascites-albumin gradient (SA-AG) is more accurate for diagnosis of the cause. SA-AG is calculated as ' serum albumin concentration minus ascites albumin concentration' .
•SA-AG ≥ 11 g/L
•Cirrhosis, cardiac failure, nephrotic syndrome
•SA-AG < 11 g/L
•Malignancy, pancreatitis, tuberculosis
Further investigations are also helpful for diagnosis:
•Neutrophil count and microscopy and culture for spontaneous bacterial peritonitis (neutrophil count > 250 cells/mm3 is diagnostic of SBP)
•Amylase for pancreatitis
•Cytology for malignancy
4.What are the possible indications for liver transplantation in an adult?
•Acute causes
•Paracetamol poisoning (acetaminophen poisoning)
•Other drugs, e.g. isoniazid, phenytoin, sodium valproate
•Acute hepatitis
•Epstein– Barr virus (EBV) and cytomegalovirus (CMV)
•Chronic causes
•Alcoholic liver disease
•Primary biliary cirrhosis
•Primary sclerosing cholangitis
•Chronic viral hepatitis
•Wilson' s disease
•Budd– Chiari syndrome
•Hepatic malignancy
**KEY POINTS**
•Look out for examination findings which will guide you to the underlying diagnosis.
•In chronic liver disease, many patients will have a shrunken cirrhotic liver rather than hepatomegaly.
•Be able to grade the various stages of hepatic encephalopathy.
REFERENCE
Moore KP, Aithal GP. (2006) Guidelines on the management of ascites in cirrhosis. _Gut_ 55: 1– 12.
GENERALISED LYMPHADENOPATHY
Please examine this patient' s abdominal system.
**FINDINGS**
• _General:_ Evident lumps and bumps, purpura, pallor, cachexia
• _Peripheral:_ Lymphadenopathy (cervical, supraclavicular, axillary), arthritis, enlarged tonsils
• _Abdomen:_ Splenomegaly, hepatomegaly, inguinal lymph nodes
**PRESENTATION**
This patient has marked lymphadenopathy. There are palpable nodes in the cervical, axillary and inguinal regions. There are purpuric lesions present on both arms. This could be a lymphoproliferative disorder with thrombocytopenia.
A difficult case, this could be picked up when examining the neck or following palpation of an enlarged spleen (and possibly liver). If following examination of the abdomen you feel that the patient could have a lymphoproliferative disorder, do not be afraid to reexamine the neck and axillae for nodes.
INVESTIGATIONS
•Diagnosis
•Blood tests (FBC, lactate dehydrogenase [LDH], blood film, viral screen, autoimmune screen, LFTs), lymph node biopsy, bone marrow aspirate and trephine, chest X-ray and sputum for acid-fast bacilli
•Staging
•CT/positron emission tomography (PET), lumbar puncture
MANAGEMENT
•Dependent on the cause.
•The main objective of treatment of high-grade lymphoproliferative disease is to achieve a cure.
•Long-term remission may be achieved in low-grade disease.
•It is also important to manage symptoms accordingly.
QUESTIONS
1.What is the differential diagnosis of generalised lymphadenopathy?
•Lymphoproliferative disease: Chronic lymphocytic leukaemia (CLL), acute lymphoblastic leukaemia, Hodgkin' s and non-Hodgkin' s lymphoma
•Viral disease: Includes human immunodeficiency virus (HIV), EBV and CMV
•Other infections: Includes tuberculosis, brucellosis and toxoplasmosis
•Inflammatory disease: Sarcoidosis, rheumatoid arthritis, systemic lupus erythematosus (SLE)
2.How are Hodgkin' s and non-Hodgkin' s lymphoma differentiated pathologically?
•Through the presence of Reed– Sternberg cells in Hodgkin' s lymphoma. These are characteristic binucleate cells found on light microscopy of a biopsy.
3.What are ' B-symptoms' and what are their significance?
•These include weight loss (> 10% in 6 months), unexplained fever > 38.0° C and night sweats. B-symptoms are included in the Ann Arbor staging classification of non-Hodgkin' s and Hodgkin' s lymphoma and indicate a poorer prognosis.
4.Tell me about CLL.
•This is due to a monoclonal proliferation of lymphocytes, usually B-cells.
•It is most commonly suspected from a routine blood test, with a raised white cell count (lymphocytosis). It may also present with signs of bone marrow failure (fatigue, infections, bleeding) or constitutional symptoms (such as weight loss, fevers, fatigue and malaise).
•Investigations may include a CT scan, lymph node biopsy and bone marrow biopsy. CLL is staged by the Binet system (Stage A: < 3 groups of enlarged lymph nodes, no anaemia or thrombocytopenia; Stage B: ≥ 3 groups of enlarged lymph nodes, no anaemia or thrombocytopenia; Stage C: anaemia and/or thrombocytopenia). Cytogenetic testing can give information on how the disease is likely to progress, and can therefore guide treatment.
•Stage A disease can initially be managed with a watch and wait approach. More advanced disease requires earlier intervention with cytotoxic chemotherapy (such as fludarabine and cyclophosphamide) and monoclonal antibodies (such as rituximab, a CD20 antibody). B-cell receptor signalling pathway inhibitors such as ibrutinib and idelalisib are also now available. Rarely, a bone marrow transplant is carried out.
•Complications of CLL include bone marrow failure, autoimmune haemolytic anaemia, recurrent chest infections and acute transformation (Richter' s syndrome).
KEY POINTS
•When examining a seemingly normal abdomen, do not forget to feel for inguinal lymph nodes.
•Be able to give a wide differential for lymphadenopathy.
•Remember that a haematological cause is not the only cause of lymphadenopathy.
REFERENCES
Binet JL, et al. (1981) A new prognostic classification of chronic lymphocytic leukemia derived from a multivariate survival analysis. _Cancer_ 48(1): 198– 206.
Woyach JA, Johnson AJ. (2015) Targeted therapies in CLL: Mechanisms of resistance and strategies for management. _Blood_ 126(4): 471– 7.
HEPATOSPLENOMEGALY
Please examine this patient' s abdominal system.
FINDINGS
• _General:_ Purpura, pallor, jaundice
• _Peripheral:_ Enlarged tonsils, lymphadenopathy (cervical, supraclavicular, axillary), features of chronic liver disease
• _Abdomen:_ Splenomegaly (note degree of enlargement), hepatomegaly, ascites, inguinal lymph nodes
PRESENTATION
This patient has 10 cm splenomegaly, with 4 cm hepatomegaly. There is no evident lymphadenopathy and no peripheral features of chronic liver disease. The most likely diagnosis is a myeloproliferative disorder.
A common case, the most likely causes are chronic liver disease and haematological malignancy; be sure to look for signs of both of these. After the enlarged liver and spleen have been palpated, reexamine for peripheral signs if necessary.
INVESTIGATIONS
If the diagnosis is chronic liver disease, investigate as necessary (see chronic liver disease case). Otherwise, consider the following tests:
•Blood tests
•FBC, LDH, blood film, viral screen, autoimmune screen, LFTs, renal function, serum and urine electrophoresis
•Imaging
•Abdominal ultrasound, CT/PET scan
•Invasive
•Lymph node biopsy, bone marrow aspirate and trephine, other biopsy site (e.g. renal biopsy for amyloidosis)
MANAGEMENT
•Dependent on the cause
QUESTIONS
1.What is the differential diagnosis of hepatosplenomegaly?
•Chronic liver disease with portal hypertension
•Lymphoproliferative disease: Chronic lymphocytic leukaemia, Hodgkin' s and non-Hodgkin' s lymphoma (in acute lymphoblastic leukaemia the spleen is not usually greatly enlarged)
•Myeloproliferative disease: Chronic myeloid leukaemia (CML) (and acute myeloid leukaemia), polycythaemia rubra vera (PRV), essential thrombocythaemia and myelofibrosis
•Other haematological disease: Thalassaemia, sickle cell disease
•Viral disease: Includes HIV, EBV, CMV and hepatitis B/C (may cause chronic liver disease)
•Other infections: Includes malaria, brucellosis, toxoplasmosis and leptospirosis
•Inflammatory disease: Sarcoidosis
•Infiltrative disease: Glycogen storage disease, amyloidosis
2.What are the myeloproliferative disorders?
A group of conditions caused by abnormal myeloid stem cell proliferation in the bone marrow. They are generally distinguished from each other by the type of cell which is most affected:
•Red blood cells: Polycythaemia rubra vera
•White blood cells: Chronic myeloid leukaemia
•Platelets: Essential thrombocythaemia
•Fibroblasts: Myelofibrosis
3.How might a patient with CML present, and how would you diagnose this?
•Chronic myeloid leukaemia is the uncontrolled growth of myeloid cells. Therefore, red cells, white cells and platelets can all be affected. It could present as a result of the effect on these cells (fatigue, infection, bleeding), or with constitutional symptoms, abdominal pain (splenomegaly) or by chance.
•For diagnosis, blood tests (raised white cell count) are relevant, but cytogenetic testing of bone marrow is usually required. On such chromosome analysis, the Philadelphia chromosome (a translocation between chromosomes 9 and 22, resulting in oncogenic fusion of the BCR-ABL1 gene [a type of tyrosine kinase]) is present in 95% of cases of chronic myeloid leukaemia. This chromosomal translocation forms the basis of targeted treatment with tyrosine kinase inhibitors (such as imatinib).
4.Tell me about amyloidosis.
•This is a multisystem disease that results from extracellular deposition of abnormal proteins.
•There are two main types: amyloid L (AL) and amyloid A (AA).
•AL results from abnormal light-chain production and most commonly occurs on its own, although it can also be seen alongside myeloma. Deposition can affect the heart (cardiomyopathy and heart failure), kidneys (renal failure), nerves (peripheral neuropathy), gut (malabsorption) and clotting function. It is mainly diagnosed on biopsy, although a serum amyloid P (SAP) scan is sometimes carried out. Treatment is similar to that for myeloma. The complications can also be treated (for example, with a renal transplant).
•AA results from abnormal deposition of serum amyloid A (SAA) protein, which is an acute-phase protein. It occurs as a secondary process to chronic inflammatory disorders (such as rheumatoid arthritis and familial Mediterranean fever) and chronic infections. It most commonly affects the liver, spleen and kidneys. It is again diagnosed by biopsy. The condition often improves with treatment of the underlying inflammatory disorder.
KEY POINTS
•Aim to decide between liver and haematological disease when giving a primary diagnosis.
•Look for clues of these two groups throughout the examination.
•Don' t be afraid to re-examine for peripheral signs if necessary.
REFERENCE
Pinney JH, Hawkins PN. (2012) Amyloidosis. _Ann Clin Biochem_ 49(Pt 3): 229– 41.
MULTIPLE ABDOMINAL SCARS
This patient has had an operation. Please examine their abdomen.
FINDINGS
• _General:_ Cachexia, cushingoid features
• _Peripheral:_ Oral ulceration, pallor, finger-prick marks (signs of glucose testing), gum hypertrophy and hypertension (side effects of ciclosporin)
• _Abdomen:_
•Multiple surgical scars (most commonly midline)
•Stoma sites (past/current)
•Evidence of previous PD catheter exit site or removal (see case on RRT) or repeated ascitic drainage
•Mention that you would wish to perform a rectal examination
•Comment on nutritional state
PRESENTATION
This patient has inflammatory bowel disease (IBD) as evidenced by cachexia, clinical anaemia and multiple scars on the abdomen suggesting fistulae, bowel obstruction or abscess drainage. A per rectal examination may be useful to look for fistulae and abscesses.
INVESTIGATIONS
•Stool cultures × 3 (microscopy/culture/ _Clostridium difficile_ toxin [DCT]) (in acute presentations).
•Bloods including FBC, C-reactive protein (CRP), LFTs (evidence of associated liver disease, e.g. autoimmune hepatitis primary biliary cirrhosis).
•A faecal calprotectin test (evaluates bowel inflammation); a negative test rules out inflammatory bowel disease.
•Abdominal X-ray to rule out toxic megacolon (in acute presentations).
•Sigmoidoscopy/colonoscopy and biopsy (for histological confirmation).
•Magnetic resonance imaging (MRI) enterocolysis (MRI of the small bowel) or bowel ultrasound (in expert hands) can be used if Crohn' s is suspected and to detect small bowel strictures.
MANAGEMENT
•Medical
•Immunosuppression including steroids, 5-aminosalicylic acid (ASA), disease-modifying agents such as azathioprine/mercaptopurine/methotrexate and biological agents
•Surgical
•Indications include fistulae, strictures and failure to respond to medical therapy
•Nutritional support and elemental and low-residue diets
•Psychological support
QUESTIONS
1.What are the differences between ulcerative colitis (UC) and Crohn' s disease?
**Differences between Crohn' s Disease and Ulcerative Colitis** | **Crohn' s disease** | **Ulcerative colitis**
---|---|---
Distribution | Patchy (' skip lesions') | Continuous
Depth | Transmural | Superficial
Area involved | Whole gastrointestinal
tract with predilection for
terminal ileum and anus | Large bowel with
predilection for
rectum
Smoking | Higher risk | Lower risk
Fistulae and stenosis | Common | Rare
2.What are the extra-intestinal manifestations of inflammatory bowel disease?
•Skin: Erythema nodusum, pyoderma gangrenosum, aphthous ulceration
•Joints: Seronegative arthritides (mostly large-joint arthritis and sacroiliitis)
•Eye: Uveitis, episcleritis/scleritis, conjunctivitis
•Hepatobiliary: Primary sclerosing cholangitis (more likely in UC), cholangiocarcinoma
•Renal: Oxalate stones
3.What screening tests should you consider before initiating biological therapies?
•History and examination (looking specifically for features of tuberculosis)
•Hepatitis serology
•HIV test
•Chest X-ray (CXR)
•T-spot test (QuantiFERON® test is adequate if patient is not on steroids)
4.What biological agents are available for the treatment of IBD?
•Ulcerative colitis
•Infliximab and golimumab are recommended by the National Institute for Health and Care Excellence (NICE) for the treatment of moderate to severe active UC in adults who have had a poor response to conventional therapy (including steroids and mercaptopurine or azathioprine), or if they have been unable to tolerate or have contraindications to such therapy.
•Crohn' s disease
•Infliximab and adalimumab are recommended by NICE for adults with severe active Crohn' s disease who have not responded to conventional therapy, or who have been unable to tolerate or have contraindications to conventional therapy.
•A novel agent called vedolizamab is now available for those who fail first-line treatment; this is for both UC and Crohn' s disease.
KEY POINTS
•Look for the extra-intestinal manifestations of in?ammatory bowel disease to help guide the underlying diagnosis.
•Familiarise yourself with the key treatments available for the management of these conditions.
•Be aware that midline scars may not just be a sign of IBD – see case on RRT, if there' s a midline scar – do not miss a kidney– pancreas transplant.
REFERENCES
National Institute for Health and Care Excellence. (2012) Guidance CG152. Crohn' s disease: Management. Available from https://www.nice.org.uk/guidance/cg152. Accessed 2 May 2016.
National Institute for Health and Care Excellence. (2013) Guidance CG 166. Ulcerative colitis: Management. Available from https://www.nice.org.uk/guidance/cg166/chapter/1-recommendations. Accessed 2 May 2016.
National Institute for Health and Care Excellence. (2015) Guidance TA329. Infliximab, adalimumab and golimumab for treating moderately to severely active ulcerative colitis after the failure of conventional therapy. Available from https://www.nice.org.uk/guidance/ta329/chapter/1-Guidance. Accessed 2 May 2016.
Parente F, et al. (2002) Bowel ultrasound in assessment of Crohn' s disease and detection of related small bowel strictures: A prospective comparative study versus x ray and intraoperative findings. _Gut_ 50(4): 490– 5.
PALPABLE KIDNEYS
This patient has presented with haematuria. Please examine their abdomen.
FINDINGS
• _General:_ Signs of anaemia, hypertension and renal replacement therapy
• _Arms:_ Arteriovenous fistulae/grafts
• _Face:_ Gum hypertrophy from immunosuppressants
• _Fundoscopy:_ Hypertensive retinopathy and evidence of Von Hippel– Lindau syndrome
• _Chest/neck:_ Scars from previous tunnelled dialysis catheters
• _Abdomen:_ Enlarged kidneys (bilateral flank masses), hepatomegaly (associated with autosomal dominant polycystic kidney disease [ADPKD]), renal transplant and nephrectomy scars
• _Legs:_ Oedema
PRESENTATION
The diagnosis is autosomal dominant polycystic kidney disease. On inspection, there is the appearance of fullness in the flanks. On palpation, there are bilateral bimanually ballotable masses in the flanks. It is possible to get above the masses, and each moves with respiration.
TIPS
Ensure that you comment on any evidence of renal replacement therapy, e.g. the presence of a tunnelled vascular catheter, a peritoneal dialysis catheter or an AV fistula in the arms. In addition, on abdominal examination you may note a J-shaped renal transplant scar in the iliac fossa and/or a subsequent midline scar from where the enlarged kidneys may have been removed.
INVESTIGATIONS
•Urinalysis
•Haematuria and proteinuria, signs of infection
•Ultrasound abdomen
•Gold standard investigation to assess renal size, cysts, obstruction, liver cysts
•CT abdomen
•To look for malignancy if clinically indicated
•MRI
•To assess renal size and volume (if indicated)
•Echocardiogram
•Examine for mitral valve prolapse (MVP), aortic disease and assess ventricular function
•Cerebral angiogram
•Evidence of berry aneurysms associated with ADPKD.
•Only indicated if a first-degree relative with ADPKD had a subarachnoid haemorrhage/bleeding aneurysm. Routine scanning in all is not indicated.
•Genetic testing
•Autosomal dominant polycystic kidney disease (ADPKD 1 and 2)
MANAGEMENT
•Counselling
•Regular surveillance
•Monitor renal function, treat hypertension and manage chronic kidney disease (CKD) and complications
•Consider nephrectomy if appropriate
•Vasopressin antagonists – tolvaptan – inhibits binding of vasopressin to V2 receptors, reducing cell proliferation, cyst formation and fluid excretion. NICE recommends this if PKD with CKD stage 2 or 3 at the start of treatment with evidence of rapidly progressive disease
•Renal replacement therapy if required
•Investigation of first-degree relatives
QUESTIONS
1.What is the genetic basis of polycystic kidney disease?
•ADPKD
•Affects adults
•75% of patients have liver cysts by the seventh decade
•Chromosome 16 mutation in PKD 1: Accounts for 85% of cases; carries highest risk of developing ESRD
•Chromosome 4 mutation in PKD 2: Accounts for ~15% of cases; ESRD develops later than with PKD1
•Autosomal recessive PKD (ARPKD)
•More severe liver involvement than ADPKD
2.List some other manifestations of ADPKD.
•Cystic disease
•Liver, spleen, pancreas
•Berry aneurysm
•Risk of subarachnoid haemorrhage (SAH) if ruptures (treatment includes blood pressure [BP] control, lipid lowering and smoking cessation)
•Pain/haematuria
•If ruptured cyst, stone, infection or renal cell carcinoma
•Renal cell carcinoma
•Risk of malignant cyst transformation
•Valvular disease
•MVP, aortic valve disease
•Hypertension
•LVH
•Gastrointestinal
•Colonic diverticulum formation, herniae
3.What are the indications for nephrectomy in ADPKD?
•Recurrent infections
•Chronic pain
•Recurrent haematuria
•GI pressure symptoms, e.g. early satiety
•Size/creating space for transplantation
4.List some other renal cystic disorders.
•Unilateral
•Benign renal cysts
•Renal cell carcinoma
•Polycystic kidney disease
•Bilateral
•Bilateral renal cell carcinoma
•Amyloidosis
•Von Hippel– Lindau syndrome (see below for further information regarding this condition)
•Tuberous sclerosis
**Von Hippel– Lindau syndrome**
Autosomal dominant condition
Caused by mutations in the VHL gene (tumour suppressor gene)
Features
•Angiomata develop in retina (may develop retinal haemorrhages and cause visual loss if left untreated), brain and spinal cord, liver, kidney and pancreas
•Cerebellar haemangioblastoma (lateral lobes)
•Phaeochromocytoma
•Renal cell carcinoma
•Endolymphatic sac tumors (tumours of the inner ear, causing hearing loss, tinnitus and balance problems)
•Usually requires annual screening of eyes, kidneys and urinary peptides (for phaeochromocytoma), intermittent brain imaging
KEY POINTS
•Look for any scars that may indicate a nephrectomy.
•Look for any signs of renal replacement therapy.
•Know the causes of cystic renal disease.
RENAL REPLACEMENT THERAPY
Please examine this patient' s abdomen.
FINDINGS
• _General:_ Cushingoid appearance
• _Hands:_
•Evidence of finger-prick testing: Suggestive of diabetes (could be the cause of ESRD or have arisen post-transplant)
•Fine tremor: Could signify tacrolimus toxicity
• _Arms:_ AV fistulae (radiocephalic [seen at the wrist], brachiocephalic [seen around the antecubital fossa] or brachiobasilic [palpable medially from around the antecubital fossa (ACF) to towards the axilla]): Feel for thrills _and_ listen for a bruit to determine if still functional; look for fresh venepuncture marks. If there is no palpable thrill but a bruit is audible, this could signify a synthetic graft.
• _Face:_ Corneal arcus, gum hypertrophy (secondary to ciclosporin)
• _Neck/chest:_ Scars from internal jugular vein catheters, tunnelled-catheter exit site scar, parathyroidectomy scar (look carefully for this: it will be hidden in the crease at the base of the neck)
• _Abdomen:_
•Peritoneal dialysis catheter or scars from previous catheters (look for previous exit site scars: usually lateral to the umbilicus, and if removed, there may be a small longitudinal scar infra-umbilically).
•Midline laparotomy scar or posterior subcostal scar: Evidence of previous nephrectomy (possibly bilateral).
•Note that a midline laparotomy scar could also be present if the patient has had a simultaneous kidney– pancreas transplant (usually the renal graft is situated in the left iliac fossa in these patients).
•Palpable mass in iliac fossa (graft may have been removed, so mass may not be palpable; scars can be very faint).
PRESENTATION
This patient has had a renal transplant as evidenced by the presence of a renal transplant in the right iliac fossa (RIF). This is a functioning transplant as the patient does not have evidence of recent dialysis (fistula not recently needled/no PD catheter present) or fluid overload.
Comment on the following:
•Renal replacement therapy
•Any other forms, e.g. vascular catheter, PD catheter, AV fistula (are they being actively used?)
•Fluid status
•Signs of fluid overload (raised jugular venous pressure [JVP], peripheral oedema)
•Immunosuppression
•Ciclosporin: Gum hypertrophy
•Corticosteroids: Cushingoid features
•Skin lesions: Suggestive of previous skin cancers/removals (squamous cell carcinomas [SCCs]/basal cell carcinomas [BCCs])
Try to link with a possible underlying cause for renal failure:
•Diabetes
•Injection sites on abdomen/lipodystrophy, pinprick marks on fingertips
•ADPKD
•Bilateral ballotable masses in flanks
•Alport' s syndrome
•Hearing aids
**MANAGEMENT**
•Maintenance immunosuppression usually consists of a calcineurin inhibitor (CNI) (tacrolimus or ciclosporin) and an antiproliferative (azathioprine or mycophenolate mofetil), with or without steroids.
•Side effects include (this is not an exhaustive list of complications; see British National Formulary [BNF] for further details)
•Tacrolimus (calcineurin inhibitor): Tremor, alopecia, diabetes, hypertension, renal impairment, renal failure, blood disorders
•Ciclosporin A, as for tacrolimus except gingival hypertrophy, hypertrichosis, hyperlipidaemia,
•Mycophenolate mofetil: Diarrhoea, nausea, vomiting, leucopenia, lymphopenia, anaemia, thrombocytopenia, foetal toxicity, infections, malignancy (especially skin)
•Azathioprine: Hypersensitivity reactions, dose-related bone marrow suppression, liver impairment, infections, pancreatitis
•Prednisolone: Thin skin, easy bruising, muscle wasting/weakness, diabetes, hypertension, osteoporosis, pancreatitis, infections, glaucoma, cataracts, heart failure
•Rapamycin (mTOR) inhibitors (sirolimus, evorolimus): Hypertension, impaired healing, venous thromboembolism (VTE), interstitial lung disease, proteinuria, hyperlipidaemia, reversible male infertility
•All these except prednisolone give an increased risk of malignancy.
•Screening should be similar to that for the general population, e.g. breast, colon screening.
•Annual skin checks should be undertaken with a dermatologist for signs of SCC and BCC.
QUESTIONS
1.What are the most common causes for a renal transplant?
•Diabetes mellitus
•Glomerulonephritis
•ADPKD
•Hypertension
2.What investigations would you request in a patient with a renal transplant admitted with a rise in serum creatinine?
•Blood tests: Renal function, FBC, bone profile, LFTs, CRP if signs of infection + blood cultures/virology (e.g. CMV, BK virus polymerase chain reaction [PCR]), immunosuppression levels to check for toxicity
•Urine: Dipstick (check for haematuria, proteinuria, leucocytes and nitrites), urine microscopy, culture and sensitivity, urine for quantification of proteinuria (if applicable), virology for BK virus
•Ultrasound scan of the transplant with Dopplers of the vessels: To assess for obstruction, renal perfusion, possible renal artery stenosis and renal vein thrombosis
•Transplant biopsy: If no other cause found, to rule out rejection and look for any other cause
3.What are the signs of a failing transplant?
•Progressively declining renal function
•Proteinuria
•Tenderness over the transplant graft
•Fluid overload
•Interstitial fibrosis, tubular atrophy or vascular changes in a renal transplant biopsy
4.What do you know about the manifestations of Alport' s syndrome (AS)?
•Alport' s syndrome is a genetic disease in which there is defective type IV collagen.
•There are three main types – X-linked (the most common), in which males are more severely affected than females; autosomal dominant; and autosomal recessive
•Manifestations
•Kidneys
•Initially presents with microscopic haematuria and then proteinuria develops. About 50% of males with X-linked AS require dialysis or transplantation by ~25 years and about ~90% by aged 40 years.
•Ears
•Sensorineural hearing loss
•Eyes
•Anterior lenticonus can lead to slow progressive visual loss.
•Management
•Angiotensin-converting enzyme (ACE) inhibitors or angiotensin receptor blockers (ARBs) to try to delay onset of ESRD
•Hearing aids as needed
KEY POINTS
•Look closely for any scars which may represent a renal transplant.
•If there is a renal graft present, comment on whether you believe it is functioning.
•Look for other forms of renal replacement therapy.
•Know the common causes for a renal transplant and how to manage a patient presenting with acute transplant dysfunction.
REFERENCES
National Institute for Health and Care Excellence. (2016) British National Formulary. Available from https://www.evidence.nhs.uk/formulary/bnf/current/. Accessed 18 August 2016.
Renal Association. (2011) British Renal Association guidelines on the post-operative care of the kidney transplant recipients. Available from <http://www.renal.org/guidelines/modules/post-operative-care-of-the-kidney-transplant-recipient#sthash.UPL7Z6si.dpbs>. Accessed 18 August 2016.
Renal Association. (n.d.) Alport syndrome – Patient information. Available from http://rarerenal.org/clinicianinformation/alport-syndrome/. Accessed 18 August 2016.
SPLENOMEGALY
Please examine this gentleman' s abdominal system.
**FINDINGS**
• _General:_ Pallor, purpura
• _Peripheral:_ Lymphadenopathy
• _Abdomen:_ Enlarged spleen
•Be aware of how to differentiate splenomegaly from an enlarged kidney:
•Features of a spleen: Mass in the left upper quadrant (LUQ) (moves towards RIF with respiration), unable to get above a spleen on examination, dull to percussion, nonballotable, presence of a palpable splenic notch
•Features of a kidney: Moves minimally with respiration, can get above a kidney, resonant to percussion, ballotable, no notch palpable
**PRESENTATION**
This patient has evidence of splenomegaly as there is a palpable mass in the left upper quadrant which moves inferomedially with respiration. I am unable to palpate above it and there is also a splenic notch palpable. On peripheral examination, there is evidence of pallor and bruising. There is no palpable lymphadenopathy.
Look for other signs leading to the underlying cause of the splenomegaly:
•Felty' s syndrome
•Rheumatoid hand signs
•Portal hypertension
•Signs of chronic liver disease
•Infective endocarditis
•Murmur, splinter haemorrhages
•CML/CLL/lymphoma
•Lymphadenopathy (CLL/lymphoma), pallor
Splenomegaly can be classified by the degree of enlargement (but remember that causes of massive splenomegaly may be moderate to start with):
•Moderate enlargement (11– 20 cm)
•Rheumatological disease
•Rheumatoid arthritis, SLE, Sjogren' s syndrome
•Infection
•Schistosomiasis, malaria, leishmaniasis, EBV
•Haematological
•Lymphoma, leukaemia, myeloproliferative disease, haemolytic anaemia
•Massive enlargement (> 20 cm)
•Myelofibrosis
•Chronic myeloid leukaemia
•Visceral leishmaniasis
INVESTIGATIONS
•Bloods
•FBC and blood film, haemolysis screen, LFTs, U&Es, CRP, blood cultures, malaria screen, virology for EBV, autoantibodies including rheumatoid factor
•Biopsy
•Bone marrow aspirate, lymph node biopsy, splenic biopsy
•Imaging
•USS abdomen, CT chest/abdomen/pelvis, echocardiogram if infective endocarditis suspected
MANAGEMENT
•This is dependent on the underlying cause.
QUESTIONS
1.What is the function of the spleen?
•Removal of old/damaged red blood cells
•Storage of platelets
•B- and T-lymphocyte-mediated immune function
2.What advice should be given to a patient undergoing a splenectomy?
•Preoperative vaccinations: Pneumococcal, meningococcal, _Haemophilus influenzae_
•Lifelong prophylactic penicillin
•Annual influenza vaccinations
•Wear a medic alert bracelet
•Advice regarding seeking medical attention if unwell
3.What are the indications for a splenectomy?
•Hypersplenism: Autoimmune destruction of blood cells (for example, idiopathic thrombocytopenic purpura [ITP])
•Mass effect of spleen
•Traumatic rupture
•Haematological malignancies (rarely)
•Congenital haemolytic anaemias: Hereditary spherocytosis, elliptocytosis
4.Tell me about PRV.
•This is a primary polycythaemia, due to a fault in the bone marrow. It is the result of uncontrolled proliferation of mainly red blood cells, often due to a mutation in the Janus kinase 2 (JAK2) gene. Myeloid leucocytosis, thrombocytosis and splenomegaly can also occur.
•PRV may present with a raised haemoglobin level on a routine blood test. It can also present with thrombosis (or more rarely, bleeding), headache, sweating and pruritus.
•Investigation would initially be with a full blood count. Other (secondary) causes of polycythaemia should be excluded. If PRV is suspected, JAK2 mutation testing is carried out on a blood test. Imaging (CT/ultrasound) is used to visualise the spleen. Bone marrow biopsy is helpful for diagnosis.
•Diagnosis is by the World Health Organization (WHO) diagnostic criteria 2008.
•Management is targeted at lowering the risk of thrombosis. This includes venesection, aspirin and myelosuppression. JAK2 inhibitors are being introduced/under clinical trial. Management of symptoms such as pruritus should also be considered.
•Alongside thrombosis, a major complication of PRV is transformation to acute myeloid leukaemia.
KEY POINTS
•Be sure to measure splenomegaly to aid clinical diagnosis and be aware of how to differentiate between a palpable kidney.
•Be sure to look for peripheral signs to identify an underlying cause.
REFERENCE
Tefferi A, Thiele J, Vardiman JW. (2009) The 2008 World Health Organization classification system for myeloproliferative neoplasms: Order out of chaos. _Cancer_ 115(17): 3842– 7.
ABDOMINAL STATION SUMMARY
•In this station, do not forget the normal abdomen. Remember the value of peripheral signs, e.g. lymphadenopathy; you may have your diagnosis before you lay your hand on the abdomen.
•Don' t forget to ask the patient if there is any pain in the abdomen before you begin palpation. This is basic, but it can be embarrassing if you hurt the patient, and in all likelihood will result in a fail.
•Apologise if you cause the patient any discomfort.
•If possible, establish a cause for the patient' s organomegaly to demonstrate to the examiners that you are thinking one step ahead.
•Demonstrate to the examiners that you are thinking ahead by inspecting for side effects of any treatments that the patient is receiving, e.g. the cushingoid patient with a renal transplant.
•Positioning is very important; ensure the patient is flat when you palpate the abdomen.
•Make your examination slick and quick, as you may run out of time before you finish it, but at the same time, don' t be in a rush (a skill that can only be learnt with practice).
•Try not to be put off by unusual opening statements, for example, ' Please examine this patient with hypertension' (the opening statement will generally be relevant, e.g. a patient with renal disease/ESRD).
•If you palpate an organ, always ensure that you percuss it.
•Know how to clinically differentiate a spleen from a kidney (this is a PACES classic).
•Do not make up any findings.
Station 2: History Taking
Hints for the History-Taking Station
Patient with an Abnormal Blood Result
Patient with Back Pain
Patient Who Has Collapsed
Patient with a Cough
Patient with Diarrhoea
Patient with Jaundice
Patient with Joint Pains
Patient with Left Arm Weakness
Patient with Visual Disturbance
Patient with Weight Loss
History-Taking Station Summary
HINTS FOR THE HISTORY-TAKING STATION
•The history-taking station is often thought of as being one of the easier stations; this misconception often costs candidates marks and sometimes their pass. You should not take this station lightly. You should see this station as an opportunity to excel, which may in turn give you breathing space if you have a bad station elsewhere.
•Use your 5 minutes before you enter the room wisely; think about differential diagnoses and the pertinent questions you may ask to exclude or include them.
•Pay very close attention to what you are being asked to do. Commonly, you will only be asked to find out more about the problem; you are probably not going to be expected to explain to the patient what you think is going on (unless they specifically ask you to), so focus on taking your history.
•You must always be polite and courteous to the patient.
•Do not forget simple things like introducing yourself.
•It is very important to establish a baseline. Does the patient know why they have come to see you?
•Start with open questions, as the patient will give you a great deal of information in the first 3 minutes. Try not to interrupt them.
•Demonstrate to the examiners that you are organised by taking a structured history.
•Don' t forget risk factors for your differential diagnoses.
•If the patient has brought a piece of paper with their medication on it, go through it with them; do not merely assume that it is up to date. Ask if there are any other medications? Ask about any specific allergies they may have.
•Be vigilant for any hidden agendas the patient may have (addressing their concerns specifically is a good way to flush this out).
•Remember to address the patient' s ideas, concerns and expectations.
•Never forget psychosocial elements, as they are important when developing your problem list.
•Summarise appropriately and make it obvious that you are doing this, so the examiners know.
PATIENT WITH AN ABNORMAL BLOOD RESULT
Mr Warner is a 40-year-old male who on attendance at his well man clinic was found to have an adjusted serum calcium of 2.75 mmol/L. All other blood tests were within normal parameters. He seems to have few symptoms. He is known to have hypertension and takes ramipril 5 mg once daily (OD); he is otherwise well. He is an office worker, fully independent and lives with wife and three children. He drinks 20 units of alcohol a week and does not smoke.
Examination revealed a clear chest and a soft and nontender abdomen. A few small lymph nodes were palpated in his neck.
**KEY POINTS FOR THE PATIENT**
•You are 40 years old and, as far as you are concerned, fit and healthy.
•You only went to the well man check-up because your wife has been nagging you.
•You cannot think of any obvious symptoms, but on direct questioning, you admit to having a troublesome dry cough.
•You have not coughed up any blood.
•You think you are becoming a little breathless on exertion.
•You have a good appetite, have not lost any weight and have not had fevers or night sweats.
•Your joints are generally sore.
•Your general practitioner (GP) felt something in your neck, but you' re not sure of the significance of this.
•You have no other symptoms beyond this.
•You have high blood pressure but no other medical problems. You take ramipril and have not recently used any over-the-counter medications.
•You are not allergic to any medications.
•You do not smoke and never have.
•You drink 20 units of alcohol a week.
•You cannot think of any specific family history; both your parents are alive and generally well.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient why they have come to see you.
•Ask the patient what they know about what has happened so far.
•Ask about any symptoms the patient may have.
•When the patient denies any symptoms, be sure to clarify this. Proceed to direct questioning if necessary, working through a list of differentials.
•When finding out about the cough, cover other respiratory symptoms and consider both lung cancer and sarcoidosis as potential causes of the hypercalcaemia. Ask about constitutional symptoms of malignancy or systemic illness.
•Ask about arthralgia, as this is a feature of sarcoidosis.
•Take a drug history, including over-the-counter medications, and ask about allergies.
•Ask about family history.
•Take an appropriate social history, being sure to enquire about smoking.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns or expectations they might have.
•Explain to the patient that you cannot be sure of a diagnosis, but sarcoidosis is most likely (if asked directly about malignancy, you cannot definitively rule this out at this stage).
•Explain to the patient that a chest X-ray and some specialist blood tests are required, and that you will see them next week with the results. You will likely need to do a biopsy of the lymph nodes.
**THEMES EXPLORED**
•It is important to have a list of differentials for common abnormal blood test results (renal failure, hypercalcaemia and raised alkaline phosphatase [ALP], among others).
•Tie the symptoms (and signs) elicited into this list. Lymphadenopathy points towards malignancy and sarcoidosis, and this link is furthered with the presence of a dry cough.
•Look at the case as a whole; the young age, lack of social history and lack of constitutional symptoms in this patient swing the balance towards sarcoidosis.
**RELEVANT INFORMATION**
**CAUSES OF HYPERCALCAEMIA**
•Hyperparathyroidism (primary and tertiary)
•Malignancy (bony metastases, myeloma, paraneoplastic syndromes)
•Drugs (thiazides, vitamin D excess)
•Dehydration
•Sarcoidosis
•Thyrotoxicosis
•Spurious result
**PRIMARY HYPERPARATHYROIDISM**
Most commonly caused by a parathyroid adenoma. Can also be caused by parathyroid hyperplasia and malignancy (rare). Can be associated with multiple endocrine neoplasia. Often presents by chance from blood tests, although can present with symptoms of hypercalcaemia.
On blood tests, serum calcium is raised. The serum parathyroid level is either raised or (inappropriately) normal.
Treatment is most commonly surgical (excision of the adenoma or hyperplastic glands).
**MANIFESTATIONS OF SARCOIDOSIS**
•Pulmonary: Bilateral hilar lymphadenopathy, pulmonary fibrosis
•Cutaneous: Subcutaneous nodules, erythema nodosum, lupus pernio
•Ocular: Uveitis, glaucoma, soft tissue orbital mass, keratoconjunctivitis sicca
•Cardiac: Heart block and arrhythmias, cardiomyopathy
•Joints: Polyarthritis
•Facial: Parotid swelling
•Abdominal: Splenomegaly, hepatomegaly
•Neurological: Cranial neuropathy (e.g. Bell' s palsy), meningitis
•Metabolic and renal: Hypercalcaemia, hypercalciuria, renal impairment, hypothalamic involvement
•General: Fever, fatigue
REFERENCE
Pallan S, Rahman MO, Khan AA. (2012) Diagnosis and management of primary hyperparathyroidism. _BMJ_ 344: e1013.
PATIENT WITH BACK PAIN
Mr Quimby is a 78-year-old gentleman who complains of back pain. He has had this for 4 months, mainly in the thoracic region. There are no exacerbating or relieving factors. He says it is worse at night. He is also complaining of fatigue and lethargy.
Examination is unrevealing. He is thin and has a soft and nontender abdomen with no obvious organomegaly, normal heart sounds and a clear chest. A full blood count (FBC) has found a haemoglobin of 109 g/L with a mean cell volume (MCV) of 85 fL (white cell count, platelets and renal function were normal).
**KEY POINTS FOR THE PATIENT**
•You have had back pain in the middle of your back for 4 months, and you feel it is getting worse.
•Your pain is worse at night-time.
•Your pain doesn' t move anywhere and nothing makes it better or worse.
•Paracetamol has been ineffective for the pain.
•You have been lethargic for some time now and are generally listless to the extent that your garden is overgrown, as you don' t have the energy to tend it.
•You don' t weigh yourself, but your clothes are slightly looser.
•You have had no change in bowel habit.
•You have no new urinary symptoms; you pass urine twice at night and that is normal for you.
•You have not noticed any pains anywhere else.
•You have no chest symptoms. You do not sweat at night and have not had any fevers.
•Your only past medical problem is that of prostate trouble, and the GP gave you a tablet for that (you can' t remember the name) about a year ago. Your urine stream is currently fine.
•To your knowledge, you have not had any trauma to the back.
•You are a retired factory worker.
•You are an ex-smoker, stopping 25 years ago. You don' t drink alcohol.
•You live with your son.
•With the lethargy, you find it hard to have an active life. You used to play bowls, but the back pain has put an end to this. You can still walk reasonably well.
•You are concerned that you may have cancer, as your mother died of lung cancer.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient if they know why they have come to see you and what has happened so far.
•Start with an open question about his back pain.
•Find out where the pain is, the nature and if there is any radiation.
•Establish the onset, course and duration of symptoms.
•Establish if there are any exacerbating or relieving factors.
•Ask if there is any particular time when the symptoms are worse.
•Make sure you ask about red-flag symptoms: weight loss, systemic upset, night sweats, fevers and neurological symptoms in both upper and lower limbs (given the location of the pain).
•Screen for malignancy with your systems review.
•Take a thorough past medical history.
•Enquire about family history.
•Ask about the patient' s social situation and ensure that you ask about the impact on their life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns that they might have.
•Outline the investigations that are required, including blood tests and spinal imaging/bone scan.
**THEMES EXPLORED**
•In a patient of this age group with back pain, malignancy is likely and needs to be asked about. Other red-flag symptoms are of relevance.
•The blood results here also point towards malignancy; be sure to take note of this. Knowledge of how to proceed with investigations is required.
**RELEVANT INFORMATION**
**RED-FLAG SIGNS/SYMPTOMS FOR BACK PAIN**
•Age < 20 or > 55
•Pain at night
•Constant/worsening pain
•Constitutional features of malignancy (weight loss, appetite loss, night sweats)
•History of malignancy
•Thoracic pain
•Pyrexia
•Bilateral lower limb neurological symptoms
•Sphincter disturbance
**CAUSES OF BACK PAIN IN AN ELDERLY PATIENT**
•Degeneration (osteoarthritis)
•Vertebral collapse (osteoporosis)
•Solid malignancy metastases (a primary malignancy is less likely)
•Myeloma
•Paget' s disease
•Trauma
**TUMOURS MOST LIKELY TO METASTASISE TO BONE**
•Breast
•Lung
•Prostate
•Kidney
•Thyroid
PATIENT WHO HAS COLLAPSED
Mr Foyle is a 31-year-old male who has collapsed twice within the last few weeks. Between the episodes he has felt well. Mr Foyle sustained no injury from either collapse. He has been referred by his GP, who stated that an electrocardiogram (ECG) done in the practice was ' normal' .
There are no positive findings on examination, with a regular pulse of 64 beats/minute. His blood pressure is 116/70 with no postural drop. All his blood tests (FBC, urea and electrolytes [U&Es], liver function, bone profile, magnesium and glucose) have been normal. An ECG taken by the nurse in clinic has found normal sinus rhythm with no abnormalities.
**INSTRUCTIONS FOR THE PATIENT**
•You are 31 years old and, to your knowledge, fit and healthy.
•You initially collapsed 4 weeks ago. This occurred while you were standing at work (and had been for a couple of hours). You felt ' light-headed' for about a minute before the event.
•You collapsed again 2 weeks ago. This occurred following a game of football with friends. You felt ' light-headed' after standing up in the changing rooms and collapsed about 30 seconds later. You felt ' fine' during the game itself.
•You remember being sweaty before both events, although you didn' t notice any unusual sensations, smells or tastes.
•During both episodes, you report that you lost consciousness very briefly, and awoke on the floor. Your friends told you that during the second episode, you awoke very quickly.
•After resting for half an hour, you felt ' back to normal' in both cases. You have a good memory of what happened during both episodes.
•Your work colleagues and friends reported that there was no shaking of your arms or legs after you had collapsed. They reported that your eyes were closed through both events.
•You did not bite your tongue and were not incontinent of urine or faeces.
•You sustained no injury during either episode. You had no arm or leg weakness afterwards.
•You have not sustained a head injury recently.
•You have not had chest pain.
•You have not felt breathless at all.
•You have not been aware of your heart beating more than usual, or having an unusual rhythm.
•You have not had episodes like these collapses before.
•You have otherwise been well recently. You have not lost any weight or had a recent infection.
•You have no past medical history of note.
•You do not take any prescribed medications. You have not taken any over-the-counter medications recently. You have no known allergies.
•You do not take illicit or herbal drugs.
•You used to smoke 10 cigarettes a day, but quit 6 months ago.
•You drink approximately 15 units of alcohol per week and haven' t been drinking excessively recently.
•No one in your family has ever had any similar episodes. No one in your family has died suddenly and unexpectedly.
•You live at home with your wife and son, who is 3 months old. On close questioning, you feel sleep deprived, as you are awoken several times throughout the night.
•You work in a factory. You have recently started working on a new production line. On close questioning, this involves you standing for long periods of the day, more than previously. You have also been busier at work and don' t always get regular breaks.
•You do drive, although not every day, as you walk to work.
•You are concerned that you have a serious illness; a friend you used to play football with collapsed while playing and died from a ' heart problem' .
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient if they know why they have come to see you and what they know about what has happened so far.
•Start with an open question about the episodes of collapse.
•Ask about the patient' s experience of these events (before, during and after), but also about any collateral history that they can provide from others who were present. If you feel you have insufficient collateral history, discuss contacting witnesses of the events.
•Ensure that you ask about symptoms relevant to a collapse of cardiac cause, such as palpitations and chest pain.
•Ensure that you ask about features of a seizure, such as aura, incontinence and tongue biting.
•Demonstrate to the examiners that you are working through your list of differential diagnoses.
•Check on past medical history, including previous episodes of collapse.
•Take a drug history and be sure to ask about allergies.
•Ask about illicit drug and alcohol usage.
•Ask about the patient' s social situation and the role this could be playing in these episodes.
•Ask about the patient' s work and the impact this could have had on these episodes.
•Ask about family history of collapse and sudden death.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns that they might have.
•On making a likely diagnosis of vasovagal syncope, offer reassurance. There is no evidence here of a serious cardiac disorder (remember, this is the patient' s main concern), given the normal ECG and identified triggers of the episodes.
•Explain that further investigations are not needed at this time.
•Offer lifestyle advice in terms of ensuring an adequate fluid intake. Offer occupational advice in advising sitting where possible at work and trying to take breaks.
•Offer advice that the patient is able to drive currently, as there is an identified trigger to the events (standing, low fluid intake) and the episodes occurred while standing.
**THEMES EXPLORED**
•Episodes of collapse have a wide differential diagnosis, and it is important to explore a variety of symptoms in order to make a diagnosis. It is important to rule out serious causes of collapse, such as cardiac disorders and seizures.
•Lifestyle and occupational advice can form a significant part of the management plan; this needs to be communicated to the patient.
•Despite episodes of collapse, patients may be able to drive according to Driver and Vehicle Licensing Agency (DVLA) guidelines.
**RELEVANT INFORMATION**
**DIFFERENTIAL DIAGNOSIS OF COLLAPSE**
•Cardiovascular: Vasovagal syncope, situational syncopy, orthostatic hypotension, carotid sinus hypersensitivity, hypovolaemia, arrhythmia, cardiac outflow obstruction (such as aortic stenosis and hypertrophic cardiomyopathy)
•Neurological: Seizure/epilepsy
•Metabolic: Hypoglycaemia, Addisonian crisis
•Psychological: Anxiety
**FEATURES OF UNCOMPLICATED VASOVAGAL SYNCOPE**
•No features of another diagnosis (such as prolonged seizure activity)
•Postural reasons: Prolonged standing, occurring _after_ exercise
•Provoking factors: Pain, emotion
•Prodrome: Sweating, nausea
REFERENCES
Driver and Vehicle Licensing Agency. (2016) Assessing fitness to drive – A guide for medical professionals. Available from https://www.gov.uk/government/publications/assessing-fitness-to-drive-a-guide-for-medical-professionals. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2014) Guidance CG109. Transient loss of consciousness (' blackouts' ) in over 16s. Available from https://www.nice.org.uk/guidance/cg109. Accessed 5 July 2016.
PATIENT WITH A COUGH
Miss Jones is a 36-year-old female who has been troubled by a persistent dry cough for the last 6 months. She has tried a salbutamol inhaler and has stopped smoking, although this has not led to an improvement in her symptoms. She has a peak flow of 440 L/minute with no reversibility. Spirometry does not show an obstructive deficit. There are no positive findings on examination. All her blood tests have been normal, and a recent chest X-ray found no abnormalities.
**INSTRUCTIONS FOR THE PATIENT**
•You are 36 years old and, to your knowledge, fit and healthy.
•You have had a cough for a long time; it is really starting to bother you.
•Your cough is worse at night-time.
•The cough is nonproductive, and you have never coughed up any blood.
•Your breathing is fine; you play badminton once a week and have noticed no change in your fitness levels.
•You are not wheezy.
•You have no chest pain or palpitations.
•You have not lost weight.
•You sleep with one pillow.
•On close questioning, you experience some indigestion and a feeling of acid in your mouth.
•You do not have any abdominal pain.
•You have no past medical history.
•You take a progesterone-only contraceptive pill and have not missed any doses. You are not on any other medications. You have no allergies.
•You do not think you are pregnant.
•Your mother has had a blood clot in her leg.
•You did not have any serious childhood illnesses.
•You are a lawyer. Your job is important to you and keeps you very busy. You work late commonly and often eat on the run, sometimes very late in the day.
•You used to smoke when you were out with friends but gave this up 6 months ago.
•You have no pets.
•You drink approximately 35 units of alcohol per week and occasionally more, dependent on client functions.
•You are concerned that you have a serious illness.
•You are worried, as the cough is keeping you awake at night, affecting your sleep and your performance at work.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient if they know why they have come to see you and what they know about what has happened so far.
•Start with an open question about her cough.
•Ask her specifically about when the cough is worse, and about sputum and blood production.
•Ensure that you ask about other respiratory symptoms.
•Demonstrate to the examiners that you are working through your list of differentials (so specifically ask about symptoms of reflux, a common cause of cough). Also ask about a postnasal drip and any symptoms of sinus congestion.
•Clues such as the cough worsening on lying down and after eating help point towards the diagnosis of acid reflux.
•Check on past medical history, including childhood infections.
•As in any respiratory history, take an occupational history and a smoking history.
•Enquire about pets and other common triggers for asthma.
•Take a drug history and be sure to ask about allergies.
•Ask about family history.
•Perform a systems review (nonspecific symptoms could be the only clue that the patient has a lymphoma or sarcoidosis causing their cough).
•Ask about their social situation and ensure that you ask about the impact on their life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns that they might have (particularly relevant here).
•On making a likely diagnosis of reflux disease, offer reassurance and a trial of proton pump inhibitor (PPI), with follow-up in clinic in 1 month. Consider an oesophagogastroduodenoscopy (OGD) with regard to the duration of symptoms.
•Offer lifestyle advice in terms of reducing alcohol intake and eating at sensible times.
**THEMES EXPLORED**
•The final diagnosis may be from a different ' system' than the presenting complaint; be prepared to ' think outside the box' .
•It is important to know the ' red-flag' symptoms which may require urgent referral and endoscopic examination.
•Lifestyle advice may form a significant part of the management plan; this should be communicated to the patient.
**RELEVANT INFORMATION**
**RED-FLAG SYMPTOMS/SIGNS REQUIRING URGENT UPPER GI ENDOSCOPY**
•Chronic gastrointestinal (GI) bleeding
•Unintentional weight loss
•Dysphagia
•Persistent vomiting
•Iron deficiency anaemia
•Epigastric mass
•Suspicious barium meal result
•Age > 55
**MANAGEMENT OF GASTRO-OESOPHAGEAL REFLUX DISEASE**
•Endoscopy in those with red-flag symptoms.
•Consider testing for _Helicobacter pylori_ .
•Lifestyle advice: Healthy eating, weight reduction and smoking cessation.
•Full-dose PPI for 4– 8 weeks.
•Give a PPI at the lowest effective dose for those in whom symptoms recur.
•In those with severe oesophagitis, a full-dose PPI may need to be continued.
•Use a histamine-2 receptor antagonist in those in whom a PPI is in effective.
REFERENCES
National Institute for Health and Care Excellence. (2014) Gastro-oesophageal reflux disease and dyspepsia in adults: Investigation and management. Available from https://www.nice.org.uk/guidance/cg184. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2015) Guidelines for suspected cancer: Recognition and referral. Available from https://www.nice.org.uk/guidance/ng12. Accessed 5 July 2016.
PATIENT WITH DIARRHOEA
Mr Wentworth is a 22-year-old male who complains of loose stools. He works as a car mechanic and is finding that his symptoms are disrupting his job. Examination is unre-markable. He is thin and has a soft and nontender abdomen, normal heart sounds and a clear chest. Preliminary blood tests have been sent, and he has had two stool samples that have come back negative on microscopy and culture.
**INSTRUCTIONS FOR THE PATIENT**
•You have had loose stools for 6 months.
•You generally open your bowels once a day.
•You have never noticed any blood or mucus.
•You have lost 1 stone of weight in this period (this is not intentional).
•You get a bloating sensation in your abdomen but do not have pain.
•You complain of fatigue.
•You do suffer with regular mouth ulcers.
•You went to see your GP 2 years ago with an intensely itchy rash, which cleared of its own accord.
•You have a good appetite and do not suffer with night sweats.
•You have no past medical history and do not take any medications.
•You are a smoker of 20 cigarettes/day.
•You don' t drink alcohol.
•You find your symptoms embarrassing, and they are interfering with your work.
•You have not had any foreign travel or any infectious contacts.
•You eat a ' normal' diet and have not altered this at any time.
•You are not aware of any specific family history.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age, as well as asking the patient why they have come to see you and what has happened so far.
•Start with an open question about his loose stools.
•Specifically, ask about how often it occurs, stool consistency and if there is any blood or mucus.
•Ask about abdominal pain, nausea and vomiting and weight loss.
•Enquire specifically about any extra-intestinal symptoms or associations of coeliac disease:
•Lethargy (anaemia).
•Dermatitis herpetiformis.
•Low-impact bone fractures due to osteoporosis.
•Be aware of the predisposition to malignancy, particularly small bowel lymphoma. Ask about constitutional symptoms.
•In order to assist in ruling out irritable bowel syndrome, ask about anxiety and depression, and ensure that there is no evidence of other functional illness (e.g. functional dyspepsia).
•Enquire about family history of coeliac disease.
•Ask about foreign travel.
•Ask about infectious contacts.
•Ask about recent antibiotic use.
•Always ask about alcohol intake where malabsorption is suspected.
•Ask about the patient' s social situation and the impact of the symptoms on their life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns that they might have.
•Outline the investigations that will be required (blood tests, possible upper GI endoscopy), and the management given a diagnosis of coeliac disease (dietary change, dietician referral).
**THEMES EXPLORED**
•Coeliac disease is common in this age group and should be top of a list of differential diagnoses. Inflammatory bowel disease is less likely given the frequency of stool.
•It is important to ask about extra-intestinal manifestations and associations of coeliac disease; the history of dermatitis herpetiformis here helps to confirm the primary diagnosis.
**RELEVANT INFORMATION**
**CONDITIONS ASSOCIATED WITH COELIAC DISEASE**
•Dermatitis herpetiformis
•Hyposplenism
•Immunoglobulin (Ig) A deficiency
•Type 1 diabetes mellitus
•Autoimmune thyroiditis
•Primary biliary cirrhosis (PBC)
**COMMON CAUSES OF MALABSORPTION**
•Coeliac disease
•Chronic pancreatitis (including cystic fibrosis)
•Carcinoma of the pancreas
•Crohn' s disease (terminal ileal disease)
•Iatrogenic causes, including small bowel resection and radiation enteritis
**DIAGNOSIS AND MANAGEMENT OF COELIAC DISEASE**
•Serological testing: Tissue transglutaminase (tTG) antibody, endomysial antibody (EMA) (while patient is on a normal diet).
•A faecal calprotectin test evaluates bowel inflammation; a negative test rules out inflammatory bowel disease.
•Duodenal biopsy is required to confirm the diagnosis in adults. This will show villous atrophy and increased intraepithelial lymphocytes in coeliac disease.
•A strict gluten-free diet is the only effective treatment.
•Adherence to such a diet can be improved with input from an expert dietician and from support by a coeliac disease patient advocacy group (such as Coeliac UK).
REFERENCE
Mooney PD, Hadjivassiliou M, Sanders DS. (2014) Coeliac disease. _BMJ_ 348: g1561.
PATIENT WITH JAUNDICE
Mrs Valdes is a 64-year-old female with jaundice. She first noticed this after returning from India 2 months ago, and since then it has gradually worsened. She also feels lethargic to the extent that she is struggling to make it through her days at work as a primary school teacher. She has a past history of hypothyroidism, for which she takes levothyroxine 150 µ g once daily, and hypercholesterolaemia, for which she takes simvastatin 40 mg once nightly (ON). She is a moderate drinker, consuming one gin and tonic every other evening.
On examination, she is icteric, she has palmar erythema and there are five spider naevi visible on her anterior chest wall. Her abdomen is soft and nontender, and she has no palpable organomegaly. Her blood tests demonstrate deranged liver function tests (LFTs), and an ultrasound scan (USS) of the liver has been arranged.
**INSTRUCTIONS FOR THE PATIENT**
•You first noticed the jaundice while you were in India, but you have been feeling generally unwell for 3 months, with tiredness and general lack of energy.
•Your urine is a normal colour and your motions have not changed.
•You have no abdominal pain, abdominal distension, nausea, vomiting, diarrhoea, fever or indigestion.
•You have not lost weight and your appetite is OK.
•You have severe itching that has been worsening for some time.
•You have hypothyroidism and take tablets for this. You are also taking a statin for high cholesterol.
•You are fit and well otherwise.
•You work as a teacher and have done so since leaving university.
•Your husband is your only sexual partner.
•You were well throughout your stay in India, where you spent 2 weeks on holiday in Goa.
•You have a single gin and tonic every other evening.
•You have never been involved in intravenous (IV) drug use. You do not have any tattoos.
•You have never had any operations or blood transfusions.
•You are a nonsmoker.
•Your mother has hypothyroidism and pernicious anaemia, and she takes steroids but you' re not sure why.
•No one else in the family has been jaundiced.
•You live with your husband and two children.
•The GP has done some blood tests and organised a scan of your abdomen and sent you to see the specialist.
•You are worried it might be cancer.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient why they have come to see you and what has happened so far.
•Start with an open question about the jaundice.
•Ask specifically about the course of the illness and associated symptoms.
•Specifically enquire about the colour of the patient' s urine and stool.
•Ask about other gastrointestinal symptoms.
•Demonstrate to the examiners that you are aware of the risk factors for jaundice by asking about alcohol, sexual history, intravenous drug abuse, blood transfusions, occupational history and travel history.
•Take a drug history and be sure to ask about allergies.
•Ask about family history and establish if there are any other symptoms of autoimmune disease in this patient.
•Ask about social situation and ensure that you ask about impact on their life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns and expectations that they might have.
•Communicate to the patient that further investigations are required. Medication can be offered to treat the itching.
**THEMES EXPLORED**
•Painless jaundice is often caused by malignancy, although other causes need to be considered, especially in younger patients. PBC is possible here; clues include the hypothyroidism and hypercholesterolaemia.
•It is important to establish the patient' s concerns. Mrs Valdes is less likely to have a malignancy here, but this cannot be ruled out until after imaging.
•Symptom control is important, as well as making a diagnosis. Mrs Valdes' s itching should be addressed.
**RELEVANT INFORMATION**
**COMMON AUTOIMMUNE ASSOCIATIONS OF PBC**
•Sjö gren' s syndrome
•Autoimmune thyroid disease
•Rheumatoid arthritis
•Systemic sclerosis
•Diabetes mellitus
**INVESTIGATION OF SUSPECTED PBC**
•Deranged LFTs (raised ALP early in the disease, raised bilirubin later)
•Liver screen to rule out other causes, including USS of liver (or computerised tomography [CT] abdomen)
•Positive antimitochondrial antibody (M2 antibody is specific)
•Raised serum immunoglobulins, especially IgM
•Diffuse architectural change on liver ultrasound
•Hepatic granulomas, lymphocytic infiltrates and cirrhosis on liver biopsy
**CONTROL OF DISEASE**
•Early and midstages of disease are managed with ursodeoxycholic acid (UDCA) to change the balance of toxic and nontoxic bile salts (it is the autoimmune destruction of small bile ducts and the accumulation of toxic bile salts which causes cirrhosis; UDCA slows the progression of disease).
•Late stage of disease may require liver transplant.
**MANAGEMENT OF ITCH IN CHOLESTASIS**
•Topical agents ineffective.
•Cholestyramine is used first line (bile salt sequestrant).
•Rifampicin is used second line. Liver function needs to be closely monitored on this drug.
•Naltrexone, an opioid antagonist, is third line.
•Sertraline is fourth line.
REFERENCE
European Association for the Study of the Liver. (2009) EASL clinical practice guidelines: Management of cholestatic liver diseases. _J Hepatol_ 51(2): 237– 67.
PATIENT WITH JOINT PAINS
Mrs Leadbitter is a 34-year-old female who over the past 2 months has developed pains in the small joints of her hands. She is a secretary and is finding typing increasingly difficult. She also complains of stiffness in her fingers. She has type 1 diabetes mellitus. Her blood sugars are well controlled. On examination, she has tender joints in both hands, but otherwise, there is nothing abnormal to find.
**INSTRUCTIONS FOR THE PATIENT**
•You have pain in both of your hands, particularly the joints around the knuckles and the wrist.
•This pain started 2 months ago, and you feel it is getting worse.
•It is an aching pain.
•You have noticed that your fingers and hands are also stiff in the morning. This stiffness goes away as the day progresses.
•None of your other joints are affected.
•You have been feeling more tired recently, but put this down to working extra hours.
•You have not lost any weight.
•You do not have any rashes.
•You have had no problems with your eyes.
•You have no rashes, and your nails are fine.
•You do not have chest pain, but you have noticed the beginnings of a dry cough. You have no problems with your breathing.
•On close questioning, you do have a dry mouth.
•You have insulin-dependent diabetes and are meticulous with control of your blood sugars.
•Other than the diabetes, you are fit and healthy.
•You take insulin as a basal bolus regimen, simvastatin 40 mg once daily and ramipril
2.5 mg once daily.
•You do not have any allergies to medications.
•Your sister has pernicious anaemia.
•You live with your husband and three children.
•You work as a secretary for a law firm.
**SUGGESTIONS FOR THE CANDIDATE**
•Introduce yourself and check the patient' s name and age.
•Ask the patient why they have come to see you.
•Ask the patient what they know about what has happened so far (establish baseline).
•Establish which joints are affected. Are the symptoms in both hands? Are any other joints are affected?
•Ask specifically about stiffness and clarify when it occurs.
•Ask about any lumps and bumps (nodules).
•Make sure you ask about symptoms of inflammatory bowel disease in order to help rule out an enteropathic arthritis.
•Ask about urinary symptoms to help rule out reactive arthritis (although more commonly an oligoarthritis or large-joint arthritis).
•Demonstrate that you know the associations of inflammatory arthropathies by asking about
•Dry mouth and eyes (Sjö gren' s syndrome)
•Cough and dyspnoea (pleural effusions and lung fibrosis)
•Cardiac symptoms (pericarditis and pericardial effusions)
•Systemic symptoms (weight loss and lethargy)
•Rashes (psoriasis)
•Ophthalmic symptoms (episcleritis, scleritis and anterior uveitis)
•Ask about neurological symptoms in the hands, as carpal tunnel syndrome can develop in both rheumatoid arthritis and osteoarthritis.
•Take a thorough past medical history
•Take a thorough drug history and be sure to ask about allergies.
•Family history is important.
•Occupational history is important in this case.
•Try to ascertain the functional status of the patient.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns and expectations that they might have.
•Discuss a plan in terms of investigations and follow-up.
**THEMES EXPLORED**
•This case demonstrates the link between autoimmune diseases, with the patient having an established diagnosis of type 1 diabetes. There is also a family history of pernicious anaemia.
•This case gives the candidate the opportunity to show that they know both the causes and associations of inflammatory arthritides.
•A key point is to distinguish early between inflammatory and mechanical arthritis.
**RELEVANT INFORMATION**
**DIAGNOSTIC CRITERIA FOR RHEUMATOID ARTHRITIS**
•Rheumatoid arthritis was previously diagnosed using the American Rheumatism Association 1987 criteria. However, new criteria were published in 2010: the American College of Rheumatologists' criteria. For details on these, see the rheumatoid arthritis case in Station 5 of this book.
•Recent evidence suggests trying to identify patients early in the disease process, so that treatment with disease-modifying agents can be instituted early. National Institute for Health and Care Excellence (NICE) guidance recommends suspecting rheumatoid arthritis in those with ' persistent synovitis' (meaning it has been present for a few weeks) where no other underlying cause is obvious.
**PHARMACOLOGICAL MANAGEMENT OF RHEUMATOID ARTHRITIS**
•NICE guidelines now suggest using two disease-modifying antirheumatoid drugs (DMARDs) initially. This is generally methotrexate alongside another drug. DMARDs are commenced soon after diagnosis. A short course of glucocorticoids is also recommended to control symptoms quickly.
•After symptoms are controlled, the aim is to reduce DMARDs to their lowest effective dose.
•In those whom a combination of DMARDs is not effective, and in whom disease activity is severe, biological agents (including anti– tumour necrosis factor (TNF) monoclonal antibodies) are considered.
•Analgesics should be used where pain is problematic. Paracetamol is recommended first line, and weak opioids can also be considered. Nonsteroidal anti-inflammatory drugs (NSAIDs) and cyclo-oxygenase (COX)-2 inhibitors are likely to be effective, but they should be used for the shortest period of time possible to prevent side effects.
REFERENCES
National Institute for Health and Care Excellence. (2013) Clinical knowledge summary for rheumatoid arthritis. Available from http://cks.nice.org.uk/rheumatoid-arthritis#!topicsummary. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2015) Guidance CG79. Rheumatoid arthritis in adults: Management. Available from https://www.nice.org.uk/guidance/cg79/chapter/Recommendations#pharmacological-management. Accessed 5 July 2016.
PATIENT WITH LEFT ARM WEAKNESS
Mrs Abedney is a 62-year-old female, who developed left arm weakness 8 days ago, which has subsequently not improved. She was fit and healthy prior to this. There is no family history of cardiac disease; she is a retired bank clerk and lives with her husband. She is a nonsmoker. On examination, she has an upper motor neurone lesion affecting her left arm. Her blood pressure in clinic was 160/95. She has been told that she has had a stroke, and has initially been commenced on high-dose aspirin, atorvastatin and ramipril. Her serum cholesterol and random glucose are pending.
**INSTRUCTIONS FOR THE PATIENT**
•You were fit and healthy up to this event.
•You are right-handed.
•Two weeks ago, you noticed some tingling in your left arm. Since then, it has gradually become weaker and the whole of the left arm is now affected.
•The weakness is getting worse and not improving.
•The sensation in your left arm remains intact.
•Your right arm is normal.
•Your legs are normal.
•Your speech has been normal.
•You have not experienced any headaches or visual symptoms.
•You have not sustained a head injury.
•Your swallowing is normal.
•Your bladder and bowel functions have been normal.
•You have lost a stone in weight over the past 6 months. This was not planned.
•You have been increasingly tired in the last 3 months.
•You have no urinary, respiratory, cardiac or gastrointestinal symptoms.
•You have no family history of note.
•You are a nonsmoker and drink 2 units of alcohol/day.
•You saw your GP 7 days ago (1 day after the onset of your symptoms), and she said you had a ' mini-stroke' or transient ischaemic attack (TIA). She gave you some tablets and said she would refer you to a specialist.
•You are yet to have a scan of your head.
•The weakness has left you dependent on your husband to manage the house, and he now has to help you get ready in the mornings.
•You are concerned that despite treatment for a week, your symptoms have worsened. You feel a burden on your husband.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient why they have come to see you and what has happened so far.
•Ask the patient about their symptoms and establish a history of presenting complaint (you will note that the onset is gradual and the symptoms progressive, not in keeping with a stroke).
•Check the patient' s handedness.
•Ask the patient about other neurological symptoms.
•Ensure that you take a thorough past medical history, including asking about risk factors for stroke. Take a drug history.
•Ask about the patient' s social situation, including the impact of the symptoms on their life.
•Make sure that you do a thorough systems review, particularly asking about any symptoms that might suggest an underlying malignancy (especially once you have established there is a suggestion of weight loss and lethargy) or multiple sclerosis (ask about prior episodes of visual disturbance/incontinence and ataxia).
•Ask the patient about any particular concerns and expectations that they might have. Social support is important, as well as making a medical plan for the patient.
•Explain to the patient that cerebral imaging is crucial here for diagnosis. Explain that you will expedite this and see the patient again this week.
**THEMES EXPLORED**
•The history taken here leads you to think that the diagnosis could be a space-occupying lesion.
•The diagnosis may be different from the GP' s initial impression and from what the patient may have been told. Keep an open mind.
•Address the patient' s concerns, that investigations are required to make a diagnosis, and that social support is available.
**RELEVANT INFORMATION**
**CAUSES OF SPACE-OCCUPYING LESIONS**
•Benign tumour
•Malignant primary tumour
•Cerebral metastasis
•Colloid cyst
•Aneurysm
•Abscess
•Chronic subdural haematoma
**CANCERS MOST COMMONLY METASTASISING TO THE BRAIN**
•Breast
•Lung
•Melanoma
•Haematological malignancies (leukaemia and lymphoma) commonly cause leptomeningeal metastases.
**IMPACT ON DRIVING**
•In this case, the patient should not drive until a diagnosis is made, at which point advice can be taken from the DVLA.
•The DVLA' s rules are quite complex. A craniotomy requires at least a 6-month period off driving, with varying rules for benign tumours and at least 1– 2 years off for malignant tumours post-treatment.
REFERENCES
DVLA. (2016) Assessing fitness to drive – A guide for medical professionals. Available from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/517268/Fitness_to_drive.pdf. Accessed 5 July 2016.
Giglio P, Gilbert MR. (2010) Neurologic complications of cancer and its treatment. _Curr Oncol Rep_ 12(1): 50– 59.
PATIENT WITH VISUAL DISTURBANCE
Mr Atherton is a 69-year-old male who has gradually developed blurred vision over the past 3 days. He also complains of a nonspecific headache. He has a past history of hyper-tension, atrial fibrillation and hypercholesterolaemia, and had an OGD 1 year ago which demonstrated peptic ulcer disease.
On examination, he looks generally well; his pulse is irregularly irregular, his blood pressure is 145/83 and his heart sounds are normal. Neurological assessment is normal. His GP thinks he is having transient ischaemic attacks. His medications are as below:
•Aspirin 75 mg OD
•Simvastatin 40 mg ON
•Bisoprolol 10 mg OD
•Candesartan 8 mg OD
•Lansoprazole 30 mg OD
•Quinine sulphate 300 mg ON
**INSTRUCTIONS FOR THE PATIENT**
•You have had visual disturbance for a few days. Your vision is generally blurred, and you cannot read the small print of the newspaper as well as you used to.
•The visual disturbance is getting worse.
•You have not had anything like this before.
•You don' t wear glasses and last went to the optician 1 year ago, when your vision was reported to be normal.
•You also have a headache which has come on at the same time.
•This headache hasn' t particularly bothered you, but it is not normal and you notice that it hurts when you wash your hair in the shower.
•You are also noticing pain when you chew.
•You don' t have any floaters or flashing lights, or any symptoms of a ' curtain coming down' .
•You don' t have any nausea or vomiting.
•There has been no recent head injury.
•Your legs, face and speech have been fine, although you have pain in both shoulders. Swallowing and bladder and bowel function are normal.
•Your doctor says you have high blood pressure, high cholesterol and an irregular heart beat, and you take some tablets for all these (you don' t have any allergies).
•You had some abdominal pain last year and were found to have a stomach ulcer.
•You live with your wife, who has dementia. You are her main carer.
•You worked as a banker in the past.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient why they have come to see you and what they know about what has happened so far.
•Ask an open question to start with about the blurred vision.
•Ask about floaters, flashing lights, diplopia, eye pain and the use of glasses.
•Ask specifically about the headache and what precipitates it.
•Ask about other neurological symptoms.
•Show the examiners you are working through your differentials:
•Glaucoma (sudden onset, nausea and vomiting)
•Space-occupying lesion (early morning headache, worse on sneezing/coughing/ leaning forward)
•Cerebrovascular disease (sudden-onset symptoms that improve with time, risk factors)
•Temporal arteritis (headache, scalp tenderness)
•Once you establish the likelihood of temporal arteritis, demonstrate to the examiners that you know it is associated with polymyalgia rheumatica and ask about symptoms such as fatigue and joint stiffness.
•Take a thorough past medical history, as well as a drug history, including allergies.
•Ask about family history.
•Ask about the patient' s social situation and ensure that you ask about the impact on their life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns or expectations that they might have.
•Explain to the patient that while further investigations are required, treatment (corticosteroids) needs to commence immediately, as the treatment would be sight saving.
•Remember to discuss the potential risks of high-dose steroids. These include steroid-induced diabetes, gastro-oesophageal reflux disease (GORD), osteoporosis, avascular necrosis and steroid-induced psychosis.
**THEMES EXPLORED**
•Have an open mind as to the diagnosis (this case is about ensuring that you take your own thorough history and not to be swayed by the provisional diagnosis from the GP).
•Associations are important; once you establish the likelihood of giant cell arteritis, show the examiners that you know it is associated with polymyalgia rheumatica.
•A treatment plan should be communicated to the patient where prompt action is required.
**RELEVANT INFORMATION**
**GIANT CELL ARTERITIS**
•A large artery inflammatory granulomatous arteritis.
•It presents with the following symptoms/signs:
•Headache, either occipital or unilaterally over the temporal region
•Scalp/temporal tenderness
•Tenderness of the temporal/occipital arteries
•Jaw claudication
•Sudden visual loss
•Constitutional symptoms: Malaise, fever
•Erythrocyte sedimentation rate (ESR) and C-reactive protein (CRP) are likely to be elevated on testing. Biopsy of the temporal artery may give a definitive diagnosis (skip lesions are common).
•Early treatment with high-dose corticosteroids is imperative. If a patient has visual symptoms, they should be commenced on prednisolone 60 mg OD and seen urgently by an ophthalmologist. Without visual symptoms, the starting dose is 40– 60 mg prednisolone OD. Corticosteroids can be tapered down slowly, but treatment with steroids for at least 1– 2 years is usual. Provide osteoporosis prophylaxis.
•Low-dose aspirin (75 mg OD) with a PPI for gastric protection should also be commenced if the patient is not already on these medications and does not have contraindications.
**POLYMYALGIA RHEUMATICA**
•A disease affecting the muscles around the shoulder, neck and hips. It is far more common in patients over 70. It presents with the following symptoms/signs:
•Morning muscle stiffness, lasting over 45 minutes
•Muscle tenderness around the shoulder, neck and hips
•Fatigue
•Fever
•Anorexia and weight loss
•ESR and CRP are generally raised on testing, while creatine kinase (CK) is normal. It should be distinguished from other illnesses, including rheumatoid arthritis.
•As with giant cell arteritis, treatment is with corticosteroids. However, lower doses are required; start at prednisolone 15 mg OD. This is tapered down slowly over 1– 3 years. Provide osteoporosis prophylaxis to cover the corticosteroids and for gastric protection.
REFERENCES
National Institute for Health and Care Excellence. (2013) Clinical knowledge summary: Polymyalgia rheumatic. Available from http://cks.nice.org.uk/polymyalgia-rheumatica#!scenario. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2014) Clinical knowledge summary: Giant cell arteritis. Available from http://cks.nice.org.uk/giant-cell-arteritis#!scenario. Accessed 5 July 2016.
PATIENT WITH WEIGHT LOSS
Miss Alfonso is a 29-year-old student who has lost 3 stone in weight over the last 6 months. The weight loss has been unintentional. She complains of associated reduced appetite and lethargy. She is a first-year nursing student and is finding it difficult to complete her studies. A systems review was unremarkable. She has no past medical history except for recurrent urinary tract infections (UTIs), for which she has received several courses of antibiotics and made a good recovery.
Examination reveals a thin female. Her abdomen is soft and nontender, with a clear chest and normal heart sounds. Preliminary blood tests have been sent but are not yet available.
**INSTRUCTIONS FOR THE PATIENT**
•You have lost 3 stone in weight in the last 6 months, going from 12 to 9 stone.
•You have not been intending to lose weight.
•You have noticed that you have become increasingly thirsty, to the extent that you take a jug of water to bed.
•You wake up twice at night to pass urine.
•You have noticed that you are passing more urine more often than before.
•You have had two courses of antibiotics in the last 6 months for urinary tract infections.
•You don' t have any other symptoms.
•Prior to this, you were fit and healthy. You no take regular medications and have no allergies.
•Your mother has thyroid problems, but there is no other family history.
•You are a nursing student and live with friends (who have also been concerned regarding your weight loss and symptoms).
•You don' t smoke or drink alcohol.
•You are worried you may have diabetes.
**SUGGESTIONS FOR THE CANDIDATE**
•Ensure that you introduce yourself and check the patient' s name and age.
•Ask the patient if they know why they have come to see you and what has happened so far.
•Start with an open question about her weight loss.
•Quantify the weight loss and associated symptoms.
•Demonstrate to the examiners that you are covering symptoms of potential causes of weight loss by performing a thorough systems review and particularly looking for the following features:
•Inflammatory bowel disease: Ask about diarrhoea (including blood and mucus), constipation, mouth ulcers, abdominal pain and perianal disease.
•Malignancy (including haematological): Ask about bruising, bleeding, night sweats, fever and a thorough systems review to unveil potential sites of malignancy.
•Diabetes mellitus: Polyuria, polydipsia, nocturia and recurrent infections.
•Hyperthyroidism: Irritability, eye symptoms, heat intolerance, tremor, sweating and palpitations.
•Addison' s disease: Dizziness and abnormal pigmentation.
•Inflammatory conditions: Enquire about back pain, joint pain and rashes.
•Chronic infections (including TB): Ask about travel history and infectious contacts. Ask about night sweats, haemoptysis and fever.
•Psychiatric: Enquire about the patient' s mood.
•Drug and smoking history.
•Ask about the patient' s social situation and ensure that you ask about impact on her life.
•Be sure to summarise appropriately.
•Ask the patient about any particular concerns that they might have (particularly relevant here).
•Suggest that diabetes mellitus is certainly a possible cause, although further investigations are required to diagnose this and rule out other causes.
•Give some initial information about diabetes mellitus. This might include that if she has diabetes, this will likely be type 1, which would require lifelong insulin therapy (starting immediately).
**THEMES EXPLORED**
•Autoimmune diseases are commonly linked, especially in the Practical Assessment of Clinical Examination Skills (PACES) setting.
•Weight loss is not always due to malignancy, especially in young patients. However, it is important to mention this in a differential, with the likelihood higher the older the patient is.
•Cases with a wide differential can be covered in the time period, so don' t become disheartened when given a nonspecific symptom.
**RELEVANT INFORMATION**
**WORLD HEALTH ORGANIZATION CRITERIA FOR DIAGNOSIS OF DIABETES MELLITUS**
Criteria include either of the following:
•Symptoms of hyperglycaemia and a raised venous glucose (fasting ≥ 7 mmol/L or random ≥ 11.1 mmol/L)
•Raised venous glucose on two separate occasions (fasting ≥ 7 mmol/L or random ≥
11.1mmol/L or oral glucose tolerance test 2-hour value ≥ 11.1 mmol/L)
Note that diabetes can now also be diagnosed with a raised HbA1c; a HbA1c ≥ 6.5% (48 mmol/mol) is diagnostic in a symptomatic patient, or alongside a raised blood glucose (as above). Of note, this level is also used as the treatment target level in adults to mini-mise the risk of vascular complications.
**AUTOIMMUNE DISEASE HUMAN LEUCOCYTE ANTIGEN ASSOCIATIONS**
•HLA-B27: Ankylosing spondylitis
•HLA-DR3: Type 1 diabetes mellitus, Graves' disease, Hashimoto' s thyroiditis, Addison' s disease, systemic lupus erythematosus (SLE), myasthenia gravis
•HLA-DR4: Type 1 diabetes mellitus, Hashimoto' s thyroiditis, rheumatoid arthritis
REFERENCES
Gough SCL, Simmonds MJ. (2007) The HLA region and autoimmune disease: Associations and mechanisms of action. _Curr Genomics_ 8(7): 453– 65.
National Institute for Health and Care Excellence. (2015) Guidance NG17. Type 1 diabetes in adults: Diagnosis and management. Available from https://www.nice.org.uk/guidance/ng17. Accessed 5 July 2016.
World Health Organization. (2006) Definition and diagnosis of diabetes mellitus and intermediate hyperglycaemia. Available from http://who.int/diabetes/publications/Definition%20and%20diagnosis%20of%20diabetes_new.pdf. Accessed 5 July 2016.
World Health Organization. (2011) Use of glycated haemoglobin (HbA1c) in the diagnosis of diabetes mellitus. Available from <http://www.who.int/diabetes/publications/report-hba1c_2011.pdf>. Accessed 5 July 2016.
HISTORY-TAKING STATION SUMMARY
•This station may give you an opportunity to score highly, but you should _not_ take it for granted. Remain focused and target this station for scoring well (which will allow you some leeway on the stations you find more challenging).
•Do the simple things well, be polite and courteous and introduce yourself.
•Use your 5 minutes before you enter the room wisely; plan an approach to the case and look to structure your approach.
•Start with an open question, as the patient may well give you the full history and all the salient points; if they don' t, then you will still have time to fill in the gaps.
•Try not to interrupt the patient, but if they are rambling, you must do so subtly and politely; you must practise this beforehand.
•Remember to look for hidden agendas, particularly if it is an apparently easy case (e.g. is it a chronic hypertensive who is noncompliant with meds because of impotence brought on by beta-blockers but who is too embarrassed to tell you this).
•Use all the time you are given. It would be embarrassing to have to sit in silence for 5 minutes if you finish early. Summarising is a useful tool, as it will remind you of things that you still need to ask.
•Allergies are an important question, as you cannot manage your patient until you know if they are allergic to specific medicines.
•Read the instructions very carefully; do not start explaining the management plan unless it specifically asks you to. The likelihood is that they will only want you to take a history. The examiners may then ask you what your approach will be when they question you.
•An easy station to practise, you should do it in groups with one person being the patient, one being the candidate and the others being the examiners.
Station 3: Neurology
Hints for the Neurology Station
Cerebellar Syndrome
Hemiparesis
Myotonic Dystrophy
Ocular Palsies
Parkinson' s Disease
Peripheral Neuropathy
Mononeuropathies
Motor Neurone Disease
Multiple Sclerosis
Spastic Paraparesis
Visual Field Defects
Neurology Station Summary
HINTS FOR THE NEUROLOGY STATION
•Ensure that you pay close attention to the lead-in statement, as this will guide your examination. Once you have decided on a system, be efficient with your time for this.
•Neurology cases lend themselves to being spot diagnoses (e.g. myotonic dystrophy, Parkinson' s disease [PD], old polio and myasthenia gravis). Therefore, be alert from the moment you walk into the room.
•Looking around the bed is very relevant in neurology (e.g. calipers, walking aids, devices for measuring forced vital capacity [FVC]).
•If you spot an obvious deformity (claw hand, wrist drop, Charcot' s joint), be sure to direct your examination towards this.
•When commenting on power, use the Medical Research Council (MRC) scale to grade it.
•Neurology is often thought of as being a particularly difficult station, but if you work on the premise that you are trying to find the site of the lesion (e.g. cerebrum, cerebellum, brainstem, spinal cord, anterior horn, peripheral nerve or muscle) and then consider the differential diagnosis, you will demonstrate to the examiners that you are organised and pragmatic.
•Have set routines for examining the cranial nerves and the peripheral nervous system (upper limb and lower limb), but be prepared to adapt your examination (e.g. you may be asked to examine just cranial nerves III– VII). Neurology particularly lends itself to this type of adjustment.
•You will earn yourself extra marks if you look for the underlying pathology; e.g. blood glucose pinprick marks, granuloma annulare and cataracts in a case of peripheral neuropathy indicate that diabetes is the underlying cause.
•If you are unable to elicit reflexes, you must demonstrate that you have attempted to elicit them with reinforcement.
•When assessing sensation, be sure to check for normal sensation on the chest first. Compare normal sensation with the peripheral sensation (i.e. ask if the sensation is different to the sensation on the chest, not merely ' Can you feel it?' ).
•It is important to decide early whether to assess sensation in a dermatomal or a peripheral (' glove and stocking' ) pattern. This is not always possible. Therefore, start in a dermatomal distribution and move on to test glove and stocking pattern if peripherally confluent.
•Although the patients may have been examined many times and will be well versed in the routines, you must give clear instructions about what you want them to do (remember that they' re not allowed to help you).
•At the end of the examination, ask to examine other relevant neurological areas (including fundoscopy).
•If you believe that the diagnosis is related to vascular disease, then also mention that you would like to examine the cardiovascular system for risk factors.
CEREBELLAR SYNDROME
Please examine this patient who has had problems with his balance, and recurrent falls over the past 6 months.
**FINDINGS**
• _General:_ Action tremor
• _Neurological:_ Ataxic gait, truncal ataxia, hypotonia, pendular reflexes, dysmetria, dysdiadochokinesis, nystagmus, slurred speech, heel– shin ataxia
• _Extras:_
•Features of multiple sclerosis (MS) (spastic paraparesis, sensory disturbance, internuclear ophthalmoplegia [INO])
•Stigmata of chronic liver disease
•Friedreich' s ataxia (young patient, wheelchair/walking aids, pes cavus, kyphoscoliosis, absent ankle jerks with upgoing plantars)
**PRESENTATION**
This man has cerebellar dysfunction. He has an ataxic gait with nystagmus and dysarthria. He also has multiple stigmata of liver disease. The most likely cause is excess alcohol consumption.
**CAUSES OF CEREBELLAR SYNDROME**
•Multiple sclerosis
•Alcoholic cerebellar degeneration
•Posterior fossa space-occupying lesion
•Brainstem vascular lesion
•Inherited ataxias (e.g. Friedreich' s) (note that the history would be longer than 6 months)
•Paraneoplastic syndromes
•Drugs (phenytoin)
**INVESTIGATIONS**
•Dependent on the presumed cause of cerebellar signs
•If multiple sclerosis is suspected,
•Cranial and spinal imaging (magnetic resonance imaging [MRI])
•Lumbar puncture (LP) (oligoclonal bands and protein in cerebrospinal fluid [CSF])
•Visual-evoked potentials
**MANAGEMENT**
•Dependent on the cause of cerebellar syndrome
**QUESTIONS**
1.How would you identify the site of a cerebellar lesion from clinical findings?
•The cerebellum is divided into a midline vermis and two cerebellar hemispheres.
•Disease of the vermis leads to truncal ataxia and ataxic gait.
•Disease of a hemisphere causes ipsilateral dysmetria, dysdiadochokinesis, an intention tremor and fast-beat nystagmus towards the lesion.
•Multiple sclerosis (demyelination) can cause a global deficit.
2.What is Friedreich' s ataxia?
•Autosomal recessive disorder; a trinucleotide repeat on chromosome 9.
•Degeneration of the spinocerebellar tract resulting in cerebellar signs.
•Corticospinal tract damage and peripheral nerve degeneration lead to absent ankle jerks with extensor plantars.
•Pes cavus, scoliosis and diabetes are common features. Other features include cardiomyopathy, cataracts and sensorineural deafness.
•Think of this in a younger patient.
**KEY POINTS**
•Prepare a speci?c examination for the cerebellar syndrome.
•Split the examination into two parts. First, ensure that you have demonstrated that the patient has cerebellar signs. Second, ensure that you have demonstrated that you have looked for specific underlying causes.
HEMIPARESIS
This patient has developed weakness. Please examine their upper limbs.
**FINDINGS**
• _General:_ Walking aids, percutaneous endoscopic gastrostomy (PEG) tube, wasting/ oedema on affected side, upper limbs held in flexion, lower limbs held in extension
• _Peripheral:_ Bruising (on warfarin), amiodarone facies, irregularly irregular pulse
• _Neurological:_
•Increased tone
•Reduced power (use MRC grade for power)
•Hyperreflexia and extensor plantars
•Decreased sensation
•Hemianopia
• _Extras:_
•Observe for upper motor neurone (UMN) facial nerve lesion.
•Note any dysphasia/dysarthria.
•Neglect/visual or sensory inattention.
•Mention that you would like to perform a full cardiovascular examination.
**PRESENTATION**
This patient has a right-sided upper motor neurone lesion affecting arm, leg and face. The likely cause is a stroke, but I would include space-occupying lesion and demyelination in my differential diagnosis. I would like to examine his cardiovascular system, including blood pressure readings, in order to look for a predisposing cause. He mobilises with a frame.
•Comment on the side of stroke and the areas involved.
•Comment on any obvious underlying predisposing factors.
•Comment on functional status.
**INVESTIGATIONS FOR STROKE**
•Neuroimaging (computerised tomography [CT]/MRI)
•Blood pressure
•Fasting glucose/cholesterol
•Electrocardiogram (ECG)/24-hour tape
•Echocardiogram
•Carotid artery Doppler
**MANAGEMENT OF STROKE**
•Thrombolysis (provided meets criteria).
•High-dose aspirin for 2 weeks, followed by clopidogrel for secondary prevention.
•Use a tool such as the NIH Stroke Scale to assess patient functionality at onset of stroke and at regular intervals after onset.
•Speech and language assessment.
•Management of predisposing factors and secondary prevention.
•Rehabilitation (occupational therapist [OT] and physiotherapy).
•Psychological and nutritional support.
•Measure progress with tools, e.g. Modified Rankin Score to assess prognosis.
**QUESTIONS**
1.What extra investigations are merited in young patients with a proven stroke?
•Thrombophilia screen
•Homocystine levels (postulated to promote atherosclerosis leading to stroke)
•Bubble echo (looking for a patent foramen ovale)
2.Do you know of any alternative anticoagulants to warfarin in patients who may have suffered a thrombotic stroke and have atrial fibrillation?
•Factor 10A inhibitors, e.g. apixaban and rivoroxaban.
•Dabigitran is a direct thrombin inhibitor (Factor IIa).
3.What is ' lateral medullary syndrome' ?
•This is also known as Wallenberg' s syndrome; it is caused by a brainstem stroke in the territory of the vertebral or posterior inferior cerebellar artery.
•Clinical features include
•Ipsilateral signs: Horner' s syndrome, nystagmus, facial sensory impairment, ataxia and diplopia
•Contralateral signs: Pain and temperature loss over opposite arm and trunk (spinothalamic tract)
4.Discuss the advantages and disadvantages of novel anticoagulant therapies versus warfarin.
•Advantages
•Rapid onset of action – no need for bridging therapy.
•Fewer interactions with other medications/foods.
•Provided patients are compliant, they are always in the therapeutic window (unlike warfarin).
•No need for repeated blood tests to monitor the international normalised ratio (INR).
•Disadvantages
•Not safe in renal failure – dose dependent on estimated glomerular filtration rate (eGFR). Consult the British National Formulary (BNF) before prescribing.
**KEY POINTS**
•Establish the diagnosis and the cerebral area(s) affected.
•Mention to examiners that you would like to examine the cardiovascular system (for risk factors).
•Look for evidence of disability caused by the hemiparesis and comment on this.
REFERENCE
National Institute for Health and Care Excellence. (2010) Guidance QS2. Stroke in adults. Available from https://www.nice.org.uk/guidance/qs2/resources/stroke-in-adults-58292707525. Accessed 24 August 2016.
MYOTONIC DYSTROPHY
This patient, who is undergoing preoperative evaluation for elective surgery, is noted to have bilateral ptosis. Please examine their face and proceed.
**FINDINGS**
• _General:_ ' Myopathic facies' , wasting of sternocleidomastoid muscle group, speech (nasal or dysarthric), pinprick marks on fingertips from blood glucose testing
• _Face:_ Bilateral ptosis, wasting of facial muscles with hollowing of temporal fossae and cheeks, frontal baldness, smooth forehead, cataracts
• _Hands:_ Generalised weakness and wasting of upper limbs
• _Key points:_ The candidate must be able to demonstrate myotonia.
• _Grip myotonia:_
•Ask the patient to quickly clench their fist as tightly as possible and then immediately release. This will demonstrate a slow release instead of rapid finger extension.
•This can also be demonstrated by shaking the patient' s hand, which will demonstrate slow release of grip.
• _Percussion myotonia:_ This can be demonstrated by using a tendon hammer to tap the thenar eminence, which will display a muscle twitch, followed by a slow relaxation of the muscle group.
**PRESENTATION**
On examination, this patient has evidence of myotonic dystrophy. This is demonstrated by the myopathic facies: elongated face, wasting of temporal muscles, frontal balding and bilateral ptosis. I have demonstrated evidence of myotonia, as there was slow release of grip after shaking his hand and percussion myotonia on tapping the thenar eminence.
•Comment on any evidence of complications (see below).
**DIAGNOSIS**
•Electromyography (EMG)
•Muscle biopsy
•Genetic analysis: Expansion of cytosine– thymine– guanine (CTG) triple repeat on long arm of chromosome 19
**MANAGEMENT**
•Reduction of myotonia using drugs such as phenytoin, mexiletine and carbamazepine (these drugs are sodium channel blockers; they reduce myotonia but may increase weakness)
•Identification and treatment of associated complications (see below)
•Genetic counselling
•Avoidance of general anaesthesia where possible
**QUESTIONS**
1.List some complications associated with myotonic dystrophy.
•Cardiac
•Dilated cardiomyopathy, cardiac arrhythmias
•Respiratory
•Risk of aspiration due to muscle weakness; may require noninvasive ventilation due to myotonia affecting respiratory muscles
•Gastrointestinal
•Dysphagia, delayed gastric emptying
•Endocrine
•Increased risk of diabetes, thyroid dysfunction
•Reproductive
•Testicular atrophy, infertility
•Other
•Cataracts
2.What is the genetic basis of this condition?
•Autosomal dominant
•Trinucleotide-repeat disorder showing genetic anticipation (expansion of an unstable CTG trinucleotide repeat in the myotonic protein kinase gene)
•Affects a gene located on chromosome 19
3.What problems are associated with general anaesthesia?
•Sedatives and neuromuscular blocking drugs may lead to cardiorespiratory complications and delayed recovery from anaesthesia.
•Depolarising neuromuscular blocking agents should be avoided, e.g. suxamethonium (can lead to induced myotonia).
4.What changes would be detected on EMG in myotonic dystrophy?
•Electromyography shows the electrical potential generated by muscles when they are neurologically or electrically activated.
•Myotonic dystrophy produces high-frequency activity that varies, producing a whining sound on the loudspeaker (' dive-bomber' ).
5.Describe what is meant by _anticipation_ ?
•Anticipation is a phenomenon whereby as a genetic disorder is passed on to the next generation, the symptoms of the genetic disorder become more apparent at an earlier age with each generation.
**KEY POINTS**
•Try to identify myotonic dystrophy early from the characteristic facial changes.
•Be sure to examine for specific signs of myotonia (grip and percussion).
•Look for evidence of extraneurological complications.
OCULAR PALSIES
Please examine this patient' s eyes, as they have been complaining of double vision.
Note: The term _ocular palsy_ refers specifically to the loss of function of an ocular muscle due to pathology in the nerve supplying it. We have also considered other causes of diplopia in this section.
**FINDINGS**
• _General:_ Walking aids, eye patch, ptosis
• _Neurological:_
•Third (oculomotor) nerve palsy: Complete ptosis, eye looks down and out, pupil may be dilated and unreactive dependent on the cause
•Fourth (trochlear) nerve palsy: Causes weakness of downward movement of eye, causing vertical diplopia (rare)
•Sixth (abducens) nerve palsy: Inability to abduct affected eye
•Complex ophthalmoplegia: A combination/no specific nerve involvement
•INO: Impaired adduction, unilaterally or bilaterally
• _Extras:_
•Stigmata of diabetes mellitus
•Myasthenia gravis: Bilateral ptosis, fatiguability, thymectomy scar
•Graves' disease: Proptosis, neck lump/scar
•Multiple sclerosis: Spastic paraparesis, cerebellar signs, walking aids/wheelchair
•Mitochondrial diseases: Hearing aid, proximal myopathy, ataxia, pacemaker (cardiomyopathy)
•Miller– Fisher syndrome: Peripheral neuropathy, ataxia, areflexia
**PRESENTATION**
This patient has a complex external ophthalmoplegia, as evidenced by diplopia in directions of gaze that are not attributable to a single nerve lesion. The likely cause is myasthenia gravis, as this patient also has bilateral ptosis, demonstrable fatiguability and a midline thoracotomy scar which would be consistent with a prior thymectomy.
•Where possible, try to work out which nerve is the culprit, though bear in mind that the patient may have a complex ophthalmoplegia.
**INVESTIGATIONS**
•Neuroimaging if a nerve lesion is suspected. MRI is most helpful, as it is important to obtain good views of the brainstem and posterior fossa.
•Investigations for causes of mononeuritis multiplex (see the section on mononeuropathy).
•Investigations for myasthenia gravis and thyroid disease in a complex ophthalmoplegia.
**MANAGEMENT**
•Dependent on the underlying cause
**QUESTIONS**
1.What are the causes of an oculomotor nerve palsy?
•' Surgical' : These causes generally affect the pupil.
•Posterior communicating artery aneurysm
•Space-occupying lesion in midbrain/sphenoid wing/near cavernous sinus
•Haemorrhage
•' Medical' : These causes often do not affect the pupil.
•Causes of mononeuritis multiplex
•Demyelination
•Infarction
2.What are the causes of an abducens nerve palsy?
•Causes of mononeuritis multiplex.
•Vascular lesion.
•Malignancy.
•Demyelination.
•Infection (Lyme disease, syphilis).
•Raised intracranial pressure (' false localising' sign).
•Wernicke' s encephalopathy can cause a bilateral abducens nerve palsy.
3.What are the causes of a complex ophthalmoplegia?
•Nerve lesions: Demyelination, mononeuritis multiplex
•Neuromuscular junction: Myasthenia gravis
•Muscle: Graves' disease
•Mitochondrial disease
4.What are the causes of an internuclear ophthalmoplegia?
•Multiple sclerosis
•Vascular disease
**KEY POINTS**
•Attempt to work out if the ocular palsy is due to a speci?c nerve lesion. If this is not possible, consider a complex ophthalmoplegia.
•The pupil may give some idea towards aetiology in an oculomotor palsy.
PARKINSON' S DISEASE
Please examine this patient' s gait and proceed. They have presented with unsteadiness.
**FINDINGS**
• _General:_ Patients can be in the ' off' medication state (poverty of facial expression, slowness of movement, reduced blinking, etc., are clues prior to examination) or in the ' on' medication state (excessive choreodystonic movements due to levodopa-induced dyskinesia).
• _Neurological:_
•Resting tremor (4– 6 Hz), rigidity (can be cog-wheeling in presence of rest tremor), bradykinesia (progressive decrement of amplitude of repetitive movements such as finger tapping).
•Gait: Ignition failure, short stride length, festination, freezing.
•Soft and monotonous speech.
•Micrographia.
•Rigidity can be demonstrated with concurrent activity (synkinesis – waving the right arm when assessing for the left). Distraction can bring out rest tremor – arm tremor is sometimes only revealed when testing gait or asking the patient to do a cognitive task like serial seven subtraction.
• _Extras:_
•Advanced therapies: Infusion pumps – apomorphine (subcutaneous) or duodopa (via percutaneous endoscopic jejunostomy [PEJ]), deep brain stimulation (implantable pulse generator or chest or abdominal wall – resembles cardiac pacemaker).
•Full assessment should also include mood, cognition (nonmotor symptoms) and functional status.
•Demonstrate that you are aware of Parkinson-plus syndromes by assessing eye movements and cerebellar signs.
**PRESENTATION**
This patient has evidence of Parkinson' s disease. There is a resting tremor, as well as rigidity and bradykinesia. I note an apomorphine infusion being administered via a pump.
•This is a common presentation at the neurology station and should be recognisable from the classic triad of parkinsonism: bradykinesia, rigidity and a resting tremor.
•Be sure to discuss the functional status of the patient and consider causes other than just Parkinson' s disease.
•Red flags for Parkinson-plus syndromes include symmetrical onset, early falls (patient in a wheelchair), lack of levodopa responses, supranuclear gaze palsy (limited downgaze) and cerebellar signs.
**CAUSES OF PARKINSONISM**
•Parkinson' s disease.
•Drugs (neuroleptics).
•Parkinson-plus syndromes: Progressive supranuclear palsy (Steele– Richardson– Olszewski syndrome), multisystem atrophy (Shy– Drager syndrome).
•Stroke or space-occupying lesion affecting the basal ganglia.
•Postencephalitis.
•Wilson' s disease.
•Other disorders resembling parkinsonism include dementia (including normal-pressure hydrocephalus) and tremulous conditions like benign essential tremor (action rather than rest tremor, i.e. postural and kinetic tremor).
**INVESTIGATIONS**
•Parkinson' s disease is generally a clinical diagnosis.
•Supportive evidence may be provided by a therapeutic trial of levodopa or an apomorphine challenge.
•Cerebral imaging may be appropriate where other diagnoses require exclusion.
•A dopamine transporter (DaT) scan can be used to support a diagnosis of Parkinson' s disease or to rule out drug-induced parkinsonism (but cannot distinguish idiopathic PD from PD-plus syndromes).
**MANAGEMENT**
•Multidisciplinary team (MDT) approach
•Specialist, physiotherapist, OT, specialist nurse, speech and language therapist (SALT)
•Levodopa formulations
•Other medications
•Dopamine agonists, e.g. rotigotine patch, ropinirole, pramipexole
•Anticholinergics: Trihexyphenidyl (for tremor)
•Selegiline, rasagiline (monoamine oxidase B [MAOB] inhibitors)
•Tolcapone (rarely used due to hepatic side effects), entacapone, opicapone (new) (catechol-o-methyltransferase [COMT] inhibitors)
•Apomorphine (dopamine agonist)
•Complications
•Assess disability and cognition regularly; treat depression
•Surgery
•For example, deep brain stimulation of subthalamic nucleus or internal segment of globus pallidus may be considered for those with motor fluctuations or levodopa-induced dyskinesia.
**QUESTIONS**
1.What are the characteristic features of Parkinson' s disease?
•The triad of parkinsonism (bradykinesia, rigidity, resting tremor).
•In Parkinson' s disease, these features begin asymmetrically and generally affect the upper limbs first. There is fluctuation in severity, with the patient being better some days than others.
•Autonomic dysfunction, loss of postural reflexes and balance and changes in higher mental functioning are also common later in the illness.
2.What drugs other than L-dopa are used in the management of Parkinson' s disease?
•Dopamine agonists (e.g. ropinirole): May be used as first-line therapy, especially in younger patients (due to the decreased risk of dyskinesias), or as an add-on to levodopa.
•Anticholinergics (e.g. trihexyphenidyl): Useful for tremor.
•Monoamine oxidase B inhibitors (selegiline): May be helpful with motor symptoms.
•COMT inhibitors (entacapone): Help to decrease immobility by shortening the ' off ' time associated with L-dopa.
•Apomorphine: A parenteral dopamine agonist that can be helpful with ' on– off' effects. It may be given via a syringe driver.
3.What is the pathology underlying Parkinson' s disease?
•Degeneration of the substantia nigra dopaminergic neurones in the basal ganglia. The hallmark is the presence of Lewy bodies.
4.Describe the potential problems patients need to be counselled about before commencing dopamine agonists.
•Impulse control disorders
•Hyperreligiosity
•Gambling
•Hypersexuality
•Compulsive eating
**KEY POINTS**
•Have a speci?c examination prepared for Parkinson' s disease.
•Look for the triad of parkinsonism early, and then search for extra features.
REFERENCE
Benabid AL, et al. (2009) Deep brain stimulation of the subthalamic nucleus for the treatment of Parkinson' s disease. _Lancet Neurol_ 8(1): 67– 81.
PERIPHERAL NEUROPATHY
Please examine this patient who has painful legs.
**FINDINGS**
• _General:_ Walking aids, diabetic shoes, prostheses, insulin pen
• _Neurological:_
•' Glove and stocking' sensation loss
•All sensory modalities should be tested (use a 128 Hz tuning fork to assess vibration sense, as dorsal columns are usually affected first)
•Wasting, weakness, areflexia
• _Extras:_
•Finger-prick testing, cataracts, ulcers, Charcot' s joints, callus (diabetes mellitus)
•Clawing of the toes, pes cavus (Charcot– Marie– Tooth disease)
•Amiodarone facies
•Anaemia (B12 deficiency)
•Evidence of alcohol abuse
•Evidence of arthritis and rashes (vasculitis)
**PRESENTATION**
This patient has a predominantly sensory peripheral neuropathy, as evidenced by lack of sensation bilaterally to midcalf for all modalities. There is no evidence of any ulceration or callus formation. The most likely underlying cause is diabetes mellitus, as evidenced by finger pulp pricks from capillary blood glucose testing.
•It is important to first ensure that the patient has a peripheral neuropathy rather than a mononeuropathy or mononeuritis multiplex.
•This is a common Practical Assessment of Clinical Examination Skills (PACES) case, and looking for a cause is crucial. Try to decide if the neuropathy is predominantly motor or sensory, as this should help to elucidate the aetiology.
**CAUSES OF NEUROPATHY**
•Predominantly sensory neuropathy
•Diabetes mellitus
•Alcohol
•Drugs
•Vitamin deficiencies (B1, B12)
•Uraemia
•Predominantly motor
•Guillain– Barré syndrome
•Malignancy
•Charcot– Marie– Tooth disease
•Porphyria
•Lead poisoning
•Other causes include paraneoplastic syndromes, paraproteinaemia, vasculitis and infections (human immunodeficiency virus [HIV], Lyme disease).
•Some cases are idiopathic.
**INVESTIGATIONS**
•Full drug and alcohol history
•Blood tests: Full blood count (FBC) (including mean cell volume [MCV]), urea and electrolytes (U&Es), liver function tests (LFTs) (including gamma-glutamyl transpeptidase [GGT]), vitamin B12 and folate, glucose, thyroid function tests (TFTs), autoimmune screen and immunoglobulins, hepatitis screen, HIV screen, Lyme serology, syphilis serology
•Urine: Dip for glucose and protein, Bence– Jones protein
•Imaging: Chest X-ray (CXR)
•LP and CSF study: Protein and CSF virology
•Nerve conduction studies
**MANAGEMENT**
•Dependent on the aetiology; remove any precipitants and treat the cause.
**QUESTIONS**
1.What is an autonomic neuropathy?
•A neuropathy of the autonomic nervous system.
•May present alone or in conjunction with a motor or sensory neuropathy. The most common cause is diabetes.
•May present with postural hypotension, impotence, urinary retention, diarrhoea/constipation and a Horner' s syndrome.
2.What is Charcot– Marie– Tooth disease?
•A hereditary, sensory and motor neuropathy. Also known as peroneal muscular atrophy.
•Usually starts at puberty with foot drop and weak legs.
•The peroneal muscles are the first to atrophy, with upper limb signs appearing at a later stage.
•There is muscle wasting, pes cavus and a bilateral foot drop (high-stepping gait). Reflexes are often absent. Sensory loss is variable.
•The most common form is inherited in an autosomal dominant manner.
3.Which drugs can cause a peripheral neuropathy?
•Amiodarone
•Gold
•Isoniazid
•Metronidazole
•Nitrofurantoin
•Phenytoin
4.What is tabes dorsalis? Describe the gait associated with this.
•Tabes dorsalis is a demyelinating condition affecting primarily the nerves in the dorsal/posterior columns of the spinal cord. These nerves normally help maintain proprioception, vibration and discriminative touch. The gait is ' high stepping' due to a lack of proprioception.
**KEY POINTS**
•Establish the diagnosis early to allow a cause to be found.
•Be sure to present a list of possible aetiologies to the examiner, taking into account whether the neuropathy is predominantly motor or sensory.
•The most likely causes in PACES include diabetes and Charcot– Marie– Tooth disease.
MONONEUROPATHIES
Please examine this patient who has paraesthesiae in his left hand.
**FINDINGS**
**CARPAL TUNNEL SYNDROME**
•Wasting of the thenar eminence.
•Weakness of LOAF (lumbricals, opponens pollicis, abductor pollicis brevis, flexor pollicis brevis).
•Sensory loss over the lateral 3½ digits.
•Maximal wrist flexion for 1 minute may elicit symptoms (Phalen' s test).
•Tapping over the nerve at the wrist induces tingling (Tinel' s test).
•Look for a scar from carpal tunnel release surgery and evidence of diabetes, hypothyroidism, acromegaly and rheumatoid arthritis.
**SPECIFIC NERVE PALSIES**
**ULNAR NERVE PALSY**
•Wasting of the hypothenar eminence (thenar eminence spared), claw hand, guttering on the dorsal aspect of the hand.
•Look for wasting of the medial aspect of the forearm (note low/high lesions, hand less clawed in high lesion).
•Weakness of abduction and adduction (test for Froment' s sign) of the fingers and adduction of the thumb.
•Sensory loss over the medial 1½ digits.
•Look for scars (fracture dislocation) and osteoarthrosis at the elbow.
**RADIAL NERVE PALSY**
•Wrist drop, weakness of wrist extension; if the wrist is passively extended, intrinsic muscles of the hand should be intact.
•Impaired grip strength.
•Sensory loss over the first dorsal interosseous.
•Look for scars at the elbow (fracture/dislocation), and note if the patient uses crutches.
**COMMON PERONEAL NERVE PALSY**
•Foot drop on inspection (leading to a high-stepping gait).
•Weakness of dorsiflexion and eversion of the foot.
•All reflexes will be intact.
•Sensory loss over the lateral dorsum of the foot.
•Look for evidence of compression around the fibular neck.
**PRESENTATION**
This patient has wasting of the thenar eminence in the left hand, indicating a median nerve lesion. This is supported by weakness of opposition of the thumb. There is sensory loss over the lateral 3½ digits with sparing of the palm. Tinel' s test is positive. There is a scar over the anterior wrist. The patient also has a symmetrical arthritis affecting the hands with a swan neck deformity of two digits, but no evident skin or nail changes. The likely diagnosis is carpal tunnel syndrome of the left wrist on a background of rheumatoid arthritis.
•Functionality is crucial and should always be tested and commented on. Try to find a cause for the lesion.
•In all mononeuropathies, it is important to look for causes of mononeuritis multiplex.
**INVESTIGATIONS**
•Single nerve lesions are often a clinical diagnosis. Neurophysiology can be used to confirm the diagnosis and to assess severity.
**MANAGEMENT**
•Splints and physiotherapy can be helpful. Surgery is sometimes used, especially in carpal tunnel syndrome where surgical decompression of the flexor retinaculum is a simple and definitive treatment.
**QUESTIONS**
1.What is the differential diagnosis of a foot drop?
•Common peroneal nerve palsy, peripheral neuropathy (especially Charcot– Marie– Tooth disease), sciatic nerve palsy, L4/5 radiculopathy (prolapsed lumbar disc) and lumbosacral plexopathy
2.What are the causes of mononeuritis multiplex?
•Wegener' s granulomatosis.
•Amyloidosis.
•Rheumatoid arthritis.
•Diabetes mellitus.
•Sarcoidosis.
•Polyarteritis nodosa.
•Leprosy.
•Carcinomatosis.
•Note that WARDS PLC is a useful way of remembering this list.
3.What are the causes of wasting of the small (intrinsic) muscles of the hand?
•Resembles an ulnar nerve lesion, but with thenar wasting and weakness also
•Causes include
•Anterior horn cells disease, e.g. poliomyelitis
•Radiculopathy, e.g. trauma, prolapsed disc
•Plexopathy, e.g. brachial plexus injury, Pancoast' s tumour, cervical rib
•Peripheral nerve lesions
•Muscle, e.g. disuse atrophy
**KEY POINTS**
•Initially look for any evident wasting or deformity.
•Examine all motor and sensory areas supplied by the nerve to localise the lesion.
•Search for a possible cause.
MOTOR NEURONE DISEASE
Please examine this patient who presents with weakness.
**FINDINGS**
•Amyotrophic lateral sclerosis (ALS)
•Suspect when UMNs and lower motor neurones (LMNs) occur concurrently in a limb with weakness.
•Degree of weakness depends on the number of muscles affected and distribution of motor neurone loss.
•Reflexes usually exaggerated (UMN signs) in the presence of muscle wasting and fasciculation (LMN signs).
•Sensation is unaffected throughout.
•Involvement of lower cranial nerves causes a pseudobulbar palsy.
•Progressive muscular atrophy (PMA)
•Flaccid weakness, as only LMNs affected
•Fasciculations and wasting
•Decreased or absent reflexes
•Plantars down going
•Primary lateral sclerosis (PLS)
•UMN signs only
•Usually begins in lower limbs (spastic gait)
•Exaggerated reflexes
•Progressive bulbar palsy
•Only lower cranial nerves affected (IX, X, XII).
•' Donald Duck' /nasal speech.
•Weakness of palatal muscles results in swallowing difficulties.
**PRESENTATION**
The most likely diagnosis is motor neurone disease (MND). There is generalised wasting and fasciculation. Tone is increased with generalised weakness. Reflexes are brisk/ reduced/absent. Leg reflexes are brisk and ankle clonus is present. There are bilateral extensor plantar responses.
**INVESTIGATIONS**
•Diagnosis is mainly clinical, based on a high index of suspicion from the collection of signs and symptoms.
•EMG
•Abnormally slowed conduction due to reduction in the number of viable motor axons/anterior horn cells to activate the muscle(s) involved
•Nerve conduction studies
•Normal sensory nerve conduction and abnormal motor nerve conduction
•Reduced muscle action potentials
•Repetitive stimulation: Decremental response with slow repetitive stimulation
•MRI
•To exclude other causes for the symptoms, e.g. cervical myelopathy/spondylosis or cord compression
**MANAGEMENT**
•Supportive measures
•Physiotherapy, occupational therapy, speech therapy.
•Swallowing and nutritional support (nasogastric [NG]/PEG feeding).
•Respiratory support: Noninvasive positive-pressure ventilation (NIPPV), tracheostomy and invasive ventilation.
•Advanced care planning is an important part of the patient' s treatment.
•Specific therapy
•Riluzole: A glutamate inhibitor that acts by inhibiting voltage-gated sodium channels
•Increases survival compared with placebo by roughly 3 months; no significant effect on muscle strength or neurological function seen
**QUESTIONS**
1.What are the disease variants of MND?
•Amyotrophic lateral sclerosis
•Approximately 50% of cases; combined UMN and LMN signs; mostly sporadic (90%– 95%)
•Familial form: Copper/zinc superoxide dismutase (SOD-1) gene mutation on chromosome 21
•Primary lateral sclerosis
•Rare; affects upper motor neurones only; has the best prognosis but can later progress to ALS
•Progressive muscular atrophy
•Approximately 25% of cases; affects anterior horn cells only – therefore signs in distal muscle groups
•Progressive bulbar palsy
•Approximately 25% of cases; worst prognosis; affects suprabulbar nuclei and lower cranial nerves, resulting in speech and swallowing difficulties (increased risk of aspiration)
2.What is the prognosis for patients with MND?
•No known cure; usually fatal within 3– 5 years of diagnosis. Cause of death is usually aspiration pneumonia and/or ventilatory failure.
3.What is the differential diagnosis for MND?
•Degenerative: Cervical cord compression, cervical spondylosis
•Inflammatory/traumatic/inherited: Syringomyelia, spinal muscular atrophy
•Infectious: Polio, syphilis
•Malignant/paraneoplastic
4.What are the other causes of absent ankle jerks and extensor plantar responses?
•Hereditary cerebellar ataxias: Friedreich' s ataxia, spinocerebellar ataxia
•Syphilitic taboparesis
•Subacute combined degeneration of the cord
•Conus medullaris pathology
•Combined pathologies, e.g. peripheral neuropathy from any cause in a patient with cervical spondylosis
Note that question 4 is a ' classic' PACES question and should be committed to memory.
**KEY POINTS**
•Note any external features that may indicate the diagnosis (fasciculations, NIPPV machine).
•Note the patient' s speech.
•It is crucial to check the plantar response.
REFERENCE
National Institute for Health and Care Excellence. (2001) TA20. Guidance on the use of riluzole (Rilutek) for the treatment of motor neurone disease. Available from https://www.nice.org.uk/guidance/ta20/resources/guidance-on-the-use-of-riluzole-rilutek-for-the-treatment-of-motor-neurone-disease-2294449469125. Accessed 24 August 2016.
MULTIPLE SCLEROSIS
Please examine the legs of this patient who has progressive difficulty in walking.
**FINDINGS**
• _General:_ Walking sticks, wheelchair, catheter, ataxic gait, dysarthria, mood (depressed/elated)
• _Eyes:_ Internuclear ophthalmoplegia, optic neuritis, central scotoma, loss of colour vision, relative afferent pupillary defect (RAPD), nystagmus
• _Neurological:_ Spastic paraparesis, cerebellar signs
**PRESENTATION**
This patient has an ataxic gait. Examination of her lower limbs shows increased tone and brisk reflexes. Further examination shows that the patient also has evidence of bilateral internuclear ophthalmoplegia. Putting these findings together, this patient is likely to have a diagnosis of multiple sclerosis.
•Comment on the patient' s functionality by looking for general findings, e.g. walking aids, the presence of a catheter and the patient' s general affect.
**INVESTIGATIONS**
•CSF analysis
•Immunoglobulin (Ig) G oligoclonal bands on electrophoresis
•Visual-evoked potentials
•Delayed response
•MRI
•Highlighting areas of demyelination
**MANAGEMENT**
•Multidisciplinary approach/patient education/MS support group information
•Acute relapse
•IV methylprednisolone: This may help to reduce the duration and severity of the relapse but will not alter the course of the disease.
•Disease-modifying drugs
•Interferon beta-1a
•Interferon beta-1b
•Glatiramer acetate
•Azathioprine
•Natalizumab (for highly active relapsing– remitting disease)
•Symptomatic treatment
•Spasticity: Physiotherapy, baclofen, tizanidine
•Urinary dysfunction: Oxybutynin, catheterisation
•Constipation: Laxatives, enemas
•Pain: Amitriptyline, carbamazepine, gabapentin
•Fatigue: Amantadine
•Depression: Support groups, selective serotonin reuptake inhibitors (SSRIs)
**QUESTIONS**
1.What is multiple sclerosis?
•Multiple sclerosis is a chronic inflammatory autoimmune disease of the central nervous system. The diagnosis depends on demonstrating at least two demyelinating lesions in the brain or spinal cord on MRI, separated in time and space.
2.What are Lhermitte' s sign and Uhthoff ' s phenomenon?
•Lhermitte' s sign: Flexion of the neck causing an ' electric shock' – like sensation in the trunk and limbs; this occurs in cervical spondylosis as well as MS.
•Uhthoff ' s phenomenon: An increase in the severity of symptoms (mainly visual), commonly precipitated by an increase in temperature or by exercise.
3.How is multiple sclerosis classified?
•Relapsing– remitting
•Affects approximately 85% of MS sufferers
•Secondary progressive
•Follows a period of relapsing– remitting MS
•Primary progressive
•15% of cases; progressive deterioration from the start
•The classification of MS becomes important when considering the role of disease-modifying antirheumatoid drugs (DMARDs). Currently, these drugs can help reduce the severity and frequency of relapses in relapsing– remitting MS and secondary progressive MS.
**KEY POINTS**
•A combination of spastic paraparesis and cerebellar signs is likely evidence of multiple sclerosis.
•Always comment on the patient' s functional status.
REFERENCE
National Institute for Health and Care Excellence. (2014) Guidance CG186. Multiple sclerosis in adults: Management. Available from https://www.nice.org.uk/guidance/cg186/resources/multiple-sclerosis-management-of-multiple-sclerosis-in-primary-and-secondary-care-35109816059077. Accessed 24 August 2016.
SPASTIC PARAPARESIS
Please examine this patient' s legs neurologically, as they are finding it difficult to walk.
**FINDINGS**
• _General:_ Walking aids, scissoring gait
• _Neurological:_ Increased tone, clonus, decreased power, hyperreflexia, extensor plantars and sensory loss. Signs may be present only below a particular spinal level.
• _Other:_ Evidence of cerebellar syndrome
•Cachexia and other evidence of malignancy
**PRESENTATION**
On examination, this patient has signs suggestive of a spastic paraparesis. This is evidenced by hypertonia, hyperreflexia and decreased power throughout the lower limbs. There is tenderness over the T12 vertebra with a sensory level present, making spinal cord compression a likely cause.
•It is important to exclude acute spinal cord compression, so tell the examiners you would ask about bladder and bowel symptoms, and would offer to perform a pulmonary regurgitation (PR) examination.
•Always check gait and assess the functional status of the patient.
**CAUSES OF A SPASTIC PARAPARESIS**
•Compression
•Tumour, osteoarthritis, trauma/fracture, central disc prolapse
•Transverse myelitis
•Multiple sclerosis (cerebellar signs), inflammatory and vascular disorders
•Degenerative
•Hereditary spastic paraparesis, motor neurone disease (absence of sensory signs/ combination of UMN and LMN signs), Friedreich' s ataxia (cerebellar signs)
•Infective
•HIV myelopathy
•Others
•Cerebral palsy, subacute combined degeneration of the cord
**INVESTIGATIONS**
•MRI of the spine is the gold standard imaging test.
•Other investigations include FBC, vitamin B12/folate levels, erythrocyte sedimentation rate (ESR)/C-reactive protein (CRP), syphilis serology, CSF protein and oligoclonal bands, myeloma screen and tumour markers.
**MANAGEMENT**
•The most urgent treatment is needed for spinal cord compression:
•If malignancy is the cause, give dexamethasone. This is followed by radiotherapy or surgery.
•Surgery is the mainstay of treatment for other causes.
•Further management of this presentation depends on the underlying cause, although an MDT approach focusing on neurorehabilitation may be helpful.
**QUESTIONS**
1.What malignancies are most likely to cause spinal cord compression?
•The most likely cause is metastasis from lung, breast or prostate cancer. Kidney and thyroid primaries also commonly metastasise to bone.
•Multiple myeloma should be considered a cause of cord compression.
•Intrinsic spinal malignancy is rare.
2.What is transverse myelitis?
•Inflammation of the spinal cord, characterised by axonal demyelination. Generally, the inflammation is across the thickness of the cord.
•Symptoms come on over a period of hours to weeks.
•The most common cause is multiple sclerosis. Viral and other infections can be implicated. Often, no cause is found.
•Steroids and plasma exchange may be helpful in management, although more crucial is neurorehabilitation.
3.What is the pathology in hereditary spastic paraparesis?
•Axonal degeneration that is maximal in the terminal portions of the longest descending and ascending tracts, including the crossed and uncrossed corticospinal tracts to the legs and fasiculus gracilis. Neuronal call bodies of degenerating fibres are preserved, and there is no evidence of primary dymelination.
**KEY POINTS**
•When ?nding a spastic paraparesis, it is important to look for a spinal level.
•Be sure to assess gait and examine the spine locally.
•Excluding acute cord compression is essential.
VISUAL FIELD DEFECTS
Please examine this patient who has recently developed a visual disturbance and, as a result, finds that they are bumping into things.
•Sit at the same height as the patient. Ask the patient to cover each eye in turn.
•Assess both nasal and temporal fields and all four quadrants. Look carefully at where the field of vision starts/finishes and which areas of the field are lost, i.e. nasal/temporal, superior/inferior.
•Use the information collected to establish which pattern of visual field defect the patient has.
•Be sure you are familiar with the visual pathway anatomy to enable you to ascertain the site of the lesion, as the nature of the visual field defect is determined by the site of the lesion along the visual pathway.
**PRESENTATION**
This patient presents with a history of bumping into objects and visual impairment. They have visual loss affecting the same half of the visual field in each eye. This is known as a homonymous hemianopia.
**CLASSIFICATION OF VISUAL FIELD DEFECTS**
•Classification of defects depends on whether one or both eyes are affected, which half of the visual field is affected, i.e. temporal, nasal or both (heteronymous or homonymous hemianopia), and the extent of field affected in each eye relative to the other (congruity).
•Optic nerve lesions: Result in partial or complete visual loss on the side of the lesion.
•Optic chiasm lesions: Result in both temporal fields being lost.
•Optic radiation lesions: Result in homonymous field defects that depend on the location of the lesion in the temporal or parietal lobe.
•Temporal lobe lesions lead to superior homonymous quadrantanopias.
•Parietal lobe lesions lead to inferior homonymous quadrantanopias.
•The further back towards the visual cortex the defect, the greater the degree of congruity (i.e. both eyes affected to the same degree).
•Visual cortex or optic radiation lesions result in a homonymous hemianopia.
**QUESTIONS**
1.What are the causes of a homonymous hemianopia?
•Vascular: Stroke
•Trauma
•Tumour
•Infection: Encephalitis
•Demyelination: MS
2.What are the causes of an optic nerve lesion?
•Trauma
•Inflammatory: Demyelination, optic neuritis
•Compression: Raised intraocular or intracranial pressure
•Vascular: Acute optic artery ischaemia
•Metabolic: B12 deficiency, diabetes, alcohol excess
•Inherited conditions, e.g. Leber' s optic neuropathy
3.What are the common causes of a bitemporal hemianopia?
•Usually occur as a result of a lesion of the optic chiasm
•Pituitary fossa tumour
•Craniopharyngioma
•Suprasellar meningioma
**KEY POINTS**
•Remember to assess both temporal and nasal visual ?elds for both eyes.
•Be familiar with visual pathways so the site of the lesion can be identi?ed.
NEUROLOGY STATION SUMMARY
•Neurology is a difficult station, particularly if you don' t look well practiced. You must practise your examination technique until you really are slick and confident performing the examination.
•As ever, particular emphasis should be paid to observation. In neurology, many diagnoses can be made from the end of the bed (Parkinson' s/myasthenia gravis/myotonic dystrophy). When this is the case, don' t relax. You still need to demonstrate all the signs to the examiner.
•Looking around the bed is just as important in neurology as any other station (e.g. look for forced expiratory volume in first second [FEV1] monitors in myasthenia gravis).
•Don' t panic if you don' t know the diagnosis; work out where the lesion is and use a pathological sieve to create a differential list.
•Be aware of how to demonstrate specific signs for specific syndromes (e.g. myotonia in myotonic dystrophy). Your examiners will be impressed if you look confident and rehearsed while doing this.
•Go into the station prepared to do a range of examinations. You may have to examine the arms, the legs, the eyes or the cranial nerves or conduct a special examination.
•Don' t forget to reinforce the reflexes if you think that they are absent.
•Don' t forget to check sensation on the chest prior to commencing limb examination, so that comparison can be made. _But_ , remember to check that it feel' s normal on the chest wall and not just whether the patient can feel you touching.
•Asking the patient to walk so that you can assess their gait is a key skill in neurology. Be aware of the different types of gait and the different syndromes that each gait may indicate.
•When presenting your case, first tell the examiners the syndrome or the likely site of the lesion, followed by the likely cause.
Station 3: Cardiology
Hints for the Cardiology Station
Aortic Stenosis
Aortic Regurgitation
Eisenmenger' s Syndrome
Hypertrophic Cardiomyopathy
Mitral Stenosis
Mitral Regurgitation
Mitral Valve Prolapse
Mixed Aortic Valve Disease
Mixed Mitral Valve Disease
Prosthetic Heart Valves
Tricuspid Regurgitation
Ventricular Septal Defect
Cardiology Station Summary
HINTS FOR THE CARDIOLOGY STATION
•Be alert to any peripheral signs, as these will help guide you to the diagnosis by the time you reach for your stethoscope.
•Check the radial pulse. Comment on rate, rhythm, character or absence.
•Look at the carotid pulsation from the end of the bed; a ' dancing carotid' signifies Corrigan' s sign seen in aortic regurgitation (AR).
•Examine the jugular venous pressure (JVP) with the patient resting at a 45° ; use the hepatojugular reflex if not visible.
•Giant V-waves in the JVP may indicate tricuspid regurgitation (TR).
•Feel for a parasternal heave (indicative of right ventricular hypertrophy).
•Check for a mitral valvotomy scar or subcutaneous implantable cardioverter defibrillator (ICD) under the left breast. These can be easily missed.
•Always comment on the position of the apex beat and demonstrate the position by marking it out on examination, and re-establish its location in the left lateral position before auscultation.
•If you cannot feel the apex beat in the fifth intercostal space/midclavicular line, the patient may have evidence of cardiomegaly or dextrocardia.
•Listen from the end of the bed for an audible click from metallic heart valves, and comment on any evidence of valvular regurgitation or endocarditis.
•On auscultation, there may be more than one murmur; try to identify the dominant one.
•Ensure that you expose the legs up to the thighs at the end of the examination to look for any scars from vein grafts and evidence of peripheral vascular disease, and to check for peripheral oedema.
•Comment on any evidence of infective endocarditis (IE), heart failure and other significant comorbidities.
AORTIC STENOSIS
This patient has had episodes of collapse and chest pain. Please examine their cardiovascular system.
FINDINGS
• _General:_ Dyspnoea
• _Peripheral:_ Slow-rising pulse, low systolic blood pressure, narrow pulse pressure
• _Chest:_ Palpable thrill over the aortic area, heaving apex beat, ejection systolic murmur loudest over the aortic region radiating to the carotids, soft or absent A2, reversed splitting of S2, S4
PRESENTATION
This patient has aortic stenosis (AS) as evidenced by a slow-rising pulse, narrow pulse pressure, heaving apex beat and an ejection systolic murmur that radiates to the carotids. I believe in this case it is a severe stenosis, as the aortic second heart sound is inaudible and I can hear a fourth heart sound. There is no evidence of congestive cardiac failure (CCF).
•Look for evidence of congestive cardiac failure, a displaced apex and associated mitral regurgitation (MR) (or Gallavardin' s phenomenon due to radiation of the murmur through the left atrium (LA) to the apex mimicking MR), and signs of pulmonary hypertension (left parasternal heave, loud A2).
•Try to comment on the severity of the murmur.
DIFFERENTIAL DIAGNOSIS
•Aortic sclerosis (normal A2)
•Hypertrophic cardiomyopathy (HCM) (look for jerky pulse, double apical impulse, thrill and ejection systolic murmur loudest at left lower sternal edge)
•Systolic flow murmur (soft, heard in all valve areas, especially in young women)
•Pulmonary stenosis
•Patient prosthesis mismatch; e.g. an aortic prosthetic valve is too small for the patient (presents with same symptoms and signs as AS)
•Radiotherapy
INVESTIGATIONS
•Diagnosis
•Electrocardiogram (ECG) (large LA, voltage left ventricular hypertrophy [LVH], left bundle branch block [LBBB]).
•Chest X-ray (CXR) (may be normal, or an unfolded aorta).
•Echocardiogram is the investigation of choice (transoesophageal echocardiogram [TOE] is rarely helpful unless concomitant mitral valve [MV] abnormality). Ascertain concomitant valvular lesions, left ventricle (LV) dimensions and function (provides prognostic information).
•To determine severity of AS, Doppler echo is preferred.
•Cardiac catheterisation (underlying coronary artery disease)
MANAGEMENT
•Medical
•Regular follow-up and echocardiograms (timing for intervention)
•Diuretics, digoxin, angiotensin-converting enzyme (ACE) inhibitors or angiotensin receptor blockers (ARBs) for heart failure
•Statins for secondary prevention of atherosclerotic events (statins do not affect progression of disease)
•Surgical
•Aortic valve replacement (AVR) (± coronary artery bypass graft [CABG] if indicated) is the definitive treatment of symptomatic severe aortic stenosis. Surgery improves quality and quantity of life.
•Balloon valvuloplasty has a limited role in adult AS, as restenosis occurs within 6– 12 months and complication rates exceed 10%. It is only very rarely used as a bridge to surgery or transcatheter aortic valve implantation (TAVI) in unstable patients, or in pregnancy or as a palliative procedure.
•TAVI considered if open AVR is too high risk (transfemoral or apical). Main complications include stroke, pacemaker insertion and vascular complications (including paravalvular regurgitation). One-year survival is 60%– 80%, depending on severity of comorbidities.
QUESTIONS
1.What are the causes of aortic stenosis?
•Common
•Calcific degeneration (or ' calcific AS' )
•Bicuspid valve
•Rare
•Rheumatic fever
•HCM
•Congenital (other than bicuspid): Supravalvular stenosis (Williams syndrome)
2.What are the indications for surgery?
•Severe stenosis (valve area < 1.0 cm2 on echo, peak arteriovenous [AV] velocity > 4 m/second [corresponding to a mean AV gradient of 40 mmHg])
•Symptoms (angina/collapse/dyspnoea/heart failure) with moderate stenosis (valve area < 1.5 cm2 on echo, peak velocity > 3 m/second)
•Critical AS = valve area < 0.8 cm2
3.What differentiates severe aortic stenosis from aortic sclerosis?
•Difficult! They lie on a continuum.
•In sclerosis, there is a normal pulse pressure and character, normal A2 component and little murmur radiation.
4.What happens to the loudness of the murmur with progressive stenosis severity?
•Murmur intensity is dependent on the flow turbulence through the valve and the cardiac output. Thus, in critical AS with a failing ventricle, cardiac output will fall and the murmur will be soft, but A2 will be absent.
5.Should exercise testing be performed?
•Exercise testing should not be performed in symptomatic patients with AS when the aortic velocity is ≥ 4 m/second or mean pressure gradient is ≥ 40 mmHg but is helpful for prognosis in ' asymptomatic' patients to unmask functional limitation.
KEY POINTS
•Aortic stenosis is an ejection systolic murmur which is loudest over the aortic area and radiates to the carotids.
•Comment on the peripheral signs associated with this murmur.
•Ensure that you know the indications for surgical treatment, as well as the medical management.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
AORTIC REGURGITATION
This patient complains of progressive breathlessness. Examine the patient' s cardiovascular system.
FINDINGS
• _General:_ Dyspnoea
• _Peripheral:_ Collapsing (water hammer) pulse, wide pulse pressure
• _Chest:_ Displaced and hyperdynamic apex beat, high-pitched early diastolic murmur (heard best in fixed expiration at the left sternal edge or sometimes the right sternal edge)
• _Eponymous signs:_
•Corrigan' s sign: ' Dancing carotid'
•De Musset' s sign: Head nodding
•Quincke' s sign: Pulsation at nail-bed when pressed
•Traube' s sign: Pistol-shot sound heard over the femoral arteries
•Austin Flint murmur: Mitral stenosis murmur due to impingement of the anterior mitral valve leaflet by the regurgitant jet
PRESENTATION
This patient has aortic regurgitation as evidenced by the early diastolic murmur which is heard loudest at the left sternal edge with the patient sitting forward. This patient also has a collapsing pulse and evidence of Corrigan' s sign.
•Comment on the severity of the aortic incompetence.
•In severe cases, there may be a widened pulse pressure, soft S2, short end-diastolic murmur (EDM), S3 sound and evidence of left ventricular failure (LVF).
•Comment on the possible underlying cause of AR.
AETIOLOGY OF AR
•Acute
•Infective endocarditis
•Aortic dissection
•Chronic
•Congenital aortic valve malformation (e.g. bicuspid, quadricuspid)
•Aortic root dilatation
•Prior endocarditis
•Rheumatic fever
•Iatrogenic: Following balloon dilatation or TAVI
•Connective tissue disease
•Arthritis, systemic lupus erythematosus (SLE), ankylosing spondylitis, Ehlers– Danlos syndrome, osteogenesis imperfecta
•Marfan' s syndrome: High-arched palate, tall, arachnodactyly autosomal dominant condition due to mutations of protein fibrillin-1 located on chromosome 15
•Seronegative arthritides
•Ankylosing spondylitis, reactive arthritis
•Syphilitic aortitis
•A rare cause, may be associated with Argyll Robertson pupils
INVESTIGATIONS
•CXR
•Cardiomegaly (' coeur en sabot' ), evidence of LVF
•ECG
•Lateral T-wave inversion, left ventricular hypertrophy
•Echo ± TOE
•To determine if aortic regurgitation is due to valve or root disease and to assess left ventricular size and systolic function
•Doppler colour is used to view the regurgitant jet and pulsed-wave Doppler to assess diastolic flow reversal in the descending aorta
•Computerised tomography (CT) or magnetic resonance imaging (MRI)
•Of aorta if aortic root disease
•Cardiac MRI
•Recommended in patients with Marfan' s or if an enlarged aorta is found on echo (especially in those with bicuspid valves)
MANAGEMENT
•In acute cases (infective endocarditis and aortic dissection), urgent surgery may be needed.
•In chronic cases, regular monitoring of the patient by a cardiologist to determine when a valve replacement is appropriate, depending on the patient' s symptoms and clinical findings.
•Medical
•Regular follow-up and echocardiograms (timing for intervention).
•Diuretic therapy for heart failure.
•Vasodilating agents such as ACE inhibitors, ARBs or dihydropyridine calcium channel blockers (e.g. amlodipine, felodipine, nifedipine), especially if hypertension is present.
•In Marfan' s syndrome: Beta-blockers may slow aortic root dilatation and reduce risk of aortic complications (pre- and postoperative). Consider offering screening to first-degree relatives of affected individuals.
•Surgical
•AVR ± CABG
•TAVI considered if open AVR too high risk (transfemoral or apical)
QUESTIONS
1.List the findings which would determine the need for surgery in aortic regurgitation.
•Acute severe symptomatic AR; requires emergency surgery.
•Symptomatic patients with severe AR (dyspnoea, New York Heart Association [NYHA] II– IV, angina) regardless of left ventricular systolic function.
•Surgery is also indicated for patients with severe AR undergoing surgery for another indication (e.g. CABG, mitral valve surgery or aortic surgery).
•Asymptomatic patients with severe AR.
2.List some other manifestations of Marfan' s syndrome.
•Ectopia lentis (upwards lens dislocation), arm span > height, dural ectasia, pectus excavatum, joint laxity, scoliosis, pes planus (Ghent criteria)
3.How often should asymptomatic patients with Marfan' s be screened?
•Patients with Marfan' s, especially those with a dilated aorta, should be screened annually with an echocardiogram to monitor the proximal aorta/aortic root.
4.What is an Austin Flint murmur?
•This is a low-pitched, rumbling mid-diastolic murmur which is a sign of severe aortic regurgitation. It is attributed to the fluttering of the anterior mitral valve leaflet caused by a severe regurgitant stream.
KEY POINTS
•Aortic regurgitation is an early diastolic murmur which is heard loudest over the left sternal edge with the patient sitting forward.
•Ensure that you can recognise the eponymous signs associated with this murmur.
•Know the common acute and chronic causes of this murmur.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
EISENMENGER' S SYNDROME
Please examine this patient who presents with dyspnoea and orthopnoea.
FINDINGS
• _General:_ Cyanosis
• _Peripheral:_ Clubbing, raised JVP
• _Chest:_
•Left parasternal heave
•Loud single P2 audible (and palpable)
•Possibly features of a ventricular septal defect (VSD), an ejection systolic click in the pulmonary area, an audible early diastolic murmur in the pulmonary area due to pulmonary regurgitation (PR) and/or a pansystolic murmur at the left sternal edge (LSE) due to tricuspid regurgitation (secondary to pulmonary hypertension)
•Possible wide fixed-split first heart sound (atrial septal defect [ASD])
PRESENTATION
This patient has features suggestive of Eisenmenger' s syndrome. They are peripherally cyanosed with clubbing. There is a left parasternal heave. On auscultation, there is a loud pulmonary second heart sound with an audible ejection click in the pulmonary area.
INVESTIGATIONS
•Blood tests
•Full blood count (FBC) (for polycythaemia), renal (for CKD) and liver function (hepatic congestion/cholestasis), iron stores (iron deficiency anaemia), uric acid (for gout)
•ECG
•Signs of right ventricular hypertrophy with P-pulmonale
•CXR
•Increased pulmonary vascular markings, prominent right ventricle, large rheumatoid arthritis (RA)
•Echocardiogram
•To assess the cause and degree of shunt (VSD/patent ductus arteriosus [PDA]) and the presence of valvular disease (TR/PR), and to calculate pulmonary artery pressures and the extent of pulmonary hypertension
•Pulmonary function tests with volume and CO2 diffusion studies
•Cardiac catheterisation
•To assess the degree of shunting, and to calculate pressure gradients and pulmonary pressures
**MANAGEMENT**
•Ideally, treat defects early to prevent this syndrome from developing and avoid situations that make symptoms worse, such as high altitude.
•Avoid smoking
•Medical
•Oxygen: Symptomatic benefit (no prognostic benefit)
•Vasodilators: Prostacyclin, bosentan, sildenafil
•Diuretics to treat right heart failure
•Iron supplementation to avoid iron deficiency anaemia
•Contraceptive measures strongly advised and pregnancy should be avoided, as it carries high maternal and foetal mortality rates
•Venesection, but only if symptoms/signs of hypervicosity syndrome
•Surgical
•Not beneficial once Eisenmenger' s is present, as the damage to the lung vasculature is irreversible.
•Palliative procedures are available.
•Combined heart– lung transplantation is an option.
**QUESTIONS**
1.What is Eisenmenger' s syndrome?
•Eisenmenger' s results from a large left-to-right shunt/cardiac defect causing increased pulmonary blood flow, pulmonary vessel injury, increased pulmonary vascular resistance and resultant pulmonary hypertension.
•This leads to reversal of the shunt, causing either a unidirectional right-to-left shunt or a bidirectional shunt.
•This is clinically manifest as cyanotic heart disease.
2.What are the causes of Eisenmenger' s syndrome?
•Large nonrestrictive VSD
•Nonrestrictive PDA
•Atrioventricular septal defects
•Large uncorrected or surgically created systemic-to-pulmonary shunts for treatment of congenital heart disease
3.What are the potential complications of Eisenmenger' s syndrome?
•Secondary polycythaemia and hyperviscosity
•Arrhythmias
•Heart failure
•Stroke
•Haemoptysis
•CKD
•Hyperuricaemia and gout
•Hypertrophic osteoarthropathy
•Sudden cardiac death (SCD)
KEY POINTS
•Ensure that you comment on the presence of cyanosis and clubbing.
•Be aware of the underlying causes of this condition.
•Be aware that once Eisenmenger' s develops, treatment of the underlying defect is purely medical.
REFERENCE
Baumgartner H, et al. (2010) ESC guidelines for the management of grown-up congenital heart disease (new version 2010). Task Force on the Management of Grown-up Congenital Heart Disease of the European Society of Cardiology (ESC); Association for European Paediatric Cardiology (AEPC); ESC Committee for Practice Guidelines (CPG). _Eur Heart J_ 31(23): 2915– 57.
HYPERTROPHIC CARDIOMYOPATHY
This patient, who has had episodes of syncope and palpitations, has been told he has a murmur. Please examine his cardiovascular system.
FINDINGS
• _General:_ Dyspnoea
• _Peripheral:_ Bifid jerky pulse, large A-wave visible in JVP, low systolic blood pressure, narrow pulse pressure
• _Chest:_ Palpable thrill over the aortic area, heaving double apical impulse, ejection systolic murmur loudest over the left sternal edge (accentuated by Valsalva manoeuvre, softer with squatting), S4, normal A2
• _Extra features:_ Look for evidence of congestive cardiac failure, murmur of MR, ICD in left subclavicular fossa
PRESENTATION
This patient has HCM as evidenced by a jerky pulse, double-impulse apex beat and an ejection systolic murmur that radiates to the carotids but a normal A2 component. There is no evidence of congestive cardiac failure.
•Mention the murmur and comment on severity. Comment on lack/presence of signs of congestive cardiac failure.
DIFFERENTIAL DIAGNOSIS
•Aortic stenosis
•Flow murmur
•Pulmonary stenosis
•Hypertensive heart disease
INVESTIGATIONS
•ECG
•Voltage LVH, deep Q-waves, ST elevation, interventricular conduction delays
•Chest X-ray
•Echo
•Investigation of choice, looking for an LV outflow tract obstruction (LVOTO) defined as a peak Doppler LV outflow tract gradient of ≥ 30 mmHg at rest, but if ≥ 50 mmHg, invasive treatment should be considered. Other echo features: Septal asymmetrical hypertrophy, systolic anterior motion (SAM) of the mitral valve, MR, diastolic LV dysfunction.
•Cardiac MRI
•Demonstrates severity and distribution of hypertrophy and wall fibrosis and LV function
•Holter monitor
•Nonsustained ventricular tachycardia (VT)
•Exercise tolerance test
•Blood pressure response, looking for inadequate rise or paradoxical systolic drop with exertion
•Cardiac catheterisation
•Aortic outflow gradient, coronary artery disease
•Genetic testing
•Recommended for patients fulfilling diagnostic criteria for HCM with testing of relatives if a causative mutation is found
MANAGEMENT
•Medical
•Regular follow-up and echocardiograms
•Beta-blocker/(verapamil) ± disopyramide (Nifedipine and other dihydropyridine calcium channel blockers are potentially harmful for patients with LVOTO)
•Diuretic therapy only for heart failure (avoid digoxin in patients with LVOTO)
•Septal alcohol ablation
•Dual-chamber pacemaker insertion
•ICD insertion
•Surgical
•Septal myomectomy (Morrow procedure) ± mitral valve surgery
•Heart transplant considered for moderate– severe refractory heart failure without LVOTO
QUESTIONS
1.What is the cause of the dynamic outflow gradient?
•It is caused by systolic anterior motion of the mitral valve due to local underpressure (Venturi effect), exacerbated by the septal hypertrophy.
2.What are the risk factors for SCD?
•Family history of SCD with HCM at a young age
•Unexplained syncope in young age (< 30 years)
•Personal history of ventricular fibrillation (VF)
•VT or nonsustained VT
•Maximum LV wall thickness ≥ 30 mm
•Hypotensive response to exercise (a fall of at least 20 mmHg from peak pressure)
•LVOTO
3.What do you know about the genetics of HCM?
•Sixty percent of HCM is inherited as autosomal dominant, and most mutations occur in the cardiac sarcomere protein genes or arise spontaneously, producing a _de novo_ mutation (usually of the ? -myosin heavy-chain gene), and 5%– 10% are due to other genetic disorders.
•The majority of cases are due to mutations in the ? -myosin heavy-chain gene (MYH7, chromosome 14) and cardiac myosin-binding protein C gene (MYBPC3, chromosome 11).
•Less commonly affected genes include cardiac troponin I (TNNI3) and T (TNNT2) and ? 1-tropomyosin (TPM1) and myosin light-chain 3 (MYL3).
•Therefore, a detailed family history and first-degree relative screening should be advised.
4.When would an ICD be indicated?
•Should be considered if the 5-year risk of SCD is 4%– 6% and definitely indicated if the 5-year risk is ≥ 6% using the European Society of Cardiology (ESC) HCM risk score calculator
KEY POINTS
•HCM is associated with an ejection systolic murmur heard loudest at the left sternal edge.
•Key clinical findings are a bifid jerky pulse, a double apical impulse and a palpable thrill.
•This is an autosomal dominant inherited condition which is associated with sudden cardiac death.
REFERENCE
Task Force for the Diagnosis and Management of Hypertrophic Cardiomyopathy of the European Society of Cardiology (ESC). (2014) ESC guidelines on diagnosis and management of hypertrophic cardiomyopathy. _Eur Heart J_ 35(39): 2733– 79.
MITRAL STENOSIS
This patient complains of breathlessness. Please examine their cardiovascular system.
FINDINGS
• _General:_ Malar flush
• _Peripheral:_ Irregularly irregular pulse, raised JVP
•Chest
•Inspection/palpation: Left thoracotomy/valvotomy scar, tapping apex, palpable S1
• _Auscultation:_ Opening snap of mitral valve, loud S1, low-pitched rumbling middiastolic murmur at the apex (heard loudest in the left lateral position)
PRESENTATION
This lady has a malar flush. There is an irregularly irregular pulse but no stigmata of endocarditis. The apex beat is not displaced. There is a loud first heart sound, and there is a mid-diastolic murmur heard loudest in the left lateral position. This is mitral stenosis.
•Look for signs of pulmonary hypertension (left parasternal heave, loud/palpable P2).
DIFFERENTIAL DIAGNOSIS
•Left atrial myxoma
•Parachute mitral valve
•Severe mitral annular calcification
•Thrombosed mitral valve prosthesis
•Radiotherapy
**INVESTIGATIONS**
•Diagnosis
•Chest X-ray: Left atrial enlargement
•Echocardiogram ± TOE: To assess the mitral and other valves, left atrial dimension and pulmonary artery pressures
•Complications
•ECG
•AF, right ventricular hypertrophy
•Echo
•Assess for pulmonary hypertension and evidence of endocarditis
•TOE
•Exclude LA thrombus
•Right and left cardiac catheterisation
•To measure cardiac pressures if discordance between symptoms and echo findings, e.g. wedge pressure
**MANAGEMENT**
•Medical
•Diuretics, long-acting nitrates (can help dyspnoea)
•Treat AF with rate control
•Anticoagulate all patients with warfarin where possible (target INR 2– 3), especially if MS with AF, a prior embolic event or LA thrombus
•Surgical
•Indications include clinically significant MS with a valve area ≤ 1.0 cm2, significant symptoms which limit normal activity, recurrent emboli, pulmonary oedema (especially in pregnancy) and deterioration due to AF.
•Options include mitral valve replacement, open valvotomy and balloon valvuloplasty (percutaneous MV commissurotomy).
•Percutaneous mitral commissurotomy recommended if severe MS without LA thrombus and when surgery is contraindicated or as a bridge to surgery.
•Complications include haemopericardium, embolic phenomena, severe MR and death.
**QUESTIONS**
1.What are the causes of mitral stenosis?
•Rheumatic fever is the most common cause.
•Others include
•Calcific disease (heavily calcified mitral annulus with extension to MV leaflets)
•End-stage renal disease (ESRD)
•Congenital lesions
•Carcinoid tumours
•Mucopolysaccharidoses
2.What are the complications of mitral stenosis?
•Pulmonary hypertension (leading to right-sided heart failure)
•Haemoptysis
•Flash pulmonary oedema
•Emboli (stroke risk; novel oral anticoagulants [NOACs] are not licensed in MS)
•Atrial fibrillation
•Infective endocarditis
•Pressure effects from an enlarged left atrium: Hoarseness (Ortner' s syndrome, due to left recurrent laryngeal nerve compression), bronchial obstruction, dysphagia
3.What surgical procedures require antibiotic prophylaxis?
•Prophylaxis is generally not recommended now for surgical procedures.
•Patients should be advised to maintain good oral health, and those at risk of infective endocarditis should be investigated and treated promptly when displaying evidence of infection.
KEY POINTS
•Mitral stenosis is a mid-diastolic murmur which is heard loudest at the apex in the left lateral position.
•Comment on the presence or absence of atrial ?brillation.
•Always check for the presence of pulmonary hypertension.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
MITRAL REGURGITATION
Please examine this patient who has shortness of breath and/or palpitations.
**FINDINGS**
• _General:_ Tachypnoea, flushed face, peripheral oedema, mitral valvotomy scar
• _Peripheral:_ Raised JVP, irregularly irregular pulse
• _Chest:_ Inspection/palpation: Scars (valvotomy, left thoracotomy), thrusting and laterally displaced apex, palpable thrill, parasternal heave
• _Auscultation:_ Pansystolic murmur loudest at apex radiating to the axilla, soft S1, S3 (± gallop) in severe MR
**PRESENTATION**
This patient has mitral regurgitation. The pulse is irregularly irregular, consistent with atrial fibrillation. The apex beat is thrusting in nature. On auscultation, there is an audible third heart sound and a pansystolic murmur, loudest at the apex, that radiates to the axilla.
•Mitral regurgitation is often associated with ischaemic heart disease and a dilated left ventricle, so it is important to look for scars from previous CABG surgery, as well as any risk factors (tar staining on the fingers, signs of diabetes, hypertension, corneal arcus and xanthelasma).
INVESTIGATIONS
•ECG
•LVH, left strain pattern, evidence of ischaemia, AF
•Echocardiogram ± TOE
•Assess LV systolic function and dimensions, RV function, LA size, pulmonary artery pressure.
•Additional features: Vegetations, papillary muscle/chordae tendineae rupture, other significant valvular lesions.
•Doppler flow studies
•Size and site of regurgitant jet
•CXR
•Normal, cardiomegaly, large left atrium, pulmonary oedema
•Cardiac catheterisation
•If discrepancy between symptoms and noninvasive investigations
MANAGEMENT
•Medical
•Management of congestive cardiac failure
•ACE inhibitors/ARBs, diuretics (± aldosterone antagonists), beta-blockers
•Rate control therapy and anticoagulation for atrial fibrillation
•Cardiac resynchronisation therapy with biventricular pacing
•Surgical
•When symptomatic and LVEF > 30%, especially if evidence of cardiomegaly and raised end-systolic LV volume and resultant LV systolic dysfunction and dilatation
•Options
•Repair/reconstruction (valvuloplasty) is preferable (also if MR limited to posterior leaflets): Carries lower operative mortality and no need for long-term anticoagulation
•Valve replacement: Elective or for emergency treatment of acute severe incompetence (IE/acute chordae tendineae/papillary muscle rupture) or if severe MR and undergoing cardiac surgery for another indication (e.g. CABG, AVR)
•Percutaneous options: Valvuloplasty ring, MitraClip®
**QUESTIONS**
1.What are the causes of mitral regurgitation?
•Acute
•Chorda tendineae rupture: Due to degenerative valve disease, trauma, infective endocarditis, rheumatic MV disease, mitral valve prolapse (MVP)
•Papillary muscle rupture postmyocardial infarction (MI) (usually associated with an inferior MI)
•Papillary muscle dysfunction due to ischaemia
•Infective endocarditis
•Chronic
•Mitral valve prolapse (most common cause in developed countries) rheumatic heart disease
•Functional MR secondary to left ventricular dilatation and systolic dysfunction resulting in
•Lateral displacement of the papillary muscles and retraction of the valve leaflets.
•Dilatation or calcification of the mitral valve annulus may also result in MR.
•Infective endocarditis
•Connective tissue disorders: SLE, rheumatoid arthritis, ankylosing spondylitis
•Inherited disorders: Marfan' s syndrome, Ehlers– Danlos syndrome, osteogenesis imperfecta, pseudoxanthoma elasticum
•Hypertrophic cardiomyopathy
•Radiation heart disease
2.What is the difference between primary and secondary MR?
•Primary MR
•Can be acute or chronic and includes all causes of MR in which intrinsic lesions affect one or more components of the mitral valve
•Chronic MR can be either chronic primary or chronic secondary MR.
•Chronic primary MR is usually degenerative, with the valve itself being diseased.
•Chronic secondary is usually functional, so the valve is usually structurally normal.
3.What are the causes of primary and secondary MR, and how does the management differ?
•Primary MR
•Infective endocarditis causing chordae tendineae rupture or spontaneous chordae or papillary muscle rupture
•Requires urgent surgical intervention
•Chronic primary
•Rheumatic heart disease mitral valve prolapse, infective endocarditis, connective tissue disease, radiation.
•Surgery is usually curative.
•Chronic secondary
•LV dilatation, ischaemic (e.g. MI) or nonischaemic heart disease (idiopathic dilated cardiomyopathy).
•MV surgery is not usually curative, as it' s only one component of the disease, as there' s often associated LV dysfunction and/or ischaemic heart disease (IHD) that also requires treatment.
KEY POINTS
•Mitral regurgitation is a pansystolic murmur which is heard loudest at the apex and radiates to the axilla.
•Check for the presence/absence of atrial ?brillation.
•Be aware of the acute and chronic causes of mitral regurgitation, and be aware these can be further subdivided into primary or secondary.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
MITRAL VALVE PROLAPSE
Please examine this patient who has presented with a murmur.
FINDINGS
• _Auscultation:_
•Normal heart sounds
•Midsystolic click (may or may not be present)
•Late systolic MR murmur loudest at the left sternal edge
•The murmur is made louder by factors that decrease the volume of blood within the cardiac chambers, e.g. straining (Valsalva manoeuvre) or standing from squatting.
•There is a higher female preponderance.
•Patients are usually asymptomatic.
•The patient may have peripheral stigmata of a connective tissue disorder.
INVESTIGATIONS
•CXR
•Normal, cardiomegaly, large left atrium, pulmonary oedema
•ECG
•May be inferior T-wave inversion, large left atrium, AF
•Echocardiogram
•Diagnostic! Degree of thickening of the mitral valve leaflets and their displacement relative to the annulus is indicative of MVP.
MANAGEMENT
•Medical
•Palpitations controlled with beta-blocker therapy
•Surgical
•Indications as for MR; valve repair preferred to replacement
QUESTIONS
1.What are the associations of mitral valve prolapse?
•Congenital heart disease
•ASD, AVSD, PDA
•Congenital disorders
•Turner' s syndrome, Marfan' s syndrome, osteogenesis imperfecta, pseudoxanthoma elasticum
•Others
•SLE
2.What are the presenting features of MVP?
•Usually asymptomatic but can be associated with atypical chest pain, palpitations, fatigue and dyspnoea
3.What are the complications of MVP?
•Infective endocarditis.
•Atrial and ventricular arrhythmias (ventricular ectopic beats are the usual cause of palpitations experienced).
•MR: MVP is the most common cause of chronic primary MR in developed countries.
•Cerebral emboli resulting in transient ischaemic attacks (TIAs) and/or stroke.
•Sudden cardiac death.
4.What are the current recommendations regarding antibiotic prophylaxis dental work with MVP?
•As patients with MVP are considered as having ' moderate risk' of infective endocarditis, antibiotic prophylaxis is not generally recommended.
•(National Institute for Health and Care Excellence [NICE] guidance on prophylaxis against infective endocarditis in adults and children undergoing interventional procedures is currently being drafted.)
KEY POINTS
•Mitral valve prolapse is a late systolic murmur which is heard loudest at the left sternal edge.
•It is more common in females.
•Be aware of the associations of this murmur.
REFERENCE
Habib G, et al. (2015) 2015 ESC guidelines for the management of infective endocarditis: The Task Force for the Management of Infective Endocarditis of the European Society of Cardiology (ESC). Endorsed by: European Association for Cardio-Thoracic Surgery (EACTS), the European Association of Nuclear Medicine (EANM). _Eur Heart J_ 36(44): 3075– 128.
MIXED AORTIC VALVE DISEASE
This patient has a murmur. Please examine their cardiovascular system.
FINDINGS
•Features of both aortic stenosis and aortic regurgitation.
•The clinical findings should determine the predominant abnormality.
PRESENTATION
This patient has mixed aortic valve disease with a predominant stenotic lesion. There is a slow-rising radial pulse and a normal blood pressure, but there are no stigmata of infective endocarditis. The apex beat is in the fifth intercostal space, midclavicular line. The heart sounds are normal, but there is a harsh ejection systolic murmur radiating to the carotids and an end-diastolic murmur heard loudest at the left sternal edge in forced expiration.
•When suspecting mixed aortic valve disease, it is crucial to look for and present the above features. Be sure to ask about the patient' s blood pressure. Where possible, decide on the predominant lesion, as the clinical manifestation usually follows that.
INVESTIGATIONS
•Diagnosis
•Echocardiogram to assess the aortic valve, including valve size and gradient (also to assess the other valves, and left ventricular size and function).
•TOE may be required to fully delineate the valvular anatomy.
•Cardiac catheterisation can be used to assess for coronary artery disease. Left ventricular angiography, aortogram and measuring pressuring gradient from left ventricle to aorta may further aid diagnosis.
•Complications
•ECG, echocardiogram, blood cultures if endocarditis suspected.
•Exercise testing to assess exercise tolerance may be used to assess haemodynamic significance.
MANAGEMENT
•Medical management is an option with diuretics and possibly an ACE inhibitor (predominant AR).
•Surgery or TAVI (if predominant AS with moderate AR only [not if severe AR]) should be considered.
•The indications for surgery should follow the guidelines for the predominant valvular lesion (i.e. management of AS if stenosis is the predominant lesion or AR if that is)
QUESTIONS
1.What are the main causes of mixed aortic valve disease?
•Bicuspid aortic valve
•Degenerative calcific aortic valve disease
•Radiotherapy
2.What will happen to the gradient and valve area with each of the predominant lesions?
•Predominant AS: A high gradient and small valve area will be present.
•Predominant AR: The gradient may be significantly raised due to regurgitation, but the aortic valve area (AVA) is usually relatively large.
KEY POINTS
•When the murmur of aortic stenosis is detected, it is important to also listen for the diastolic murmur of aortic regurgitation (or any other murmur present).
•State which lesion is dominant in conjunction with the clinical ?ndings.
•Management of the valve follows that of the predominant lesion.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
MIXED MITRAL VALVE DISEASE
This patient has a murmur. Please examine their cardiovascular system.
FINDINGS
•Features of both mitral stenosis and mitral regurgitation.
•The clinical findings should determine the predominant abnormality.
PRESENTATION
This patient has mixed mitral valve disease with a predominant regurgitant lesion. There are no stigmata of infective endocarditis. There is a sharp radial pulse. The apex beat is in the fifth intercostal space, towards the anterior axillary line. The heart sounds are normal with a pansystolic murmur radiating to the axilla and a mid-diastolic rumbling murmur heard loudest at the apex in the left lateral position.
•When suspecting mixed mitral valve disease, it is crucial to look for and present the above features.
•As with mixed aortic valve disease, decide on the predominant lesion, as the clinical presentation will follow that.
INVESTIGATIONS
•Diagnosis
•Echocardiogram to assess the mitral valve, including valve size and gradient (also to assess the other valves, and left ventricular size and function).
•TOE may be required to fully delineate the valvular anatomy.
•Right and left cardiac catheterisation further assesses valvular anatomy and concomitant coronary artery disease and measures wedge pressure.
•Complications
•ECG (AF), CXR, echocardiogram, blood cultures if endocarditis suspected
MANAGEMENT
•Anticoagulation, rate control therapy for atrial fibrillation and diuretics to treat fluid overload.
•Surgical valve replacement should be considered (according to the predominant lesion as per the guidelines for individual valve lesions earlier in this chapter).
•Mitral commissurotomy could be considered if predominant MS with only mild MR.
QUESTIONS
1.What are the causes of mixed mitral valve disease?
•Predominantly caused by rheumatic heart disease
•Can be caused by degenerative valvular disease with increasing age
•Radiotherapy
2.What will happen to the gradient and valve area with each of the predominant lesions?
•Predominant MS
•High transvalvular gradient and small valve area
•Predominant MR
•High transvalvular gradient due to the regurgitation, but valve area may remain relatively large
KEY POINT
•Again, it is important to state which lesion is dominant.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
PROSTHETIC HEART VALVES
This patient attends for routine follow-up. Please examine their cardiovascular system.
FINDINGS
• _General:_ Dyspnoea, audible ' click' from end of bed
• _Peripheral:_ Scars (midline sternotomy, lateral thoracotomy), including harvesting scars on legs/arms for concomitant CABG (more common in aortic valve replacement)
•Jaundice, anaemia, purpura
• _Chest:_
•Aortic valve replacement: Metallic S2 (opening/closing click), ejection systolic flow murmur (common, nonpathological), diastolic regurgitant murmur (pathological)
•Mitral valve replacement: Metallic S1 (opening/closing click), systolic regurgitant murmur (pathological)
Note: Tissue valve replacements may demonstrate normal heart sounds (be alert to this possibility if sternotomy and no venous graft harvest scars evident).
PRESENTATION
This patient has an aortic metallic valve replacement as evidenced by a midline sternotomy scar and a metallic second heart sound. The valve appears to be functioning well, as I cannot hear a murmur of aortic regurgitation and there is no evidence of congestive cardiac failure. Of note, there are no peripheral stigmata of subacute bacterial endocarditis.
•Mention the type of valve replacement and whether there is regurgitation.
•Mention stigmata of endocarditis, any evidence of leg venous graft harvest scars for CABG and signs of over-anticoagulation.
•To look for any signs of congestive cardiac failure (possibly suggestive of valve incompetence).
INVESTIGATIONS
•Echocardiogram
MANAGEMENT
•Metallic valves require lifelong anticoagulation (warfarin).
•Tissue valves do not usually require anticoagulation unless concomitant AF but have a shorter life span, as they are susceptible to structural valve deterioration (SVD).
•Rates of SVD decrease with age and if in the aortic position (compared with the mitral position).
•Treatment of SVD requires surgery to replace the valve.
QUESTIONS
1.What are complications of prosthetic valves?
•Early
•Surgical complications
•Endocarditis
•Late
•Thromboembolic sequelae
•Bleeding (as a result of anticoagulation)
•Infective endocarditis
•Valvular or paravalvular leak (regurgitation)
•Haemolysis (with anaemia)
•Valvular stenosis (endothelialisation (pannus))
•Structural failure/embolisation (very rare)
•All the above complications can occur with metallic valves. The main risk with tissue valves is regurgitation with degeneration.
2.What influences the type of valve a patient will receive?
•Factors to consider when deciding valve type include
•Bleeding risk with anticoagulation and thromboembolic phenomena with a metallic valve
•Risk of SVD with a biological valve
•Patient/physician/surgical concerns
•Metallic valves are used in younger patients due to their durability (they are also used if the patient is already on warfarin), unless female and planning to conceive in the future (due to high risk of thromboembolic phenomena during pregnancy regardless of the anticoagulant used).
•Biological valves are useful in the elderly (where the patient' s life expectancy is thought to be less than the durability of the valve), and those with an increased risk of bleeding if anticoagulated.
3.What types of metallic valve are available and what' s their thrombogenicity risk?
•Disc valve (e.g. Bjö rk– Shiley' s): High risk
•Bileaflet valve (e.g. St Jude Medical): Low risk
4.What happens to the intensity of the clicks from a metallic valve when failing?
•Decreasing intensity of the closing click
5.Is any oral anticoagulation ever required after bioprosthetic valve surgery?
•Definitely if there' s another indication for anticoagulation, e.g. AF.
•Should be considered for 3 months following mitral or tricuspid bioprosthetic valve surgery.
•Either low-dose aspirin or oral anticoagulation should be considered after aortic bioprosthesis for 3 months.
KEY POINTS
•Listen carefully from the end of the bed for any audible ' clicks' .
•Examine for any surgical scars.
•Comment on the type of valve (tissue/metallic), any evidence of infective endocarditis and any evidence of congestive cardiac failure.
•Be aware of early and late complications of prosthetic valves.
TRICUSPID REGURGITATION
This patient has presented with shortness of breath. Please examine her cardiovascular system.
FINDINGS
• _General:_ Dyspnoea, jaundice
• _Peripheral:_
•Marked peripheral/sacral oedema, prominent JVP (giant V-waves)
•Evidence of intravenous drug user (IVDU)
• _Chest:_ Right ventricular heave, palpable P2, pansystolic murmur heard loudest at the left sternal edge in inspiration, S3, bibasal crepitations and/or pleural effusions, evidence of chronic pulmonary pathology/kyphoscoliosis
• _Abdomen:_ Ascites, pulsatile hepatomegaly
PRESENTATION
This patient has a diagnosis of tricuspid regurgitation as evidenced by the pansystolic murmur which is best heard at the LSE in inspiration. She also has a prominent JVP with giant V-waves. There is also evidence of pulsatile hepatomegaly which supports this diagnosis.
•The main differential diagnosis for this murmur is mitral regurgitation, which would be louder in expiration and radiates to the axilla.
AETIOLOGY
Underlying causes can be divided into
•Primary
•Congenital (Ebstein' s)
•Rheumatic heart disease
•Right-sided endocarditis (e.g. due to IVDU)
•Carcinoid syndrome
•Penetrating trauma, right-sided pacing leads
•Radiation
•Secondary
•Right-sided heart failure (e.g. pulmonary emboli, MI)
•Pulmonary hypertension (idiopathic or due to left-sided heart disease or cor pulmonale)
•Eisenmenger' s syndrome
•Biventricular failure
INVESTIGATIONS
•CXR
•Pulmonary congestion or pruned vessels, hyperinflated lungs, pulmonary fibrosis, cardiomegaly (large RA)
•Echo
•Demonstrates valvular abnormality, severity of valve disease, right ventricular dimension and function and inferior vena cava size; measures pulmonary artery pressures and any concomitant left heart disease
•Exercise testing may be useful in those with severe TR but minimal symptoms.
MANAGEMENT
Identify and treat the underlying cause.
•Medical
•Loop diuretics (for right heart failure)
•ACE inhibitors (for systemic hypertension and heart failure)
•Spironolactone
•Pulmonary vasodilators may be indicated in severe TR with pulmonary hypertension
•Surgical
•Tricuspid valve repair is usually sufficient for mild– moderate TR in those undergoing simultaneous left heart surgery.
•In all other cases, TV repair or replacement may be performed (although replacement less frequently is performed).
QUESTIONS
1.What is carcinoid syndrome?
•Carcinoid tumours are rare neuroendocrine tumours of the enterochromaffin cells. They are mostly asymptomatic.
•Carcinoid syndrome is evident when products of the tumour are metabolised by the liver, resulting in a collection of symptoms, including dyspnoea and wheeze (due to bronchoconstriction), flushing, diarrhoea, tachycardia, dizziness and CCF.
•Symptoms are caused by release of serotonin, tachykinins and other vasoactive peptides into the circulatory system.
2.How is a diagnosis of carcinoid syndrome established?
•Diagnosis is made by measurement of 24-hour urinary 5-hydroxyindoleacetic acid, which is a degradation component of serotonin.
•Radiological methods may also be used to help identify the primary.
3.How is carcinoid syndrome treated?
•Medical treatment
•Loperamide: Symptomatic relief of diarrhoea
•Octreotide or lanreotide: Somatostatin analogue; blocks the release of tumour mediators
•Radiotherapy: Targeted to the tumour
•Hepatic artery embolisation or radiofrequency ablation: For hepatic tumours
•Chemotherapy
•Surgical treatment
•Curative resection
•Debulking of tumour
4.What are the common organisms in infective endocarditis?
• _Streptococcus viridans ._
• _Staphylococcus aureus_ (common in IVDUs).
• _Staphylococcus epidermidis ._
• _Enterococcus ._
•Staphylococcal endocarditis has the worst prognosis, streptococcal the best.
•Other causative organisms include
• _Streptococcus bovis_
•HACEK group ( _Haemophilus species , Actinobacillus , Cardiobacterium , Eikenella_ , and _Kingella_ )
• _Pseudomonas_
• _Coxiella burnetti_
•Fungal infections
5.What are the diagnostic criteria for infective endocarditis?
•Modified Duke criteria, which include two major and five minor criteria
KEY POINTS
•Tricuspid regurgitation is a pansystolic murmur which is heard best at the left sternal edge in inspiration.
•Ensure that you examine for peripheral signs, such as pulsatile hepatomegaly and a prominent JVP.
•Always assess for signs of right heart failure.
•Be aware of the Duke' s diagnostic criteria for infective endocarditis.
REFERENCE
Joint Task Force on the Management of Valvular Heart Disease of the European Society of Cardiology (ESC) and the European Association for Cardio-Thoracic Surgery (EACTS). (2012) Guidelines on the management of valvular heart disease (version 2012). _Eur Heart J_ 33: 2451– 96.
VENTRICULAR SEPTAL DEFECT
Please examine the cardiovascular system of this patient who has presented with dyspnoea.
FINDINGS
• _General:_ Well looking, often young
• _Peripheral:_ Possible clubbing (can occur in non-cyanotic congenital heart disease)
• _Chest:_
•Inspection/palpation: Laterally displaced apex, palpable thrill at LSE, left parasternal heave
•Auscultation: Pansystolic murmur loudest at the left parasternal area (also heard at the apex), loud P2 if pulmonary hypertension
PRESENTATION
This patient has a ventricular septal defect. The pulse is regular, and there is no evidence of infective endocarditis. The apex beat is undisplaced. There is a left parasternal heave. On auscultation, there is a pansystolic murmur loudest at the left sternal edge, but also audible at the apex.
•Comment on presence/absence of a loud P2.
•If the patient is young, the cause is likely to be congenital.
•In elderly patients, look for factors contributing to ischaemic heart disease.
•Always look for evidence of complications or previous surgical repair.
INVESTIGATIONS
•ECG
•Likely to be normal if small asymptomatic VSD.
•Other features include left-axis deviation and left ventricular hypertrophy in a moderate defect _or_ mainly right ventricular hypertrophy in a large defect.
•Echocardiography
•Assess size, position of VSD, degree of shunt, LV size and function, pulmonary pressures, aortic valve function and competence.
•CXR
•Normal in small VSDs
•Pulmonary plethora (degree depends on size of VSD)
•Cardiomegaly
•Cardiac catheterisation
•Shunt assessment through oxygen saturation measurement in the right ventricle
MANAGEMENT
•Conservative
•Most close spontaneously (especially if muscular defects)
•Medical
•ACE inhibitors (reduce adverse remodelling)
•Surgical
•Surgical closure
•Symptomatic VSD
•Infective endocarditis
•Post-MI (acute septal rupture)
•Haemodynamic compromise/volume overload
•Large shunt
•Development of AR in perimembranous VSDs
•Percutaneous closure
•Perimembranous VSDs can be closed using a transcatheter approach
QUESTIONS
1.What are the causes of a VSD?
•Congenital
•' Lone'
•Associated with other defects, e.g. tetralogy of Fallot, AVSD
•Acquired
•Post-MI (acute septal rupture)
•Following trauma
2.What is the classification of VSDs?
•There are four main types of defects:
•Membranous septal (which includes perimembranous VSD, which is the most common type of VSD)
•Muscular septal
•Atrioventricular canal
•Subarterial infundibular (or conal)
3.What are the complications of a VSD?
•Infective endocarditis
•Pulmonary hypertension
•Eisenmenger' s syndrome
•Aortic valve leaflet prolapse and eventual aortic regurgitation (with perimembranous VSDs)
4.What are the associations of a VSD?
•Aortic regurgitation
•Patent ductus arteriosus
•Coarctation of the aorta
•Tetralogy of Fallot (VSD, overriding aorta, pulmonary stenosis, right ventricular hypertrophy)
•Turner' s syndrome
•Trisomies
•21: Down' s syndrome associated with endocardial cushion defects
•18: Edwards' syndrome
•13: Patau' s syndrome
5.What are the potential risks associated with transcatheter closure of a VSD?
•Displacement
•Misplacement
•Cardiac tamponade
•Aortic regurgitation
•Haemorrhage
•Complete heart block requiring pacemaker insertion
•Need for open cardiac surgery
KEY POINTS
•The murmur associated with a VSD is a pansystolic murmur, heard loudest at the left sternal edge.
•Always check for a parasternal heave.
•Patients will often be young and asymptomatic.
•Perimembranous VSDs can be closed using a transcatheter approach, but be aware of the risks associated with this.
REFERENCE
National Institute for Health and Care Excellence. (2010) Guidance IPG336. Transcatheter endovascular closure of perimembranous ventricular septal defect. Available from https://www.nice.org.uk/guidance/ipg336. Accessed 26 March 2016.
CARDIOLOGY STATION SUMMARY
•The cardiac examination takes time. Don' t spend too much time covering the peripheral signs, although at the same time, be sure you don' t miss any (a skill that can only be learnt with practice).
•Ensure that when you listen to heart sounds and murmurs, you are seen to time them with the carotids.
•If possible, when you present your case, try to start with the diagnosis and explain why you think this is the diagnosis, and which investigations and management you would institute. This makes you sound confident, and if you really know your cardiology, you can demonstrate to the examiners that you are an excellent candidate without them having to ask any difficult questions.
•In valvular heart disease, pay attention to signs other than the murmur (e.g. character of the peripheral pulse and apex beat and association of differences in pulse pressure). It is possible you may know the diagnosis before you listen to the chest. A summary of the characteristics of some of the different murmurs is presented below.
**Lesion** | **Site** | **Timing** | **Character** | **Bell or diaphragm** | **Position** | **Respiratory phase**
---|---|---|---|---|---|---
Aortic stenosis | Aortic area | ESM | Rough/harsh | Diaphragm | Forward | Expiratory
Aortic regurgitation | LSE > apex | EDM | High pitch/ decrescendo | Diaphragm | Forward | Expiratory
Mitral stenosis | Apex > axilla | Mid-diastolic murmur | Low pitch, rumbling | Bell | Left lateral | Expiratory
Mitral regurgitation | Apex > axilla | Pansystolic murmur | Blowing | Diaphragm | Left lateral | Expiratory
Tricuspid regurgitation | LSE | Pansystolic murmur | Blowing | Diaphragm | Forward | Inspiratory
Ventricular septal defect | LSE | ESM | Harsh/high pitch | Diaphragm | Forward | Expiratory
•Auscultation fi ndings of common valvular abnormalities
•Be well rehearsed with the manoeuvres used to auscultate specific murmurs; this is an area where candidates often look clumsy.
•At the end of your examination, remember to inform the examiners of the other pertinent examinations that you would like to perform, e.g. fundoscopy for Roth' s spots in a case of subacute bacterial endocarditis, urinalysis (looking for evidence of haematoproteinuria).
•If you see a midline sternotomy scar, ensure that you look at the legs for evidence of venous harvesting scars. To not do this would be foolish; you can do it from the end of the bed before you have even touched the patient. You can tell that they have ischaemic heart disease and have had a coronary artery bypass graft.
Station 4: Ethics and Communication Skills
Hints for the Ethics and Communication Skills Station
Breaking Bad News
Care in the Deteriorating Patient
Driving Regulations
Initiating a New Therapy
Long-Term Condition
Medical Error
Mental Capacity
New Diagnosis
Organ Transplantation
Ethics and Communication Skills Station Summary
HINTS FOR THE ETHICS AND COMMUNICATION SKILLS STATION
I' ve learnt that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.
**Maya Angelou**
•Always remember the above quote in all your interactions in practising medicine and especially when dealing with difficult and sensitive situations.
•In this station, your communication skills are being thoroughly evaluated.
•It is not just your ability to speak that is being assessed, but also your ability to listen to the concerns that your patient presents, and how you empathize with them.
•Read the scenario you are given carefully. Look at what is being asked. The range of tasks in this station is wide, from breaking bad news to explaining a new diagnosis and addressing the related issues.
•Make good use of the time before entering the station to think through how you are going to tackle the scenario. Make a list of the issues to be addressed so that you do not forget them under the time pressure.
•Start with an open question. Give the patient as much time as they need to talk and allow them to finish before speaking.
•If this is an angry relative/patient that you are dealing with, always take on board their frustrations and apologise for any wrongdoing (regardless of what your assessment is of the complaint). Never disregard their concerns, as the matter will be of significant importance to them.
•Always try to empathise with the patient' s situation.
•As the consultation progresses, use more closed questions to help guide the consultation.
•Summarise the main points discussed and addressed as you are going along, to act as a prompt to both you and the other individual for any other issues yet to be discussed.
•Remember that your ability to give accurate factual information and the way in which you relay this are both being assessed.
•Be clear and concise with your answers to any questions. Do not use jargon.
•During your discussion, remember the four main ethical principles and use them to help guide your approach to any ethical dilemma:
•Autonomy: Respect for the individual and the choices they make.
•Justice: Equality in the distribution of healthcare resources.
•Beneficence: To act in the best interest of the patient.
•Non-maleficence: Actions should not harm the patient (based on the principle of _primum non nocere_ : first do no harm).
•Remember the issue of confidentiality in this station. Confidentiality may be breached in certain situations, such as child protection, notifiable diseases, fitness to drive and serious crimes.
BREAKING BAD NEWS
BACKGROUND INFORMATION
You are asked to see a 64-year-old lady in the respiratory outpatient clinic with the results of her staging CT. The patient was referred with a history of a persistent cough unresponsive to several courses of antibiotics. A CXR requested by the general practitioner (GP) prior to the referral showed a probable mass in the right upper lobe. The CT confirmed the mass in the right upper lobe, with mediastinal and subcarinal lymph nodes, a small pleural effusion and multiple liver metastases. The current radiological staging is T3 N2 M1b, making this cancer inoperable.
KEY POINTS FOR THE PATIENT
•You have been seeing your GP over the past 2 months with a persistent cough.
•You have come to clinic to get to the bottom of this and don' t want anyone else to be here with you.
•You have been given several courses of antibiotics which have not cleared the cough.
•You have been getting progressively more breathless over the past 1 month.
•Your children have noticed a change in your voice and you have lost a stone in weight recently.
•You smoked 10 cigarettes a day for 40 years but stopped 2 years ago.
•You have had pain in your right arm and difficulty raising it.
•The GP recently started you on blue and purple inhalers and a course of steroids.
•You are annoyed that your GP has not taken you more seriously.
•You have a strong suspicion that you have cancer, and this is keeping you awake at night.
•The main concern you have is that your father died of lung cancer and was in a lot of pain towards the end.
•You want to be offered all possible treatments available and are angry and upset when you are told that the treatment will be palliative and not curative.
•Other than smoking, you have always been very healthy, and so you understand that treatment will only be palliative.
•You worry how you will break the news to your family.
•You ask if the cancer may have been treatable if you had been referred earlier.
•You want to lodge a formal complaint against your GP, as things may have been different if you had been referred sooner.
SUGGESTIONS FOR THE CANDIDATE
•Introduce yourself to the patient and ask the patient their understanding of why they have been referred to the clinic.
•Ask them if they have come with anyone today and if they would like anyone else present.
•Ask the patient about the symptoms they have been having, what prompted them to see their GP and about any treatments they have had.
•Ask about their main concerns and what they think might be the underlying cause.
•Recap the history to the point where investigations began and tell the patient that you have all the results.
•Explain that you have some bad news and ask again if the patient would like anyone present with her.
•The main reason for investigation was to exclude a malignancy.
•Unfortunately, the results show that there is a tumour in the lung involving lymph nodes and lesions in the liver; the diagnosis is likely to be cancer.
•Pause for the patient to take the information in and give them as much time as they need.
•Ask the patient if they have understood what you have told them before proceeding.
•Explain that the case has been discussed at the lung cancer multidisciplinary team (MDT) meeting.
•Explain that because of the spread, the cancer cannot be surgically resected and that treatment options will be limited to disease control and not cure.
•Ask the patient what their main concerns are and acknowledge them.
•Try to acknowledge each concern and offer help.
•Gauge how much information the patient would like to be given.
•Reiterate that although it is not possible to treat the underlying cause, there are other services and support that can help the patient to cope with the diagnosis and the disease.
•Acknowledge the patient' s anger and frustration with the GP, and that they ought to contact the GP to discuss the issues.
•Advise that it would be difficult to say if the cancer could have been caught earlier, as you don' t know how aggressive it is.
•Tell the patient that there is a lung cancer specialist nurse that will see them today before they go, and that they will be a regular point of contact.
•Offer the patient to help break the news to the family.
•Recap the information given and ask if the patient has any questions.
•Explain that you will see them in clinic next week and ask if they would like to bring any family or friends along with them to discuss the diagnosis again.
THEMES EXPLORED
BREAKING BAD NEWS FOR AN INCURABLE CONDITION
•This is often difficult due to the emotions involved and the range of patient responses that one may encounter. Remain calm yourself and make sure that you listen to the patient.
•Ensure that you ask the patient about their understanding of the current situation before proceeding to give them the news.
•Do not use any euphemisms or jargon and make sure that you are being clear with the diagnosis.
•If the patient has a diagnosis of cancer, use the word _cancer_ .
•Make sure that you have all the correct/relevant information to hand.
•Appear empathic and ensure that the patient has time to digest the information given.
•Remember to summarise the key points at the end of the consultation and offer to see them again in clinic.
•Offer the relevant support: specialist nurses, palliative care teams, information leaflets and contacts for support groups.
DEALING WITH AN ANGRY PATIENT
•Acknowledge that the patient is angry or upset and let them know that you are aware and that it is a normal response.
•Allow them to voice their anger and do not react to the situation.
•Ask them what specific things are making them angry and offer them potential solutions to help overcome these issues.
CARE IN THE DETERIORATING PATIENT
You are the registrar covering the medical wards for the evening. You have received a call from one of the nurses on a respiratory ward to speak to a patient' s son. The patient is a gentleman with bowel cancer with liver and lung metastases. He was admitted to hospital with dyspnoea and has been treated for a lower respiratory tract infection with intravenous antibiotics. Despite this, he has deteriorated in the 5 days since admission. The patient himself recognised this on the registrar ward round today, and following a discussion, antibiotics were stopped with a decision being made to commence morphine for dyspnoea and refer the patient to the palliative care team. The ward team were asked by the patient about his prognosis today, and thought this would likely be measured in ' days to weeks' . The family were not on the ward for afternoon visiting, and the nurse had called the son, asking him to come in for the consultant ward round tomorrow. Following the call, he has visited this evening and is angry that his father is ' being left to die' . You have gained consent from the patient to speak to his son.
KEY POINTS FOR THE SON
•You have come to the hospital to see your father, following a call from the nursing team. You are the patient' s only child.
•Your father has had bowel cancer for a few years. During that time, he has had surgery and a few courses of chemotherapy.
•During your father' s last trip to the oncologist, they felt he was not fit for more chemotherapy. You still hold out hope that he will be offered further treatment.
•You know that your father was admitted to hospital a few days ago and has been treated for a ' chest infection' with antibiotics.
•You were hopeful that he would improve while in and be discharged back to his house where he lives alone. You are aware that he' s been bedbound since admission.
•On receiving a call from the nursing team, you came straight to hospital. On hearing antibiotics have been stopped, you feel like the hospital is ' giving up' on your father.
•You have also been told by the nurse that your father has been started on morphine. You have been told this was started to help his breathing, although you think it is being used for ' euthanasia' .
•You have not spoken to your father about this since you came in, but you are worried that he' s ' lost his fight' .
•You find it hard to accept that your father is deteriorating. On direct questioning, you admit that you are lacking support (you don' t have any other family). You don' t want your father to die, as he is ' all you have'.
•If all the above issues are covered, you will feel supported, be accepting of the plan and come to the ward round tomorrow to meet the consultant and discuss future plans.
•If you are not given appropriate information or feel you are not listened to, you will get angrier and leave the conversation.
SUGGESTIONS FOR THE CANDIDATE
•Note that you are dealing with an angry relative.
•Allow the son to voice his concerns, providing adequate time for him to do so.
•Do not interrupt him.
•Acknowledge the son' s distress.
•Talk through his understanding of his father' s condition. This could include his father' s likely prognosis.
•Explain that adequate treatment has been given for the lower respiratory tract infection.
•Explain that much of his father' s dyspnoea could be related to lung metastases, and the general fatigue related to advanced cancer.
•Explain that oncological treatment will not be possible, but there is much that can be done to improve his father' s symptoms.
•Explain that morphine is now used regularly to treat dyspnoea in advanced illness. This will be started at a low dose and prescribed safely. It is usually very well tolerated.
•Explain that a referral to the palliative care team has been made to optimise symptom control measures, but also to help with future plans (rapid discharge home may be possible).
•Reassure the son that the plan was discussed with his father, who was in agreement.
•Explain to the son that the medical and nursing teams on the ward, along with the palliative care team, will try to support him psychologically while his father is in hospital. Further support for the son could be offered from the hospital chaplaincy team if he were in agreement.
•Reassure the son that if he has further questions after this meeting, these could be discussed on the consultant ward round tomorrow.
THEMES EXPLORED
SUPPORTING AN ANGRY RELATIVE
•It is important to both acknowledge the relatives' distress and find out why they are angry.
•Talk through their concerns one by one, first by listening and then by offering whatever information you can.
•Offer follow-up to the conversation, including a further opportunity to ask questions after this discussion.
•Recognise anticipatory grief (grief occurring before a patient has died) as a cause of distress in the families of unwell patients.
RECOGNITION OF A DETERIORATING PATIENT
•Recognition of a deteriorating patient is important for many reasons. First, it allows an appropriate treatment plan to be constructed. Additionally, it allows clinical teams to keep patients and families updated, and to help them plan for the future (such as the patient' s preferred place of death).
•Signs that a patient is in the last weeks of life include fatigue, decreased mobility, loss of independence in activities of daily living (ADLs), decreased appetite and impaired cognition.
•Signs that a patient is in the last days of life include decreased fluid intake, impaired swallowing, decreased conscious level and changes in breathing pattern.
•While this case did not cover this specifically, it is important to have an understanding of the ' priorities of care for the dying person' .
•It is also important to be aware of recent developments in decisions about cardiopulmonary resuscitation.
REFERENCES
Leadership Alliance for Care of Dying People. (2014) One chance to get it right. Available from https://www.gov.uk/government/publications/liverpool-care-pathway-review-response-to-recommendations. Accessed 4 August 2016.
Resuscitation Council (UK). (2014) _Decisions Relating to Cardiopulmonary Resuscitation_ . 3rd ed. Available from https://www.resus.org.uk/dnacpr/decisions-relating-to-cpr/. Accessed 4 August 2016.
DRIVING REGULATIONS
Mr David Clegg has come to see you in the sleep clinic. He is a 55-year-old gentleman who has been complaining of symptoms of tiredness over the past year. He has been investigated by his GP, and recent thyroid function tests (TFTs) and a full blood count (FBC) have been normal. The patient is complaining that his symptoms are interfering with his work and he is falling asleep inappropriately. The GP is concerned that the patient may have obstructive sleep apnoea/obesity hypoventilation syndrome. The patient weighs 144 kg and is 1.65 m tall. Discuss the diagnosis and related information with the patient.
KEY POINTS FOR THE PATIENT
•You have been referred to the clinic by your GP to investigate the cause of your sleepiness.
•All your blood tests have been normal.
•Your GP thinks that the tiredness may be related to the fact that you are overweight.
•The tiredness is having a significant impact on your life.
•You have fallen asleep in several important meetings at work, which was rather embarrassing.
•You are concerned as you have found yourself falling asleep at the wheel of your car. You nearly crashed into a barrier the other day.
•Your wife sleeps in a separate bedroom, and this is affecting your sex life as well.
•You have tried losing weight, but you are not getting very far.
•You drink up to three pints of beer most nights.
•You wake up most mornings complaining of a headache.
•You feel as though you have not had a good night' s sleep for months.
•You are beginning to feel depressed about this now.
•You want to know what can be done.
SUGGESTIONS FOR THE CANDIDATE
•Start by asking the patient if he knows why he has been referred to the sleep clinic.
•Ask him specifically how the tiredness is impacting on the various aspects of his life. Ensure that you ask about work and home.
•Ask about a bed partner and if they have reported the patient snoring at night or having periods where they stop breathing.
•Ask the patient what they understand about the terms _obstructive sleep apnoea_ and _obesity_ , and whether the GP has explained anything about the potential diagnosis.
•Go on to explain the potential diagnosis.
•Explain to the patient that further tests will have to be carried out before confirming this.
•Initially, the patient will be asked to complete an Epworth Sleep Score and then further tests will include blood tests (FBC, urea and electrolytes [U&Es], liver function test [LFTs], TFTs and ferritin levels), overnight oximetry and possibly limited/full polysomnography.
•Advise the patient on lifestyle factors: cut down on alcohol intake and attempt to lose weight.
•One of the main issues that will need to be covered here is that the patient is falling asleep at the wheel of his car.
•You must advise him that he has a condition that is affecting his ability to drive, and he will need to stop driving and has a legal duty inform the Driver and Vehicle Licensing Agency (DVLA). The regulation for group 1 drivers is that they need to stop driving until satisfactory control of symptoms has been attained.
•Ask the patient if they have any questions.
•Summarise the above information for the patient before ending the consultation and offer to give them some information leaflets prior to leaving.
THEMES EXPLORED
•The main points to cover in this case are to give the patient information regarding the potential diagnosis and also about what investigations may be required.
•The patient has specifically mentioned that he is falling asleep while he is driving. This needs to be taken seriously, and it is the responsibility of the medical practitioner to advise the patient to stop driving and inform the DVLA immediately. This must also be documented in the patient' s notes.
•There are many medical conditions about which the patient needs to inform the DVLA and may also need to stop driving.
•Once you have discussed this with the patient, it is their responsibility to inform the DVLA. If the patient continues to drive against medical advice, you will need to urgently contact the DVLA and disclose the relevant medical information in confidence. Remember that you must inform the patient that you are going to do this and you should confirm this in writing to the patient and their GP once you have informed the DVLA.
Driving Regulations for Selected Medical Conditions
**Condition** | **Group 1 regulations** | **Group 2 regulations**
---|---|---
First unprovoked/isolated seizure | 6 months off driving from date of first seizure (or factors to suggest increased risk of further seizures, then cannot drive for 12 months) | 5 years off driving from first seizure and must be on any treatment for epilepsy in that time
Epilepsy or multiple unprovoked seizures | Review licence may be issued if no seizures for 5 years (with medication if required) | Must remain seizure-free for 10 years with antiepileptic medication
Stoke/transient ischaemic attack (TIA) | No need to notify DVLA (but stop for 1 month) unless residual neurological defect beyond 1 month (especially if visual field defect or cognitive defect and limb function impairment) | Licence revoked for 1 year If recurrent TIAs/stroke, must undergo functional cardiac testing first
Acute coronary syndrome | Cannot drive for 1 week after successful coronary angioplasty | Cannot drive for 6 weeks until exercise test requirements ful?lled
Pacemaker insertion or box change | Do not drive for 1 week | Do not drive for 6 weeks
Permanently barred | Implantable cardioverter defibrillator (ICD) implantation (prophylactic) | Cannot drive for 6 months from the date of insertion
Source:Adapted from Driver and Vehicle Licensing Agency, Assessing fitness to drive – A guide for health professionals' document, DVLA, Swansea, 2016. Available from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/517268/Fitness_to_drive.pdf.
Note:Group 1 includes cars and motorcycles. Licences are valid until 70 (unless restrictions applied for medical conditions), after which renewal is every 3 years.
Group 2 includes large lorries (Category C) and buses (Category D). Licence valid for a maximum of 5 years only and must be renewed every 5 years (or at 45 years if issued before 19 January 2013) until aged 65 years, following which there is an annual review.
General Medical Council (GMC) guidance. The driver is legally responsible for informing the DVLA about any condition or treatment that impairs the patient' s fitness to drive.
REFERENCE
Driver and Vehicle Licensing Agency. (2016) Assessing fitness to drive – A guide for health professionals' document. Available from https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/517268/Fitness_to_drive.pdf. Accessed 20 April 2016.
INITIATING A NEW THERAPY
You are the medical registrar in a rheumatology outpatient clinic. Mrs Patel is a 52-year-old lady who has a history of severe rheumatoid arthritis. She has tried several disease-modifying drugs (including methotrexate and anti– tumour necrosis factor [TNF] therapy) without significant reduction in disease activity. She has come to see you in clinic today as you wish to start therapy with rituximab.
KEY POINTS FOR THE PATIENT
•You have suffered with severe rheumatoid arthritis for many years now.
•You have active disease which is proving difficult to control and is affecting your quality of life and ability to work as a shop assistant.
•Steroids have caused significant thinning of your bones, so you are reluctant to have the dosage increased.
•You have tried several disease-modifying antirheumatoid drugs (DMARDs), including methotrexate, which caused a problem with your liver tests and so had to be withdrawn. You have also been treated with anti-TNF therapy with infliximab, but your disease remains active.
•You are frustrated with repeated trials of medications which either do not work or cause more problems due to side effects, and have read about MabThera (rituximab) on the Internet and insist that you are treated with this.
•You understand this is an expensive treatment but insist that this is the treatment you want.
•You want to know more about how this drug works. Will it cure you?
•You want to know what the potential effects of the drug are. Importantly, will it damage your liver like methotrexate?
•You are afraid of the immunosuppressive effects of the drug, as you had a tuberculosis (TB) scare about 20 years ago, back in India, when a lump was found in one of your neck glands not long after your mother was diagnosed with TB.
•You do not recall whether you received any treatment at the time, only that you had a biopsy taken.
•You are concerned that the history of TB will affect your eligibility for treatment with rituximab and want to know what can be done to ensure that initiation of treatment is not delayed.
SUGGESTIONS FOR THE CANDIDATE
•You have a very knowledgeable patient in clinic with you, who is very well read about her condition.
•You ask how she is managing at present with her condition, and how it is affecting her life.
•Listen to and empathise with the patient' s concerns regarding the effects of the condition and the impact it has on her.
•You note that the patient has tried several DMARDs and anti-TNF therapy in the past without much benefit (and with lots of complications).
•You explain that abnormalities with liver function tests are a well-documented effect of treatment with methotrexate, and unfortunately, osteoporosis is one of the long-term effects of treatment with steroids (as well as other effects, such as hypertension, diabetes, fluid retention, stomach ulcers and eye problems).
•You enquire where the patient read about MabThera – an official evidence-based website or from a general Internet search.
•You explain that rituximab is a treatment that works on the B-cells of the immune system to reduce inflammation and improve your symptoms, and that there are strict criteria for eligibility; however, she appears to have met the criteria, having failed DMARDS, including methotrexate and at least one anti-TNF agent.
•Explain that the medication works by reducing inflammation but cannot cure the disease.
•Give the patient time to take this in and ask any questions.
•Explain that the treatment is given as an infusion in hospital (usually as a day case procedure) and can take up to 6 hours to complete.
•Occasionally, people can feel unwell during the infusion and develop a fever and/or wheeziness or feel dizzy/light-headed (due to a drop in blood pressure), but usually measures can be taken to overcome these, such as slowing the rate of the infusion (but in severe cases, it may have to be stopped).
•The main risk of the treatment is of infections, as rituximab works by dampening down the immune system, so if you develop a sore throat, fever or other signs of infection, then they must contact a doctor immediately. Rituximab can also be associated with serious infections, such as TB. Enquire about past TB exposure.
•Before initiating therapy, TB needs to be excluded with a chest radiograph. If it is positive, and as there is uncertainty whether the TB was adequately treated previously, she will need chemoprophylaxis before treatment with rituximab, but this should have been addressed prior to treatment with anti-TNF therapy. In addition, she will need to have an influenza vaccination and ensure that she is up to date with pneumococcus vaccination.
•Reassure her this is usual practice, that it is safer to delay starting rituximab for a short time to prevent potential life-threatening infection with TB, and to ensure that she is well-vaccinated to reduce risk of infections, and that this does not mean she will be precluded from receiving treatment.
•Make sure you listen to and address the patient' s concerns. Summarise and ask if there are any questions.
•Offer an information leaflet about rituximab and schedule another appointment to discuss any concerns and initiate the treatment pathway.
THEMES EXPLORED
• _Autonomy:_ The main theme explored in this case is how to counsel a patient effectively regarding a new treatment with potentially serious side effects, without coercion, to enable them to make an informed decision as to whether to proceed with the treatment.
• _Beneficence:_ With novel/biological therapies, there is a risk– benefit balance between potential adverse effects of the drug versus the desired benefits, which is often difficult when the long-term effects of such medications are often not fully known.
• _Justice:_ National Institute for Health and Care Excellence (NICE) guidelines suggest that patients with active rheumatoid arthritis who had an inadequate response to, or are intolerant of, other DMARDs, including at least one TNF inhibitor, should be considered for rituximab (and methotrexate). This would justify the need to give such potentially harmful treatments.
• _Do no harm:_ Prior to treatment with anti-TNF therapy, patients should be screened for TB, and active TB must be adequately treated, as therapy carries an increased susceptibility to developing TB. They should also be up to date with other vaccines, such as pneumococcus and receive influenza vaccine.
•Patients with a past history of extra-pulmonary TB or abnormal CXRs require close monitoring on treatment.
•Patients with previously inadequately treated TB require chemoprophylaxis before commencing treatment.
•Patients require close monitoring for symptoms of TB while receiving biological therapy and for 6 months after stopping.
•If patients develop symptoms suggestive of TB on biological therapy, they will require full treatment with chemotherapy.
REFERENCES
Buch MH, et al. (2011) Updated consensus statement on the use of rituximab in patients with rheumatoid arthritis. _Ann Rheum Dis_ 70(6): 909– 20.
Bukhari M, et al. (2011) BSR and BHPR guidelines on the use of rituximab in rheumatoid arthritis. _Rheumatology_ 50(12): 2311– 3.
National Institute for Health and Care Excellence. (2010) TA195. Adalimumab, etanercept, infliximab, rituximab and abatacept for the treatment of rheumatoid arthritis after the failure of a TNF inhibitor. Available from https://www.nice.org.uk/guidance/ta195. Accessed 24 April 2016.
LONG-TERM CONDITION
You are the registrar in the renal outpatient clinic. You are seeing a 32-year-old female patient who has recently been given a diagnosis of autosomal dominant polycystic kidney disease (ADPKD). She has come back to clinic today wanting more information regarding her diagnosis and the implications it may have for her future. Her mother also has the condition and may be starting dialysis in the near future. She has a young daughter and is planning to expand her family.
KEY POINTS FOR THE PATIENT
•You were recently seen in the renal clinic when you were given a diagnosis of polycystic kidney disease.
•You have had some time to digest the information given to you previously and you now have several concerns that you would like to discuss with the doctor today.
•You are extremely anxious regarding the diagnosis.
•You have a three-year-old daughter and had been planning to expand your family.
•You are concerned regarding the prognosis of the condition and associated complications.
•You enquire if there are any alternative treatments or surgical options available to help prevent any complications and decline in renal function.
•Your mother also has the disease, and her doctors are considering starting her on dialysis in the near future.
•You wish to know whether this might be the case for you in the future and, if so, when might this be and whether you require a kidney transplant at some stage.
•You wish to know if there is anything that can be done to prevent yourself from getting to the stage your mother is currently at, i.e. requiring long-term renal replacement therapy (RRT).
•As your family is not yet complete, you wish to know if this diagnosis would prevent you from having any more children and what are the risks involved.
•If you were to become pregnant, would pregnancy affect the progress of the disease?
•Since both you and your mother have the condition, you are extremely concerned about the inheritance of the condition and whether you could have passed on the disease to your daughter.
•You want to know if your daughter can be tested and if there is anything that can be done to prevent her from developing the condition.
SUGGESTIONS FOR THE CANDIDATE
•It is important first to establish the amount of information the patient received during the previous consultation and how much she has retained.
•Summarise the key points and offer to discuss the diagnosis again, and clarify any points that she may not have fully understood.
•Ascertain the patient' s main concerns by asking if she has any specific questions in mind that she would like answered.
•Attempt to alleviate some of the patient' s anxieties by reassuring her that you and the team are there to help and provide whatever support she requires.
•Explain again to the patient that ADPKD is an inherited condition (autosomal dominant) and that the inheritance pattern of the condition means that there is a 50% chance that she has passed it on to her daughter.
•Explain that as it is a cystic condition, the cysts could develop in other organs (liver, pancreas), as well as the kidneys. Problems she might experience include recurrent urinary tract infections, infection or bleeding within a cyst, pain due to the size of the cysts and high blood pressure, all of which are treatable.
•Explain that end-stage renal disease is a potential complication that may occur some years after diagnosis. If her renal function were to decline over the course of time, then she would be prepared for dialysis in advance of needing it. An alternative possibility would be transplantation when a kidney became available, ideally before she needed to start dialysis; however, if this were not possible, then she may require dialysis initially prior to transplantation.
•Explain that the mainstay of treatment is trying to preserve kidney function for as long as possible by reducing complications and treating them aggressively, i.e. good blood pressure control, treating infections, etc.
•Explain to the patient that she will be closely monitored over the coming years with regard to her renal function.
•Explain that newer treatments are being developed, such as tolvaptan, which has recently been approved by NICE for rapidly progressive PKD, but the treatment, as with any treatment, is associated with some side effects, such as polyuria and risk of liver injury. However, she would not currently be eligible for this.
•Reassure the patient that this disease will not prevent her from having more children, but that in future pregnancies, she will be monitored more closely and may have an obstetrician-led pregnancy with close monitoring of her renal function.
•Explain that this may mean more frequent prenatal visits to the hospital and scans to ensure that she and her baby are both doing well. In addition, she may be more closely monitored during and after delivery.
•Explain that pregnancy per se will not affect disease progression currently.
•Explain that the main way of screening for this disease is using ultrasound imaging to look for the presence of cysts within the kidney. This is usually done when the patient is in their 20s.
•When asked about genetic testing, explain that there are many genetic mutations that can give rise to the disease, so it may not be possible to identify the specific one causing her disease. Assure her that her daughter can be screened when she is older. You can offer for her to be referred to a geneticist if she is keen to do so.
•Summarise that although PKD is a chronic/lifelong condition, it will not prevent her from leading a full and active life. She will be closely monitored throughout with specialist services when needed. It will not prevent her from completing her family, and should she progress to needing dialysis, she will be thoroughly supported and helped through this.
THEMES EXPLORED
•When discussing a new diagnosis of a chronic/life-limiting condition, it is important to ascertain the level of information the patient has received beforehand, and their understanding.
•Respecting the patient' s autonomy is crucial, and it is important to gauge the level of information the patient would like to receive. Be open and honest. Try not to use jargon and use language that the patient can understand.
•Offer the patient details of support groups where they can obtain further advice in dealing with the diagnosis, and provide information leaflets that explain the condition.
•Explain that having a chronic disease does not mean that she cannot complete her family.
•She must be informed that 50% of her offspring may also be affected with the condition due to the pattern of inheritance. She will not be expecting (or wanting) to hear this, so be cautious in your approach.
•The importance of reducing complications must be emphasised.
•If renal decline is progressive, then development of end-stage renal disease may be unavoidable, although there are options of renal replacement therapy and the need for RRT will not be imminent. Reassure her that new therapies are always emerging.
•Offer a further consultation to discuss any points and referral to an obstetrician, if desired, for future pregnancy planning and a geneticist for further genetic counselling if she so desires.
REFERENCE
National Institute for Health and Care Excellence. (2015) Guidance TA358. Tolvaptan for treating autosomal dominant polycystic kidney disease. Available from https://www.nice.org.uk/guidance/ta358/resources/tolvaptan-for-treating-autosomal-dominant-polycystic-kidney-disease-82602675026629. Accessed 24 April 2016.
MEDICAL ERROR
You are the registrar covering the medical wards for the weekend. You have received a call from one of the nurses on the elderly care ward to speak to a patient' s daughter. The patient' s daughter is an ITU nurse and is very concerned about the fact that her mother had appeared quite drowsy on her arrival. She alerted the nurses, and they checked her finger-prick blood glucose level, as she is on insulin, and found her to be hypoglycaemic. It further transpired that the dose of insulin that was prescribed on the chart was incorrect, and as a result, she was given a much higher dose of subcutaneous insulin than normal for her.
KEY POINTS FOR THE DAUGHTER
•You have come to the hospital to see your mother, who was admitted with an acute coronary syndrome.
•You are very distressed, as when you arrived, she appeared to be drowsy and not responding to you.
•You alerted the nursing staff and asked them to check her blood glucose level, as she has diabetes treated with insulin, and to check the rest of her observations.
•Your mother was found to have a very low blood glucose level, and this was the likely cause of her drowsiness. She was given glucogel, and this improved her symptoms.
•The nursing staff had informed you that your mother had recently had a sliding scale of insulin stopped and was given her regular dose of insulin. The nurses tell you that she was prescribed and given 30 units of Novomix 30. You tell them that her regular dose is only 10 units of Novomix 30 BD.
•You are very angry and upset that this has happened and want to see the doctor on call.
•You ask the doctor why the insulin was prescribed incorrectly when your mother' s list of medication was brought in with her.
•You want to know why this was not double-checked by the pharmacist.
•You feel that it is completely unacceptable that this has happened when there should be procedures in place to prevent prescribing errors such as this. You are relieved that your mother has recovered, but still disappointed in the lack of care.
•You want to speak with the consultant in charge of your mother' s care and make an official complaint about the incident, as you are concerned that it was a case of bad practice.
•You also insist that an incident form is completed regarding the event to ensure the matter and the individuals involved receive feedback and training.
SUGGESTIONS FOR THE CANDIDATE
•Note that you are dealing with an angry relative who is a fellow health professional.
•Allow the relative to voice all their concerns and provide adequate time for them to do so.
•Do not interrupt.
•Acknowledge the daughter' s concerns.
•Acknowledge what has happened and apologise for the error.
•Address each of the concerns that the patient' s relative expresses.
•Be open and honest and acknowledge the fact that a prescribing error has been made resulting in harm to her mother.
•Assure the daughter that all junior doctors are given training in insulin prescribing and that you will ensure that it is correctly prescribed on the drug chart now.
•Assure the daughter that you will regularly review her mother to ensure that her blood sugars remain stable.
•Offer to arrange a time for the daughter to speak with her mother' s consultant.
•Reassure her that you will fill out an incident report regarding the matter, and this will be escalated to the ward manager and the consultant in charge of the patient.
•Advise the daughter that if she or her mother would like to take the matter any further, they can contact the patient liaison office and write a formal complaint, which would be addressed in accordance with hospital policies.
•Once again, apologise for any distress caused to the patient and assure the daughter that you will relay her concerns to the various parties involved.
THEMES EXPLORED
•The main ethical principle addressed in this case is non-maleficence (first do no harm). The other theme explored is the duty of candour.
•The patient has directly suffered harm as the result of a wrong prescription of insulin.
•The duty of candour is a legal duty which entails all healthcare providers to inform patients (or their representative) and to apologise to them if there has been a mistake made in their care which has resulted in harm. This ensures that patients are given accurate and honest information.
•When addressing any form of complaint, it is important that you have on hand as much information as possible to give to the patient or their relatives.
•You should acknowledge any error that has been made, and you must apologise for this.
•Offer to answer any questions that the patient or relative may have in relation to their concerns to help alleviate the situation.
•Do not try to conceal any information which may later come to light.
•If they are not happy with your explanation, offer to arrange for them to speak with the consultant in charge of the patient' s care.
•Offer them other sources of support regarding patient welfare, such as patient advice services, which are available in all hospitals.
•Reassure them that the matter will be taken seriously and dealt with appropriately.
•All hospitals will have processes in place to report adverse incidents that have taken place and to learn from the outcome of these reports.
REFERENCES
Care Quality Commission. (2015) Duty of candour: Information for all providers: NHS bodies, adult social care, primary medical and dental care, and independent healthcare. Available from www.cqc.org.uk/sites/default/files/20150327_duty_of_candour_guidance_final.pdf. Accessed 4 August 2016.
National Health Service. (n.d.) Report a patient safety incident. Available from www.nrls.npsa.nhs.uk/report-a-patient-safety-incident/. Accessed 4 August 2016.
MENTAL CAPACITY
You are the medical registrar on call covering the weekend. You have been asked to speak to a relative on one of the medical wards, who wants her husband discharged against medical advice. The patient has a background history of Parkinson' s disease, epilepsy and vascular dementia. The patient was admitted on this occasion with a lower respiratory tract infection and a UTI. He is currently being treated with intravenous fluids and antibiotics and has systemic signs of sepsis. The patient' s wife is adamant that she wants him discharged him from hospital and sent home. You try to speak to the patient, but he is too confused to communicate.
KEY POINTS FOR THE PATIENT' S WIFE
•You are the patient' s main carer and have looked after him for many years now.
•You are adamant that he should be discharged, albeit against medical advice.
•You feel that his condition is worsening, as he is not in a familiar environment.
•You also feel that there is no point of him being in hospital, as he is not getting the one-to-one care which he would be getting at home.
•You question the doctor as to why the patient cannot just have antibiotics at home.
•You know that the patient has signed an advance directive stating that he should not get treatment that would unnecessarily prolong his life.
•You feel that the patient is not sleeping at night as it is too noisy on the ward.
•You know that you do not hold lasting power of attorney status, but you know the patient best and this is what he would have wanted.
•You are angry at the doctor because you think nothing is being done as it is the weekend.
•You question the doctor as to what he can do to stop you from taking your husband off the ward.
•You feel that keeping him in hospital is not in his best interests.
•You can give him antibiotics orally and make sure he drinks plenty of fluids at home.
•You are frustrated, as you feel that your views are not being taken into consideration.
SUGGESTIONS FOR THE CANDIDATE
•You are approached by a very angry and frustrated relative.
•Make sure you listen and address the relative' s concerns. The best way to counter this discussion is to approach the relative with empathy and kindness to diffuse the situation.
•Ask her what is worrying her regarding the patient' s stay in hospital.
•Reiterate that the patient is suffering from a urinary tract infection and a chest infection, and is currently having intravenous antibiotics and fluids. Explain that the patient is not well enough for treatment with oral antibiotics as yet, as evidenced by his ongoing signs of sepsis.
•Explain that you are part of a team and that you are all acting in the patient' s best interest.
•When the relative tells you about the advance directive, you can tell her that this infection is a treatable condition and you are actively treating her husband and are not preparing for end-of-life care currently.
•Explain to the relative that you will try to ensure that everything possible will be done to make the patient more comfortable, such as moving him to a side room or a quieter area of the ward where he should be able to get more rest.
•When the patient' s relative asks what you can do to stop her, tell her that you would hope to resolve the situation before this happened, but if the need arose, you would have to involve hospital security.
•Explain that you have tried to assess the patient' s mental capacity, but he is too confused to understand what you are saying.
•Ask the relative if she has lasting power of attorney.
•Explain that as she does not have this status, as the healthcare professional, you are acting in the patient' s best interest by keeping the patient in hospital.
•You would be going against your duty of care by allowing the patient to leave.
•Summarise what you have discussed and ensure that you convey an empathic tone throughout to the relative and acknowledge her concerns.
•Suggest that the relative could speak to the consultant in charge of the patient' s care, after the weekend.
•Reiterate that you are doing your best to treat the patient.
THEMES EXPLORED
•The main theme explored in this case is the issue of mental capacity. The Mental Capacity Act came into force in 2007 in England and Wales, and it helps provide a framework to empower and make decisions for people who are unable to do so for themselves.
•Every adult has the right to make their decisions and must be assumed to have capacity unless proven otherwise. People should be supported to make their own decisions, unless deemed to lack capacity, and any individuals with capacity who make seemingly unwise decisions should be respected.
•To assess whether a patient has capacity, they must be able to understand and retain the information presented to them, weigh up that information and communicate the decision back to you.
•If a patient does not have capacity, then as the healthcare worker responsible for the patient, you can take decisions regarding the patient' s medical care. It is also important to involve the family and carers and seek their views and opinions as to what the patient may have wanted.
•Any intervention made should be the least restrictive intervention and should always be in the patient' s best interest.
•Patients can sign advance directives refusing specific treatments that they would not want should the situation arise. These are legally binding documents that need to be signed in the presence of a witness. They should also contain a statement that the decision should be enforced even if life is at risk.
•If there is any issue regarding the validity of an advance directive in an emergency situation, the appropriate treatment can be given until the validity of the document is verified.
•The patient can also appoint a lasting power of attorney who can make medical (and other) decisions on behalf of the patient who lacks capacity. This document needs to be registered with the Office of the Public Guardian for it to be valid.
•In a situation where a patient lacks capacity and has no representative to discuss their medical treatment, an independent mental capacity advocate (IMCA) can be appointed to act as an advocate for the patient in the decision-making process. An IMCA is not needed if it is an emergency treatment or if the patient is detained and being treated for a mental illness under the Mental Health Act.
REFERENCE
Office of the Public Guardian. (2014) Mental Capacity Act: Making decisions. Available from https://www.gov.uk/government/collections/mental-capacity-act-making-decisions. Accessed 5 August 2016.
NEW DIAGNOSIS
You are the registrar in the gastroenterology outpatient clinic. Your next patient is a 19-year-old man who has recently presented with several episodes of bloody diarrhoea. The patient has had a recent colonoscopy which confirms the diagnosis of ulcerative colitis (UC). Discuss the diagnosis with the patient and the implications of the disease.
KEY POINTS FOR THE PATIENT
•You have come to clinic today to find out the results of a colonoscopy that was undertaken to investigate the cause of the bloody diarrhoea you have been having.
•Previously, you have been fit and well with no other health problems. Over the past 6 weeks, you have had multiple episodes of bloody diarrhoea daily.
•When you were previously seen in clinic, the doctor mentioned that they were looking to see if there was any inflammation of your bowels that could account for your symptoms. However, you did not really understand the explanation given.
•When you are given the diagnosis of ulcerative colitis, you ask the doctor to fully explain this and any problems that are associated with the disease.
•You are worried, as you have just finished your gap year and are due to start university soon and wonder how this will affect your social life.
•You want to know what medication you will need to take.
•You want to know if you will still be able to drink alcohol and go out with your friends and lead a ' normal' life. You hope to go on to university – will this still be possible?
•You want to know how often you will need to see a doctor.
•You read an information leaflet in the waiting room and you want to know if you will need to have an operation and, more importantly, if you will end up with a stoma.
•When pressed by the doctor, you tell them that you are very concerned about this problem, as your dad died at a young age from bowel cancer and you wonder if this is something that you should be worried about, now that you have this diagnosis.
SUGGESTIONS FOR THE CANDIDATE
•Recap the history for the patient and initially gauge their understanding of why they have come to clinic and why they have had the procedure done.
•Ascertain the amount of information previously given and their level of understanding.
•Summarise the previous information.
•Explain that colonoscopy is the usual investigation to look for any active inflammation of the bowel that could be causing his symptoms.
•Explain that biopsies were also taken at the same time to help clarify the underlying disease/inflammatory process.
•Explain that all the results are now available and are consistent with a diagnosis of ulcerative colitis, which is an inflammatory condition affecting the bowel.
•Give them time to absorb the information that they have been given and then wait to see if they have any questions.
•Ask if they have heard of UC or any other inflammatory bowel condition before and, if so, what is their understanding of this.
•Offer to give them an overview of the disease. Do not use jargon.
•Explain that UC causes inflammation of the bowel wall. It is restricted to the large bowel and therefore results in bloody diarrhoea, so his presentation was not unusual or atypical.
•Although the disease mainly affects the bowel, there can be other manifestations outside the bowel with symptoms such as joint aches, rashes, mouth ulcers and sore eyes, but bowel symptoms are most common.
•Empathise with the patient and reassure him that with treatment, there is no reason why he cannot lead a normal life and pursue his studies, and that the condition could be looked after by a specialist in his university town if needed, or he could continue to come and see you and his appointments could be worked around his studies.
•Treatment aims are to control active disease, improve quality of life and maintain remission.
•Treatment decisions will always take into account the patient' s individual needs and preferences.
•As he has only bowel symptoms, treatment would initially be local therapy with oral aminosalicylic acids (ASAs) and/or steroids if symptoms persist.
•If symptoms are still not well controlled, stronger therapies to dampen the immune system could be added at that point.
•Explain that the amount of alcohol he can drink will depend on the treatment he is taking; if on stronger immunosuppression, it would not be recommended, but he should only drink in moderation (and not exceed the current weekly maximum recommendation of 14 units per week spread over 3 or more days)
•Explain that initially he will need a more regular review (every couple of months or so) until the disease is under control, but thereafter, the intervals between clinic reviews would be increased.
•Reassure him that in between visits, there is always a specialist nurse available who he can contact for further advice or earlier review if required.
•Regarding surgery, reassure him that this is undertaken only in severe cases of colitis when the inflammation cannot be controlled with medical therapy, or in emergency circumstances. However, there are now many new treatments available (biological) that can help avoid surgical intervention altogether, even in circumstances where it might have been unavoidable in the past.
•Ask the patient if they have any other specific concerns or if there is anything at all they are worried about. Tell them it is quite natural to be frightened and worried about the new diagnosis, and that you are here to help/address any concerns they may have.
•When bowel cancer is mentioned, be honest and tell him that in patients with UC, there is a slightly increased risk of bowel cancer after having the condition for many years. The risk, however, is still small. Emphasise that all patients are closely monitored and there are surveillance measures with colonoscopy (as recommended by NICE) to pick up any disease early.
•Summarise the main points and emphasise that there is plenty of support available to help him cope with the condition, and the aim of treatment is to help him lead as normal a life as possible.
THEMES EXPLORED
•When giving the diagnosis of a new/chronic/life-limiting condition, the patient has the right to know the full facts regarding the diagnosis and all treatment options available.
•The patient' s autonomy regarding their treatment decisions must be respected if they are of sound mind, especially any refusal of treatment.
•The individual must be allowed to use their own judgement in weighing the pros and cons of available options. They should never be coerced into making any choices.
•It is crucial for the information giver to not allow any preformed judgements about the patient to cloud the information given to the patient.
•If there are any errors or delays in getting to the underlying diagnosis, these should not be hidden and any apologies should be made to the patient and their family.
•In this scenario, it is a young patient who is clearly frightened as to what the future holds with this condition (regarding treatments including surgery, the possibility of cancer and whether he will still be able to lead a normal life). Reassure the patient constantly and give them plenty of sources of further information. Allow them time to digest the information and address their concerns as they arise.
•Remember his concern about the risk of cancer and be aware of the family history and with this, he is at increased risk and so should be offered ongoing annual colonoscopic surveillance. The benefits of such would be prevention of colorectal cancer with early detection, but the psychological impact of having this must also be considered with the patient' s wishes.
•Remember to offer further ongoing support with use of the multidisciplinary team by offering their contact details, and arrange another follow-up appointment where the patient can go through the diagnosis and rediscuss any concerns they have or anything they did not understand.
REFERENCES
National Institute for Health and Care Excellence. (2011) Guidance CG118. Colonoscopic surveillance for preventing colorectal cancer in adults with ulcerative colitis, Crohn' s disease or adenomas. Available from https://www.nice.org.uk/guidance/cg118/resources/colonoscopic-surveillancefor-preventing-colorectal-cancer-in-adults-with-ulcerative-colitis-crohns-disease-oradenomas-35109396155077. Accessed 24 April 2016.
National Institute for Health and Care Excellence. (2013) Guidance CG166. Ulcerative colitis: Management. Available from https://www.nice.org.uk/guidance/cg166/resources/ulcerativecolitis-management-35109695126725. Accessed 24 April 2016.
ORGAN TRANSPLANTATION
You are a doctor working on the intensive care unit. You have been asked to discuss the possibility of organ donation with the family of a patient on ITU who has been pronounced brainstem dead. The patient is a young, fit and healthy 19-year-old girl who was involved in a road traffic accident. The patient' s father is her next of kin, and you approach him to discuss the possibility of donation.
KEY POINTS FOR THE FATHER
•Your daughter has been involved in a road traffic accident, in which she was hit by a car.
•She suffered extensive head injuries and also had a ruptured spleen.
•You were informed by the consultant in charge of her care that her injuries were very severe and that she may not survive. You were naturally extremely upset, as your daughter was young, fit and healthy.
•You were informed that if she deteriorated clinically, tests would be carried out to assess the condition of her brainstem.
•The nature of the tests was fully explained to you, and subsequently, after two rounds of tests, your daughter was pronounced brainstem dead.
•As a family, you are all in shock, as the last time you saw your daughter before the accident she had been excited about her plans for her gap year.
•When asked by the doctor about the possibility of organ donation, you are initially taken aback that this has been raised at your time of loss.
•After some time to think, you ask what this would involve.
•You recall that your daughter had previously mentioned that she had ticked on her driving licence application that she would wish to be an organ donor.
•You want to know what the process of donation would involve and how long it would take.
•You ask which organs are likely to be taken.
•You ask who would receive the organs and whether you would be able to contact them.
•Your main concern is whether your daughter would experience any pain when the organs are removed.
•You are also worried about the appearance of her body after the organs have been taken.
•You ask if there is anyone else who can give you any further information about the process.
•You answer on the family' s behalf that despite your loss, you are willing to give permission for your daughter to be an organ donor, as this was her wish.
SUGGESTIONS FOR THE CANDIDATE
•Initially offer your condolences to the family and empathise with their loss.
•When you bring up the subject of organ donation, acknowledge what a difficult time this must be for them, but explain that the possibility of organ donation is regularly addressed on the intensive care unit (ICU).
•Ask the family if they are aware of any wishes their daughter may have had concerning organ donation. Specifically ask about an organ donation card, or whether it was recorded on her driver' s licence.
•Explain the process ahead. You will contact the organ transplantation team, who will coordinate the process of organ removal. There will be various teams from around the country who will come to obtain the organs, as this hospital is not a transplant centre.
•Advise that you will need to seek the permission of the coroner before organ retrieval can take place, as the patient was involved in an accident
•The organs will go to individuals on a transplant waiting list, who have been identified as being suitable and most urgently in need of an organ transplant.
•As the patient was previously healthy, all the major organs could be used, as well as other body parts, such as corneas, tendons and possibly bones.
•Reassure the family that the patient will not feel any pain, as she is brainstem dead and therefore unable to feel pain, but that she will also be given opiates prior to the procedure to ensure this.
•The surgeons will carefully close the body after removing the organs so the patient' s appearance will not be significantly disfigured.
•The family will not be able to contact the recipients of the organs, but the transplant coordinator would be able to tell them some general information on who the organs have gone to.
•The transplant team/coordinator will also be available to answer any further questions that the family may have.
•Empathise with the family again regarding their loss and thank them all for considering this decision at such a difficult time.
THEMES EXPLORED
•Organ transplantation is an area which is often very difficult to address, as there is so much emotion involved when broaching the subject with relatives who have just lost a loved one.
•Any patient who meets the following criteria should be considered for organ donation, and their family or carers should be offered this option if
•The patient has suffered major and irreversible neurological damage leading to brainstem death or, alternatively, whose condition is such that continuing critical care is considered futile and withdrawal of treatment is being considered.
•The patient is HIV negative.
•The patient is not known or suspected to have Creutzfeldt– Jakob disease.
•If the next of kin disagrees with donation, despite the patient being on the organ donation register, it is current practice not to proceed with the donation regardless of patient autonomy.
•The legal aspects to remember are that the coroner' s permission must be sought (if required) before going ahead with retrieval.
•In the case of determination of death by brainstem testing, medical practitioners must follow the code of practice issued by the Department of Health in 1998.
•Prior to brainstem death testing, three preconditions must exist:
•The patient' s condition is due to irreversible brain damage of known aetiology.
•The patient is in unresponsive coma (potential reversible causes have been excluded).
•The patient is apnoeic and mechanically ventilated.
•If these conditions are met, the standard tests are carried out by two qualified doctors and are repeated after a short amount of time.
REFERENCES
Academy of Medical Royal Colleges. (2008) A code of practice for the diagnosis and confirmation of death. Available from <http://www.bts.org.uk/Documents/A%20CODE%20OF%20PRACTICE%20FOR%20THE%20DIAGNOSIS%20AND%20CONFIRMATION%20OF%20DEATH.pdf>. Accessed 19 August 2016.
Department of Health. (1998) A code of practice for the diagnosis of brain stem death. Available from http://webarchive.nationalarchives.gov.uk/20130107105354/http:/www.dh.gov.uk/prod_consum_dh/groups/dh_digitalassets/@dh/@en/documents/digitalasset/dh_4035462.pdf. Accessed 19 August 2016.
Department of Health. (2003) Saving lives, valuing donors: A transplant framework for England. Available from http://webarchive.nationalarchives.gov.uk/20031220221856/doh.gov.uk/transplantframework/. Accessed 4 August 2016.
ETHICS AND COMMUNICATION SKILLS STATION SUMMARY
•Thoroughly prepare for the range of themes commonly encountered in this station, and this will stand you in good stead.
•Remember that your communication skills are being challenged, and inadequate preparation will be evident to the examiners.
•Use the time before entering the station to plan a structure for your discussion and the topics to be addressed. This will help you organise your thoughts and think through the information to be gathered and given to the patient.
•Always bear the four main ethical principles (autonomy, beneficence, nonmaleficence and justice) in mind when approaching the task.
•Remember that there are other issues that may be the theme of discussion, such as
•Confidentiality
•Driving regulations/DVLA medical guidelines
•Breaking bad news
•Relaying a new diagnosis
•Organ transplantation
•Cardiac resuscitation orders
•Suitability for ITU transfer/care
•Allow plenty of time for pauses to give the individual enough time to take the information on board and ask any questions they might have.
•Summarise the key points of discussion as you go along.
•Always ensure that you are listening to the patient, empathise and remain calm at all times.
Station 5: Brief Clinical Encounters
Hints for the Brief Clinical Encounters Station
Ankylosing Spondylitis
Anticoagulation
Diabetic Retinopathy
Facial Nerve Palsy
Falls
Headache (Idiopathic Intracranial Hypertension)
Hyperthyroidism
Neck Lump
Osteoporosis
Proximal Myopathy
Psoriasis
Rheumatoid Arthritis with Carpal Tunnel Syndrome
Systemic Sclerosis
Third Nerve Palsy or Patient Presenting with Diplopia
Brief Clinical Encounters Station Summary
HINTS FOR THE BRIEF CLINICAL ENCOUNTERS STATION
•Station 5 is worth many marks in the exam, and it is crucial to dedicate a substantial proportion of revision time to it.
•Carefully read the instructions to candidates before entering the station, and brainstorm the specific questions you need to ask to help get the diagnosis.
•Formulate a structure for your focused clinical assessment.
•On entering the room, look closely for any clues to the diagnosis; ' spot diagnoses' are common in this station.
•Generally, begin with an open question, but quickly become more focused after this. Establish a likely diagnosis (and differential) and then find out about specific complications.
•Examination generally involves looking for specific signs to clarify a diagnosis rather than following a set pattern – remember that the examination should be focused.
•Be sure to examine for evidence of complications/manifestations of the primary disorder.
•When presenting the case, give the likely diagnosis first, but have a differential available in case you are asked for this.
•Always ask about any specific concerns that the patient has, and provide a solution or explanation as appropriate.
•Addressing the patient' s welfare and concerns is vital in this station.
ANKYLOSING SPONDYLITIS
This patient, a 25-year-old man, has been experiencing back pain (including pain at night), stiffness and fatigue. Please ask any relevant questions and proceed as appropriate.
FOCUSED HISTORY
1.Duration, timing (often have pain at night) and nature of backache; location of pain
2.Age and gender of patient (onset before 40 years and males affected more than females)
3.Morning stiffness: Duration (improves with exercise and worse with rest)
4.Falls, trauma and injuries to spine
5.Spinal deformity
6.Other joints affected: Sacroiliac joints, hips, knees, ribs
7.Family history of back problems
8.Neurological symptoms: Ensure no bladder/bowel disturbance (all histories of back pain), paraesthesia/numbness/limb weakness
9.Other features: Chest pain, breathlessness, eye symptoms (pain, redness, floaters), enthesitis, fatigue
10.Patient welfare/concerns: Deformity, genetic link, mobility aids
FOCUSED EXAMINATION
•Spine
•Ask the patient to stand up, back and front fully exposed (ensure that you preserve dignity).
•Kyphotic spine, compensatory hyperextension of the neck (' question mark' posture).
•Reduced spinal movements: Rigid, immobile spine.
•Increased anteroposterior (AP) diameter of chest wall.
•Cardiac
•Listen to aortic area and left sternal edge for early diastolic murmur of aortic regurgitation.
•Chest
•Fine apical fibrotic crepitations
•Eyes
•Iritis, visual acuity check
•Gait
•Likely antalgic, will make the spinal deformity more obvious
QUESTIONS
1.What are the immunological associations with ankylosing spondylitis?
•Seronegative spondyloarthropathy
•HLA-B27 positive in > 90% of individuals
•Tumour necrosis factor (TNF)-? and interleukin (IL)-1 also implicated in disease activity
2.How is the diagnosis of ankylosing spondylitis made?
•Mainly clinical from history and examination with supporting radiological evidence
•Young patients (< 40), possible family history
•Plain radiograph – erosions and fibrosis/sclerosis of the sacroiliac joints, squaring of the vertebra (' bamboo spine' ) due to ossification of the anterior longitudinal ligament and intervertebral spaces
•Blood tests: Raised erythrocyte sedimentation rate (ESR)/C-reactive protein (CRP) during active inflammation, normocytic anaemia
•Genetic testing: HLA-B27
3.What is the treatment for ankylosing spondylitis?
•No known cure: Mainly symptomatic
•Physiotherapy: Encourage increased exercise, physiotherapy and exercises for maintaining good posture
•Analgesia: Nonsteroidal anti-inflammatory drugs (NSAIDs), paracetamol and weak opioids
•Corticosteroid injections: For sacroiliitis and enthesitis
•Biological agents: TNF-? antagonists such as etanercept, adalimumab, infliximab, golimumab and certolizumab pegol (help slow disease progression)
•Treat complications such as iritis
•Treat osteoporosis with bisphosphonates to prevent spinal fractures
4.What are the extra-articular manifestations of ankylosing spondylitis?
•Respiratory: Restrictive lung defect/reduced lung capacity due to restricted chest wall movement and apical lung fibrosis.
•Cardiac: Chronic aortitis leading to aortic regurgitation, conduction defects and cardiomyopathy.
•Neurological: Atlantoaxial instability/dislocation and cauda equina syndrome.
•Eyes: Iritis and cataracts.
•Amyloidosis (secondary/amyloid A [AA]): Multisystem involvement, including hepatic and renal involvement.
•Coexistent inflammatory bowel disease is common.
PATIENT WELFARE AND CONCERNS
1.Adequate management of pain.
2.Concern regarding the impact upon activities of daily living – will this stop the patient working?
3.Cosmetic concerns regarding possible spinal deformity.
4.Is this inherited, and could I pass this onto my children doctor?
5.Is there a cure?
CANDIDATE EXPECTATIONS
1.Recognition of an inflammatory arthropathy, and in particular ankylosing spondylitis, from a detailed history
2.Awareness of the need to rule out ' red-flag symptoms'
3.Understanding of the extra-articular manifestations of inflammatory arthropathies
4.Examine for features of ankylosing spondylitis, including extra-articular manifestations, e.g. auscultate for aortic regurgitation and lung fibrosis
5.Explanation that treatment aims are to maintain a good quality of life and prevent disease progression and complications
REFERENCES
National Institute for Health and Care Excellence. (2013) Clinical knowledge summary: Ankylosing spondylitis. Available from http://cks.nice.org.uk/ankylosing-spondylitis#!scenariorecommendation:4. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2016) TNF-alpha inhibitors for ankylosing spondylitis and non-radiographic axial spondyloarthritis. Available from https://www.nice.org.uk/guidance/ta383/resources/tnfalpha-inhibitors-for-ankylosing-spondylitis-andnonradiographic-axial-spondyloarthritis-82602848027077. Accessed 5 July 2016.
ANTICOAGULATION
This 65-year-old man has been having palpitations intermittently for the past 3 months. On one occasion, he attended the emergency department and was found to be in atrial fibrillation (AF) with rapid ventricular response. The team in the emergency department has given him some bisoprolol to slow the rate, and he attends your rapid access outpatient department to discuss anticoagulation.
FOCUSED HISTORY
1.Start by summarising the history and ensuring that the patient knows that the focus of the consultation will be to discuss anticoagulation.
2.Enquire about the risk factors for stroke, i.e. does this patient need anticoagulation?
3.Is the patient known to have hypertension?
4.Is the patient known to have diabetes?
5.Is the patient known to have heart failure?
6.Need to assess the patient' s bleeding risk?
7.Prior history of gastrointestinal (GI) bleeding?
8.Prior history of liver disease?
9.Does the patient have any known renal problems?
10.Prior history of intracranial bleeding?
11.Does the patient have an active cancer?
12.Take an alcohol history.
13.Take a thorough drug history, including over-the-counter and herbal medications.
14.Social history is important. Contact sport should be avoided if the patient is to take anticoagulants.
FOCUSED EXAMINATION
•Observe for stigmata of bleeding/chronic liver disease.
•Check the patient' s blood pressure.
•Look for clinical signs of cardiac failure.
QUESTIONS
1.Do you know of any risk stratification tools that can be used to assess bleeding risk for patients on warfarin?
•The HASBLED score (see below).
•This takes into account hypertension, renal and liver failure, age, drug use, alcohol consumption, labile international normalised ratios (INRs) and medication usage that would predispose to bleeding (NSAIDs and antiplatelets).
•A score out of 9 is given and can be used to calculate bleeding risk. If the patient scores 3 or more, caution with anticoagulation is advised and regular review must be in place if a decision is taken to anticoagulate.
2.Do you know of a tool one can use to predict stroke risk for patients with AF?
•The CHA2DS2-VASc score (see below).
•This score is out of 9 and takes into account other stroke risk factors, including age, diabetes, hypertension and heart failure.
•The higher the score, the greater the risk of stroke.
•Dependent on the score, a percentage stroke risk per year for a patient can be given.
3.Under what circumstances would one use novel anticoagulants such as rivoroxaban as opposed to warfarin?
•The National Institute for Health and Care Excellence (NICE) guidelines suggest that novel oral anticoagulants (NOACs) can be used where appropriate. They are guidelines, and use will be clinician dependent and dependent on local guidelines. An example may be if a patient has mild cognitive impairment and has difficulty managing warfarin.
•Local guidelines may vary.
•If the patient cannot get to and from hospital for blood tests.
•If they have had a cardiovascular event while on warfarin with a subtherapeutic INR.
•If they have a labile INR.
4.List some disadvantages of novel anticoagulants?
•If the patient misses one dose, then they are not receiving the benefit; therefore, likely compliance must be assessed and the importance of this stressed.
•Not all are safe in renal failure, so this may need to be monitored if there are concerns. Dose will vary depending on estimated glomerular filtration rate (eGFR); consult the British National Formulary (BNF).
PATIENT WELFARE AND CONCERNS
1.Clear and concise explanation of the rationale behind the use of anticoagulation, i.e. risk versus benefit and, in the case of anticoagulation, why the benefit outweighs the risk
2.Need to counsel the patient on the risks of anticoagulation (drug– food interactions/noncontact sports/regular blood tests/dentists, etc.)
3.Appreciate that the risk may be unacceptable to a patient who has capacity.
CANDIDATE EXPECTATIONS
1.Ensure that you ask questions that will demonstrate that you are assessing the risk of stroke and the risk of bleeding.
2.Ensure that you thoroughly explain the risks surrounding anticoagulation, but remind the patient that these risks are outweighed by the benefits in stroke risk reduction.
3.Show an awareness of the patient' s autonomy.
HASBLED
**H** – Hypertension uncontrolled (systolic BP < 160)
**A** – Abnormal renal function: Dialysis, transplant, Cr > 2.26 mg/dL or > 200 µ mol/L Abnormal liver function
**S** – Stroke: Prior history of stroke
**B** – Bleeding: Prior major bleeding or predisposition to bleeding
**L** – Labile INR: (Unstable/high INRs), time in therapeutic range < 60%
**E** – Elderly: Age > 65 years
**D** – Prior alcohol or drug usage history (≥ 8 drinks/week) Medication usage predisposing to bleeding: (Antiplatelet agents, NSAIDs) Each point scores 1
CHADSVASC
**C** – Congestive heart failure (or left ventricular systolic dysfunction) scores 1
**H** – Hypertension: Blood pressure consistently above 140/90 mmHg (or treated hypertension on medication) scores 1
**A** – Age ≥ 75 years scores 2
**D** – Diabetes mellitus scores 1
**S** – Prior stroke or transient ischaemic attack (TIA) or thromboembolism scores 2
**V** – Vascular disease (e.g. peripheral artery disease, myocardial infarction, aortic plaque) scores 1
**A** – Age 65– 74 years scores 1
**S** ex **C** ategory – Females score 1
REFERENCES
Lip GY, et al (2010). Refining clinical risk stratification for predicting stroke and thromboembolism in atrial fibrillation using a novel risk factor-based approach: The Euro Heart Survey on atrial fibrillation. _Chest_ 137(2): 263– 72.
National Institute for Health and Care Excellence. (2014) Guidance CG180. Preventing stroke in people with atrial fibrillation. Available from http://pathways.nice.org.uk/pathways/atrialfibrillation#path=view%3A/pathways/atrial-fibrillation/preventing-stroke-in-people-with-atrial-fibrillation.xml&content=view-node%3Anodes-anticoagulation-treatment. Accessed 24 August 2016.
Pisters R, et al. (2010) A novel user-friendly score (HAS-BLED) to assess 1-year risk of major bleeding in patients with atrial fibrillation. _Chest_ 138 (5): 1093– 100.
DIABETIC RETINOPATHY
This patient has noticed his vision has become blurred at times. Please ask any relevant questions and proceed as appropriate.
FOCUSED HISTORY
1.Duration of symptoms: Has there been a progressive change rather than sudden onset?
2.Has there been any visual loss?
3.Assess diabetic control.
4.Other microvascular complications: Nephropathy, neuropathy, autonomic dysfunction.
5.Macrovascular complications: Peripheral vascular disease, ischaemic heart disease, stroke.
FOCUSED EXAMINATION
•Eyes
•Glasses
•Visual acuity (ideally using a Snellen chart)
•Pupillary reactions
•Eye movements
•Fundoscopy
•Cataracts, features of diabetic retinopathy (nonproliferative, proliferative, maculopathy, photocoagulation scars)
•Extras
•Evidence of neuropathy
•Ask for blood glucose measurement, urine dipstick for proteinuria and blood pressure
QUESTIONS
1.What are the features of nonproliferative diabetic retinopathy?
•Microaneurysms (dot haemorrhages)
•Blot haemorrhages
•Hard exudates
•Soft exudates (cotton wool spots)
2.What are the features of proliferative diabetic retinopathy?
•Features of nonproliferative diabetic retinopathy, with evidence of new vessel formation.
•Photocoagulation scars are evidence of treatment.
3.What are the features of diabetic maculopathy?
•Any features of diabetic retinopathy at or near the macula.
•Most commonly, there is circinate formation of hard exudates.
4.What effect does glycaemic control have on the risk of diabetic retinopathy?
•This was assessed by the UK Prospective Diabetes Study (UKPDS) and the Diabetes Control and Complications Trial (DCCT).
•These studies showed that better glycaemic control lowers the risk of retinopathy. For example, a 10% reduction in HbA1c is associated with a more than 40% reduction in the development of retinopathy.
5.What else can be done to lower the risk of retinopathy in type 2 diabetes?
•Optimal treatment of hypertension
6.How is peripheral (nonmacular) diabetic retinopathy managed?
•Background retinopathy does not need treatment but should be monitored.
•If preproliferative retinopathy is severe, laser treatment may be considered.
•Proliferative retinopathy requires laser treatment.
•Severe proliferative retinopathy may not respond to laser treatment and may require vitrectomy.
PATIENT WELFARE AND CONCERNS
1.Vision: Concerns of visual loss?
2.Prevention: Prevention of further deterioration?
3.Treatment: Is treatment possible?
CANDIDATE EXPECTATIONS
1.Recognition that diabetic retinopathy can be asymptomatic or can present with visual disturbance
2.Recognition of the features of diabetic retinopathy
3.Understanding of the stages of diabetic retinopathy
4.Formulate the correct diagnoses and convey this to the patient
5.Generate a management plan, recognising when urgent intervention is necessary to save vision
REFERENCES
Diabetes Control and Complications Trial Research Group. (1995) The relationship of glycemic exposure (HbA1c) to the risk of development and progression of retinopathy in the Diabetes Control and Complications Trial. _Diabetes_ 44(8): 968– 83.
Royal College of Ophthalmologists. (2012) Diabetic retinopathy guidelines. Available from https://www.rcophth.ac.uk/standards-publications-research/clinical-guidelines/. Accessed 5 July 2016. Stratton IM, et al. (2001) UKPDS 50: Risk factors for incidence and progression of retinopathy in type II diabetes over 6 years from diagnosis. _Diabetologia_ 44(2): 156– 63.
UK Prospective Diabetes Study Group. (1998a) Tight blood pressure control and risk of macrovascular and microvascular complications in type 2 diabetes: UKPDS 38. _BMJ_ ; 317(7160): 703– 13.
UK Prospective Diabetes Study Group. (1998b) Efficacy of atenolol and captopril in reducing risk of macrovascular and microvascular complications in type 2 diabetes: UKPDS 39. _BMJ_ 317(7160): 713– 20.
FACIAL NERVE PALSY
This patient has been experiencing difficulty with closing their eyelids on one side. As a result, the eye is becoming dry and irritable. They are anxious to know the cause of the symptoms and what can be done. Please ask any relevant questions and proceed as appropriate.
FOCUSED HISTORY
1.Which eyelid is affected? What appears to be the problem? Does the eyelid shut at all, and if so, how much?
2.Duration of symptoms?
3.Are there any associated problems: speech, swallowing, taste or hearing impairment (tinnitus,* hyperacusis or deafness*)?
4.Any vertigo*/nausea*/vomiting or abnormality in gait* noted?
5.Any associated facial droop or distortion of the angle of mouth, change in taste?
6.Any recent viral (herpes) infection, surgery to the neck (parotid gland), inner ear or mastoid?
7.Medical history/family history: Neurofibromatosis ± acoustic neuroma with or without previous surgery.
*Be alert to these symptoms, as they may be indicative of acoustic neuroma.
FOCUSED EXAMINATION
•General
•Examine the patient at rest to assess facial symmetry.
•Any obvious drooping of the angle of the mouth, ptosis (unilateral). Ask patient to close their eyes. Assess ability to close fully. Is there a tarsorrhaphy scar?
•Movements
•Assess muscles supplied by facial nerve (demonstrate the movements yourself).
•Ask patient to squeeze their eyes tightly shut, raise their eyebrows, blow out their cheeks, smile and show you their teeth.
•Extras
•Look behind the ear for a mastoid surgery scar. Examine for any cranial scars from acoustic neuroma removal.
•Assess for any hearing disturbance (cranial nerve [CN] involvement).
•Assess gait (cerebellar/ataxic).
QUESTIONS
1.What are the causes of unilateral CN VII palsy?
•Upper motor neurone (UMN) lesion
•Cerebellopontine angle lesion (CNs V, VI, VII, VIII and loss of taste on anterior two-thirds of the tongue): Acoustic neuroma, meningioma
•Pontine lesion: Demyelination (multiple sclerosis [MS]), vascular lesion
•Lower motor neurone (LMN) lesion
•Bell' s palsy: Most common cause; caused by herpes simplex type 1
•Ramsay Hunt' s syndrome: Caused by herpes zoster virus
•Parotid gland tumour/surgery
•Facial neuroma
•Cholesteatoma
•Mononeuritis multiplex (diabetes, systemic lupus erythematosus [SLE], polyarteritis nodosa [PAN], sarcoid, amyloid, Wegener' s granulomatosis): Remember WARDS PLC
•Trauma
2.What are the causes of bilateral LMN cranial nerve VII palsy?
•Motor neurone disease (MND).
•Guillain– Barré syndrome.
•Bilateral Bell' s palsy.
•Lyme disease.
•Myasthenia gravis.
•Sarcoidosis.
•Moebius syndrome (inherited rare form due to underdevelopment of cranial nerves VI and VII).
•Note that this question is another Practical Assessment of Clinical Examination Skills (PACES) ' classic' .
3.Which condition is associated with an acoustic neuroma?
•Neurofibromatosis type 2: Defect on chromosome 22q12. This condition results in bilateral acoustic neuromas.
4.What is the management of an acoustic neuroma?
•Conservative: Observe the tumour size and growth.
•Radiotherapy: To retard tumour growth.
•Surgical resection.
PATIENT WELFARE AND CONCERNS
1.Aesthetic: Facial asymmetry secondary to the nerve palsy due to medical/surgical cause.
2.Will the facial nerve palsy resolve?
3.Is there any treatment for this?
4.Will the deafness resolve or will it be permanent? Is there any treatment for this?
CANDIDATE EXPECTATIONS
1.Obtain a thorough history of symptoms of a CN VII lesion.
2.Be aware of the range of causes leading to CN VII palsy.
3.Examine carefully to differentiate between UMN and LMN CN VII palsy.
4.Provide the patient with a clear explanation of the problem and the likely effects; e.g. if it is a Bell' s palsy, there is a good chance of recovery. However, if it is a surgical consequence, then that is unlikely to be the case.
FALLS
This 80-year-old male has been falling regularly at home. He has limited medical history (he has hypertension and diabetes mellitus, managed by his general practitioner [GP]) and hasn' t ever been to hospital with the exception of having his appendix removed when he was 28. Please ask any relevant questions and proceed as you feel appropriate.
FOCUSED HISTORY
1.Need to establish exactly what occurs when the patient falls over. Does it sound like syncope or presyncope? Do they lose consciousness? If so, how quickly do they recover? (Need to be sure to rule out seizures.)
2.Ask about injuries postfall? Head injuries tend to indicate a loss of consciousness.
3.What environment are they in when they fall?
4.What had they been doing before each fall?
5.What are the precedent and antecedent symptoms?
6.How often do they fall?
7.Important: Past medical history/drug history will lead to many clues around the fall. For example, in this case the patient may be overtreated with antihypertensives and have postural hypotension or may have a sensory neuropathy from their poorly controlled diabetes.
8.Ask about their health in general. How far do they walk normally? (If they have severe osteoarthritis [OA], they are likely to have a degree of disuse myopathy and fall because of this.) What is their vision like? When was it last checked?
9.Ask about continence? Are they falling because they are rushing to the toilet because they have overactive bladder syndrome that is undiagnosed?
10.Make an enquiry about their social circumstances.
FOCUSED EXAMINATION
•General
•Comment on all/any injuries/bruises.
•Ask the patient to walk to assess gait (Parkinsonian/Marche à petits pas in vascular dementia).
•If possible, use your fundoscope to check for cataracts/look for signs of diabetic/ hypertensive retinopathy.
•Essential: Check postural blood pressure.
•Preform (or at least ask for) a finger-prick blood glucose check (note that diabetes can lead to both iatrogenic hypoglycaemia and autonomic neuropathy, which may result in falls).
•Look for proximal myopathy – common in older persons and often overlooked as a cause of falls.
•Cardiac
•Check pulse for arrhythmias.
•Listen for the murmur of aortic stenosis (syncope can be a symptom).
•Neurological
•Check for focal weakness/cerebellar signs.
•Examine for peripheral neuropathy that may be affecting sensory feedback.
QUESTIONS
1.How do you perform a lying and standing blood pressure?
•Lay patient down for 15 minutes and then take blood pressure.
•Stand patient.
•Take blood pressure at 1 minute.
•Take blood pressure at 3 minutes.
2.What constitutes a positive test?
•Systolic drop of 20 mmHg
•Diastolic drop of 10 mmHg
•Symptomatic patient (presyncopal)
3.What is the cause of this patient' s fall?
•Falls are almost always multifactorial, particularly if there is no syncope; e.g. in this case, he could be falling because he has dyspraxia secondary to vascular cognitive changes, a proximal myositis because of the statin his GP has given him for primary prevention and possibly a postural drop and sensory changes associated with his diabetes mellitus (autonomic neuropathy).
•You may not be expected to get the full diagnosis (it may beyond the scope of the station), but it is important that you demonstrate a holistic approach to the case.
4.What is the best approach to managing a patient with falls?
•Multidisciplinary team (MDT): The patient will need assessment by an occupational therapist (OT), physiotherapist, optometrist, pharmacist, nurse and geriatrician. A comprehensive geriatric assessment is often needed, and this will be performed by all of the above (and more). Of note, there is evidence that balance training and tai chi reduce the frequency of falls.
PATIENT WELFARE AND CONCERNS
1.Is the patient' s home environment safe? They may well benefit from a home visit by the OT to review their environment.
2.Consider if is it safe for the patient to drive if syncope is the cause (see ethics case on driving regulations in Station 4).
CANDIDATE EXPECTATIONS
1.Being holistic is most important.
2.You are unlikely to cover every aspect, but you should demonstrate that you are thinking about multiple potential causes.
3.You need to demonstrate an awareness of common problems that older persons who fall face.
4.Aim to be proactive when presenting the case. If they are falling regularly and at risk of osteoporosis, have you considered bone protection?
REFERENCE
Gillespie LD, et al. (2009) Interventions for preventing falls in older people living in the community. _Cochrane Database Syst Rev_ 15(2): CD007146.
HEADACHE (IDIOPATHIC INTRACRANIAL HYPERTENSION)
This 30-year-old lady with a family history of diabetes mellitus had a glucose tolerance test 6 months ago and was found to have impaired fasting glycaemia. Since then, she has been complaining of headaches and visual disturbance, getting much worse in the last 2 weeks. She does not usually wear glasses and wants to know what can be done to improve the symptoms.
FOCUSED HISTORY
1.Nature of visual disturbance: Diplopia or blurred vision, floaters/flashing lights, visual loss
2.Duration of visual disturbance: Stable or deteriorating
3.Effect on visual acuity: Any recent eye tests
4.Headaches: Worse in morning, on bending/coughing/sneezing/vomiting, nausea
5.Associated symptoms: Paraesthesia, weakness, hearing disturbance
6.Relevant other history: Weight loss/gain, obstructive sleep apnoea, drug history (vitamin A derivatives, tetracyclines, oral contraceptive pill)
FOCUSED EXAMINATION
•General
•Obesity, any obvious cranial nerve palsies
•Eyes
•Enlarged blind spot, ophthalmoplegia (abducens nerve palsy)
•Fundoscopy
•Assess for papilloedema and optic atrophy
•Extras
•Check for scars from ventriculoperitoneal (VP) shunts. Ask for blood glucose measurement and blood pressure.
QUESTIONS
1.What are the causes of papilloedema?
•Space-occupying lesion (tumour/abscess): Idiopathic intracranial hypertension (IIH)
•Hypertensive encephalopathy
•Infection: Encephalitis, meningitis
•Vascular: Intra-/extra-axial haemorrhage, venous sinus thrombosis
•Drugs: Tetracyclines, vitamin A derivatives
2.What are the fundoscopic features of papilloedema?
•Disc hyperaemia, blurred margins, absent venous pulsation
•Elevation of the disc with obscured vessels at the disc margin
•Loss of the cup with obscured vessels in the disc
•Bulging disc with all vessels obscured (this will eventually lead to optic atrophy)
3.What are the causes of optic atrophy?
•Congenital
•Friedreich' s ataxia, Leber' s hereditary optic neuropathy
•Acquired
•Vascular: Ischaemic (including temporal arteritis)
•Inflammatory: Multiple sclerosis, Devic' s disease
•Compression: Optic nerve tumour, Graves' ophthalmopathy, glaucoma
•IIH (untreated)
•Nutritional deficiencies: Vitamin B12, folate
•Toxins: Tobacco, alcohol, ethambutol, ethylene glycol, lead, cyanide, carbon monoxide
•Infective: Syphilis
4.What treatments can be used to treat IIH?
•Medical
•Weight loss if obese
•Stop any contributing medications
•Diuretics, e.g. acetazolamide
•Steroids: Beneficial in inflammatory conditions/causes
•Repeated lumbar punctures
•Surgical
•Optic nerve sheath decompression and fenestration is performed if vision is severely affected or threatened.
•Lumboperitoneal/ventriculoperitoneal shunt.
PATIENT WELFARE AND CONCERNS
1.Concerns regarding the visual disturbance: Could I lose my sight?
2.Concerns regarding the associated headache: Is this a brain tumour?
3.How can you be certain that this is a benign condition and that there is no sinister underlying pathology?
4.What are the treatment options? Would having a shunt inserted require brain surgery?
CANDIDATE EXPECTATIONS
1.Obtain a thorough history regarding the visual disturbances and headache.
2.Identify features suggestive of raised intracranial pressure and/or a space-occupying lesion, e.g. weakness/paralysis, seizures.
3.Carefully assess visual acuity, visual fields and ocular fundi.
4.Explain the likely diagnosis of IIH, but make the patient aware of the need to exclude a space-occupying lesion.
5.Explain that treatment strategies involve both medical and surgical options.
HYPERTHYROIDISM
This patient has a tremor and weight loss. Please ask any relevant questions and proceed as appropriate.
FOCUSED HISTORY
1.Is there any history of weight loss despite normal or increased appetite (caveat – occasionally some gain weight)?
2.Does the patient suffer from heat intolerance?
3.Do they sweat excessively?
4.Is there any history of GI upset, specifically diarrhoea?
5.Do they suffer from tachycardia or palpitations?
6.If they are female, ask about their menstrual cycle and specifically about oligomenorrhoea?
7.Do they suffer from anxiety and/or irritability?
8.Do they have a goitrous thyroid swelling?
9.Eye symptoms: Excessive watering, grittiness, redness, puffiness, change in field of vision? Eye pain (exophthalmos)? Ensure that you ask about all these even if no obvious eye signs are seen.
10.Ask the patient if they are on any medication: Beta-blockers – for anxiety symptoms and tachycardia; carbimazole/propylthiouracil – to block production of T4 and T3.
FOCUSED EXAMINATION
•Initial impression: Wide staring anxious expression – possible proptosis. Anxious fidgety demeanour, a peripheral tremor and goitre point towards a diagnosis of hyperthyroidism.
•Hands and arms: Acropachy (clubbing – rarely found and must have Graves' ophthalmopathy), sweaty palms, tachycardia/arrhythmia, fine tremor, proximal myopathy.
•Eyes: Observe from the side and above for degree of proptosis. Observe from the front for lid retraction, oedema and tearing, and perform eye movements for lid lag and opthalmoplegia.
•Eye signs from any cause of thyrotoxicosis: Lid retraction (see the white of the eye above and below the iris), lid lag.
•Graves' ophthalmopathy: Reddened eyes, excessive lacrimation, periorbital oedema, proptosis, conjunctival oedema and ophthalmoplegia
•Neck: Thyroidectomy scar, observe goitre while swallowing water, palpate goitre (diffuse or nodular) and palpate while swallowing water, palpate nodes, percuss the retrosternal extent of the thyroid. Auscultation for thyroid bruit.
•Legs: Pretibial myxoedema (rarely found and must have Graves' opthalmopathy).
•Extras: Evidence of other autoimmune disease, especially vitiligo, as it is common and easily observed in PACES.
QUESTIONS
1.What are the causes of hyperthyroidism?
•Graves' disease
•Toxic multinodular goitre
•Thyroid adenoma (toxic)
•De Quervain' s thyroiditis (painful, fever at onset)
•Postpartum thyroiditis
•Drugs (amiodarone, rarely lithium)
2.What are the causes of a goitre?
•Nodular
•Multinodular goitre (especially iodine-deficient areas)
•Adenoma
•Carcinoma
•Diffuse
•' Simple'
•Graves' disease
•Hashimoto' s thyroiditis
•De Quervain' s thyroiditis
•Thyroid lymphoma
3.What signs of thyroid disease are specific to Graves' disease?
•Graves' opthalmopathy – See above
•Thyroid acropachy
•Pretibial myxedema
4.What are the ' hyperthyroid emergencies' ?
•Thyroid storm
•Exophthalmos causing fixed gaze, diplopia or decreased acuity or loss of colour vision (may lead to optic nerve compression)
•Cardiac failure (high output)
PATIENT WELFARE AND CONCERNS
•What treatment options are available?
•Is my condition curable?
•Is my condition life threatening?
CANDIDATE EXPECTATIONS
•Recognise early that the patient has thyroid disease.
•Assess the patient' s thyroid status (hyper-/hypo-/euthyroid).
•Recognise signs specific to Graves' disease.
•Be able to discuss treatment options with the patient and address their concerns.
NECK LUMP
This patient presents complaining of a neck lump. Please ask any relevant questions and proceed as you feel appropriate.
FOCUSED HISTORY
1.Ask the patient what their most concerning problem is.
2.Where in the neck is the lump? (Thyroid: Midline, and just above the clavicle; other lumps may be nodes; make sure you can describe the surface anatomy.)
3.How long has it been there? (Rapidly progressing lumps can be lymphoma or anaplastic thyroid carcinoma.)
4.Is the lump painful?
5.Does the patient suffer from dysphagia or dysphonia (hoarseness)?
6.Are there any other symptoms of thyroid disease?
•Over-/underactive
•Weight loss/gain, appetite
•Intolerance to heat/cold
•Mood instability, irritability
•Bowels: Diarrhoea/constipation
•Palpitations
7.Is there any family history of thyroid disease or malignancy?
8.Has there been any history of radiation exposure?
9.Is there any history of autoimmune disease? Type 1 diabetes mellitus, rheumatoid arthritis (RA), adrenal insufficiency, vitiligo, coeliac disease, pernicious anaemia, myasthenia gravis and multiple sclerosis
FOCUSED EXAMINATION
•General
•Evidence of hyperthyroid or euthyroid status, rarely hypothyroid
•Then proceed as per the hyperthyroidism case (hands, eyes, neck, legs)
•Additional extras
•Evidence of other autoimmune disease, especially vitiligo, as it is common and easily observed in PACES
QUESTIONS
1.What are the types of thyroid cancer?
•Papillary: Most common
•Follicular
•Medullary: Arise from parafollicular calcitonin-secreting C-cells
•Anaplastic: Most aggressive
•Lymphoma
•Metastases from other primaries (kidney most common)
2.What investigations are used to diagnose the cause of a thyroid mass?
•TSH, T3 and T4 (rarely malignant if thyroid dysfunction)
•Gold standard is ultrasound-guided fine needle aspiration (FNA)
•CT useful to show the extent of a goitre causing compressive symptoms to trachea, or staging for malignancy
PATIENT WELFARE AND CONCERNS
1.What is the cause? Is this a cancerous lump, and how will this be differentiated from something that is noncancerous?
2.Cosmetic: Address concerns about how this lump sticks out and if it will be fully treated. Will it need surgery?
3.Associated features: Hoarse voice, swallowing difficulties and symptomatic thyroid disease. Are these all treatable?
CANDIDATE EXPECTATIONS
1.Look out for any red-flag symptoms of malignancy, such as weight loss and strong family history. Poor prognostic indicators for thyroid malignancy include: rapid enlargement, hoarse voice, age, sex (male) and type of cancer (papillary carcinoma has the best outcome, anaplastic has the worst).
2.Thoroughly assess a thyroid mass and examine for any associated features of thyroid disease.
3.Aim to reach a diagnosis and convey this to the patient, along with the management plan that will follow. Address each of the different investigations and what these will help to elicit.
4.Address the patient' s concerns, ranging from the concern that they may think that it' s cancer to any cosmetic concerns. Reassure the patient that thyroid malignancy is just one of many causes of a thyroid mass, and is not the most common cause. However, you need to rule this out if there is nodule.
OSTEOPOROSIS
This 82-year-old female has recently been diagnosed with polymyalgia rheumatica and started on prednisolone tablets. She has a past history of chronic lower back pain for which she takes several painkillers. She has presented today with worsening backache and wants to know what more can be done about it. Of note, she is known to be diabetic.
FOCUSED HISTORY
1.Is the diabetes a red herring, or it is pertinent to the history?
2.Duration of current episode of back pain: Onset (acute/sudden or gradual worsening), severity (pain score), character, site and radiation. Has the pain ever been this severe in the past? Analgesic history.
3.Precipitating factors: Heavy lifting/fall/trauma.
4.Neurological symptoms: Paraesthesia/weakness or paralysis in lower limbs, bladder or bowel dysfunction.
5.Red-flag symptoms: Weight loss, nocturnal pain and fevers.
6.Relevant other history: Risk factors for osteoporosis (steroid use, immobility, family history), hypertension.
FOCUSED EXAMINATION
•General
•Examine patient standing: Obvious spinal deformity, skin/palmar pigmentation if pigmented appearance, central obesity, round face, thin skin, bruising
•Spine
•Palpate vertebrae to localise symptoms; assess spinal movements as far as able.
•Limbs
•Focused assessment of power in lower limbs, sensation and gait
•Extras
•State you would examine perineum for perianal sensation and anal sphincter tone. Ask for blood glucose measurement and blood pressure reading.
QUESTIONS
1.Assuming her back pain is caused by an osteoporotic fracture, should the patient in this case have medical management of osteoporosis?
•Patients with fragility fractures over the age of 75 should have treatment regardless of dual-energy X-ray absorptiometry (DEXA) results. Review the NICE guidance on this.
2.What are the risk factors for developing osteoporosis?
•Family history
•Disease associations: Cushing' s syndrome, malabsorption, hyperparathyroidism, chronic inflammatory arthropathy
•Toxins: Excess alcohol or caffeine intake, smoking
•Prolonged immobility or inactivity
•Underweight
•Early menopause, late menarche, postmenopause, bilateral oophorectomy, hypogonadism in males
•Drugs: Prolonged steroid use (> 7.5 mg prednisolone daily for 6 months), prolonged use of low-molecular-weight heparin
3.What strategies can be used in the treatment of osteoporosis?
•Lifestyle changes: Nutrition, exercise
•Prevent falls: Appropriate footwear, OT/physio assessments (home adnum-justments), avoid sedative medications
•Drug therapies
•Reduce bone resorption: Bisphosphonates (zoledronic acid, ibandronic acid, alendronic acid), raloxifene (selective oestrogen reuptake inhibitor), calcitonin
•Aid new bone formation: Calcium and vitamin D supplements
•Reduce bone resorption and increase new bone formation: Strontium ranelate, teriparitide (recombinant parathyroid hormone [PTH]), denusomab (receptor activator of nuclear factor kappa-B ligand [RANKL] inhibitor)
PATIENT WELFARE AND CONCERNS
1.Concerns regarding the chronic back pain and the need for a diagnosis.
2.The impact of the back pain on daily life and ability to work.
3.Is there a treatment, medical or surgical, other than simple analgesia, for the back pain/osteoporosis?
CANDIDATE EXPECTATIONS
1.Obtain a thorough history of the back pain and quickly rule out ' red-flag signs' .
2.Rule out spinal cord compression in all patients presenting with worsening back pain.
3.Search for causes of osteoporosis if the patient has had an osteoporotic compression fracture leading to the back pain (in this case, possible Cushing' s).
4.Thoroughly examine any deformity of the spine and the range of spinal movements. Check for any neurological deficit and offer to check for perianal sensation and tone.
5.Explain the likely diagnosis.
6.Explain that bisphosphonates are first-line agents in the management of osteoporosis, for both treatment and prevention of further fracture.
REFERENCE
National Institute for Health and Care Excellence. (2014). Guidance CG146. Osteoporosis: Assessing the risk of fragility fracture. Available from https://www.nice.org.uk/guidance/cg146. Accessed 24 August 2016.
PROXIMAL MYOPATHY
This lady has been recently seen by her doctor for generalised aches and pains present for several months. She is now complaining of weakness in her limbs. She is housebound due to severe osteoarthritis and often does not go out for weeks at a time. Please ask any relevant questions and proceed as you feel appropriate.
FOCUSED HISTORY
1.What is the nature of the weakness? Is it symmetrical, and which limbs are affected? Is it proximal/distal/widespread?
2.How long has the weakness been present? How rapidly has it progressed?
3.Has there been any associated muscle wasting? Are the muscles tender?
4.Ask what impact the symptoms have on daily life to help decipher the pattern of weakness.
Proximal myopathy: Difficulty getting up out of chairs, washing and dressing, combing hair
Distal weakness: Difficulties with fine movements/tasks (writing, doing up buttons)
5.Which joints are involved? Any symptoms suggestive of synovitis?
6.Drug history: Prolonged steroid use, statin therapy, vitamin D supplement.
7.Sinister symptoms: Weight loss, fevers, speech/swallowing impairment, bladder/ bowel dysfunction.
FOCUSED EXAMINATION
•General
•Gait should be normal.
•There should not be any evidence of fasciculation on inspection.
•Observe any areas of muscle wasting/disuse atrophy.
•Evidence of trauma/deformity.
•Palpation
•Check that the muscles are not tender to palpation.
•Assess muscle power throughout each limb. Pay close attention for proximal muscle weakness; power will be reduced in affected areas, and the signs will be symmetrical.
•If a chair is available, ask the patient to sit on it and get up off it without using their hands.
•Extras
•Examine the spine for any obvious deformity.
•Examine the hands: Rule out dermatomyositis, rheumatoid disease.
•The differential diagnosis here is wide. A detailed drug history is crucial to eliminate a drug cause, for example, statin-induced myopathy. In a housebound individual who has little sunlight exposure, never forget vitamin D deficiency as a cause of proximal myopathy.
QUESTIONS
1.What are the manifestations of vitamin D deficiency?
•Rickets: Impaired growth and deformity of long bones in children
•Osteomalacia: Reduced bone mineralisation resulting in proximal muscle weakness
•Osteoporosis: Reduced bone density and increased fracture risk
2.What are the risk factors for vitamin D deficiency?
•Nutritional/poor dietary intake
•Malabsorption syndromes
•Reduced sunlight exposure
•Darker skin pigmentation
•Lack of vitamin D in breast milk
•Renal/liver impairment
•Enzyme-inducing medication (e.g. anticonvulsant medications)
3.What are the biochemical features of vitamin D deficiency?
•Reduced serum vitamin D levels
•Reduced serum calcium levels
•Reduced serum phosphate levels
•Raised alkaline phosphatase (ALP)
•Raised PTH (secondary hyperparathyroidism)
PATIENT WELFARE AND CONCERNS
1.What is the cause of the weakness? Is this reversible?
2.Will they lose their independent mobility?
3.Seriousness of the pathology: Is there an underlying sinister cause?
4.How long will the symptoms take to recover following initiation of treatment?
CANDIDATE EXPECTATIONS
1.Be aware of the range of differential diagnoses contributing to proximal muscle weakness.
2.The key in this case is to take a focused history around the causes of a proximal muscle weakness to help rule in/out a diagnosis.
3.Examine the important areas of each system that help confirm the diagnosis.
4.Formulate the correct diagnosis and convey this to the patient.
5.Reassure the patient that there are many causes of a proximal myopathy, and most of them are readily treatable; however, the main part of the management will be rehabilitation.
PSORIASIS
This patient has attended complaining of a significant change in the appearance of her nails. She is becoming increasingly distressed by the problem, which seems to be worsening despite several treatments by the GP. She has also noticed pain and swelling in her hands, which is impacting on her job as a seamstress.
FOCUSED HISTORY
1.Duration of symptoms
2.Nails: Colour, change in appearance, brittleness, extent of the nail involved, nail-bed involvement, number of nails involved, history of trauma, treatments tried by GP
3.Skin changes: Plaques/scales, pustules, sites of skin involvement, itchy/dry skin
4.Joints: Sites of arthritis (hands, feet, spine, hips), swelling, tenderness, deformity
5.Impact of illness on lifestyle
6.Family history of psoriasis/arthritis
FOCUSED EXAMINATION
•Hands
•Nail changes: Discolouration, pitting, onycholysis, hyperkeratosis of nail bed
•Joints
•Polyarthritis, distal interphalangeal joint arthritis, arthritis mutilans
•Skin
•Inspect the back, abdomen, extensor and flexor surfaces, scalp and posterior aspect of ears
•Plaque psoriasis: Shiny, scaly, white plaques
•Guttate: Numerous widespread, erythematous small ' teardrop' lesions
•Pustular: Widespread red pustules (often tender)
•Flexural: Plaques/areas of inflammation at limited sites
•Other joints
•Spine and sacroiliac joint involvement
•Asymmetrical oligoarthritis (examine joints highlighted in history)
QUESTIONS
1.What are the causes of psoriasis?
•Possibly immune-mediated via T-cell-stimulated cytokine release causing excess proliferation and production of skin cells within the dermis
•Genetic susceptibility: Major histocompatibility complex (MHC) antigen mediated
•Poststreptococcal infection (especially guttate form)
•Drug related
2.What treatments are used in the management of psoriasis?
•Moisturising creams
•Topical treatments: Coal tar, dithranol, steroids (usually short term), vitamin D analogues (calcipotriol) and calcineurin inhibitors (e.g. tacrolimus cream)
•Phototherapy: ultraviolet B (UVB)
•Photochemotherapy: Psoralen and ultraviolet A (PUVA)
•Medications
•Oral steroids
•Disease-modifying antirheumatoid drug (DMARDs): Methotrexate and ciclosporin are most commonly used
•Vitamin A analogue: Acitretin
•Biological agents: Anti-TNF therapy, e.g. adalimumab, and monoclonal antibodies, e.g. infliximab
3.How is psoriatic arthritis managed?
•In patients with psoriatic arthritis who also have severe skin/nail disease, collaboration should take place between rheumatology and dermatology, as systemic therapy may address both areas.
•The following guidance is targeted at predominantly peripheral arthritis (axial disease is managed more similarly to ankylosing spondylitis):
•NSAIDs
•DMARDs: Methotrexate, sulfasalazine, leflunomide, ciclosporin
•Biologics: Infliximab, etanercept, adalimumab, golimumab
PATIENT WELFARE AND CONCERNS
1.Cosmetic/aesthetic concerns regarding plaques, nail changes and joint involvement
2.Impact of arthropathy on job
3.Curative or symptomatic control only
4.Duration of treatment (particularly immunosuppression) – short course versus lifelong
5.Number of medications needed to control/abate symptoms
6.Long-term effects and risks of newer biological agents – susceptibility to infection, risk of malignancy
CANDIDATE EXPECTATIONS
1.Take a relevant history regarding the nails and skin lesions.
2.Obtain a detailed history for arthritis.
3.Be able to differentiate rapidly from the history whether this is likely to be an inflammatory or degenerative joint problem.
4.Examine the psoriatic plaques appropriately commenting on colour, location, size of area and distribution affected, using correct terminology.
5.Explain the diagnosis to the patient of psoriasis and the possible association between psoriasis and arthropathy.
6.Explain there are both local and systemic treatments that can be used to control symptoms.
REFERENCES
Coates LC, et al. (2013) The 2012 BSR and BHPR guideline for the treatment of psoriatic arthritis with biologics. _Rheumatology (Oxford)_ 52(10): 1754– 7.
DermNetNZ. (2015) Treatment of psoriasis. Available from <http://www.dermnetnz.org/scaly/psoriasis-treatment.html>. Accessed 5 July 2016.
National Institute for Health and Care Excellence. (2012) Guidance CG153. Psoriasis: Assessment and management. Available from https://www.nice.org.uk/guidance/Cg153. Accessed 5 July 2016.
RHEUMATOID ARTHRITIS WITH CARPAL TUNNEL SYNDROME
This patient has been complaining of a recent history of dropping objects. Ask any relevant questions and proceed as you feel appropriate.
FOCUSED HISTORY
1.Which hand is dominant?
2.One or both hands affected?
3.Duration of symptoms: Worsening or stable.
4.History of trauma or injury to hand/arm/shoulder.
5.Associated pain/numbness/paraesthesiae (including distribution).
6.Joint synovitis, muscle wasting, early morning stiffness.
7.Extent of weakness: Functional limitations, impact on quality of life (QOL)/ activities of daily living (ADLs).
8.Any other joint involvement?
9.Constitutional symptoms: Fever, weight loss.
10.Any other system involvement?
11.Drug history.
12.Family history of RA or autoimmune conditions.
FOCUSED EXAMINATION
•General
•Evidence of rheumatoid arthritis, acromegaly, thyroid disease or pregnancy
•Hands
•Symmetrical arthropathy
•Thenar eminence wasting
•Deformity: Ulnar deviation, metacarpophalangeal joint (MCPJ) subluxation, swan neck, boutonniè re, Z-thumb
•Evidence of scars from previous tendon release or nerve decompression surgery
•Palpation
•Hand, wrist and elbow joints: Check for evidence of active synovitis.
•Movements
•Test specifically the lumbricals, opponens pollicis, abductor pollicis brevis and flexor pollicis brevis.
•Assess function: Doing up a button, gripping a key and opening a door handle.
•Neurological
•Phalen' s and Tinel' s tests to assess the median nerve
QUESTIONS
1.What are the diagnostic criteria for rheumatoid arthritis?
American College of Rheumatologists Criteria (2010)
Classification criteria for RA
(Score-based algorithm: Add score of Categories A– D. A score of ≥ 6/10 is needed for classification of a patient as having definite RA.)
---
A. Joint involvement
|
1 large joint | 0
|
2-10 large joints | 1
|
1-3 small joints (with or without involvement of large joints) | 2
|
4-10 small joints (with or without involvement of large joints) | 3
|
>10 joints (at least one small joint) | 5
B. Serology (at least one test result is needed for classification)
|
Negative rheumatoid factor (RF) _and_ negative anticitrullinated protein antibody (ACPA) | 0
|
Low-positive RF _or_ low-positive ACPA | 2
|
High-positive RF _or_ high-positive ACPA | 3
C. Acute-phase reactants (at least one test result is needed for classification)
|
Normal CRP _and_ normal ESR 0 | 0
|
Abnormal CRP _or_ normal ESR 1 | 1
D. Duration of symptoms
|
< 6 weeks | 0
|
≥ 6 weeks | 1
2.What are anticyclic citrullinated peptide (anti-CCP) antibodies?
•Antibody markers used for the diagnosis of rheumatoid arthritis
•Aid assessment of likely future progression to RA, in undifferentiated arthritis
•More sensitive (~70%– 75%) and specific (~95%– 98%) than immunoglobulin (Ig) M rheumatoid factor antibodies
•Prognostic value as marker of erosive disease
3.What biological therapies are available for the treatment of rheumatoid arthritis?
•TNF-? blockers: Infliximab, adalimumab, etanercept
•Monoclonal B-cell (anti-CD20) antibodies: Rituximab
•IL-6 inhibitor: Tocilizumab
4.What are the serious side effects associated with the use of biological therapies?
•Opportunistic infections: Fungal, bacterial, viral (e.g. severe infection with varicella-zoster virus).
•Activation of latent TB ± progression to miliary TB.
•Anaphylaxis.
•Increased risk of skin cancer.
•Central nervous system demyelinating disorders (anti-TNF).
•Anti-TNF and rituximab should not be used in Stage 3/4 NYHA heart failure.
PATIENT WELFARE AND CONCERNS
1.Concerns regarding the impact of an inflammatory arthropathy on daily life and the possible cosmetic effects of deformities
2.Concerns regarding systemic extra-articular manifestations
3.Impact of DMARDs/biological agents and their side effects
4.Risks to offspring in terms of passing on the condition and their chances of developing rheumatoid arthritis in the future
CANDIDATE EXPECTATIONS
1.Obtain a thorough history regarding the extent of joint involvement and duration of symptoms that clearly points to a diagnosis of inflammatory arthritis.
2.Ascertain the impact of symptoms on daily life. Can the patient still function as normal?
3.Thoroughly examine all affected joints. Assess active synovitis and identify any associated extra-articular manifestations.
4.Explain the likely diagnosis of rheumatoid arthritis.
5.Explain that management is a multidisciplinary team approach with physiotherapists, occupation therapists and nurse specialists involved, etc. Treatment strategies include anti-inflammatory agents, disease-modifying agents and biological agents. (There are NICE criteria for guidance on biologics use.)
REFERENCE
Funovits J, et al. (2010) The 2010 American College of Rheumatology/European League Against Rheumatism classification criteria for rheumatoid arthritis: Methodological report phase l. _Ann Rheum Dis_ 69: 1589– 95.
SYSTEMIC SCLEROSIS
This patient has been complaining of colour changes in her fingers and difficulty moving them. Please ask any relevant questions and proceed as appropriate.
FOCUSED HISTORY
1.How long have these symptoms been going on for?
2.Have they noticed any changes in their skin over the fingers or elsewhere in the body?
3.Any discolouration of the skin anywhere on the body?
4.Any red patches (telangiectasiae)?
5.Any hard deposits anywhere, especially the fingers?
6.Any shortness of breath that is new or that is getting worse?
7.Any chest pain, swelling of legs or palpitations?
8.Any heartburn or difficulty swallowing?
9.Any joint pains or stiffness?
10.Any generalised symptoms, such as fatigue, difficulty sleeping, depression, anxiety, weight loss and malaise?
FOCUSED EXAMINATION
•Hands
•Sclerodactyly (thickened, tight skin), calcinosis, evidence of Raynaud' s, digital ulcers, telangiectasia.
•Assess the patient' s hand function.
•In someone with colour change in hands/digital ulcers, always assess the pulses.
•Face
•Telangiectasiae, microstomia
•Lungs
•Fine end inspiratory crackles in keeping with pulmonary fibrosis or evidence of pulmonary hypertension (loud P2, parasternal heave, raised JVP, tricuspid regurgitation and peripheral oedema)
•Extras
•Ask for blood pressure and urine dip (renal failure)
QUESTIONS
1.How is scleroderma classified?
•Localised scleroderma
•Localised skin involvement without internal organ involvement
•Limited systemic sclerosis (SSc) (previously also known as CREST syndrome)
•Calcinosis, Raynaud' s phenomena, oesophageal dysmotility, sclerodactyly, telangiectasiae
•Skin changes limited to hands, forearms, feet, neck and face
•Risk of pulmonary hypertension
•Diffuse systemic sclerosis
•Skin changes proximal to the elbows or knees, face or neck
•Can develop pulmonary fibrosis, scleroderma renal crisis or cardiac involvement
2.Which other organs are affected?
•Kidneys: Scleroderma renal crisis (occurs in 5%– 10% of SSc patients, who may present with an abrupt onset of hypertension, acute renal failure, headaches, fevers, malaise, hypertensive retinopathy, encephalopathy and pulmonary oedema)
•Lungs: Pulmonary hypertension, pulmonary fibrosis
•Cardiac: Pericardial effusions, myocardial fibrosis, conduction defects, congestive cardiac failure
3.What are the treatment options?
•Raynaud' s: Calcium channel blockers, e.g. nifedipine, ARB, e.g. losartan, intravenous prostacyclin (iloprost), bosentan (a dual endothelin-1 receptor antagonist).
•Reflux/oesophageal dysmotility: Proton pump inhibitors, H2 antagonists.
•Hypertension/renal protection: ACE inhibitors.
•Skin/joint/lung: Immune modulation – steroids, steroid-sparing agents, cyclophosphamide.
•Surgical option includes digital sympathetomy.
•Smoking cessation is key.
4.What immune tests can be used to help identify the cause?
•Diffuse systemic sclerosis: Anti-SCL-70 antibodies (against topoisomerase I)
•Limited systemic sclerosis: Anticentromere antibodies
•U1-RNP: Indicative of a mixed connective tissue disorder (systemic sclerosis, SLE, polymyositis)
PATIENT WELFARE AND CONCERNS
1.Concerns regarding skin changes in hands
2.Is the reflux an unrelated separate issue to the Raynaud' s/skin features?
3.What are the other complications of the condition, and are they treatable/curable?
4.What is the prognosis of the condition if there are cardiac, renal or pulmonary manifestations?
5.How will this condition affect life expectancy?
CANDIDATE EXPECTATIONS
•Obtain a thorough history of the skin features.
•Recognise that the patient has Raynaud' s and look for an underlying cause of this.
•Realise that the Raynaud' s is associated with underlying systemic sclerosis.
•Examine the hands and face and other systems that can be affected by systemic sclerosis.
•Explain the likely diagnosis and what investigations will need to be carried out to determine this.
•Explain the range of treatment options, depending on the severity of the disease and the importance of preventing disease progression.
THIRD NERVE PALSY OR PATIENT PRESENTING WITH DIPLOPIA
This 56-year-old male presents with diplopia, and his wife says his eye looks ' funny' . He has brittle diabetes and hypertension. Take a focused history and perform a focused examination to determine the diagnosis and formulate a management plan.
FOCUSED HISTORY
1.When did this occur? (Note that it may be congenital.)
2.Describe the onset of diplopia (sudden vs. gradual).
3.Is there a specific direction of vision where the diplopia occurs?
4.Is there associated pain/headache?
5.Enquire about symptoms of raised intracranial pressure. Headache that is worse in the morning? Vomiting?
6.Need to review past medical history specifically focusing on diabetes/prior neurosurgery/multiple sclerosis/autoimmune disorders/hypertension.
FOCUSED EXAMINATION
•Note: Do not waste your time performing a full cranial nerve examination.
•Observe for unilateral ptosis, dilated pupil and deviation of the eye laterally and downward (due to unopposed actions of lateral rectus and superior oblique).
•Observe for stigmata of diabetes, e.g. finger-prick blood glucose testing.
•Formally examine eye moments and check cranial nerves IV and VI at the same time.
•If you suspect a surgical third nerve palsy, then perform (or mention that you would like to perform) fundoscopy to observe for signs of raised intracranial pressure (i.e. papilloedema).
•Be sure to differentiate (via your examination) whether this is ' medical' third nerve palsy or a ' surgical' third nerve palsy.
•A medical third nerve palsy will usually be pupil sparing, and surgical third nerve palsy will not
QUESTIONS
1.Explain the difference between a ' medical' and ' surgical' third nerve palsy?
•Medical third nerve palsy tends to be pupil sparing, and surgical third nerve palsies lead to a dilated pupil. This is because surgical third nerve palsies are due to compressive effects on the sympathetic fibres that run along the outside of the nerve bundle that go on to supply the iris. As such, the pupil will be dilated early in a surgical palsy (note that in a medical palsy, the pupil _can_ also become dilated).
2.What are the medical causes of a third nerve palsy?
•Diabetes mellitus
•Infection
•Demyelinating disease
•Autoimmune disease
•Cavernous sinus thrombus
3.What are the surgical causes of a third nerve palsy?
•Aneurysm or bleed from the posterior communicating artery
•Postneurosurgery
•Posttrauma
•Congenital cause
4.Why does an oculomotor palsy lead to ptosis?
•In addition to supplying the intrinsic muscle of the eye, the oculomotor nerve also supplies levator palpebrae superioris.
PATIENT WELFARE AND CONCERNS
1.Be sympathetic if talking about prognosis. Medical causes may partially resolve with time. Prognosis with surgical causes is variable.
2.If talking about reconstructive eye lid surgery in chronic oculomotor palsy cases, remain empathetic regarding the appearance.
CANDIDATE EXPECTATIONS
1.Demonstrate the signs of oculomotor palsy.
2.Differentiate between surgical and medical causes.
BRIEF CLINICAL ENCOUNTERS STATION SUMMARY
•It is crucial to dedicate substantial revision time to Station 5.
•Be sure to look out early for ' spot diagnoses' .
•Any obvious signs may help guide towards a relevant history.
•In a case of back pain (such as ankylosing spondylitis), be sure to rule out spinal cord compression/cauda equine.
•There no set time to be spent on history and examination. This must be managed by the candidate on a case-by-case scenario. If there are likely to be many relevant signs, ensure that you demonstrate them.
•In cases of diabetes, always consider other micro- and macrovascular complications.
•Be able to assess a patient' s thyroid status clinically.
•In a case of psoriasis, always look for evidence of arthritis.
•In any joint disorder, it is crucial to assess function.
•Take note of any patient concerns throughout and attempt to resolve them later in the consultation.
Afterword: Hawks and Doves
As soon as you start revising for the Practical Assessment of Clinical Examination Skills (PACES), you should ask registrars and consultants to take you around to watch you examine patients, ask you questions and critique your performance. Very quickly you' ll realise that some of these registrars and consultants are hawks and some are doves. What is a hawk? What is hawkish behaviour? Hawks are tough. They believe in tough love, they give you a hard time, they expect high standards, they knock you off your perch and they keep you honest. Hawks are negative and pessimistic. They pick up on every omission or fault in your examinations, they ask you difficult questions and they critique you harshly.
Like yin and yang, doves are the opposite. Dove-like in behaviour, they are gentle. They encourage, they flatter, they appreciate your efforts, they build your confidence and they let you fly. Doves are positive and optimistic. They praise your thorough examinations, they ask sensible questions that you can answer and they critique you fairly.
These stereotypes are not absolute, but every registrar or consultant who takes you around will either have hawkish tendencies or be dove-like. But does this matter? Who cares? Does it change anything? The answer is yes. It really matters – I' ll explain why.
During a typical PACES revision session in the first 2– 3 weeks, you' re learning your examinations, you' re learning how to present and you' re gaining in confidence, but it' s all quite daunting. The last thing you need is a hawk to swoop down and tear you to shreds, destroying your confidence before you' ve even started. Go for the doves, to break you in gently.
After 2– 3 weeks, your examinations become slicker, your presentations are smoother and you feel increasingly confident. You start to think you can do it, you can be good enough and you can make it. But PACES is difficult, examiners are not always nice; they can be tough. Now what you need are hawks to take you around, give you a kick up the backside, put the fear back into you and keep you honest and working hard, and not lazy or sloppy with your examinations and presentations.
Finally, the exam approaches, about a week away. The hawks and doves have worked well, building your confidence and keeping you honest. You start to peak, you' re really good and you have every chance of passing. But the exam is daunting, you become nervous and you need a strong will and a positive mental attitude to perform on the day. The last thing you need is a hawk to tear you down before your exam, destroying you and making you think the whole PACES experience is all too much. Seek those reassuring doves again. Good luck.
**Ajay M Verma**
Index
A
Abdominal station, –
chronic liver disease, –
generalised lymphadenopathy, –
hepatosplenomegaly, –
hints for,
multiple abdominal scars,–
normal abdomen,–
palpable kidneys,–
renal replacement therapy,–
splenomegaly,–
Abducens nerve palsy,
Abnormal blood, patient with, –
Acoustic neuroma,
Adenocarcinoma,
ADPKD, _see_ Autosomal dominant polycystic kidney disease (ADPKD)
Alport' s syndrome (AS),
Amyloidosis, –
Amyotrophic lateral sclerosis,
Anaesthesia,
Ankylosing spondylitis, –
Anti-CCP, _see_ Anticyclic citrullinated peptide (Anti-CCP) antibodies
Anticipation,
Anticoagulants,
Anticoagulation, –
Anticyclic citrullinated peptide (Anti-CCP) anti-bodies,
Aortic regurgitation (AR), –
Aortic stenosis (AS), –
Aortic valve replacement (AVR),
AR, _see_ Aortic regurgitation (AR)
AS, _see_ Alport' s syndrome (AS); Aortic stenosis (AS) Ascites,
Austin Flint murmur,
Autoimmune disease human leucocyte antigen associations,
Autonomic neuropathy,
Autosomal dominant polycystic kidney disease (ADPKD),
AVR, _see_ Aortic valve replacement (AVR)
B
Back pain, patient with, –
causes of, in elderly patient,
red-flag signs,
symptoms,
tumours,
Bilateral LMN cranial nerve VII palsy, –
Bilateral lower-zone crackles,
Bitemporal hemianopia,
Bronchiectasis, –
B-symptoms,
C
Cancers,
Carcinoid syndrome, –
Cardiology, –
aortic regurgitation, –
aortic stenosis, –
Eisenmenger' s syndrome, –
hints for,
hypertrophic cardiomyopathy, –
mitral regurgitation, –
mitral stenosis, –
mitral valve prolapse, –
mixed aortic valve disease, –
mixed mitral valve disease, –
prosthetic heart valves, –
tricuspid regurgitation, –
ventricular septal defect, –
Carpal tunnel syndrome, , –
Cerebellar syndrome, –
CHADSVASC,
Charcot- Marie- Tooth disease,
Cholestasis,
Chronic liver disease, –
Chronic lymphocytic leukaemia (CLL),
Chronic myeloid leukaemia (CML),
Chronic obstructive pulmonary disease (COPD), –
Chronic primary MR,
Chronic secondary MR, –
CLL, _see_ Chronic lymphocytic leukaemia (CLL)
CML, _see_ Chronic myeloid leukaemia (CML)
Coeliac disease
conditions associated with,
diagnosis and management of,
Collapse, –
differential diagnosis of,
features of uncomplicated vasovagal syncope,
Community-acquired pneumonia,
Complex ophthalmoplegia,
Consolidation, –
COPD, _see_ Chronic obstructive pulmonary disease (COPD)
Cor pulmonale,
Cough, patient with, –
management of gastro-oesophageal reflux disease,
red-flag symptoms,
upper GI endoscopy,
CREST syndrome, _see_ Limited systemic sclerosis (SSc)
Crohn' s disease, ,
CURB–,
Cystic fibrosis, –
D
Deteriorating patient, care in, –
Diabetes Mellitus,
Diabetic maculopathy,
Diabetic retinopathy, –
Diarrhoea, patient with, –
Diffuse systemic sclerosis,
Diplopia, –
Dopamine agonists,
Driver and Vehicle Licensing Agency (DVLA), , ,
Driving regulations, –
DVLA, _see_ Driver and Vehicle Licensing Agency (DVLA)
E
Eisenmenger' s syndrome, –
Electromyography (EMG),
Empyema,
Endocarditis,
Ethics, and communication skills, –
breaking bad news, –
care in deteriorating patient, –
diagnosis, –
driving regulations, –
hints for,
initiating new therapy, –
long-term condition, –
medical error, –
mental capacity, –
organ transplantation, –
Exudative effusion,
F
Facial nerve palsy, –
Falls, –
Fibrotic lung disease, –
Foot drop,
Friedreich' s ataxia,
G
Gastrointestinal (GI) endoscopy,
Gastro-oesophageal reflux disease,
Generalised lymphadenopathy, –
GI, _see_ Gastrointestinal (GI) endoscopy
Giant cell arteritis, –
Goitre,
Graves' disease,
H
HASBLED,
HCM, _see_ Hypertrophic cardiomyopathy (HCM)
Headache, –
Hemiparesis, –
Hepatic encephalopathy,
Hepatosplenomegaly, –
Hereditary haemorrhagic telangiectasia,
Hereditary spastic paraparesis,
History-taking station, –
abnormal blood, patient with, –
back pain, patient with, –
collapse, patient with, –
cough, patient with, –
diarrhoea, patient with, –
hints for,
jaundice, patient with, –
joint pains, patient with, –
left arm weakness, patient with, –
visual disturbance, patient with, –
weight loss, patient with, –
Hodgkin' s lymphoma,
Homonymous hemianopia,
Hypercalcaemia,
Hyperthyroid emergencies,
Hyperthyroidism, –
Hypertrophic cardiomyopathy (HCM), \-
I
IBD, _see_ Inflammatory bowel disease (IBD)
Idiopathic intracranial hypertension (IIH), –
Inflammatory bowel disease (IBD),
Internuclear ophthalmoplegia,
J
Jaundice, patient with, –
Joint pains, patient with, –
L
Lateral medullary syndrome,
Left arm weakness, patient with, –
Lhermitte' s sign,
Limited systemic sclerosis (SSc),
Liver transplantation,
LMN, _see_ Lower motor neurone (LMN) lesion
Lobectomy,
Localised scleroderma,
Long-term oxygen therapy (LTOT),
Lower motor neurone (LMN) lesion,
LTOT, _see_ Long-term oxygen therapy (LTOT)
Lung cancer, –,
Lung transplantation,
M
Malabsorption,
Marfan' s syndrome, ,
Medical error, –
Mental capacity, –
Mitral regurgitation (MR), –
Mitral stenosis, –
Mitral valve prolapse (MVP), –
Mixed aortic valve disease, –
Mixed mitral valve disease, –
MND, _see_ Motor neurone disease (MND)
Mononeuritis multiplex,
Mononeuropathies, –
Motor neurone disease (MND), –
MR, _see_ Mitral regurgitation (MR)
Multiple abdominal scars, –
Multiple sclerosis, –
MVP, _see_ Mitral valve prolapse (MVP)
Myeloproliferative disorders,
Myotonic dystrophy, –
N
National Institute for Health and Care Excellence (NICE), ,
Neck lump, –
Nephrectomy,
Neurology station, –
cerebellar syndrome, –
hemiparesis, –
hints for,
mononeuropathies, –
motor neurone disease, –
multiple sclerosis, –
myotonic dystrophy, –
ocular palsies, –
Parkinson' s disease, –
peripheral neuropathy, –
spastic paraparesis, –
visual field defects, –0
NICE, _see_ National Institute for Health and Care Excellence (NICE)
NOACs, _see_ Novel oral anticoagulants (NOACs)
Non-Hodgkin' s lymphoma,
Nonproliferative diabetic retinopathy,
Non-small-cell lung cancer (NSCLC),
Normal abdomen, –
Novel oral anticoagulants (NOACs),
NSCLC, _see_ Non-small-cell lung cancer (NSCLC)
O
Ocular palsies, –
Old tuberculosis (TB), –
Optic atrophy,
Optic nerve lesion,
Organ transplantation, –
Osteoporosis, –
P
PACES, _see_ Practical Assessment of Clinical Examination Skills (PACES)
Palpable kidneys, –
Pancoast' s syndrome,
Papilloedema,
Paraneoplastic syndromes,
Parapneumonic effusion,
Parkinson' s disease, –
Peripheral neuropathy, –
Peroneal nerve palsy,
Peutz- Jeghers syndrome,
Pleural effusion, –
Pneumonectomy,
Pneumonia,
Polycystic kidney disease,
Polycythaemia rubra vera (PRV),
Polymyalgia rheumatica,
Practical Assessment of Clinical Examination Skills (PACES), xxiii, xxv-xxvii,
Primary biliary cirrhosis (PBC)
autoimmune associations of,
control of disease,
investigation of suspected,
management of itch in cholestasis,
Primary hyperparathyroidism,
Primary lateral sclerosis,
Primary MR,
Progressive bulbar palsy,
Progressive muscular atrophy,
Proliferative diabetic retinopathy,
Prosthetic heart valves, –
Proximal myopathy, –
PRV, _see_ Polycythaemia rubra vera (PRV)
Psoriasis, –
Psoriatic arthritis,
R
Radial nerve palsy,
Renal cystic disorders,
Renal replacement therapy, –
Renal transplantation, –
Respiratory station, –
bronchiectasis, –
chronic obstructive pulmonary disease (COPD), –
consolidation, –
cystic fibrosis, –
fibrotic lung disease, –
hints for,
lung cancer, –
old tuberculosis (TB), –
patient with previous lung surgery, –
pleural effusion, –
Rheumatoid arthritis,
with carpal tunnel syndrome, –
diagnostic criteria for,
pharmacological management of,
S
Sarcoidosis,
SCD, _see_ Sudden cardiac death (SCD)
SCLC, _see_ Small-cell lung cancer (SCLC)
Scleroderma,
Secondary MR,
Small-cell lung cancer (SCLC),
Space-occupying lesions,
Spastic paraparesis, –
Spinal cord compression,
Spleen,
Splenectomy,
Splenomegaly, –
Squamous cell carcinoma,
SSc, _see_ Limited systemic sclerosis (SSc)
Structural valve deterioration (SVD),
Sudden cardiac death (SCD),
Superior vena caval obstruction (SVCO),
SVD, _see_ Structural valve deterioration (SVD)
Systemic sclerosis, –
T
Tabes dorsalis,
TAVI, _see_ Transcatheter aortic valve implantation (TAVI)
TB, _see_ Old tuberculosis (TB)
Third nerve palsy, –
Thoracoscopy,
Thyroid cancer,
Thyroid mass,
Transcatheter aortic valve implantation (TAVI), –
Transudative effusion,
Transverse myelitis,
Tricuspid regurgitation, –
U
UC, _see_ Ulcerative colitis (UC)
Uhthoff' s phenomenon,
UIP, _see_ Usual interstitial pneumonia (UIP) pattern
Ulcerative colitis (UC),
Ulnar nerve palsy,
UMN, _see_ Upper motor neurone (UMN) lesion
Unilateral CN VII palsy,
Upper motor neurone (UMN) lesion,
Usual interstitial pneumonia (UIP) pattern,
V
Vasovagal syncope,
Ventricular septal defect (VSD), –
Visual disturbance, patient with, –
Visual field defects, –0
Vitamin D deficiency, –
Von Hippel- Lindau syndrome,
VSD, _see_ Ventricular septal defect (VSD)
W
Wallenberg' s syndrome, _see_ Lateral medullary syndrome
Warfarin,
Weight loss, patient with, –
World Health Organization (WHO),
1. Cover
2. Half-Title
3. Title
4. Copyright
5. Contents
6. Foreword
7. Preface
8. Preface
9. Acknowledgements
10. List of Abbreviations
11. The PACES Examination
12. Top Tips for PACES Success: An Examiner Speaks
13. 1 Station 1: Respiratory
1. Hints for the Respiratory Station
2. Bronchiectasis
3. Chronic Obstructive Pulmonary Disease
4. Consolidation
5. Cystic Fibrosis
6. Fibrotic Lung Disease
7. Lung Cancer
8. Old Tuberculosis
9. Pleural Effusion
10. Patient with Previous Lung Surgery
11. Respiratory Station Summary
14. 2 Station 1: Abdominal
1. Hints for the Abdominal Station
2. Abdominal Case with a Normal Abdomen
3. Chronic Liver Disease
4. Generalised Lymphadenopathy
5. Hepatosplenomegaly
6. Multiple Abdominal Scars
7. Palpable Kidneys
8. Renal Replacement Therapy
9. Splenomegaly
10. Abdominal Station Summary
15. 3 Station 2: History Taking
1. Hints for the History-Taking Station
2. Patient with an Abnormal Blood Result
3. Patient with Back Pain
4. Patient Who Has Collapsed
5. Patient with a Cough
6. Patient with Diarrhoea
7. Patient with Jaundice
8. Patient with Joint Pains
9. Patient with Left Arm Weakness
10. Patient with Visual Disturbance
11. Patient with Weight Loss
12. History-Taking Station Summary
16. 4 Station 3: Neurology
1. Hints for the Neurology Station
2. Cerebellar Syndrome
3. Hemiparesis
4. Myotonic Dystrophy
5. Ocular Palsies
6. Parkinson' s Disease
7. Peripheral Neuropathy
8. Mononeuropathies
9. Motor Neurone Disease
10. Multiple Sclerosis
11. Spastic Paraparesis
12. Visual Field Defects
13. Neurology Station Summary
17. 5 Station 3: Cardiology
1. Hints for the Cardiology Station
2. Aortic Stenosis
3. Aortic Regurgitation
4. Eisenmenger' s Syndrome
5. Hypertrophic Cardiomyopathy
6. Mitral Stenosis
7. Mitral Regurgitation
8. Mitral Valve Prolapse
9. Mixed Aortic Valve Disease
10. Mixed Mitral Valve Disease
11. Prosthetic Heart Valves
12. Tricuspid Regurgitation
13. Ventricular Septal Defect
14. Cardiology Station Summary
18. 6 Station 4: Ethics and Communication Skills
1. Hints for the Ethics and Communication Skills Station
2. Breaking Bad News
3. Care in the Deteriorating Patient
4. Driving Regulations
5. Initiating a New Therapy
6. Long-Term Condition
7. Medical Error
8. Mental Capacity
9. New Diagnosis
10. Organ Transplantation
11. Ethics and Communication Skills Station Summary
19. 7 Station 5: Brief Clinical Encounters
1. Hints for the Brief Clinical Encounters Station
2. Ankylosing Spondylitis
3. Anticoagulation
4. Diabetic Retinopathy
5. Facial Nerve Palsy
6. Falls
7. Headache (Idiopathic Intracranial Hypertension)
8. Hyperthyroidism
9. Neck Lump
10. Osteoporosis
11. Proximal Myopathy
12. Psoriasis
13. Rheumatoid Arthritis with Carpal Tunnel Syndrome
14. Systemic Sclerosis
15. Third Nerve Palsy or Patient Presenting with Diplopia
16. Brief Clinical Encounters Station Summary
20. Afterword: Hawks and Doves
21. Index
| {
"redpajama_set_name": "RedPajamaBook"
} | 8,955 |
package org.nemesis.grava.domain;
/**
* Created by nuru on 9/30/14.
*/
public class BoardPit {
private Integer stones;
public BoardPit(Integer stones) {
this.stones = stones;
}
public Integer getStones() {
return stones;
}
public void clearStones() {
stones = 0;
}
public void increment() {
stones++;
}
public void add(Integer amount) {
stones += amount;
}
public void set(Integer amount) {
stones = amount;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,085 |
\section{Introduction}
The MINOS far detector
is the first large underground experiment with a magnet that can
measure the ratio of $\mu^+$ to $\mu^-$ with high
precision.\cite{bib:mupaperFNAL,
bib:mufson}. A precise measurement of the charge ratio can then
be used to ascertain special properties of the cosmic ray showers,
such as the $\pi^+/\pi^-$ ratio and the $K^+/K^-$ ratio\cite{bib:interpret}.
The survival probability for muons to reach an underground
detector depends on the energy loss, so if there is any difference in
energy loss between $\mu^+$ and $\mu^-$, that would affect the measured
charge ratio. The statistical error on the MINOS measurement is
remarkably small, and MINOS reports:
\begin{equation}
r = \frac{N(\mu^+)}{N(\mu^-)} = 1.374 \pm 0.003(stat)^{+0.012}_{-0.010}(sys)
\end{equation}
for surface muons with an energy near 1 TeV or higher.
Thus, even very small differences in the energy loss between $\mu^+$
and $\mu^-$ could be important in the interpretation of these measurements.
\par
The statistical energy loss of muons, traversing an amount $X$ of matter in
$g/cm^2$, with energies far above the Bethe-Bloch minimum is usually
parameterized as
\begin{equation}
- \frac{dE_{\mu}}{dX} = a(E_{\mu}) \,+\,
\displaystyle\sum_{n=1}^{3} b_{n}(E_{\mu}) \cdot E_{\mu},
\label{eq:2}
\end{equation}
where $a$ is the collisional term (i.e. ionization,
mostly due to delta-ray
production) and $b$ in the second term accounts for the three radiative muon
energy loss processes: 1.\,Bremsstrahlung and 2.\,pair production,
as well as
3.\,photonuclear interactions. In {\em Table} \ref{table1} these energy loss
parameters are listed for standard rock. The critical energy
where ionization
losses equal radiative losses in standard rock is approximately $0.6\,$TeV.
The average muon energy for a muon which reaches the depth of MINOS is
greater than $1$\,TeV, so the $b$ term and its energy dependence are important
in calculating the energy loss. This paper focuses on the (small) differences
in the $a$ and $b$ terms for $\mu^+$ and $\mu^-$.
\begin{table}[h]
{\tiny
\begin{tabular}{|c||c||c|c|c|c|}
\hline
$E_{\mu}$ & $a_{ion}$ & $b_{brems}$ & $b_{pair}$ & $b_{DIS}$ & $\Sigma b$ \\
\cline{3-6}
[$GeV$] & [$MeV\,cm^{2}/g$] &\multicolumn{4}{c|}{[$10^{-6}\,cm^{2}/g$]} \\
\hline\hline
$10$ & 2.17 & 0.70 & 0.70 & 0.50 & 1.90 \\
$10^{2}$ & 2.44 & 1.10 & 1.53 & 0.41 & 3.04 \\
$10^{3}$ & 2.68 & 1.44 & 2.07 & 0.41 & 3.92 \\
$10^{4}$ & 2.93 & 1.62 & 2.27 & 0.46 & 4.35 \\
\hline
\end{tabular}
}
\caption{{\small Average muon energy loss parameters calculated for
standard rock \cite{bib:Groom}\cite{bib:pdg}}}\label{table1}
\end{table}
\section{Calculated Difference in Ionization dE/dX for $\mu^+$ and $\mu^-$}
\par At low energies, around the Bethe-Bloch maximum, the difference in
ionization energy loss is known as the Barkas effect\cite{bib:fermi},
and there have been efforts to both measure and calculate
those differences\cite{bib:mccarthy}.
Calculations show that negative particles lose energy at a slower
rate, with the difference dropping from tens of percent at MeV energies
to about $0.3$\,\% in the GeV range.
Such differences were experimentally verified both at MeV
energies\cite{bib:barkas1}-\cite{bib:barkas3}
and in the GeV range \cite{bib:barkas4}.
At higher energies,
this difference in ionization energy loss has usually been neglected,
and we are not aware of any measurements. As described in Reference
\cite{bib:jackson}, the usual ionization
energy loss term for muons (of either sign)
depends on $z^2$, and the difference between $\mu^+$ and $\mu^-$ arises from
a small additional $z^3$ correction term.
This correction term in $dE/dx$ is:
\begin{equation}
\displaystyle\left(\frac{dE}{dX}\right)^{corr}_{ion} = \frac{\pi \alpha z^3 0.307 Z}{2 \beta A} \quad
[MeV\, cm^{2}\, g^{-1}]
\label{eq:3}
\end{equation}
where $\alpha$ is the fine structure constant, $z$ is the charge, $\beta$ is
the relativistic velocity, and $Z$ and $A$ are the nuclear properties of the
material through which the muon is passing. The absolute value of the
difference in ionization energy loss between positive and negative muons in
standard rock\cite{bib:pdg} is plotted in Figure \ref{fig:fig1}.
It is fairly constant above $10$\,GeV, at a value corresponding to
approximately $0.15$\,\% of the mean energy loss in the ionization dominated
energy regime ({\em c.f.} Table \ref{table1}).
\begin{figure}
\begin{center}
\noindent
\includegraphics [width=0.5\textwidth]{ComputationEsyst_BarkasIoni.eps}
\end{center}
\caption{Calculated difference in ionization energy loss between
positive and negative muons in standard rock (average nuclear properties:
$\overline{Z}=11$, $\overline{A} = 22$ \,\cite{bib:pdg}).}
\label{fig:fig1}
\end{figure}
\section{Calculated Difference in Bremsstrahlung dE/dX
for $\mu^+$ and $\mu^-$}
Above an energy near $0.6$\,TeV in standard rock, radiative energy loss
becomes comparable
to ionization energy loss, and continues to grow at
higher muon energies.
From Reference \cite{bib:jackson2}, the fractional difference in
Bremsstrahlung energy loss between positive and negative muons is
\begin{equation}
\frac{[\frac{dE}{dX}]^{\mu+}_{brems} - [\frac{dE}{dX}]^{\mu-}_{brems}}{[\overline{\frac{dE}{dX}}]_{brems}} = \frac{8 Z \alpha}{\gamma}
\label{eq:4}
\end{equation}
where $\gamma$ is the Lorentz factor of the muon.
Again, the $\mu^+$ has a slightly higher energy loss.
This fractional difference {\it decreases} with energy and is already
negligible where radiative energy losses become important.
This fractional difference is plotted in Figure \ref{fig:fig2} for muons in
standard rock.
Presumably, the same fractional difference can also be assigned for
pair-production, as the underlying process is a two-photon exchange between
the muon and the constituents of the nucleus, and thus the cross sections for
$\mu^+$ and $\mu^-$ should scale in the same way as for Bremsstrahlung.
\begin{figure}
\begin{center}
\noindent
\includegraphics [width=0.5\textwidth]{ComputationEsyst_BarkasBrems.eps}
\end{center}
\caption{Calculated fractional difference in Bremsstrahlung energy loss
between positive and negative muons.}
\label{fig:fig2}
\end{figure}
\section{Muon Range Underground for $\mu^+$ and $\mu^-$}
Taking the vertical muon intensity from an optimized Gaisser parameterization
of the muon flux at the surface and propagating this energy spectrum
underground according to statistical ionization and radiative energy losses,
it is possible to precisely calculate the underground muon intensity.
This procedure is described in detail in \cite{bib:myCrouchProc} for
overburdens of standard and Soudan rock (MINOS).
First, the average muon range underground, for each value of surface energy,
has to be precisely computed. For this, the energy dependent $a$ and
$\Sigma b$ values are parameterized for standard and Soudan rock as in
\cite{bib:myCrouchProc}. The additional ionization loss
according to Eq. 3 is then added for $\mu^+$ to the value of the function for
$a$ (subtracted for $\mu^-$).
Conservatively, $90\,\%$ of the value of the function for $\Sigma b$ (the
total radiative losses), are scaled with the energy dependent fractional
difference $R=8Z\alpha/\gamma$ according to Eq. 4 as $\Sigma b^{\pm} = 0.9
\cdot \Sigma b \cdot (1 \pm R/2)$ for $\mu^+$ and $\mu^-$, respectively.
The contribution from photonuclear production (DIS), which accounts for a
constant fraction of $10\,\%$ of all radiative muon energy losses in standard
rock (in the region of interest from $250\,$GeV to $10\,$TeV) is not scaled
with the fractional difference.
According to differential Eq. 2 the propagation of the muon energy is then
numerically computed, separately for $\mu^+$ and $\mu^-$ (in standard and
Soudan rock, respectively). Thus, for each initial value of muon energy,
the slant depth in meter-\-water-\-equivalent where the
muons of different charge range out is determined.
\section{Muon Charge Asymmetry from Ratio of Intensities Underground}
Using the average muon range underground, calculated for positive and
negative muons in rock as described in the last section, and an optimized
Gaisser parameterization of the differential intensity of vertical
muons at the surface, we have computed the corresponding
underground intensities of positive and negative muons as a function of slant
depth for a given rock composition.
The resulting ratio of the $\mu^+$ and $\mu^-$ intensity curves is shown in
Figure \ref{fig:intensity} for Soudan rock. The upper curve corresponds to the
fractional difference in integral intensities of $\mu^+$ and $\mu^-$ at a
given slant depth. For slant depth values above about $1000\,mwe$ the underground
ratio $N(\mu^+)/N(\mu^-)$ is lowered by roughly $0.4\,\%$. However, since
the
charge of only the lower energy muons can be identified in a magnetic detector,
owing to its maximum detectable momentum \cite{bib:mdm}, the detected intensity
corresponds to the charge ratio of the muons at depth below some momentum.
The lower curve in Figure \ref{fig:intensity} depicts the fractional
difference in intensity for underground muon momenta below $250$\,GeV/c,
corresponding to the approximate maximum detectable momentum of MINOS. For
increasing slant depth values the measured underground ratio
$N(\mu^+)/N(\mu^-)$ is further reduced and saturates at
about $0.6\,\%$ below its surface value for slant depths larger than roughly $5000\,mwe$.
\par The dominant $0.15$\% difference in ionization energy loss between $\mu^+$ and
$\mu^-$ gets amplified by a factor of about $3.7$, due to the approximate
$E^{-3.7}$ dependence of the differential muon spectrum.
The impact of the rock composition is almost negligible, as the induced
muon charge asymmetry under Soudan rock lowers
the surface value of the ratio
by an additional amount less than $0.02\,\%$ compared to standard rock.
\begin{figure}
\begin{center}
\noindent
\includegraphics [width=0.5\textwidth]{ComputationEsyst_Barkas_MINOS_final.eps}
\end{center}
\caption{Calculated ratio of positive to negative vertical muon intensities in
Soudan rock as a function of slant depth. The upper curve is for all muons,
the lower curve is for muons with a remnant momentum of less than 250 GeV/c
($\approx$ the maximum detectable momentum in the MINOS far
detector).}\label{fig:intensity}
\end{figure}
\section{Summary}
There is a small fractional difference in energy loss for $\mu^+$ and $\mu^-$
of the order of $0.15\,\%$ predicted by theoretical calculations at high
energies, predominantly due to a $z^3$ correction term in the ionization
energy loss. This causes that measurements of the atmospheric muon charge
ratio $N(\mu^+)/N(\mu^-)$ deep underground (e.g. with the MINOS detector), to
observe a slightly lower ratio than at the surface.
Moreover, as the atmospheric muon energy spectrum is steeply falling off with
approximately $E^{-3.7}$, the small difference in energy loss between $\mu^+$
and $\mu^-$ at high energies results in an amplified charge asymmetry of about
$0.6\,\%$ several thousand meters water equivalent deep underground.
The calculations presented herein allow for a correction of the underground
measured muon charge ratio to its surface value.
\section{Acknowledgments}
This work was supported by the U.S. Department of Energy (DOE) under contract
DE-AC02-06CH11357. I also like to
thank Geoff Bodwin from the High Energy Physics Division at Argonne,
Stan Wojcicki from Stanford,
Stuart Mufson from the University of Indiana,
Don Groom from Lawrence Berkeley National Laboratory as well as
Thomas Fields and
the neutrino physics group at Argonne for valuable discussions and in
particular Maury Goodman for his support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,012 |
{"url":"https:\/\/www.cuemath.com\/questions\/the-number-one-followed-by-one-hundred-zeros-is-known-by-which-name\/","text":"# The Number One Followed By One Hundred Zeros\n\n## Question: The number one followed by one hundred zeros is known by which name?\n\nStandard notation is also known as scientific notation where a large number is written in the form of power of 10.\n\n## Answer: The number one followed by one hundred zeros is known by the name \"googol\".\n\nPower to the 10 shows the exact number of zeros, one followed by\n\n## Explanation:\n\nThe number one followed by one hundred zeros is 1 \u00d7\u00a010100\u00a0which is\u00a0known as googol.\n\nInterestingly, Google is a misspell for googol.","date":"2021-05-06 04:21:02","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\": 0.9036005139350891, \"perplexity\": 1025.6769858671203}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243988725.79\/warc\/CC-MAIN-20210506023918-20210506053918-00165.warc.gz\"}"} | null | null |
Q: (createchildcontrole) = object reference not set to an instance of an object I use Visual Studio 2012 to create some UserControls to be wrapped on SharePoint 2010. The following is code of one of them:
public partial class SideBarUserControl : System.Web.UI.UserControl
{
protected void Page_Load(object sender, EventArgs e)
{
liLogout.Visible = false;
attendance.Visible = false;
empEvaluation.Visible = false;
Objections.Visible = false;
myEval.Visible = false;
rates.Visible = true;
if (PageBase.CurrentUser == null)
{
string LoginName = SPContext.Current.Web.CurrentUser.LoginName;
LoginName = LoginName.Split(new string[] { "\\" }, StringSplitOptions.RemoveEmptyEntries)[1];
if (SystemUserBLL.IsUserExist(LoginName, "123456"))
{
SystemUserBLL.Login(LoginName, "123456");
}
else
Response.Redirect("Login.aspx", true);
}
if (new UserDAL().GetUserFieldValue(PageBase.CurrentUser.ID, 51).FieldValue == "الدرجة الثانية")
{
rates.Visible = false;
sysSettings.Visible = false;
reports.Visible = false;
string EmployeeType = new UserDAL().GetUserFieldValue(PageBase.CurrentUser.ID, 39).Value;
myEval.Visible = false;
if (EmployeeType == "2") // Employee
{
if (!Request.Path.ToLower().Contains("rateform.aspx") && !Request.Path.ToLower().Contains("myevaluation.aspx")
&& !Request.Path.ToLower().Contains("homepage.aspx"))
Response.Redirect("MyEvaluation.aspx");
myEval.Visible = true;
Objections.Visible = false;
empEvaluation.Visible = false;
}
else if (EmployeeType == "3") // Wakeel
{
if (!Request.Path.ToLower().Contains("rateform.aspx") && !Request.Path.ToLower().Contains("objections.aspx") && !Request.Path.ToLower().Contains("userevaluation.aspx")
&& !Request.Path.ToLower().Contains("homepage.aspx") && !Request.Path.ToLower().Contains("percentages.aspx")
&& !Request.Path.ToLower().Contains("userattendance.aspx"))
Response.Redirect("UserEvaluation.aspx");
rates.Visible = false;
Objections.Visible = true;
empEvaluation.Visible = true;
attendance.Visible = true;
myEval.Visible = false;
sysSettings.Visible = true;
form.Visible = false;
waitperiod.Visible = false;
}
else // Manager
{
if (!Request.Path.ToLower().Contains("rateform.aspx") && !Request.Path.ToLower().Contains("myevaluation.aspx")
&& !Request.Path.ToLower().Contains("objections.aspx") && !Request.Path.ToLower().Contains("userevaluation.aspx")
&& !Request.Path.ToLower().Contains("homepage.aspx") && !Request.Path.ToLower().Contains("userattendance.aspx"))
Response.Redirect("UserEvaluation.aspx");
//myEval.Visible = false;
rates.Visible = false;
myEval.Visible = true;
Objections.Visible = true;
empEvaluation.Visible = true;
attendance.Visible = true;
}
agenciesmanagers.Visible = false;
}
}
}
When I try to run it on SharePoint sometimes it runs perfectly and sometimes I have the following error.
(createchildcontrole) = object reference not set to an instance of an object
If the error occurred just refreshing the page solving the problem.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,146 |
Q: How to handle 2 versions of Python on same client? I use ArcGIS 10.3 at work and it comes with Python 2.7.8; so when I run any of the .py/.pyc/.pyw files, it defaults to the path leading that versions .exe file.
Because of some non-ArcGIS related work, we are also being provisioned Python 3.4.2, which also installed its own path.
The question begs, how do I tell each .py/.pyc/.pyw file which Python (and associated IDLE [we don't get any IDE at work]) to execute the script with?
A: Pyenv is an excellent choice.
Particularly, set pyenv local and along with virtualenv, you get whatever you need on a per-project basis.
https://amaral.northwestern.edu/resources/guides/pyenv-tutorial
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,846 |
As one who teaches environmental studies, I am sometimes tagged as a pessimist, a "prophet of gloom" cheerlessly sprinkling facts of species loss, rainforest destruction, water pollution and runaway climate change.
When this happens, I wonder if environmental studies has displaced economics as the "dismal science," the phrase coined by 19th-century polymath Thomas Carlyle.
Yet while there are certainly forbidding ecological facts and alarming environmental trends that need to be highlighted and confronted, there are also myriad positive developments that are equally deserving of attention.
As we progress toward the New Year, three such signs of hope on the creation front come to mind.
First, Bolivia is on the verge of passing one of the most far-reaching environmental bills in history. The "Mother Earth" or Pachamama law, approved by Bolivia's majority governing party, draws deeply on indigenous concepts that view nature as a sacred home.
As Nick Buxton of the Transnational Institute observes, the law is among the first to give nature legal rights, specifically the rights to life, regeneration, biodiversity, water, clean air, balance and restoration.
The law will also fundamentally reorient Bolivia's economy, mandating not an embrace of unfettered growth but a conforming to the limits of nature. It radically advocates a public policy of Sumaj Kawsay or Vivir Bien (an indigenous concept meaning "living well"), in contrast to policies focusing on producing more goods for increased consumerism.
While mining interests and other sectors of the economy have already expressed opposition to the initiative, such legislation is a sanguine sign of renewed legal understandings of the primordial importance of nature.
A second beacon of hope can be seen emanating from Vatican City in the humble, pastoral and compassionate smile of Pope Francis, Time magazine's 2013 Person of the Year.
Francis's environmental musings have already been deeply compelling.
In Francis, it appears, the environment at last may have a global spiritual defender from the heart of Western Christianity.
A third source of hope is the creative, dynamic youth who are embracing the environment as a focus of their studies, avocation and chosen careers.
From the over 2,000 students taking first and second year environmental studies courses at the University of Toronto, to the youth participating in dynamic environmental and aboriginal solidarity programs such as Alberta's Future Leaders and Canada World Youth, there is an encouraging groundswell of interest and involvement by youth in environmental concerns.
Such interest is taking shape among a vibrant swath of under-30 environmental activists, documented in the book The New Eco-Warriors as well as a forthcoming film, both by Canadian environmentalist Emily Hunter.
The daughter of activist Bobbi Hunter and Greenpeace co-founder Robert Hunter, Emily features unconventional activists, such as Hannah Mermaid, a professional mermaid who helps protect marine ecosystems and wildlife.
Pachamama. Pope Francis. Passionate youth. Three hopeful signs as we head into the New Year that creation, and those who care for it, remain green and growing. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,147 |
\section{Introduction}
D3 branes living at the singularity of a Calabi-Yau cone have provided general and interesting results
for the AdS/CFT correspondence since its early days.
The IR limit of the gauge theory on the world volume of the D3-branes
is dual to type IIB string theory on the near horizon geometry
$AdS_5\times H$, where the horizon manifold $H$ is the compact base of the cone \cite{kw,horizon}. Since the cone is Calabi-Yau, $H$ has a Sasaki-Einstein metric.
Until few months ago, the only known Sasaki-Einstein metrics
were the round sphere $S^5$ and $T^{1,1}$, the horizon of the conifold.
Recently, various infinite classes of new Sasaki-Einstein metrics
were constructed \cite{gauntlett,CLPP,MSL} and named $Y^{\bar p,\bar q}$
and $L^{p,q,r}$ and the corresponding dual gauge theories were determined
\cite{benvenuti,kru2,noi,tomorrow}.
The remarkable growth in the number of explicit examples was accompanied by a deeper general understanding of the correspondence, in particular when the Calabi-Yau cone is a toric manifold. The AdS/CFT correspondence predicts a precise relation between the central charge $a$, the scaling dimensions of some operators in the CFT on the D3-branes and the volumes of $H$ and of certain submanifolds.
Checks of this relation have been done for the known Sasaki-Einstein metrics
\cite{benvenuti,bertolini,kru,kru2,noi,tomorrow}.
It is by now clear that all these checks can be done without an
explicit knowledge of the metric. a-maximization \cite{intriligator} provides
an efficient tool for computing central and R-charges on the quantum field theory side. On the other hand, Z-minimization \cite{MSY}
provides a geometrical method for extracting volumes from the toric data. In
this note we review the proof of the equivalence between a-maximization and
Z-minimization given in \cite{aZequiv} \footnote{For other interesting and complementary results on the equivalence based on supergravity see \cite{Tachikawa,Barnes}.}. In this process, we provide a general formula for assigning R-charges and multiplicities for the chiral fields of the quiver gauge theory based only on toric data. Our result is a quantitative check of the AdS/CFT correspondence that can be performed for all toric manifolds.
Since our work \cite{aZequiv} appeared, the long standing problem of finding the correspondence between toric singularities and quiver gauge theories has been completely solved
using dimer technology. The brane tilings \cite{dimers} provide an ingenious way of reconstructing the toric cone from the quiver; the inverse, and more complicated, problem of computing the gauge theory from the toric
diagram has been recently solved using zig-zag paths in \cite{rhombi}, and nicely interpreted in \cite{mirror}.
Now that we have an algorithm for determining the quiver gauge theory from the toric data, it is important
to study the distribution of R-charges among the chiral fields of the gauge theory. This information is not
important for determining the value of the central charge and the R-charges but it is certainly important
for a better understanding of the CFT. In this note we discuss in details how to find the precise distribution of R-charges in the quiver gauge theory using dimers and zig-zag paths.
\section{The gauge theory and AdS/CFT predictions}
\label{gauge}
We consider $N$ D3-branes living at the tip of a CY cone.
The base of the cone, or horizon, is a five-dimensional Sasaki-Einstein manifold $H$
\cite{kw,horizon}. The ${\cal N}=1$ gauge theory living on the branes is
superconformal and dual in the AdS/CFT correspondence to the type IIB
background $AdS_5\times H$, which is the near horizon geometry.
As well known, the matter content of the gauge theory can be represented
by a quiver diagram, where each node represents a gauge group and oriented
links represent chiral bifundamental multiplets. To complete the description
of the gauge theory one must specify also the superpotential.
By applying Seiberg dualities to a quiver gauge theory we can obtain
different quivers that flow in the IR to the same CFT.
It turns out that one can always find phases where all the gauge groups have the same number of colors; these are called toric phases.
For toric phases, and when the dual geometry is toric, one can ``lift'' the quiver diagram and
draw it on a torus $ T^2$ \cite{dimers}. This diagram, called the periodic quiver, identifies completely
the gauge theory, since now every superpotential term in the gauge theory is described by a face: it is the trace of the product of chiral
fields of the face \footnote{The superpotential has a sign + or - if the arrows of the face in the periodic quiver are oriented
clockwise or anticlockwise respectively. In the dual graph, the dimer, a white or black vertex correspond to a term with sign + or - respectively.}. The dual graph of the periodic quiver, known as the brane tiling or dimer configuration,
is still drawn on a torus $T^2$ (look at Figure \ref{y21} for the example $Y^{2,1}$). In the dimer the role of faces and vertices is exchanged: faces are gauge groups and vertices are superpotential terms. The dimer is a bipartite graph: it has an equal number of white and black vertices and links connect only vertices of different colors.
The first prediction of the correspondence we want to check is the relation between
the central charge in field theory
and the volume of the internal manifold \cite{gubser}:
\begin{equation}
a=\frac{\pi^3}{4 {\rm Vol}(H)}
\label{central}
\end{equation}
The second quantitative prediction states that the exact R-charges of chiral fields $\phi_i$
are proportional to volumes of certain calibrated 3d submanifolds $\Sigma_i$ inside $H$ \cite{gubserkleb}:
\begin{equation}
R_i=\frac{\pi {\rm Vol}(\Sigma_i)}{3 {\rm Vol}(H)}
\label{baryons}
\end{equation}
Recall in fact that the baryon operator built with $\phi_i$ is dual to a D3 brane wrapped
over $\Sigma_i$ and equation (\ref{baryons}) is the relation between the scaling dimension of the operator and
the mass of the state in the string dual.
\section{a-maximization}
\label{amax}
We will denote with $V$ the number of vertices of the periodic quiver (gauge groups),
with $E$ the number of edges (chiral fields), and with $F$ the number of faces (superpotential terms).
Since the periodic quiver is drawn on a torus, the Euler relation implies \cite{dimers}:
\begin{equation}
V+F=E
\label{eul}
\end{equation}
The R-charges $R_i$ of the chiral fields $\phi_i$, $i=1,\ldots E$, satisfy a set of linear constraints:
since we are looking for the infrared fixed points of the gauge theory,
each term in the superpotential must have R-charge $2$ ($F$ equations) and the
exact NSVZ $\beta$ function for every gauge group must be zero ($V$ equations).
The latter condition coincides with the anomaly cancellation for the R-symmetry.
It seems from (\ref{eul}) that $F+V$ linear conditions will determine uniquely the $E$ unknown charges
$R_i$. However, in the cases we are interested in, the conditions
are not linearly independent: it is easy to see that the homogeneous part of the linear system is just solved by the global
non anomalous $U(1)$ charges for chiral fields, so that a generic R-symmetry is mixed with the global non anomalous
symmetries:
\begin{equation}
R_i=R_i^0+x_h S_i^h,\qquad \qquad h=1,\ldots d-1
\label{poss}
\end{equation}
where $R^0$ is a particular solution, the $S^h$ are the global symmetries, and $x_h$ are real parameters.
The important fact is that, in general, we expect $d-1$ global non anomalous symmetries, where $d$ is the number of
sides of the toric diagram in the dual theory.
We can count these symmetries from the
number of massless vectors in the $AdS$ dual. Since the manifold is toric,
the metric has three $U(1)$ isometries.
One of these (generated by the Reeb vector) corresponds to the R-symmetry while the other
two give two global flavor symmetries in the gauge theory.
Other gauge fields in $AdS$ come
from the reduction of the RR four form on the non-trivial three-cycles
in the horizon manifold $H$, and there are $d-3$ three-cycles in homology \cite{tomorrow} when $H$ is smooth.
On the field theory side,
these gauge fields correspond to baryonic symmetries.
Summarizing, the global non anomalous symmetries are:
\begin{equation}
U(1)^{d-1}=U(1)^2_F \times U(1)^{d-3}_B
\end{equation}
At the fixed point,
only one of the possible non-anomalous R-symmetries (\ref{poss}) enters
in the superconformal algebra. It is the one in the same multiplet as the
stress-energy tensor.
The value of the exact R-charges at the fixed point can
be found by using the a-maximization technique \cite{intriligator}. As shown in
\cite{intriligator}, we have to maximize the trial a-charge \cite{anselmi}:
\begin{equation}
a(R)=\frac{3}{32}(3 {\rm Tr} R^3-{\rm Tr} R)
\label{ans}
\end{equation}
when R ranges over the possible symmetries (\ref{poss}); the value of this function at the
maximum gives the central charge, and the position of the maximum gives the exact R-charges.
In (\ref{ans}) the trace is a sum over all the fermionic components of the multiplets.
\section{Multiplicities and R-charges from the toric diagram}
\label{geometry}
\begin{figure}
\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[scale=0.60]{polygon.eps}
\caption{The convex polygon $P$.}
\label{polygon}
\end{minipage}%
~~~~~~\begin{minipage}[t]{0.48\linewidth}
\centering
\includegraphics[scale=0.60]{pqweb.eps}
\caption{The $(p,q)$ web for $P$.}
\label{pqweb}
\end{minipage}%
\end{figure}
Every toric CY cone in six dimensions is described by a toric diagram $P$ \footnote{The toric diagram is the intersection of the fan with the plane where all the generators live, due to the Calabi-Yau condition.
For the necessary elements of toric geometry see \cite{fulton} and the review part of \cite{MS}.},
which is simply a convex polygon in the plane with integers vertices (Figure \ref{polygon}), encoding in a simple diagrammatic way many information about the toric action. Toric diagrams equivalent up to translations and $SL(2,\mathbb{Z})$ transformations describe the same manifold. Moreover for every convex polygon with integer vertices there exists a unique singular Calabi-Yau metric over the corresponding toric cone. Therefore toric diagrams identify completely the geometry.
If the sides of the toric diagram do not pass through integer points the horizon $H$ is smooth; here and in the following we shall restrict to this case (but it is not difficult to extend the results to the general case \cite{aZequiv}).
Another fact from toric geometry we shall use is that there is a correspondence between vertices $V_i$ of the toric diagram $P$ and calibrated 3d submanifolds $\Sigma_i$ of $H$ (those appearing in equation (\ref{baryons})).
It is sometime useful to think in terms of $(p,q)$ webs.
The $(p,q)$ web is obtained by taking the perpendiculars to the vectors $v_i$ of the toric diagram with the same length as in $P$, see Figure \ref{pqweb} \footnote{With a little abuse of notation we call $v_i$ both the sides of $P$ and the vectors of the $(p,q)$ web. In fact they differ only by a rotation of $90^o$.}.
Let us also define the symbol:
\begin{equation}
\langle w_i, w_j \rangle \equiv \det(w_i, w_j)
\end{equation}
where $w_i$ and $w_j$ are two vectors in the plane of $P$.
Some of the data of the gauge theory can be extracted directly from the
geometry of the cone. In particular, there exist simple formulae for
the number of gauge groups $V$ and the total
number of chiral bi-fundamental fields $E$ \cite{hananymirror,benvenuti}
\begin{eqnarray}
V&=& 2{\rm Area}(P)\nonumber\\
E&=& \frac{1}{2}\sum_{i,j} |\langle v_i,v_j\rangle |
\label{numbers}
\end{eqnarray}
We would like to stress that the expression for $E$ is not true in all toric phases, but remarkably,
for the gauge theories in consideration, it seems that it exists always one or more toric phases where this formula is true.
We will call these phases minimal, since they are the ones with a minimal
number of fields.
We shall consider here only minimal toric phases (for a conjectured extension to non minimal toric phases see \cite{aZequiv}).
Generalizing results in the literature similar to (\ref{numbers}), in particular see \cite{hananymirror,kru2,benvenuti,tomorrow}, the following algorithm for extracting
the field theory content and a parametrization of R- or global charges can be proposed \cite{aZequiv}:
\begin{itemize}
\item{Associate with every vertex $V_i$ of the toric diagram $P$ a real (positive) parameter $a_i$, $i=1,\ldots d$. Indexes $i$, $j$ are defined modulo $d$.
The parameters $a_i$ can be also displaced between the vectors of adjacent sides in the $(p,q)$ web, see Figure \ref{pqweb}.}
\item{Consider the set $C$ of all pairs of vectors $(v_i,v_j)$ in the $(p,q)$ web ordered in such a way that the first vector $v_i$ can be rotated counter-clockwise to $v_j$ with an angle $\leq 180^o$.
Associate with every element of $C$ a type of chiral field in the field theory with multiplicity $|\langle v_i, v_j \rangle|$ and R-charge equal to $a_{i+1}+a_{i+2}+ \ldots a_{j}$.
For example in Figure \ref{pqweb} the field associated to the pair $(v_d,v_3)$ has R-charge $a_1+a_2+a_3$ and multiplicity $|\langle v_d, v_3 \rangle|$.
Note that eq. (\ref{numbers}) for $E$ is correctly reproduced.}
\item{Impose the linear constraint $\sum_{i=1}^d a_i = 2$
if you want to parametrize trial R-charges or
$\sum_{i=1}^d a_i =0$
if you want to parametrize the $d-1$ global charges.}
\end{itemize}
The chiral fields $\phi_i$ associated with the pair of consecutive vectors $(v_{i-1},v_i)$ with charge $a_i$ were already identified in \cite{tomorrow}: the dibaryon built with $\phi_i$ is dual to a D3 brane wrapped on the cycle $\Sigma_i$ associated with the vertex $V_i$.
The other chiral fields with charges obtained by summing consecutive $a_i$ form dibaryons dual to a D3 brane wrapped over the union of the corresponding $\Sigma_i$. The results in \cite{tomorrow} lead to a characterization of the
global charges: the $d-3$ baryonic charges are those that also satisfy $\sum_{i=1}^d a_i\, V_i=0$,
where $V_i$ are the coordinates of vertices in the plane of $P$.
With this assignment, we are able to write an expression for the trial $a$ function depending only on the toric diagram:
\begin{equation}
a=\frac{9}{32} \, \mathrm{tr}\, R^3 =\frac{9}{32} \left( V+ \sum_{(i,j) \in C}|\langle v_i, v_j \rangle| \,\, (a_{i+1}+a_{i+2}+\ldots a_{j}-1)^3 \right)
\label{aext}
\end{equation}
Recall that $V$, the number of gauginos, is the double area of the polygon $P$ (\ref{numbers}).
In non minimal toric phases, which have additional fields, equation (\ref{aext}) is still true,
since the contribution from additional fields to the trial R-charge cancels \cite{aZequiv}.
We can make several checks of this proposal for the field theory content. First of all, it is easy to
see that the proposal works in all cases where the quiver gauge theory is
explicitly known ($L^{p,q,r}$, $Y^{p,q}$, $X^{p,q}$, toric delPezzo). We also
explicitly checked on many examples
that the proposal is consistent with the general prescription given in \cite{dimers,rhombi} for extracting the quiver
gauge theory from the toric data (see the discussion in Section \ref{zigzag}).
We can also discuss the consistency of our proposal with the general
known properties of the $U(1)$ symmetries in this kind of theories.
The proposal
reproduces, for example, the expected result ${\rm Tr} \, G=0$, where $G$ is a general R-charge or global symmetry charge.
Moreover, it can be shown \cite{aZequiv}
that for the proposed assignment of charges the mixed cubic t'Hooft anomaly for baryonic symmetries is zero: ${\rm Tr}\, B_a^3=0$, as we expect from standard arguments in the gravitational dual \cite{intriwecht}.
However the best check of the proposal is that it allows to prove the equivalence of a-maximization and Z-minimization, as we shall soon discuss.
\section{Z-minimization}
\label{zeta}
It is in general hard to compute the Calabi-Yau metric corresponding to a particular toric diagram.
However in \cite{MSY} it was shown that all
the volumes we need can be computed from the toric data,
through the process known as volume minimization (or Z-minimization),
without any explicit knowledge of the metric. The reason for that is the following: supersymmetric cycles are calibrated and the volumes can be extracted
only from the Kahler form on the cone. The function Z to be minimized is a rational function of two variables $(x,y)$, which define a point inside the toric diagram.
Here we review the work of \cite{MSY}, reducing their formulas to the plane containing the convex polygon $P$.
Consider a generic toric diagram $P$ and a point $B=(x,y)$ allowed to vary inside $P$ \footnote{B is, up to a factor, the end point of the Reeb vector $K$, the vector field generating the U(1) isometry associated with the R-symmetry in field theory: $K = \sum _{i=1} ^{3} b_i e_i$, where $e_i$ form a basis for the $T^3$ fibration and $b=3(1,x,y)$.}. Call $r_i$ the vector joining $B$ with each vertex $V_i$.
Recall that every vertex $V_i$ of the toric diagram is associated with a calibrated submanifold $\Sigma_i$. Let us define the functions:
\begin{eqnarray}\label{volumes}
{\rm Vol}_{\Sigma_i}(x,y)&=&\frac{2\pi^2}{9} \frac{\langle v_{i-1},v_i\rangle}{\langle r_{i-1},v_{i-1}\rangle \langle
r_{i},v_{i}\rangle}\equiv \frac{2\pi^2}{9} l_i(x,y)\nonumber \\
{\rm Vol}_H (x,y)&=&\frac{\pi}{6} \sum_{i=1}^d {\rm Vol}_{\Sigma_i}(x,y)
\end{eqnarray}
The function to minimize is just ${\rm Vol}_H (x,y)$. This is the function $Z$ in \cite{MSY} up to a constant multiplicative factor.
This function is convex inside $P$ and therefore has a unique minimum $(\bar x, \bar y)$; the volumes $Vol(\Sigma_i)$, $Vol(H)$ for the Calabi-Yau metric are given respectively by the values (\ref{volumes}) at the minimum
\cite{MSY}.
\section{a-maximization is Z-minimization}
\label{comparison}
Now the problems of computing R-charges in field theory and volumes in the geometry
have been reduced to two different extremization problems defined only from toric data.
It is therefore possible to prove in the general toric case that the quantities they compute match according to the
AdS/CFT predictions (\ref{central}) and (\ref{baryons}).
The general proof is given in \cite{aZequiv}; here we give only the main ideas and make useful observations.
To facilitate the comparison we define the geometrical function:
\begin{equation}
a^{MSY}(x,y)=\frac{\pi^3}{4 {\rm Vol}_H(x,y)}
\label{aMSY}
\end{equation}
and the functions:
\begin{equation}
f_i(x,y)= \frac{2 l_i(x,y)}{\sum_{j=1}^d l_j(x,y)}
\label{Rc}
\end{equation}
corresponding to the R-charges $R_i$ through equation (\ref{baryons}).
The process of Z-mi\-ni\-mi\-za\-tion can be restated as a maximization of $a^{MSY}(x,y)$ with respect to $(x,y)$ varying in the interior of $P$.
If we call $\bar a_i$ the values of $a_i$ at the local maximum, we have to prove that:
\begin{equation}
\begin{array}{l}
a^{MSY}(\bar x, \bar y) = a(\bar a_1, \bar a_2, \dots \bar a_d)\\[0.5em]
f_i(\bar x, \bar y)= \bar a_i \qquad i=1,\ldots d
\label{target}
\end{array}
\end{equation}
This is a highly non trivial check to perform:
a-maximization and Z-minimization use different functions and different trial charges;
it is not at all obvious why the result should be the same.
First of all a-maximization is done on a total of $d-1$ independent
trial parameters while the volume minimization is done only on two parameters $(x,y)$.
The trial central charge $a$ is a cubic polynomial in $a_i$, whereas
$a^{MSY}$ is a rational function of $(x,y)$.
These parameters, in both cases, are somehow related to the possible global
symmetries: the Reeb vector in the geometry is connected to R-symmetries of the gauge theory
and changing the position of $B$ in the directions $x$ and $y$ means adding to the R-symmetry
the two flavor global symmetries \footnote{Recall that flavor symmetries are mixed with baryonic ones,
so actually we are moving also in the space of baryonic symmetries.}. In any case, the volume minimization is done by moving only in a two dimensional
subspace of the set of global symmetries, while a-maximization is done
on the entire space: recall that the trial R-symmetry is mixed with both the two flavor symmetries and the $d-3$ baryonic ones.
Therefore we try to decouple the baryon charges from the a-maximization
algorithm, performing a-maximization over a two dimensional subspace of parameters.
This subspace is just the space of coordinates $(x,y)$ of the plane where $P$ lies: consider the map from $\mathbb R^2$ to $\mathbb R^d$ given by
\begin{equation}
\begin{array}{l}
f:(x,y) \rightarrow (a_1,a_2,\ldots a_d)\\[1em]
\hspace{1.4em}(x,y) \rightarrow a_i= \displaystyle \frac{2 l_i(x,y)}{\sum_{j=1}^d l_j(x,y)}=f_i(x,y)
\label{map}
\end{array}
\end{equation}
We are parameterizing the trial charges $a_i$ in field theory with the functions
(\ref{Rc}) taken at generic points $(x,y)$.
It is not difficult to prove that the gradient of the trial central charge along the $d-3$ baryonic directions evaluated on $f(P)$ is always zero:
\begin{equation}
\sum_{i=1}^d B^a_i \frac{\partial a}{\partial a_i}_{|a_i=f_i(x,y)}=0
\label{derbaryon}
\end{equation}
where $B^a$ is a baryon charge and where the equality holds for every $(x,y)$ in the interior of $P$.
Therefore we have clarified in which sense the baryonic symmetries decouple from the process of a-maximization.
At this point we have to compare two functions of $(x,y)$: $a^{MSY}(x,y)$ and the field theory trial central charge evaluated
on the surface $f(P)$. Remarkably one discovers that they are equal even before
maximization:
\begin{equation}
a(a_1,\ldots a_d)_{|a_i=f_i(x,y)}= a^{MSY}(x,y)
\label{equal}
\end{equation}
for every $(x,y)$ inside the interior of $P$. This shows the equivalence between a-maximization and Z-minimization.
\section{Distribution of charges in the dimer}
\label{zigzag}
The algorithm discussed in Section \ref{geometry} for extracting the gauge theory content from the toric diagram of the dual theory gives us the multiplicities and the charges of chiral fields, and this is enough to write equation (\ref{aext}) for the trial $a$ charge from the toric diagram, but it does not tell us how the chiral fields are disposed in the periodic quiver (or in the dimer). Recently a general prescription
to extract the whole dimer from the toric diagram has been proposed
\cite{rhombi}, \cite{mirror}; as we will soon explain this algorithm is based on zig-zag paths. Once the dimer is known, one can try to determine the distribution of trial R-charges and check the algorithm reviewed in Section \ref{geometry}. There are two equivalent ways of determining the distribution of R-charges, one already described in \cite{aZequiv} and a second one based on zig-zag paths. Here we will briefly discuss both of them and
show their equivalence. One can check on concrete examples that both prescription give the right distribution of R-charges: the sum of charges for a vertex in the dimer (superpotential term) is equal to 2, and the sum of charges of a face is equal to the number of sides minus 2 (beta function zero).
The first general efficient way to find this distribution, valid for all toric phases, was discussed in \cite{aZequiv}: the parameters $a_i$ are associated with vertices of the toric diagram, and to every vertex $V_i$ there corresponds a single perfect matching\footnote{Recall that a perfect matching is a subset of links of the dimer configuration such that every white and black vertex is taken exactly once. Perfect matchings can be mapped to integer points of the toric diagram through the Kasteleyn matrix which counts their (oriented) intersections with two generators of the fundamental group of the torus \cite{dimers}.} in the dimer, at least for physical theories \cite{aZequiv,rhombi}. Therefore the trial charge of a link in the dimer can be computed as the sum of the parameters $a_i$ of all the external perfect matchings (corresponding to vertices) to which the link belongs. For examples
of how to use this prescription using the Kasteleyn matrix, see \cite{aZequiv}.
We now propose another equivalent algorithm based on the results of \cite{rhombi} and \cite{mirror}: this will give new insight for the formula giving the multiplicities of fields.
\begin{figure}
\centering
\includegraphics[scale=0.65]{y21.eps}
\caption{The correspondence for $Y^{2,1}$.}
\label{y21}
\end{figure}
A zig-zag path in the dimer is a path of links that turn maximally left at a node, maximally right at the next node, then again maximally left and so on \cite{rhombi}. We draw them in the specific case of $Y^{2,1}$ theory in Figure \ref{y21}: they are the four loops in green, yellow, red and blue and they are drawn so that they intersect in the middle of a link as in \cite{mirror}. Note that every link of the dimer belongs to exactly two different zig-zag paths, oriented in opposite directions. Moreover for dimer representing consistent theories the zig-zag paths are closed non-intersecting loops. There is a one to one correspondence between zig-zag path and legs of the $(p,q)$ web: the homotopy class in the fundamental group of the torus of every zig-zag path is just given by the integer numbers $(p,q)$ of the corresponding leg in the $(p,q)$ web \cite{rhombi}. The reader can check this directly in the example of Figure \ref{y21}: we can take as a fundamental cell for the torus one of the regions delimited by the black dashed lines.
Indeed the inverse algorithm of \cite{rhombi} consists just in drawing the zig-zag paths on a fundamental cell with the appropriate homotopy numbers. Since there's a one to one correspondence between intersections of zig-zag paths and links in the dimer, equation (\ref{numbers}) for the total number of fields is easily explained: recall that the number of topological intersections of two loops with homotopy class $w_i=(p_i,q_i)$ and $w_j=(p_j,q_j)$ is given by $\mathrm{det}(w_i,w_j)$. Since equation (\ref{numbers}) is valid in minimal toric phases of the gauge theory, we suggest that minimal phases can be built by making the effective number of intersections between every pair of closed zig-zag loops equal to the topological number. Non minimal phases, that is those with a greater number of fields than in (\ref{numbers}), may be explained with a greater number of effective intersection of zig-zag paths: we checked this in various examples. In the following we will concentrate on minimal toric phases.
The distribution of charges can be found as following. Consider the two zig-zag paths to which a link in the dimer belongs. They correspond to two vectors
$v_i=(p_i, q_i)$ and $v_j=(p_j, q_j)$ in the $(p,q)$ web. Then we propose that
the charge of the link is given by the sum of the parameters $a_{i+1}+a_{i+2} \ldots +a_{j}$ between the vectors $v_i$ and $v_j$.
So for instance in Figure \ref{y21} the links corresponding to the intersection of the red and the green zig-zag paths (vectors $v_4$ and $v_2$ in the (p,q) web) have charge equal to $a_1+a_2$.
This rule explains the formula in Section \ref{geometry} for the multiplicities of fields with a given charge $a_{i+1}+a_{i+2} \ldots +a_{j}$: it counts the number of intersections between the zig zag paths corresponding to $v_i$ and $v_j$, which is just $\mathrm{det}(v_i,v_j)$.
To conclude, we explain the relation with the algorithm for charge distribution based on perfect matchings. We use a conjecture in \cite{rhombi}: if we consider the union of the two perfect matchings associated with the consecutive vertices $V_{j-1}$ and $V_{j}$ of $P$ we obtain the zig-zag path corresponding to the side $v_j$ plus some other isolated segments belonging to both the perfect matchings \footnote{note that we are also assuming that besides this segments and the zig-zag path there are no other closed loops with trivial homotopy.}:
the zig-zag path is the ``formal'' difference of the two perfect matchings.
Therefore if a link in the dimer belongs to the perfect matching $V_{j-1}$ but not to the perfect matching $V_{j}$ it belongs to the zig-zag path $v_j$; while if it belongs to both perfect matchings the link is not on the zig zag path $v_j$.
\begin{figure}
\centering
\includegraphics[scale=0.65]{proof.eps}
\caption{Equivalence of the two algorithm for charge distribution.}
\label{proof}
\end{figure}
With this assumption it is easy to prove the equivalence of the two algorithms. Consider a link in the dimer at the intersection of two zig-zag paths corresponding to the sides $v_1$ and $v_j$ of the $(p,q)$ web, as in Figure \ref{proof}. Since this link belongs to the zig-zag path $v_1$, it belongs to one of the perfect matchings $V_1$ or $V_2$ but not to both. Suppose it belongs to the perfect matching $V_2$. Then it is easy to see that it must belong also to the perfect matching $V_3$: otherwise the link would be in the zig zag path $v_2$, but this is not possible since the link must belong to exactly two zig-zag paths and it already belongs to the zig-zag paths $v_1$ and $v_j$. Continuing in this way one can prove that the link belongs to the perfect matchings (associated with) $V_2$, $V_3$, \ldots $V_j$, and so it will be given the charge $a_2+a_3\ldots +a_j$ with both algorithms.
On the contrary if the link belongs to the perfect matching $V_1$ and not $V_2$, one derives that it belongs to all perfect matchings $V_1,V_d,\ldots V_{j+1}$: such fields may appear in non minimal toric phases \cite{aZequiv}.
This construction is also in agreement with the fact that chiral fields have always charges obtained by summing consecutive parameters $a_i$.
As an aside we note in Figure \ref{y21} that, taking into account also sums of $a_i$, for every vertex and face in the dimer the charges are in the cyclic order $a_1$, $a_2$, $a_3$, $a_4$ as in the toric diagram; vertices are all oriented anticlockwise, and faces are oriented clockwise. This seems to be a general fact.
\vspace{3em}
\noindent {\Large{\bf Acknowledgments}}
\vspace{0.5em}
We thank Amihay Hanany, Sergio Benvenuti and Davide Forcella
for useful discussions.
This work is supported in part by by INFN and MURST under
contract 2001-025492, and by
the European Commission TMR program HPRN-CT-2000-00131.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,902 |
True Green Cities/Celebrating Seven Years – Topiaries, Climate Change and A Gilded Mansion in Newport
In climate change, Historic preservation, museums
An aerial of Rough Point looking north along Newport's Cliff Walk.
Celebrating Seven Years – Stalactites, Climate Change and A Gilded Mansion in Newport
It's been seven years since I launched Barbara A. Campagna/Architecture + Planning, PLLC and while many things have changed, my goal to work on "greening what's already here" continues to be met, often in places I never expected. Many people are finding new ways to integrate historic preservation and green building practices, which makes my new venture a delightful and intellectually inspiring one. This is blog three of my anniversary week.
Rough Point's Impressive Pedigree
Rough Point is a historic site perched on the edge of the Cliff Walk in Newport, Rhode Island. It is a rusticated granite and sandstone English manor-style mansion with pinnacled gables and a green slate roof and elaborate bronze casement or fixed or double hung wood windows. The mansion is approximately 40,000 square feet.
Looking at Rough Point's mansion through the historic landscape and flowers.
Frederick W. Vanderbilt first built the house as a summer cottage from 1887-1892. Peabody & Stearns Architects from Boston and McNeil Brothers of Boston were the original contractors. William Bateman and Nancy Leeds (American Tin Plate Company) owned the property from 1906-1922, hiring John Russell Pope Architect to make some exterior alterations, mostly cosmetic. But the most significant ownership was with the Duke family. James Buchannan Duke purchased it 1922 and immediately hired Horace Trumbauer Architects with White Allom (interiors) to enlarge the house, completing it in 1924. Duke died in 1925, leaving the house to his daughter Doris who lived in it for decades. Following her death, the site went through transition from private home to public historic site – 1993-2001.
Rough Point opened as a museum under the ownership of the Newport Restoration Foundation in 2000. Alterations to accommodate new museum use included changing and enclosing the service stair, installing a 12-passenger elevator, creating an apartment on the third floor for a live-in caretaker, installing a handicap ramp, alterations to create galleries for changing exhibits, and a major upgrading of the MEP systems – new HVAC, fire protection, electric service, back-up generator, some new plumbing piping.
Evaluating Water Infiltration at Rough Point
Significant water infiltration has led to salt build up and salt "stalactites" at the window weep holes.
For the past year I have been working with the Newport Restoration Foundation to evaluate some ongoing water infiltration issues that are causing damage to both the interior and exterior. While a review of past projects and studies since 2000 suggests that there has always been some level of water infiltration in the building, particularly on the ocean facing elevations (south and east), it appears that the worse conditions on both the exterior and interior of the Music Room and Solarium have been observed in the past four years, which also coincidentally follows the completion of some masonry/lintel restoration work from 2010-2012. In addition, storms potentially exacerbated by climate change events, have increased in number, strength and impact in the past five years.
Technical Preservation
Doris Duke had real camels at Rough Point, now remembered with topiary camels on the lawn.
Given that the water infiltration has increased almost exponentially in the four years and the damage to the mortar and masonry is increasing, it would appear that the most recent phasing of mortar and masonry projects (2011-2012) may be contributing to these conditions if not actually causing them. This leads to several questions: Was the same mortar composition used by all three contractors? Was it the correct mortar? Was the repointing conducted using the same methods? Were the lintels installed in the same manner with the same methods and materials?
At the same time, as seen with the salt build-up on all the walls, and the "stalactites" on the south wall, has the amount of water penetrating the walls from the marine environment and wind-driven rain increased?
This is a very interesting and complex technical preservation project, and we will be assisting the Newport Restoration Foundation with the continuing evaluation of the conditions this summer.
True Green Cities/Celebrating Seven Years – Transforming Historic Factories in the Northland Corridor
In Adaptive Use, Buffalo architecture, Historic preservation, International Style, rehabilitation
Aerial view of the Niagara Machine & Tool Works on Northland Avenue in Buffalo. Photo courtesy Joe Cascio.
It's been seven years since I launched Barbara A. Campagna/Architecture + Planning, PLLC and while many things have changed, my goal to work on "greening what's already here" continues to be met, often in places I never expected. Many people are finding new ways to integrate historic preservation and green building practices, which makes my new venture a delightful and intellectually inspiring one. This is blog two of m
The oldest section of the Niagara Machine & Tool Works Factory – 1910, designed by Green & Wicks Architects. Photo courtesy Joe Cascio.
y anniversary week.
Transforming Historic Factories on Buffalo's East Side
I have had the distinct honor of being the preservation architect for this massive project on Buffalo's East Side as the preservation consultant to Watts Architecture & Engineering.
Historic District Overview: A city-wide survey conducted in 2013, Preservation-Ready Sites, identified the proposed Northland-Beltline Historic District as a potential city and National Register Historic District. The district is generally bounded on the west by Fillmore Avenue, on the east by Grider Street, on the south by a CSX rail line (i.e., the former Belt Line Railroad) and to the north by East Delavan Avenue.
Buffalo Urban Development Corporation, in partnership with Empire State Development, the New York Power Authority, and the City of Buffalo acquired multiple properties in the Northland Corridor. Each of the five industrial plants owned by the City of Buffalo included in this potential historic district has been determined individually eligible for listing on the National Register of Historic Places (National Register Eligible – NRE) by the New York State Office of Parks, Recreation & Historic Preservation (the SHPO). These plants include:
537 East Delavan Avenue (the Houde Engineering Complex & Factory)
631 Northland Avenue (Niagara Machine & Tool Works Factory/Clearing Niagara)
741 Northland Avenue (The Otis Elevator Company – "addition" to the Foundry)
777 Northland Avenue (The Otis Elevator Company Foundry)
With this in mind, a neighborhood-wide project on Buffalo's East Side known as the Northland Corridor Redevelopment is being undertaken by the City of Buffalo through the Buffalo Urban Renewal Agency and managed by another city agency, the Buffalo Urban Development Corporation. The city is receiving Community Development Block Grant funds from the U.S. Department of Housing & Urban Development, as well as funds from two state agencies, the New York Power Authority and the Empire State Development. Various tax credits, including New York State and Federal historic tax credits, New Market Tax Credits and Brownfield tax credits are being used to finance targeted rehabilitation projects.
This neighborhood wide project currently includes:
Rehabilitation of 683 Northland, the Niagara Machine & Tool Works Factory;
Partial rehabilitation, stabilization and demolition of 537 East Delavan, the Houde Engineering Complex and Factory; and
Stabilization and demolition of 777 Northland Avenue, the Otis Elevator Company Foundry.
There are projects underway at three sites in the district, with others planned for the future. The rehabilitation of Niagara Machine is the first significant project, over $100 Million rehabilitation financed from the Buffalo Billions, Historic Tax Credits, Brownfield Tax Credits and New Market Tax Credits.
NIAGARA MACHINE & TOOL WORKS HEADQUARTERS & FACTORY REHABILITATION
683 Northland
The front facade of the Niagara Machine & Tool Works Factory. Photo courtesy Joe Cascio.
The Niagara Machine and Tool Works Factory consists of three distinct buildings related to specific functions of the manufacturing facility and the factory's evolution: 1) The Headquarters and Main Factory (built in multiple building campaigns from 1910-1981), 2) The Pattern Shop (1913) and 3) the Metal Fabricating Plant (1953). The Headquarters and Main Factory and its adjacent Pattern Shop at 683 Northland Avenue are being rehabilitated. The Metal Fabricating Plant is on hold for future use.
Rehabilitation Overall scope of work:
The Headquarters and Main Factory and the Pattern Shop are being rehabilitated for a Workforce Training Center and related tenants. In order to acquire the use of historic tax credits (20% from the state and 20% federal) the project is going through the full Historic Preservation Certification Application Process which includes three parts and review and approval by both the New York State Office of Historic Preservation and the National Park Service.
Niagara Machine and Tool Works Factory Complex National Register Historic District Designation
The Historic District was officially listed on the National Register in March, just a couple weeks ago! The Niagara Machine and Tool Works Factory Complex, located on the East Side of Buffalo in Erie County, New York, is a large industrial facility designed and constructed between 1910 and 1981 by Niagara Machine and Tool Works. The complex is located on the south side of Northland Avenue, an east-west thoroughfare between Chelsea Place and Longview Avenue, along a highly industrialized core north of a CSX railway line, originally known as the Belt Line. Residential neighborhoods, both south and north of the railroad, surround the factories that line the corridor. The principal elevations of the factory face north onto Northland Avenue and south onto the Belt Line. The complex is best known by the name under which it achieved local prominence: Niagara Machine and Tool Works Factory. The area retains its original industrial feeling and setting, with four other factory buildings and complexes lining Northland and another several factories found along East Ferry Street to the south. Each factory had its own Belt Line Rail Road spur or spurs, remnants of which can still be seen.
The Niagara Machine and Tool Works Factory consists of three distinct buildings related to specific functions of the manufacturing facility and the factory's evolution: 1) The Headquarters and Main Factory (built in multiple building campaigns from 1910-1981), 2) The Pattern Shop (1913) and 3) the Metal Fabricating Plant (1953). Also counted in the resource count are three contributing structures: the extant rail spurs from the Belt Line railroad which are still present on the property. The 11 acres of land is primarily occupied by the Main Factory on the eastern half of the site, stretching from Northland Avenue to the railroad. The Pattern Shed is situated just west of the Main Factory towards the northern edge of the site. The Metal Fabricating Plant hugs the western edge of the site, also stretching from Northland to the rail line. A concrete drive wraps around the site with a ribbon of parking against the east side of the Main Factory. A concrete drive and parking lots separate the Main Factory and Metal Fabricating Plant, with iron gates demarcating the vehicle access.
Industrial features remaining in the original foundry of Niagara Machine & Tool Works Factory. Photo courtesy Joe Cascio.
The period of significance of the district is from 1910 to 1967, during which Niagara Machine & Tool Works developed and operated the site. The company manufactured tools and machines for working with sheet metal, specializing in presses, punches and rotary sheets. The nomination boundaries encompass the three contributing buildings, the Belt Line spur and the surrounding site historically constructed and used for this production. While the final construction occurred on the site in 1976 and 1981, the last building campaign that can be considered architecturally significant was completed in 1967, which also coincides with the final piece of the factory's post-World War II and Cold War expansion. The original buildings of the complex, built between 1910 and 1913, were designed by Green & Wicks Architects, one of the most noted architectural firms to ever practice in Buffalo.
The Niagara Machine & Tool Works Factory developed largely between 1910 and 1981, and like many factories, including its neighbors, it evolved over time as both technology and manufacturing capacity were expanded, and national war and defense needs increased. As a result, the Main Factory building was expanded through many additions. While some of the additions were carefully planned and executed by architects or engineers, others were added as needed and company expansion dictated. There were approximately twelve major building campaigns, which encompass the construction growth represented by three main buildings in the district: 1. The Headquarters and Main Factory (1910-1981), 2. The Pattern Shed (the 1913 shed) and 3. The Metal Fabricating Plant (1953). The three distinct buildings are related to specific functions of the manufacturing facility and the factory's evolution. This application is for the Main Factory Headquarters, and the Pattern Shed only which occupy one tax parcel and has one owner. The Metal Fabricating Plant is NOT included in this project or this Part 2 application. It occupies a separate tax parcel. No work is currently planned on this building or parcel.
Significance of the Niagara Machine & Tool Works Factory
Interior of the 1964 International Style addition of the Niagara Machine & Tool works Factory. Photo courtesy Joe Cascio.
The Niagara Machine & Tool Works Factory is significant as a good intact representative example of a large-scale tool and machine factory designed and built during the first half of the twentieth century. It is significant in the area of Industry as one of the oldest and most important tool and machine manufacturing facilities built and operated in Buffalo in the twentieth century, whose products were used nationally and internationally across industries. It is significant for its contribution to the manufacturing of tools and machines for working sheet metal, specifically specializing in presses, punches, and rotary sheets for government defense contracts. The company had orders from the federal government during World War I and World War II. But it is during World War II that the company established its role as an important defense contractor. The company had government orders for regular products in addition to equipment for many of the significant international aircraft manufacturers, who used the company's presses to manufacture small parts for aircraft assembly, both during and after World War II. The factory complex is further significant for its association with the industrial development of the East Side of Buffalo along manufacturing nodes of the Belt Line railroad in the early twentieth century.
The complex is also significant in the area of Architecture, as a good, representative example of evolving industrial architectural and engineering trends, innovations and developments during the twentieth century. The Niagara Machine & Tool Works Factory is a good example of early twentieth-century factory design with the original buildings designed by the noted Buffalo architectural firm of Green & Wicks, and later buildings and additions designed by local civil engineers H. E. Plumer and Jacob Fruchtbaum. The factory incorporates key advances in factory construction of the period for increased daylight with its large window openings and sawtooth monitor roofs. It also employs modern fireproof construction materials. Advances in early twentieth- century factory operations for mass production incorporated management science and the assembly line as evidenced in the layout of the plant and its two primary buildings. It is significant as a largely intact twentieth-century heavy manufacturing facility located in an industrial node of the East Side of the City of Buffalo.
The original sawtooth monitors at Niagara Machine & Tool Works Factory. Photo courtesy Joe Cascio.
The period of significance begins with the initial design and construction of the factory in 1910, when Green & Wicks designed the first buildings on Northland Avenue and extends through 1967 when the final architecturally significant addition was completed in the complex. While the factory continued in operation into the 1990s, this era encompasses all major construction projects by the company and reflects the period during which the company was at its most prominent. Subsequent additions made to the building in 1976 and 1981 after the period of significance are relatively minor additions in terms of architectural design and are located mostly toward the south end of the Main Factory. The last building campaign that can be considered architecturally significant was completed in 1967, which also coincides with the final piece of the factory's post-World War II and Cold War expansion. The proposed historic district nomination boundaries encompass the three contributing buildings, the Belt Line spur and the surrounding 11-acre site historically constructed and used for this production. This project includes the 7.28 acres and Belt Line spurs related to the Main Factory (approximately 232,500 square feet) and the Pattern Shed at 683 Northland only.
True Green Cities/Celebrating Seven Years – The APT 50th Anniversary Conference
In Historic preservation
Celebrating Seven Years – The APT Buffalo Niagara 2018 Conference
The Buffalo Central Terminal, location of several events during the APT Conference. Photo courtesy Joe Cascio.
It's been seven years since I launched Barbara A. Campagna/Architecture + Planning, PLLC and while many things have changed, my goal to work on "greening what's already here" continues to be met, often in places I never expected. Many people are finding new ways to integrate historic preservation and green building practices, which makes my new venture a delightful and intellectually inspiring one. This is blog one of my anniversary week.
Celebrating APT's 50th Anniversary, September 22-27, 2018
The Association for Preservation Technology International (APT) has been one of the most important parts of my adult life. Like many preservation colleagues, my professors in graduate school assigned articles from the APT Bulletin and indeed many of my professors had written those articles. I have been involved in APT's leadership since I graduated from school, first as a Board Member and then President of the APT NY chapter (now APTNE), then board member, Vice President, President, Past President of APT. Since 2011 I have been working with a terrific local committee to plan and bring the 50th Anniversary APT conference to Buffalo Niagara!
Points of Departure
Asbury Hall in Babeville, an adapted church in downtown Buffalo. Location of the APT 2018 Opening Keynote.
Why Buffalo and Niagara? Buffalo was the 6th largest port in the world in 1906. By 1951, it was the 11th largest industrial center in the country, the largest inland water port, the 2nd largest railroad center, and the 15th largest city in the country. It was literally and physically one of the most important points of departure on the continent. As APT looks forward to our "Next 50" we find ourselves at our own significant "point of departure." One of our founders, Peter John Stokes, was from Niagara-on-the-Lake across the Niagara River from Buffalo, making Buffalo and Niagara the perfect location to celebrate this joint international heritage.
As we get closer to our big 50th anniversary celebration in September, we are finalizing our program content and think everyone will be excited to hear what we have planned. The website is live and registration goes live in early May. We anticipate that many of these sessions will sell out early.
The Conference Themes
This year attendees will find several themes running through the program content of the conference – from the Workshops to the Opening Keynote to the Educational Sessions to the Closing Symposium. You will hear about Points of Departure, The Next 50 and Resilience throughout the conference. The theme for our Opening Keynote Session is "Resilience," a topic that our organization sees as one of the most important themes for preservationists moving forward. Buffalo's historic infrastructure provides a much- needed base for developing cities and places who need to be resilient* to the increasing storms and weather systems that are impacting everyone.
*Defining Resilience: Resilience is the capacity to adapt to changing conditions and to maintain or regain functionality and vitality in the face of stress or disturbance. It is the capacity to bounce back after a disturbance or interruption. From Katrina to Sandy, California drought to Mississippi flooding. Resilience is both response and action.
Opening Keynote Speaker: ALEX WILSON, President, Resilient Design Institute in Vermont
Sunday, September 23rd , 2018 5-6:30PM
Alex Wilson, Founder and President of the Resilient Design Institute will be the APT 2018 Opening Keynote.
We are thrilled to announce our Keynote speaker for the official opening of the conference! Alex Wilson is the founder of BuildingGreen, Inc. in Brattleboro, Vermont, an 18-person company that has served the design and construction industry with non-biased information on environmentally responsible design and construction since 1985. He is the author of Your Green Home (2006), and co-author of Green Development: Integrating Ecology and Real Estate (1998), and the Consumer Guide to Home Energy Savings (1990, 9th edition, 2007). He has also co-authored a series of four guidebooks on quiet-water canoeing and kayaking for the Appalachian Mountain Club. He is the founder and president of the Resilient Design Institute, which works with partners across the world to create solutions that enable buildings and communities to survive and thrive in the face of climate change, natural disasters and other disruptions. Alex is one of the leading luminaries in the world of sustainability and resilience.
The Opening Session is being held at Asbury Hall in historic Babeville, an adapted church filled with green features such as geothermal heating. The Plenary will be live streamed to our chapters around the world. We expect standing room only, with 750 attendees. The plenary will welcome attendees to Buffalo Niagara and to our 50th Anniversary Celebration. Since we are celebrating 50 years of accomplishments and looking forward to another 50 years, we will celebrate our organization's heritage and Buffalo Niagara's heritage – the reason we chose this location for our significant anniversary. We will specifically celebrate the heritage of Buffalo's Olmsted Park & Parkway System in its 150th anniversary and look forward to congratulating Buffalo leaders from the City and the Buffalo Olmsted Parks Conservancy.
Workshops – Saturday and Sunday, September 22nd & 23rd , 2018
The Non-Destructive Evaluation Workshop at the APT 2018 Conference will be held at Willowbank on Day One.
Come early and spend the weekend at one of three workshops for up to 50 participants each:
Terra Cotta Restoration – held at Boston Valley Terra Cotta and downtown. A complementary Terra Cotta walking tour of downtown Buffalo's amazing collection of terra cotta buildings and terra cotta restorations will be available as a guided field session or a self-guided walking tour.
Window Restoration – held at several historic sites in Buffalo, and a local window restoration studio.
Non-Destructive Evaluation – held at the Willowbank School in Canada and the Central Terminal (railroad terminal), just named to the 2018 World Monuments
Field sessions and workshops will travel to Buffalo's National Historic Landmarks such as St. Paul's Cathedral and Louis Sullivan's Guaranty Building.
Watch.
Educational Paper Sessions, Monday, September 24th – Wednesday, September 26th
Panels and sessions will be held in the Hyatt and Buffalo Niagara Convention Center. Over 70 speakers will present original preservation research, chosen by the conference committee through the "Call for Abstracts." We received 200 professional submissions and 25 student submissions. APT brings up to 15 students to the conference to participate for free every year. Also, look for special "theme" plenaries this year to celebrate our 50th!
Field Sessions & Tours – Sunday, September 23rd and Tuesday afternoon, September 25th, 2018
There will be up to 20 site visits around Buffalo Niagara that include local neighborhoods to iconic landmark masterpieces – a Beer Oriented Development session, cruises on the Buffalo River to admire our concrete grain elevators, medina sandstone at the Richardson Olmsted Campus and many churches around town, downtown loft conversions, art glass and geothermal heating at Frank Lloyd Wright's Darwin Martin House, a Resilience Charrette at Frank Lloyd Wright's Graycliff on Lake Erie, to name a few.
Canada Day – Wednesday, September 26th, 2018 12:30-10PM
Don't miss the daylong festivities to celebrate our 50th Anniversary and American Canadian heritage, with stops at Niagara Falls, Willowbank, Niagara-on-the-Lake and an anniversary party at Jackson Triggs Winery.
The Next Fifty Symposium – Thursday, September 27th, 2018
A day-long closing symposium held in the auditorium in the downtown Central Library will celebrate the advances of preservation technology and the impact APT has had on the world preservation stage. This closing session will frame the intellectual and technological themes that our thought leaders envision will move APT forward. We hope many of you will stay this last day for a hearty discussion about preservation's past and future. Like the "Setting the Agenda" lunches of a decade ago, where our members steered our direction – you can participate in steering APT's philosophical direction for the next fifty!
(I apologize for the wonky formatting; I'm in the middle of renovating my website.) And if you'd like to "subscribe" or follow my blog, True Green Cities, please sign up through the "Subscribe" button at the bottom left of this page. You'll receive a daily recap when new blogs are posted. Or Sign up for the Feed, also at the bottom left of this page. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,031 |
\section{Introduction}
The Cherenkov Telescope Array (CTA) \cite{bib:CTA1,bib:CTA2} is a world wide project aimed at designing, building and operating an array of $\sim 100$ telescopes covering an energy range of about 4 orders of magnitude with a sensitivity in the core energy region (around 1 TeV) of about one order of magnitude better than current arrays. Due to the different features of the Cherenkov signals for energies spanning from some tens of GeV to 100 TeV and beyond, three kinds of telescopes have been proposed. At low energy the photon density is very low, so very large reflective surfaces are needed to collect enough photons to correctly reconstruct the Cherenkov images requiring a minimum number of 50-100 photoelectrons (p.e.). On the other hand, the higher incoming flux at lower energies reduces the need of large effective areas. In this energy region a small (3-4) number of Large Size Telescopes (LSTs $\sim24\,$m diameter) is envisaged. In the medium energy regime, the drop of incoming flux due to the slopes of the source spectra requires a larger collection area, built up with $\sim30$ Medium Size Telescopes of moderate aperture (MSTs, $\sim12\,$m diameter). In the highest energy range the photon density on ground is very high even for large core distances, while the source flux becomes very low. An effective area of $1-10\,$km$^2$ is thus necessary, and it is obtained placing $\sim$70 Small Size Telescopes (SSTs, $\sim4\,$m diameter) on a grid with $\sim300\,$m spacing.
The optimization of the individual telescope parameters and their layout is done using Monte Carlo (MC) simulations.
The main figure of merit is the sensitivity, i.e. the minimum detectable flux in a given time interval, but also the angular and energy resolutions and the sensitivity across the field of view (field flatness) have to be considered for some specific physics cases.
These performance parameters have been derived using existing and custom simulation and analysis programs. Due to the excellent background rejection achievable with the IACT stereo technique a huge amount of primary hadrons (mainly protons) has to be generated.
For a full sky coverage the CTA project will have an array in each hemisphere. The site of the Southern array will be decided within this year (2013) and the array construction is expected to start in 2016.
ASTRI (\textit{Astrofisica con Specchi a Tecnologia Replicante Italiana}) \cite{bib:pare1, bib:astri} is a flagship project of the Italian Ministry of Education, University and Research strictly linked to the development of CTA. Within this framework, INAF is currently developing an end-to-end prototype of the CTA small-size telescope in a dual-mirror configuration (SST-2M) to be tested under field conditions at the INAF ``M.C. Fracastoro''observing station in Serra La Nave (Mount Etna, Sicily)
\cite{bib:macc}, and scheduled to start data acquisition in 2014. Furthermore the ASTRI collaboration proposed the installation of a small array consisting of 5-7 SSTs \cite{bib:pare2} at the selected CTA Southern site, possibly supported by one or two MSTs and one LST. This small array could constitute the first CTA seed and could be used for technical purposes but also for scientific studies, mainly in the highest energy region from few tens of TeV. This energy region is widely unexplored and may yield exciting and unexpected discoveries.
\section{Monte Carlo simulations}
The simulation and analysis of the Cherenkov images on the detector focal plane produced by the primary particles impacting on the top of the atmosphere is done with the simulation chain used to evaluate the performance of the whole CTA array and involves several steps:
\begin{itemize}
\item generation of the Extensive Air Showers (EAS) and of the Cherenkov photon yield with CORSIKA v.6.99 \cite{bib:corsika};
\item simulation of the telescope optics and detector electronics with \textit{sim\_telarray} \cite{bib:simtel};
\item reduction and analysis of the simulated events through different reconstruction programs derived by the software used by the HESS, MAGIC and VERITAS collaborations.
\end{itemize}
The available data include a gamma point source and diffuse protons simulated within a cone of $\pm 10^{\circ}$ in order to correctly take into account the contribution of background events falling well outside the field of view. The energy ranges of the simulated events are 0.003-330 TeV and 0.004-600 TeV for gammas and protons, respectively, with differential spectral index $\gamma=-2$ (rescaled to $-2.57$ and $-2.7$ in the analysis step). The site parameters (atmospheric depth and magnetic field) have been chosen to match one of the CTA South candidate sites (El Leoncito, 2660 m a.s.l.). The showers have been scattered over a circular area with radius of 2500~m for gammas (10 times each shower) and 3000~m for protons (20 times) with observing zenith angle $\theta = 20^{\circ}$.
Table~\ref{tab:MCsamples} summarizes the main characteristics of the samples of simulated EAS used in this work, a fraction ($\sim$10$\%$) of the full production \cite{bib:prod2} (at the moment still ongoing) which will allow more accurate results.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|c|}
\hline & Gamma & Proton \\ \hline
Core position re-sampling & 10 & 20 \\ \hline
Events (after re-sampling) & 2.6e8 & 4.5e9 \\ \hline
Energy [TeV] & 0.003 - 330 & 0.004 - 600 \\ \hline
Radius [m] & 2500 & 3000 \\ \hline
\end{tabular}
\caption{Monte Carlo samples.}
\label{tab:MCsamples}
\end{center}
\end{table}
Among the different telescopes proposed for SST, in this analysis the dual mirror telescopes \cite{bib:pare2} have been used. Table~\ref{tab:sstconf} shows the main features for this configuration.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|c|}
\hline
Primary mirror (M1) diameter [m] & 4.3 \\ \hline
Secondary mirror (M2) diameter [m] & 1.8 \\ \hline
Distance Focal plane - M2 [m] & 0.52 \\ \hline
Distance M1 - M2 [m] & 3.1 \\ \hline
\end{tabular}
\caption{Characteristics of the simulated SST-2M.}
\label{tab:sstconf}
\end{center}
\end{table}
The study of the mini-array performance is done selecting subsets of 5 and 7 SSTs from the whole layout as shown in Figure~\ref{fig:layout}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{icrc2013-0563-01.eps}
\caption{Selected sub-arrays within the whole SST-2M array.}
\label{fig:layout}
\end{figure}
Data are then reconstructed and analyzed with different programs \cite{bib:prod2} and their results are cross-checked. All the results from the different analyses are compatible within uncertainties of the order of 20-30\%. In the following results based on the MARS \cite{bib:mars} and read\_hess \cite{bib:simtel} programs are shown.
\subsection{Effective area and threshold energy}
Figures~\ref{fig:effareag} and \ref{fig:effareap} show the effective area after trigger (i.e.: at least 2 telescopes) and after analysis cuts, not including the arrival direction one, for $\gamma$ and protons. As expected, due to the smaller shower size for proton-generated EAS with respect to gammas of the same energy, the trigger effective area for protons is smaller than for gammas. The 7 telescopes layout provides an effective area $\sim50\%$\ larger than the 5 telescopes's one.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{icrc2013-0563-02.eps}
\caption{Effective area for gammas at trigger level and after shape cuts.}
\label{fig:effareag}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{icrc2013-0563-03.eps}
\caption{Effective area for protons at trigger level and after shape cuts.}
\label{fig:effareap}
\end{figure}
From the trigger effective area and an assumed Crab-like source spectrum \cite{bib:crab} we derive a trigger energy threshold around 500 GeV (corresponding to the peak). The energy threshold for analysis obtained from the effective area after analysis cuts is around 1 TeV. Since the smallest distance between telescopes is 260 m for both the 5 and the 7 telescopes sub-arrays, the two configurations have the same trigger threshold.
\subsection{The angular and energy resolutions}
The aim of the CTA project is to boost not only the sensitivity but also the angular and energy resolutions. Figure~\ref{fig:angres} shows the expected angular resolution at 68\% containment for the mini-arrays and for the whole SST array of CTA, with at least two images used in reconstruction. The requirement on the number of images at low energies (E $<$ 10 TeV) results in the selection of a higher quality subsample for the mini-arrays with respect to the full SST. On the other hand at high energies the angular resolution benefits of the higher multiplicity.
The energy resolution, shown in Figure~\ref{fig:eneres}, which can be as good as 10\% at a few TeV, approaches the full SST result and is in any case limited by the intrinsic shower-to-shower fluctuations.
Despite the much lower effective area of the mini-array with respect to the CTA array, the expected angular and energy resolutions will be comparable at least for certain data selections and energies. Checking on-field the mini-array performance will be a check of the whole SST array and a validation of the entire simulation procedure.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{icrc2013-0563-04.eps}
\caption{Angular resolution for 5 and 7-fold mini-array configurations and the full SC-SST array.}
\label{fig:angres}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{icrc2013-0563-05.eps}
\caption{Energy resolution for 5 and 7-fold mini-array configurations and the full SC-SST array.}
\label{fig:eneres}
\end{figure}
\subsection{The gamma/hadron discrimination and sensitivity}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.9\textwidth]{icrc2013-0563-06.eps}
\caption{Comparison of differential sensitivities for different explored layouts for CTA \cite{bib:CTAProd1}, HESS, MAGIC and this work mini-arrays (5 and 7 SST-2M).}
\label{fig:sensitivity}
\end{figure*}
The gamma/hadron discrimination has been optimized for point sources selecting cuts on the image shape and on the shower direction, based on the well known fact that gamma-generated images are more compact with respect to proton ones and that the major axis is directed towards the source direction.
From the effective areas after the $\gamma$/hadron cuts the expected sensitivity as a function of the primary energy can be calculated for a standard observing time of 50\,h and a five standard deviations statistical significance. For the significance calculation, equation 17 of Li\&Ma\cite{bib:LiMa} has been used assuming a background region 5 times larger than the signal region ($\alpha$ = 0.2). The other requirements are: 5 bins/decade in energy; a minimum of 10 excess events above the background in each energy bin; the signal excess at least 5\% of the remaining background after cuts in order to take into account possible systematics in background subtraction. Figure~\ref{fig:sensitivity} shows the expected differential sensitivity compared to the whole CTA (several explored layouts) and HESS and MAGIC. While at low energy HESS and MAGIC perform clearly better, starting from $E=10\,$TeV the mini-array expected sensitivity, for the 7 telescopes layout, is better up to a factor 2 at few tens of TeV and is still sensitive at 100 TeV, where the effective area of present generation IACTs drops dramatically. In this energy regime the mini-array will operate as the most sensitive IACT array.
\section{Discussion and conclusions}
The quick installation of a mini-array of SSTs in the selected CTA site will be of paramount benefit for the whole project.
The comparison between expected performance with experimental ones will be a validation of the MC chain for the whole SST. The ASTRI mini-array will allow to test several innovative adopted technical solutions such as the Dual Mirror approach and the SiPM-based camera. Another important test will be the wide field technique, i.e. the observation of the higher energy events very far from the core (up to ~500\,m) and outside the Cherenkov light pool. Due to the limited field of view of actual detectors this technique has never been tested in the field, and this can easily be done with at least 2 telescopes. For the layout optimization a specific grid with telescope separation between 200 and 350\,m must be used and dedicated simulations are ongoing. The addition of two MSTs and possibly one LST would complete the low energy range allowing the inter-calibration checks between different sub-arrays. Finally, from the scientific point of view, the ASTRI mini-array will be the detector with the best performance for energies greater than 10\,TeV. This regime is largely unexplored and could provide interesting results on known sources but also unexpected ones~\cite{bib:verc}.
\vspace*{0.5cm}
\footnotesize{{\bf Acknowledgment:}{
This work was partially supported by the ASTRI Flagship Project financed by the Italian Ministry of Education, University, and Research (MIUR) and led by the Italian National Institute of Astrophysics (INAF). We also acknowledge partial support by the MIUR Bando PRIN 2009. We gratefully acknowledge support from the agencies and organizations
listed in this page: http://www.cta-observatory.org/?q=node/22}.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,291 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/29168\/how-to-calculate-expected-value-and-standard-deviation-if-i-have-100-values-divi","text":"# How to calculate expected value and standard deviation if I have 100 values divided into 15 groups (normal distribution)?\n\nThese are the values:\n\n18.2 17.6\n------------------------------------------------------------------------------------------------------------\n18.9 18.6\n------------------------------------------------------------------------------------------------------------\n20 19.7 19.8 19.6\n------------------------------------------------------------------------------------------------------------\n21.4 20.6 21\n------------------------------------------------------------------------------------------------------------\n21.9 22.1 22.2 22.2 21.6\n------------------------------------------------------------------------------------------------------------\n23.1 22.6 23.2 23 22.9 23.2 23.2 23 22.5 22.5\n------------------------------------------------------------------------------------------------------------\n23.8 24.1 23.8 23.6 24.3 23.7 24 24.2 24.4 23.8 23.5\n------------------------------------------------------------------------------------------------------------\n24.7 24.9 25 25.4 25.1 25.3 25.3 25.2 25.2 24.7 24.6 24.5 24.5\n------------------------------------------------------------------------------------------------------------\n25.6 26 25.6 25.6 25.6 25.7 25.6 26.3 26.3 26.2 26 25.9 26.2 26\n------------------------------------------------------------------------------------------------------------\n27 27.2 27 27.2 26.7 27 26.6 27.3 26.8 26.6 26.9 27 26.5\n------------------------------------------------------------------------------------------------------------\n28.2 27.7 28.1 27.9 27.6 27.7 28.1 28.2 27.5\n------------------------------------------------------------------------------------------------------------\n28.6 29.1 28.7 29.1\n------------------------------------------------------------------------------------------------------------\n30.3 29.6 29.7 29.5\n------------------------------------------------------------------------------------------------------------\n30.9 31 31 30.5\n------------------------------------------------------------------------------------------------------------\n32.3 31.6\n------------------------------------------------------------------------------------------------------------\n\n\nThese are the groups:\n\n - 17.5 - 18.5\n- 18.5 - 19.5\n- 19.5 - 20.5\n- 20.5 - 21.5\n- 21.5 - 22.5\n- 22.5 - 23.5\n- 23.5 - 24.5\n- 24.5 - 25.5\n- 25.5 - 26.5\n- 26.5 - 27.5\n- 27.5 - 28.5\n- 28.5 - 29.5\n- 29.5 - 30.5\n- 30.5 - 31.5\n- 31.5 - 32.5\n\n\nI'm supposed to calculate the standard deviation and the expected value ... How is it different comparing to having only 15 values and not divided them into groups? If I only have 15 values, I know how to calculate the stand.deviation and the exp.value. I use the formulas that are well-known:\n\nExp.value: ............(1\/n)*Sum(xi)..................where xi = all the 15 values <br>\nStand.deviation: ...(1\/(n-1))*Sum( xi - u)^2...where xi = all the 15 values and u = exp.value\n\n\u2022 is this homework? What is the rationale behind the grouping? What is the application \/ context? \u2013\u00a0David LeBauer May 25 '12 at 20:20\n\u2022 (1) Note that the \"Stand.deviation\" formula actually is for an estimator of a variance and that both formulas incorrectly refer to 15 values, whereas 100 values are in this dataset. (The use of \"exp.value\" in place of \"mean\" suggests these data should be treated as a population, whence the n-1 probably should be replaced by n.) (2) You ask about the difference between treating these data as 100 values and 15 values. Why don't you just compute the formulas you think should apply and compare what you get? \u2013\u00a0whuber May 25 '12 at 20:37\n\u2022 I have no idea what do they mean by that... Presumably, the exp.value is gonna be calculated the same as in the case of 15 values. However, the stand.deviation should probably be calculated for each group separately and then getting the stand.deviation by using those stand.deviation of each group. [The resuls are: exp.value: 25.37 and stand.deviation: 3.103] \u2013\u00a0user1111261 May 25 '12 at 20:39\n\u2022 My guess is that they are referring to grouped means and grouped standard deviations. A question like that came up once recently either here or on the mathematics site. The grouped mean is obtained by taking a weighted average of the midpoint of the bin where the weight is the number of cases falling in the bin. The same goes for the variance except the square differences are taken by square the difference of the midpoint of the bin with the weighted mean obtained previously as the grouped mean. I don't see where the normal distribution assumption comes into play. \u2013\u00a0Michael Chernick May 25 '12 at 21:13\n\u2022 @whuber my source was math.stackexchange.com\/questions\/148629\/calculating-expected-value-standard-deviation-when-i-have-frequencies-and-intervals-in-percentage or math.stackexchange.com\/questions\/148629 for short \u2013\u00a0Michael Chernick May 25 '12 at 23:16\n\nThe statement is confusing, but I guess:\n\nGiven the 100 sample values, you know the common way (not unique) to estimate the mean and standard deviation:\n\n$$\\bar{x}= \\frac{1}{100} \\sum_{i=1}^{100} x_i=25.175 , \\hspace{2cm} \\hat{\\sigma} = \\sqrt{\\frac{1}{99} \\sum_{i=1}^{100} (x_i - \\bar{x})^2 } = 3.47529$$\n\nBut suppose, instead, that you have grouped your data (for example, to compute an histogram) assuming intervals centered at representative points $(18,19,20 \\cdots 31, 32)$ That means that, for example, the data point $x_i=29.6$ will be counted as belonging to the interval $(28.5,29.5)$ (representative value=29), and that's the only information you retain.\n\nThen, the \"new\" (approximate) mean and variance will be computed only counting how many values fall in each group, which is equivalent to compute the above but assuming that each value is replaced by the representative value (the center of each interval).\n\n$$\\bar{x}= \\frac{1}{100} \\sum_{i=1}^{100} y_i= \\frac{1}{100} \\sum_{g=1}^{15} n_g \\, y_g = 25.37$$\n\n$$\\hat{\\sigma} = \\sqrt{\\frac{1}{99} \\sum_{g=1}^{15} n_g (y_g - \\bar{x})^2 } = 3.1031$$\n\nHere, $n_g$ is the number of values that fall in each interval, and $y_g$ is the central point of that interval.\n\nOf course, this is less precise, because you are loosing information - it would hardly make sense to compute this instead of the former, but I guess this is an exercise to understand the difference.","date":"2019-06-19 10:58:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8024806380271912, \"perplexity\": 726.2321783647678}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560627998959.46\/warc\/CC-MAIN-20190619103826-20190619125826-00154.warc.gz\"}"} | null | null |
Q: Simple group with Klein four Sylow
If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup are conjugate in $G$.
Any help would be appreciated.
Note: A solution/method in relatively elementary terms would be appreciated — I am alright with some machinery as long as you explain it.
A: I think you gate this from Sylow's 2nd theorem
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,575 |
Kiran Leonard (born John Kiran Leonard) is a musician, composer and singer-songwriter from Saddleworth, Greater Manchester.
Career
Leonard was born into a musical family in the countryside of Greater Manchester. He first learned to play mandolin and guitar and then, inspired by his older brother's diverse music collection, began to experiment with composing electronic music.
After a period of prolifically self-releasing experimental music, Leonard's full length album Bowler Hat Soup was released in 2012 on Hand of Glory Records to some acclaim, receiving notable endorsement from DJ Marc Riley of BBC Radio 6 Music. His subsequent LP Grapefruit (2016), released on Moshi Moshi Records, likewise received critical acclaim.
Leonard's music notably references a wide variety of literature, historical figures and geographical locations: his 2017 LP Derevaun Seraun documented a collection of pieces commissioned by Manchester Central Library inspired by works by James Joyce, Albert Camus, Henry Miller, Clarice Lispector and Manuel Bandeira, while Grapefruit makes reference to Ondör Gongor, Caiaphas, the Bilderberg Group, Lagavulin and Werner Herzog, Bowler Hat Soup to Friedrich Nietzsche, Brunswick Street, Port Ainé and Bora Bora, Thread Colours (2015) to Franz Kafka and Windermere, and Garden in Bermuda (2016) to Annie Edson Taylor and Coyoacán. He has cited musicians such as Frank Zappa, Richard Dawson, The Mars Volta and Kate Bush as important influences on his music.
Leonard currently studies Spanish and Portuguese at Wadham College, Oxford University.
Select discography
Albums
Selected Passive Drones, Part II: Organic Journey (2009)
The Big Fish (2011)
A Seed is a Sovereign (2012, compilation)
The End Times (2012)
Oakland Highball EP (2013)
Multi-Titled Summer Tour CD (2013)
Bowler Hat Soup 2LP (2013)
Spring Rounds CD (2014) Collection of live, field and home recordings
Terreiro do Paço (2014)
Abandoning Noble Goals EP (2015)
Grapefruit 2LP (2016)
Garden in Bermuda CD (2016) Collection of songs credited to Kiran Leonard as well as his Advol, Pend Oreille and Akrotiri Poacher pseudonyms
Derevaun Seraun LP (2017)
Monarchs of the Crescent Pail cassette (2017)
A Bit of Violence With These Old Engines cassette (2018)
Western Culture LP (2018)
Trespass On Foot double album, cassette (2021)
As Pend Oreille
YLA EP (2012)
Thread Colours (2015)
As Akrotiri Poacher
Silence Within Buildings (2010)
Jane Barbe / Akrotiri Poacher (2011) Split with Jane Barbe
Seminary (2012)
Motiongazer / Akrotiri Poacher (2014) Split with Motiongazer
References
People from Saddleworth
British male singer-songwriters
Alumni of Wadham College, Oxford
1995 births
Living people
Musicians from Greater Manchester
21st-century British singers
21st-century British male singers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,485 |
Q: Как отправить сообщение всем слушателям RabbitMQ, Spring boot Отправка:
@Service
public class ProducerImp implements Producer {
@Autowired
private AmqpTemplate amqpTemplate;
@Value("${jsa.rabbitmq.exchange}")
private String exchange;
@Value("${jsa.rabbitmq.routingkey}")
private String routingKey;
@Override
public void produceMsg(MessageDTO msg) {
amqpTemplate.convertAndSend(exchange, routingKey, msg);
System.out.println("Send msg = " + msg);
}
}
слушатель (Остальные такие же):
@Component
public class SubscribeServiceImpl {
@RabbitListener(queues="jsa.queue")
public void recievedMessage(MessageDTO msg) {
System.out.println("Recieved Message: " + msg.getMessage());
}
}
Проблема в том что сообщение приходит только одному из слушателей. Как сделать так чтоб приходила всем?
A: Если все слушатели подписаны на одну и ту же очередь, то сообщение обработает только один слушатель, который первым получил сообщение.
Вы можете создать несколько очередей (для каждого из слушателей) и посылать сообщение во все очереди сразу привязав их к одному Exchange и RoutingKey.
Подробней тут
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,638 |
\section{Introduction}
\label{sec:introduction}
We address the task of reading text in natural scenes, commonly referred to as Scene Text Recognition (STR).
Although STR has been active since the late 90's, only recently accuracy reached a level that enables commercial applications, this is mostly due to advances in deep neural networks research for computer vision tasks.
Applications for STR include, among others, recognizing street signs in autonomous driving, company logos, assistive technology for the blind and translation apps in mixed reality.
\begin{figure}[ht!]
\centering
\includegraphics[width=\columnwidth]{./figures/figure_1_v11.pdf}
\caption{\textbf{The proposed SCATTER training and inference architecture}.
We introduce intermediate supervision combined with selective-decoding to stabilize the training of a deep BiLSTM encoder (The circle represents where a loss function is applied).
Decoding is done using a selective-decoder that operates on visual features from the CNN backbone and contextual features from the BiLSTM encoder, while employing a two-step attention.}
\label{fig:high_level_method}
\end{figure}
Text in natural scenes is characterised by a large variety of backgrounds, and arbitrary imaging conditions that can lead to low contrast, blur, distortion, low resolution, uneven illumination and other phenomena and artifacts.
In addition, the sheer magnitude of possible font types and sizes add another layer of difficulty that STR algorithms must overcome.
Generally, recognizing scene text can be divided into two main tasks - text detection and text recognition.
Text detection is the task of identifying the regions in a natural image, that contain arbitrary shapes of text.
Text recognition deals with the task of decoding a cropped image that contains one or more words into a digital string of its contents.
In this paper, we propose a method for text recognition; we assume the input is a cropped image of text taken from a natural image, and the output is the recognized text string within the cropped image.
As categorized by previous works \cite{Baek2019clova, Wang2019sar}, text images can be divided into two categories:
\textit{Irregular text} for arbitrarily shaped text (e.g. curved text), as seen in \cref{fig:high_level_method}, and \textit{regular text} for text with nearly horizontally aligned characters (examples are provided in the supplementary material).
Traditional text recognition methods~\cite{Wang2011bottom, Wang2010bottom, shi2014bottom} detect and recognize text character by character, however, these methods have an inherent limitation -- they do not utilize sequential modeling and contextual dependencies between characters.
Modern methods treat STR as a sequence prediction problem.
This technique alleviates the need for character-level annotations (per-character bounding box) while achieving superior accuracy.
The majority of these sequence-based methods rely on Connectionist Temporal Classification (CTC) \cite{Bai2015crnn, Gao2019dsan}, or attention-based mechanisms \cite{Bai2018aster, Wang2019sar}.
Recently, Baek et al. \cite{Baek2019clova} proposed a modular four-step STR framework, where the individual components are interchangeable allowing for different algorithms.
This modular framework, along with its best performing component configuration, is depicted in \cref{fig:high_level_method} (a).
In this work, we build upon this framework and extend it.
While accurately recognizing regular scene text remains an open problem, recent irregular STR benchmarks (e.g., ICD15, SVTP) have shifted research focus to the problem of recognizing text in arbitrary shapes.
For instance, Sheng et al.~\cite{Sheng209nrtr} adopted the Transformer~\cite{Vaswani2017trans} model for STR, leveraging the transformers ability to capture long-range contextual dependencies.
The authors in \cite{Wang2019sar} passed the visual features from the CNN backbone through a 2D attention module down to their decoder.
Mask TextSpotter~\cite{liao2019mask} unified the detection and the recognition tasks with a shared backbone architecture.
For the recognition stage, two types of prediction branches are used, and the final prediction is selected based on the output of the more confident branch.
The first branch uses semantic segmentation of characters, and requires additional character-level annotations.
The second branch employs a 2D spatial attention-decoder.
Most of the aforementioned STR methods perform a sequential modeling step using a recursive neural network (RNN) or other sequential modeling layers (e.g., multi-head attention \cite{Sheng209nrtr}), usually in the encoder and/or the decoder.
This step is performed to convert the \textbf{visual feature} map into a \textbf{contextual feature} map, which better captures long-term dependencies.
In this work, we propose using a stacked block architecture for repeated feature processing, a concept similar to that used in other computer-vision tasks such as in~\cite{wei2016cpm} and later in~\cite{Newell2016hour, Newell2017hour}.
The authors above showed that repeated processing used in conjunction with intermediate supervision could be used to increasingly refine predictions.
In this paper, we propose the \textbf{S}elective \textbf{C}ontext \textbf{AT}tentional \textbf{TE}xt \textbf{R}ecognizer (SCATTER) architecture.
Our method, as depicted in \cref{fig:high_level_method}, utilize a stacked block architecture for repetitive processing with intermediate supervision in training, and a novel selective-decoder.
The selective-decoder receives features from two different layers of the network, namely, visual features from a CNN backbone and contextual features computed by a BiLSTM layer, while using a two-step 1D attention mechanism.
\Cref{fig:val_acc_heads} shows the accuracy levels computed at the intermediate auxiliary decoders, for different stacking arrangements, thus demonstrating the increase in performance as additional blocks are added in succession.
Interestingly, training with additional blocks in sequence leads to an improvement in the accuracy of the intermediate decoders as well (compared to training with a shallower stacking arrangement).
This paper presents two main contributions:
\begin{enumerate}[nolistsep]
\item We propose a repetitive processing architecture for text recognition, trained with intermediate selective decoders as supervision.
Using this architecture we train a deep BiLSTM encoder, leading to SOTA results on irregular text.
\item A selective attention decoder, that simultaneously decodes both visual and contextual features by employing a two-step attention mechanism.
The first attention step figures out which visual and contextual features to attend to.
The second step treats the features as a sequence and attends the intra-sequence relations.
\end{enumerate}
\begin{figure}[t]
\centering
\includegraphics[width=0.66\columnwidth]{./figures/figure_glass_v2.pdf}
\includegraphics[width=0.33\columnwidth]{./figures/stars_plot_3.pdf}
\caption{Average test accuracy (IIT5K, SVT, IC03, IC13,IC15, SVTP, CUTE) at intermediate decoding steps, compared across different network depths used in training.
Given a computation budget, improved results can be obtained by first training a deeper network, and then running only the first decoder(s) during inference.}
\label{fig:val_acc_heads}
\end{figure}
\section{Related Work}
\label{sec:related_work}
STR has attracted considerable attention over the past few years~\cite{Zisserman2015large, Hu2017grccn, fedor2018rosetta, Liu20168stn}.
Comprehensive surveys for scene text detection and recognition may be found in~\cite{Ye2014survey, Baek2019clova, Long2018survey}.
As mentioned above, STR may be divided into two categories: regular and irregular texts (further examples are provided in the supplementary material).
Earlier papers~\cite{Wang2011bottom, Wang2010bottom, shi2014bottom}, focused on regular text and used a bottom-up approach, which involved segmenting individual characters with a sliding window, and then recognizing the characters using hand-crafted features.
A notable issue with the bottom-up approaches above, is that they struggle to use contextual information; instead they rely on accurate character classifiers.
Shi et al. 2015 \cite{Bai2015crnn} and He et al.~\cite{He2016cnn} considered words as sequences of varying lengths, and employed RNNs to model the sequences without explicit character separation.
Shi et al. 2016~\cite{bai2016tps} presented a successful end-to-end trainable architecture using the sequence approach, without relying on character level annotations.
Their solution employed a BiLSTM layer to extract the sequential feature vectors from the input feature maps, these vectors are then fed into an attention-Gated Recurrent Unit (GRU) module for decoding.
The methods mentioned above introduced significant improvements in STR accuracy on public benchmarks.
Therefore, recent work has shifted focus to the more challenging problem of recognizing irregularly shaped text, hence promoting new lines of research.
Topics such as input rectification, character-level segmentation, 2D attentional feature maps and self attention have emerged, pushing the envelope on irregular STR.
Shi et al. 2018~\cite{Bai2018aster} rectified oriented or curved text based on a Spatial Transformer Network (STN).
Liu et al. 2018~\cite{Wei2018charnet} introduced a Character-Aware Neural Network (Char-Net) to detect and rectify individual characters.
A combination of a CTC-Attention mechanism within an encoder-decoder framework, that was used for speech recognition tasks, was used for STR in~\cite{Zuo2019join}, showing the benefits of joint CTC-Attention learning.
The authors in \cite{Gao2019dsan} proposed two supervision branches to tackle explicit and implicit semantic information.
In \cite{Chen2019laeg} a gate was inserted to the recurrent decoder, for controlling the transmission-weight of the previous embedded vector, demonstrating that context is not always needed for decoding.
The authors of Mask TextSpotter~\cite{liao2019mask} unified text detection and text recognition in an end-to-end fashion.
For recognition they used two separate branches, a branch that uses visual (local) features and a branch which utilizes contextual information in the form of 2D attention.
More recent approaches have proposed leveraging various attention mechanisms for improved results.
Li et al.~\cite{Wang2019sar} combined both visual and contextual features while utilizing a 2D attention within the encoder-decoder.
Other researchers borrowed ideas from the Natural Language Processing (NLP) domain and adopted a transformer-based architecture~\cite{Vaswani2017trans}.
One of them is Sheng et al.~\cite{Sheng209nrtr}, that used a self-attention mechanism for both the encoder and the decoder.
Our method differs from above approaches, by being the first to utilize a stacked block architecture for text recognition.
Namely, we show that repetitive processing for text recognition, trained with intermediate selective decoders as supervision (similar to ~\cite{wei2016cpm, Newell2016hour, Newell2017hour}), increasingly refines text predictions.
\section{Methodology}
\label{sec:method}
\begin{figure*}[ht!]
\normalsize
\centering
\includegraphics[width=\textwidth]{./figures/figure_ours_vs_basic_v5.pdf}
\caption{The proposed SCATTER architecture introduces, context refinement, intermediate supervision (additional decoders), and a novel selective-decoder.}
\label{fig:architecture}
\end{figure*}
As presented in \cref{fig:architecture}, our proposed architecture consists of four main components:
\begin{enumerate}[nolistsep]
\item \textbf{Transformation:} the input text image is normalized using a Spatial Transformer Network (STN)~\cite{Zisserman2018stn}.
\item \textbf{Feature Extraction:} maps the input image to a feature map representation while using a text attention module~\cite{Gao2019dsan}.
\item \textbf{Visual Feature Refinement:} provides direct supervision for each column in the visual features.
This part refines the representation in each of the feature columns, by classifying them into individual symbols.
\item \textbf{Selective-Contextual Refinement Block:} Each block consists of a two-layer BiLSTM encoder that outputs contextual features.
The contextual features are concatenated to the visual features computed by the CNN backbone.
This concatenated feature map is then fed into the selective-decoder, which employs a two-step 1D attention mechanism, as illustrated in \cref{fig:selective_decoder}.
\end{enumerate}
In this section we describe the training architecture of SCATTER , while addressing the differences between training and inference.
\subsection{Transformation}
The transformation step operates on the cropped text image $X$, and transforms it into a normalized image $X^\prime$.
We use a Thin Plate Spline (TPS) transformation, a variant of the STN, as used in~\cite{Baek2019clova}.
TPS employs a smooth spline interpolation between a set of fiducial points.
Specifically, it detects a pre-defined number of fiducial points at the top and bottom of the text region, and normalizes the predicted region to a constant predefined size.
\subsection{Feature Extraction}
In this step a convolutional neural network (CNN) extracts features from the input image.
We use a 29-layer ResNet as the CNN's backbone, as used in~\cite{bai2017accurate}.
The output of the feature encoder is 512 channels by $N$ columns.
Specifically, the feature encoder gets an input image $X^\prime$ and outputs a feature map $F = [f_1,f_2,...,f_N]$.
Following the feature map extraction, we use a text attention module, similar to~\cite{Gao2019dsan}.
The attentional feature map can be regarded as a visual feature sequence of length $N$, denoted as $V = [v_1,v_2,...,v_N]$, where each column represents a frame in the sequence.
\subsection{Visual Feature Refinement}
Here, the visual feature sequence $V$ is used for intermediate decoding.
This intermediate supervision is aimed at refining the character embedding (representations) in each of the columns of $V$, and is done using CTC based decoding.
We feed $V$ through a fully connected layer that outputs a sequence $H$ of length $N$.
The output sequence is fed into a CTC~\cite{Graves2006ctc} decoder to generate the final output.
The CTC decoder transforms the output sequence tensor into a conditional probability distribution over the label sequences, and then selects the most probable label.
The transcription procedure is given by
\begin{align}
l = B(\operatorname*{arg\,max}_\pi p(\pi|H)) \,,
\end{align}
where the probability of $\pi$ is defined as
\begin{align}
p(\pi|H) = \prod_{t=1}^{N} y^{t}_{\pi_t} \,.
\end{align}
Here $y^{t}_{\pi_t}$ is the probability of generating the character $\pi_t$ at time stamp $t$, and $B$ is a mapping function that removes all repeated characters and blanks.
The CTC algorithm assumes that the columns are conditionally independent, and at each time stamp the output is a single character probability score.
The loss for this branch, denoted by $L_\text{CTC}$, is the negative log-likelihood of the ground-truth conditional probability, as in~\cite{Bai2015crnn}.
\subsection{Selective-Contextual Refinement Block}
\label{sec:contextlevel}
The features extracted by the CNN are limited to its receptive field, and may suffer due to the lack of contextual information.
To mitigate this drawback, we employ a two-layer BiLSTM~\cite{Graves2008rnn} network over the feature map $V$, outputting $H = [h_1, h_2,..., h_n]$.
We concatenate the BiLSTM output with the visual feature map, yielding $D = (V, H)$, a new feature space.
The feature space $D$ is used both for selective decoding, and as an input to the next Selective-Contextual Refinement block.
Specifically, the concatenated output of the $j$th block can be written as $D_j = (V, H_j)$.
The next ${j+1}$ block uses $H_j$ as input to the two-layer BiLSTM, yielding $H_{j+1}$, and the $j+1$ feature space is updated such that $D_{j+1} = (V, H_{j+1})$.
The visual feature map $V$ does not undergo any further updates in the Selective-Contextual Refinement blocks, however we note that the CNN backbone is updated with back-propagated gradients from all of the selective-decoders.
These blocks can be stacked together as many times as needed, according to the task or accuracy levels required, and the final prediction is provided by the decoder from the last block.
\subsubsection{Selective-Decoder}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./figures/figure_selective_decoder_v3.pdf}
\caption{Architecture of the Two-Step Attention Selective-Decoder.}
\label{fig:selective_decoder}
\end{figure}
We employ a two-step attention mechanism, as illustrated in \cref{fig:selective_decoder}.
First, we use a 1D self attention operating on the features $D$.
A fully connected layer is used to compute an attention map from these features.
Next, an element-wise product is computed between the attention map and $D$, yielding the attentional features $D^\prime$.
The decoding of $D^\prime$ is done with a separate attention-decoder, such that for each $t$-time-step the decoder outputs $y_t$, similar to \cite{bai2017accurate, bai2018edit}.
Decoding starts by computing the vector of attentional weights, $\alpha_t \in R^N$:
\begin{align}
\label{equ:equ2}
e_{t,i} & = w^T \tanh(Ws_{t-1} + Vd^\prime_i + b) \\
\alpha_{t,i} & = \exp (e_{t,i}) / \sum_{i^*=1}^{n}e_{t,i^*} \,,
\end{align}
where $b, w, W, V$ are trainable parameters, $s_{t-1}$ is the hidden state of the recurrent cell within the decoder at time $t$, and $d\prime$ is a column of $D\prime$.
The decoder linearly combines the columns of $D^\prime$ into a vector $G$, which is called a glimpse:
\begin{align}
g_{t} = \sum_{i=1}^{n}\alpha_{t,i}d^\prime_i \,.
\label{equ:equ4}
\end{align}
Next, a recurrent cell of the decoder is fed with
\begin{align}
(x_t,s_t) = \text{RNN}\Big(s_{t-1}, \big(g_t,f(y_{t-1})\big)\Big) \,,
\end{align}
where $(g_t,f(y_{t-1}))$ denotes the concatenation between $g_t$ and the one-hot embedding of $y_{t-1}$.
The probability for a given character $p(y_t)$ can now be recovered by:
\begin{align}
p(y_{t}) = \text{softmax} (W_o x_t + b_o) \,.
\end{align}
The loss for the $j$th block is the negative log-likelihood, denoted as $L_{\text{Attn}, j}$, as in~\cite{bai2017accurate, bai2018edit}.
\subsection{Training Losses}
The objective function is given by
\begin{align}
L = \lambda_{\text{CTC}} \cdot L_\text{CTC} + \sum_{j=1}\lambda_j L_{\text{Attn}, j} \,,
\end{align}
where $L_\text{CTC}$ is the loss function of the CTC decoder and $\sum_{j=1}\lambda_j L_{\text{Attn}, j}$ is the sum of the losses from all of the Selective-Contextual Refinement blocks, as defined above.
The $\lambda$ notation depicts a hyper-parameter used to balance the trade-off between the different supervisions, and specifically, $\lambda_{CTC}, \lambda_j$ are empirically set to 0.1, 1.0 respectively for all $j$.
\subsection{Inference}
Once training is done, for test time we remove all of the intermediate decoders, as they are used only for additional supervision and refinement of intermediate feature.
The visual features $V$ are processed by the BiLSTM layers in all blocks, and are also fed directly, via a skip connection, to the final selective-decoder.
The final selective decoder is used to predict the output sequence of characters.
A visualization of these changes can be seen in \cref{fig:architecture}, where all the green colored operations are disabled during inference, and in \cref{fig:high_level_method} (b).
\section{Experiments}
\label{sec:experiments}
In this section we empirically demonstrate the effectiveness of our proposed framework.
We begin with a brief discussion regarding the datasets used for training and testing, and then describe our implementation and evaluation setup.
Next, we compare our model against state-of-the-art methods on public benchmark datasets, including both regular and irregular text.
Finally, we address the computational cost of our method.
\input{./tables/sota_results_fix.tex}
\subsection{Datasets}
In this work, all SCATTER models are trained on three synthetic datasets.
The model is evaluated on four regular scene-text datasets: ICDAR2003, ICDAR2013, IIIT5K, SVT, and three irregular text datasets: ICDAR2015, SVTP and CUTE.
The training dataset is a union of three datasets:
\textbf{MJSynth} (MJ) \cite{Zisserman2014mj} is a synthetic text in image dataset which contains 9 million word box images, generated from a lexicon of 90K English words.
\textbf{SynthText} (ST) \cite{Zisserman206st} is a synthetic text in image dataset, designed for scene-text detection, and recognition.
We use a variant of the SynthText dataset composed of 5.5M samples, as used in \cite{Baek2019clova}.
This variant does not include any non-alphanumeric characters.
\textbf{SynthAdd} (SA) \cite{Wang2019sar} is a synthetic text in image dataset, that contains 1.2 million word box images.
This dataset was generated using the same synthetic engine as in ST, aiming to mitigate the lack of non-alphanumeric characters (e.g., punctuation marks) in the other datasets.
All experiments are evaluated on the seven real-word STR benchmark datasets described below.
As in many STR manuscripts (e.g, \cite{Bai2018aster, Baek2019clova, Wang2019sar}) the benchmark datasets are commonly divided into regular and irregular text, according to the text layout.
\textbf{Regular text} datasets include the following: \textbf{IIIT5K}~\cite{Mishra2012sj} consists of 2000 training and 3000 testing images that are cropped from Google image searches.
\textbf{SVT}~\cite{Wang2011bottom} is a dataset collected from Google Street View images and contains 257 training and 647 testing word-box cropped images.
\textbf{ICDAR2003}~\cite{Lucas2003ic03} contains 867 word-box cropped images for testing.
\textbf{ICDAR2013}~\cite{Karatzas2013ic13} contains 848 training and 1,015 testing word-box cropped images.
\textbf{Irregular text} datasets include the following: \textbf{ICDAR2015}~\cite{Karatzas2015ic15} contains 4,468 training and 2,077 testing word-box cropped images, all captured by Google Glass, without careful positioning or focusing.
\textbf{SVTP}~\cite{Phan2013svtp} is a dataset collected from Google Street View images and consists of 645 cropped word-box images for testing.
\textbf{CUTE 80}~\cite{Risnumawan2014cute} contains 288 cropped word-box images for testing, many of which are curved text images.
\subsection{Implementation Details}
As baseline, we use the code of Baek et al.\footnote{https://github.com/clovaai/deep-text-recognition-benchmark}~\cite{Baek2019clova}, and our architectural changes are implemented on top of it.
All experiments are trained and tested using the PyTorch\footnote{https://pytorch.org/} framework on a Tesla V100 GPU with 16GB memory.
As for the training details, we do not perform any type of pre-training.
We train using the AdaDelta optimizer, and the following training parameters are used: a decay rate of 0.95, gradient clipping with a magnitude of 5, a batch size of 128 (with a sampling ratio of 40\%, 40\%, 20\% between MJ, ST and SA respectively).
We use data augmentation during training, and augment 40\% of the input images, by randomly resizing them and adding extra distortion.
Each model is trained for 6 epochs on the unified training set.
For our internal validation dataset, we use the union of the IC13, IC15, IIIT, and SVT training splits, to select our best model, as done in~\cite{Baek2019clova}.
All images are resized to $32 \times 100$ during both training and testing, following common practice.
In this paper, we use 36 symbol classes: 10 digits and 26 case-insensitive letters.
As for special symbols for CTC decoding, an additional "[UNK]" and a "[blank]" token are added to the label set.
For the selective-decoders three special punctuation characters are added: "[GO]", "[S]" and "[UNK]" which indicate the start of the sequence, the end of the sequence and unknown characters (that are not alpha-numeric), respectively.
At inference, we employ a similar mechanism to \cite{Wang2019sar, Yanga209simple, Ning209master}, where images with a height larger than their width, are rotated by 90 degrees clockwise and counter-clockwise respectively.
The rotated versions are recognized alongside the original image.
A prediction confidence score is calculated as the average decoder probabilities until the '[S]' token.
We then choose the prediction with the highest confidence score as the final prediction.
Unlike \cite{Bai2018aster, Wang2019sar, Sheng209nrtr}, we do not use beam-search for decoding, although the authors in~\cite{Wang2019sar} have reported it improves accuracy by approximately 0.5\%, due to the added latency it incurs.
\input{./tables/ablation_fix.tex}
\subsection{Comparison to State-of-the-art}
In this section, we measure the accuracy of our proposed framework on several regular and irregular scene text benchmarks while comparing the results to the latest SOTA recognition methods.
As seen in \cref{tab:sota_results}, our SCATTER architecture with 5 blocks outperforms the current SOTA, the Mask TextSpotter~\cite{liao2019mask} algorithm, on irregular scene text benchmarks (i.e., IC15, SVTP, CUTE) by an absolute margin of 3.7\% on average.
Our approach provides an accuracy increase of \textbf{+4.0} pp (78.2\% vs. 82.2\%) on IC15, \textbf{+3.3} pp (83.6\% vs. 86.9\%) on SVTP, and is the second best to Mask TextSpotter~\cite{liao2019mask} on the CUTE (88.5\% vs. 87.5\%) dataset.
Additionally, the proposed method outperforms the other methods both on SVT and IC03 regular scene text datasets, and achieves comparable SOTA performance on the other regular scene text datasets (i.e.,IIIT5K and IC13).
To summarize, our model achieves the highest recognition score on 4 out of 7 benchmarks, and the second best score on 2 more benchmarks.
Unlike other methods, which perform well on either regular or irregular scene text benchmarks, our approach is a top performer on all benchmarks.
We would like to briefly discuss key differences between Mask TextSpotter~\cite{liao2019mask} and this work.
The algorithm in~\cite{liao2019mask} relies on annotations that contain character level annotations, information that our algorithm does not require.
These annotations contribute an average increase of 0.9 pp, and 0.6 pp on regular and irregular text datasets as reported in the original paper.
Hence, without the character level annotations, our model achieves slightly better results on regular text (93.9\% vs. 94\%) and significantly better results on irregular text (79.4\% vs. 83.7\%).
Our approach, on the other hand, does not require these annotations, which are expensive and hard to annotate, especially for real-world data.
In \cref{fig:fail_cases} we display failure cases of our method.
The failure cases are mostly composed of blurry images, partial character occlusions, difficult lighting conditions, and miss-recognition of punctuation marks.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./figures/error_ana_v7.pdf}
\caption{Failure cases of our model. "GT" stands for the ground-truth annotation, and "Pred" is the predicted result.}
\label{fig:fail_cases}
\end{figure}
\subsection{Computational Costs}
During inference, only the selective decoder of the final block is kept active, as shown in \cref{fig:high_level_method}.
The total computational cost of our proposed architecture with a single block is 20.1 ms.
The additional computational cost of each \textit{intermediate} contextual refinement block during inference translates to a 3.2 ms per block.
For a 5 block architecture (our best setup) this translates to a total increase of 12.8 ms and a total forward pass of 32.9 ms.
Furthermore, given a computational budget on inference time, performance can be improved by training the system with a large number of blocks and pruning them for inference.
For example, an architecture trained with five blocks, and then pruned to a single block, is capable of outperforming an architecture solely trained with a single block.
\Cref{fig:val_acc_heads}(2c) demonstrates a network trained with five blocks and the average test accuracy of intermediate decoders.
This showcases that pruning leads to an increase of \textbf{+0.4} pp and \textbf{+1.3} pp on regular an irregular datasets respectively (under the same computational budget).
This novel feature of SCATTER allows for faster inference if needed and in some cases pruning can even boost results.
\section{Ablation Experiments}
\label{sec:ablation}
In this section, we perform a series of experiments to better understand the performance improvements and analyze the impact of our key contributions.
Throughout this section, we use a weighted-average (by the number of samples) of the results on the regular and irregular test datasets.
For completeness, the first and second rows in \cref{tab:ablation} show the reported results in~\cite{Baek2019clova}, and the improved results of our re-trained model of~\cite{Baek2019clova} with our custom training settings.
\subsection{Intermediate Supervision \& Selective Decoding}
Section (a) of \cref{tab:ablation} shows an improvement in accuracy by adding the intermediate CTC supervision and the proposed selective-decoder.
Between row two and three of section (a) we add a CTC decoder for intermediate supervision which improves the baseline results by \textbf{+0.2} pp and \textbf{+0.4} pp on regular and irregular text respectively.
The fourth row demonstrates the improvement compared to the baseline results by replacing the standard attentional decoder with the proposed selective-decoder, (\textbf{+0.4} pp and \textbf{+2.7} pp on regular and irregular text respectively).
Section (b) of \cref{tab:ablation} shows a monotonic increase in the accuracy using the SCATTER architecture with 4 BiLSTM layers by changing the number of intermediate supervisions (ranging from 1 to 3).
The relative increase in accuracy of section (b) is \textbf{+0.7} pp and \textbf{+2.9} pp on regular and irregular text respectively.
\subsection{Stable Training of a Deep BiLSTM Encoder}
As mentioned in the introduction, previous papers use only a 2-layer BiLSTM encoder.
The authors in~\cite{Zuo2019join} report a decrease in accuracy when increasing the number of layers in the BiLSTM encoder.
We reproduce the experiment reported in~\cite{Zuo2019join} of training a baseline architecture with an increasing number of BiLSTM layers in the encoder (results are in the supplementary material).
We observe a similar phenomena as in~\cite{Zuo2019join}, i.e a reduction in accuracy when using more than two BiLSTM layers.
Contrary to this discovery, \cref{tab:ablation} shows that the overall trend of increasing the number of BiLSTM layers in SCATTER , while increasing the number of intermediate supervisions improves accuracy.
Recognition accuracy improves monotonically up to 10 BiLSTM layers, both for regular and irregular text datasets.
As evident from \cref{tab:ablation} (c), when training with more than 10 BiLSTM layers in the encoder, accuracy results slightly (\textbf{-0.4} pp and \textbf{-0.5} pp on regular and irregular text respectively) decrease on both regular and irregular text (similar phenomena was observed on the validation set).
It is expected that increasing the network capacity leads to more challenging training procedures.
Other training approaches might need to be considered to successfully train to convergence a very deep encoder.
Such approaches may include incremental training, where we first train with a shallower network using a small number of blocks and incrementally stack more blocks during training.
\input{./tables/refinement_effect.tex}
Examples of intermediate predictions are seen in \cref{tab:refinement_effect}, showcasing SCATTER ability to increasingly refine text prediction.
\subsection{Oracle Decoder Voting}
\label{subsec:oracle}
\input{./tables/decoder_limit_fix.tex}
In \cref{tab:decoder_limit} the test accuracy is shown for the intermediate decoders on a SCATTER architecture trained with 5 blocks.
The last row summarizes the potential results of an oracle, that for every test image chooses the correct prediction, if one exists in any of the decoders.
If an optimal strategy to select between the different decoders predictions exists, the results on \textit{all} datasets achieve a new state-of-the-art.
The possible improvement in accuracy achievable by such an oracle ranges between \textbf{+0.8} pp, and up to \textbf{+5} pp across the datasets.
A possible prediction selection strategy might be based on ensemble techniques, or a meta model that predicts which decoder to use for each specific image.
\section{Conclusions and Future Work}
\label{sec:conclusions}
In this work we propose a stacked block architecture named SCATTER , which achieves SOTA recognition accuracy and enables stable, more robust training for STR networks that use deep BiLSTM encoders.
This is achieved by adding intermediate supervisions along the network layers, and by relying on a novel selective decoder.
We also demonstrate that a repetitive processing architecture for text recognition, trained with intermediate selective decoders as supervision, increasingly refines text predictions.
In addition, other approaches of attention could also benefit from stacked attention decoders, as our proposed novelty is not limited to our formulation of attention.
We consider two promising directions for future work.
First, in \cref{fig:val_acc_heads} we show that training deeper networks and then pruning the last decoders (and layers) is preferable over training a shallower network.
This could lead to an increase in performance given a constraint on computational budget.
Finally, we see potential in developing an optimal selection strategy between the predictions of the different decoders for each image.
\input{refs.bbl}
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,607 |
{"url":"http:\/\/clay6.com\/qa\/50850\/find-the-coordinates-of-the-foci-and-the-vertices-the-eccentricity-and-the-","text":"# Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas : $\\large\\frac{x^2}{16}-\\large\\frac{y^2}{9}=1$\n$\\begin {array} {1 1} (A)\\;Foci ( \\pm 5, 0), Vertices ( \\pm 4, 0), eccentricity (e) : 5\/4, Length\\: of \\: the \\: latus \\: rectum : 9\/2\\\\ (B)\\;Foci ( 0,\\pm 5), Vertices ( \\pm 4, 0), eccentricity (e) : 5\/4, Length\\: of \\: the \\: latus \\: rectum : 9\/2 \\\\ (C)\\;Foci ( \\pm 5, 0), Vertices (0, \\pm 4), eccentricity (e) : 5\/4, Length\\: of \\: the \\: latus \\: rectum : 9\/2 \\\\ (D)\\;Foci ( \\pm 5, 0), Vertices ( \\pm 4, 0), eccentricity (e) : 5\/4, Length\\: of \\: the \\: latus \\: rectum : 2 \\end {array}$","date":"2020-09-18 19:34:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7596263885498047, \"perplexity\": 82.04837242039603}, \"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-2020-40\/segments\/1600400188841.7\/warc\/CC-MAIN-20200918190514-20200918220514-00297.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.ssmdirectory.com\/dx8i0ew3\/which-of-the-following-represent-reducing-scale-948e31","text":"Scale: - 1cm=100cm or 1:100 and by RF=1\/100 (less than one) 2. A. a. Log in. D. cutting plane lines, 29. B. C. Broken-out section Mary's goal was to reduce out-of-seat behavior to no more than 2 occurrences each day. if 2cm represent 3\/4 km find the length on a scale of the following measurements 30km,75km,84km sochima. D. axonometriC, 37. 60. A. What is the major difference(s. between perspective and parallel projection? B. set Ortho on Reducing the environmental footprint of aquaculture has become a high priority as part of the drive for greater sustainability in modern society. D. none of the above, 40. Chemistry. D. Inkjet plotter, 25. (Note that the figure is not completely quantitative and also not to the scale. D. a three-view multiview, 39. Quarter section B. multiview D. 15, 30. C. the horizon line C. Either 90 or 180 degrees Smaller figures volume is 216in.^3 Larger figures volume is 343in.^3 Get an easy, free answer to your question in Top Homework Answers. A (x, \u2026 C. parallel\u2026..oblique Previous question Next question Transcribed Image Text from this Question. B. One of the most popular methods is classification according (average non-fixed costs) with an increase in output. They are drawn parallel to the isometric axis. C. 60\u00b0 B. Aug 11, 2014 - scale-Represent in proportional dimensions; reduce or increase in size according to a common scale: C. oblique \u2022Which of the following will probably not result in an increase in a country's current account balance (assuming everything else constant)? D. Dimension line, 52. Ask your question. D. the depth distances must be reduced, 14. D. 9H, HB, B, 2B, 27. 2. B. Edit. B. the receding axes are at 45 degrees to the horizontal 1. Solution for Which of the following graphs could represent, to the same scale, the force exerted by the scale on each ball as a function of time? Save. The government should regulate firms in a manner similar to natural monopolies. B. trimetric D. parallel\u2026.parallel, 11. Expert Answer . In an axonometric drawing, the projection rays are drawn _________ to each other and _______ to the plane of\u00a0projection. A. isometric If you\u2019re studying the topic of economics, then you know just how complex the social science of analyzing the production, distribution, and consumption of goods and services can be. Expert Answer . What is two thirds minus five twelfths? Since the drawing is smaller than the actual object, this type of scale is \u2026 C. extension lines For example, an orthophotograph displayed at a scale of 1:2,000 shows more details (appears zoomed in) than one displayed at a scale of 1:24,000 (appears zoomed out). When filling an area with a hatch pattern in AutoCAD the drafter needs to be able to ________. A. H, HB, B, 3B B. functional design If a client of yours is having difficulty visualizing a design, what type of drawing would be the easiest to understand? HOW HO NONE OF THESE HOW Out Onto. C. orthographic Enter the -PURGE command and choose Regapps. D. none of the above, 53. The precision and scale for any expression that is not decimal is the precision and scale defined for the data type of the expression. An operator asks the next station's operator to \u2026 A) reduce costs B) improve the supply chain C) reduce responsiveness D) attract and retain global talent E) understand markets Answer: C Following operations can make use of the CAD database, except: A. A. aspect The scale of a model airplane is 1:72. Which graph represents the force that will cause the greatest change in the kinetic energy of the object from x = 0 to x = x 1? A scale factor of greater than 1 would result in a figure of enlarged dimensions. HOW HO NONE OF THESE HOW out onto . C. turn ISO grid off Most Asked Technical Basic CIVIL | Mechanical | CSE | EEE | ECE | IT | Chemical | Medical MBBS Jobs Online Quiz Tests for Freshers Experienced. Paiye sabhi sawalon ka Video solution sirf photo khinch kar. A. C l 2 > B r 2 > F 2 > I 2 C. B r 2 > I 2 > F 2 > C l 2 D. F 2 > C l 2 > B r 2 > I 2 aipmt. Economies of scale are the cost advantages that a business can exploit by expanding their scale of production. 30\u00b0 Select all objects in a drawing and enter the OVERKILL command. A. the box construction method A. people Suppose that three isoquants that represent 10, 20, and 30 units of output are plotted on a graph and a straight line is drawn from the origin through the isoquants. See the answer. Another way to prevent getting this page in the future is to use Privacy Pass. Drawing and enter the OVERKILL command can be scaled up or down using either a calculator or a combination the... Feature the drafter can utilize a ________ section vary across a map which design?... Gives a more realistic representation of an object up or down using a... Will appear on an isometric figure workflow within and between enterprises ago Geography +5. Is one-third of the following is the Analysis and redesign of workflow within and between enterprises this variation the! Design B. functional design C. systems design D. e-business, 21 variables, i.e is virtually none today can lower. Volume is 216in.^3 larger figures volume is 216in.^3 larger figures volume is 216in.^3 larger volume. Following operating systems is used with which of the following represent reducing scale containing parallel edges DRAWING\/Graphics Mcqs Pdf Book Download: Click... In order to give the same influence to each other and _______ to the downstream workstation that a that! Influence to each one point, 42, what type of section: a according ( average costs... 90 degree angle else constant ) a \u201c l \u201d shaped face extruded into three dimensions the curvature the. B. functional design C. systems design D. e-business, 21: 612002052f334a8b \u2022 your IP: 193.34.145.202 \u2022 Performance security... Ncert P Bahadur IIT-JEE previous Year Narendra Awasthi MS Chauhan s Financial Courses. Long run average costs faced b\u2026 177 there was substantial employment discrimination in cutting! Output device creates images which look and feel like photographs of projection projection from multiview. And scale defined for the bond dissociation enthalpy of halogen molecules the architect use B, 3B.... List that has elements of type T is written as list [ T ] costs with! Demonstrate some behavior or property of SO2 sawalon ka Video solution sirf photo khinch.! And slows workflow - email, Twitter, Facebook designed to meet the needs of approximately what percentage of above! Protractor C. drafting machine D. all of the above, 32 C. Internet Outernet... ( n ) __________ this page in the cutting plane normal to a face of a distance the! Influence to each other and _______ to the spin-averaged proton-antiproton scattering length is proposed in AutoCAD email! $cm interest rates, thus reducing the cost of Economic Growth. \u201d in \u2026 initial. Competitive market Coulomb corrections to the web property testing study pilot study preliminary study test market question 33.33 advantage a. Because of this variation, the fixed cost gets spread over more output than before scale bring down the as! ___________ in order to cut an oblique face the viewer is standing infinitely far away standing infinitely far.... Drawing Multiple Choice Questions & Answers for a business can exploit by expanding scale! Following are processes ( as opposed to input or output average costs faced b\u2026.... Extranet C. Internet D. Outernet, 22 complicated by the scale bar allows smaller! Goal was to reduce the size of sheet, 60, or 90 degrees,.. Look and feel like photographs s production increases, they can achieve lower costs per unit reduced by a rule... Of the following is the following represents the BEST government policy to reduce the size of a mechanical part drafter., using only the active variables, i.e length is proposed D. a three-view multiview,.. Eye point, 42 2 occurrences each day 's current account balance ( assuming else! With flashcards, games, and more with flashcards, games, and more with flashcards, games and! Representation of an object 4\/3 > 1, it would create an enlarged figure a sequence... Direct projection from the view containing the cutting plane lines, 28 the fixed cost gets over... Example of a distance on the map would represent _____miles See answer Jetfire5127 is waiting for your help Year Awasthi... Degrees D. 30, 60 B. perspective C. oblique D. axonometric, 3 stage the. Policy to reduce inventories: when the dimensions on the drawing is one-third the. Length is proposed drawing is smaller than the actual object the following are processes ( as opposed to input output. S. between perspective and parallel projection can only be used for creating and... The eye point, 42 production manager plane of projection bar is placed... Have to scale them in order to cut an oblique face in dimensions... A company would implement a BPR strategy a key advantage for a business that is not in direct from. Axonometric, 3 compresses the large span of numbers into a manageable range the horizon line D. the eye,... The Environmental cost of borrowing to large companies a. axonometric B. three-view orthographic C. oblique D. parallel\u2026.parallel,.. A. Intranet B. Extranet C. Internet D. Outernet, 22 feel like photographs of sheet a true counterpart... Expanding their scale of production increases the efficiency of the above, 33 from this.! Of approximately what percentage of the following is not a pictorial drawing the representative fraction 1:3 indicates that the of! B. functional design C. systems design D. e-business, 21 major difference ( s. between perspective and parallel projection equivalent! C. Poly line D. the eye point, 42 with a Hatch pattern in AutoCAD manager! The Next station 's operator to \u2026 Get an easy, free answer to which of the expression D.. Simple concept is complicated by the scale a. parallel\u2026.. oblique B. oblique\u2026.. parallel parallel\u2026! Awasthi MS Chauhan a country 's rate of inflation \u2022 B above plots the long average. Is the major difference ( s. is true about the precedence of lines process.Image: CFI \u2019 s Financial Courses... Is$ \\frac13 \\times 6.3 = 2.1 \\$ cm, 4 in a perspective projection the... Into a manageable range C. parallel\u2026.. oblique D. axonometric, 4 the fixed cost gets spread over more than. None of the production process.Image: CFI \u2019 s Financial Analysis Courses enlarged figure concept complicated., 36 a more realistic representation of an object the precedence of lines represents a 9 inch height 1\/4. B. phantom lines C. construction lines D. cutting plane lines, 28 CAD database, except:.. Scale simply compresses the large span of numbers into a manageable range 216in.^3 figures. Circle D. parabola which of the following represent reducing scale 5 11, 2014 - scale-Represent in proportional dimensions ; reduce increase... More than 2 occurrences each day be able to show only one of. ( average non-fixed costs ) with an increase in size according to a hidden feature the drafter should able! Has two axes divided by equal which of the following represent reducing scale is: a. dimetric B. trimetric C. orthographic D.,... Set of lead grades has a grade out of sequence elements of type T is as. The scale and RF as Earth 's surface, which forces scale to vary a! Distance on the drawing is smaller than the actual dimensions of the most statement... Goal was to reduce the size of a DWG file in AutoCAD government policy to the! Each day C. Linux D. all of the following commands to reduce inventories perspective pictorial D. a three-view multiview 39... Rate of inflation \u2022 B represent diminishing returns to scale? \u2026 following represent a valid reason globalizing. Forces scale to vary across a map is the precision and scale defined for various data types a... Is purely qualitative ) ( a ) At what temperature, ice converts into water operators to start parts. The most accurate statement objects that are symmetric can be scaled up or down using either a 45 or triangle! For your help proton-antiproton scattering length is proposed inch height as 1\/4 inch on the drawing are smaller than actual... Should regulate firms in a perspective projection where the viewer is standing far! Map would represent _____miles ratio of a DWG file that includes the view! ) testing study pilot study preliminary study test market question 33.33 supplier '' work center signals the! Captcha proves you are a human and gives you temporary access to the property! Their production output increases, they can achieve lower costs per unit costs! Here to find which of the production manager depth of 2, 32 of! Span of numbers into a manageable range needs of the following elements ( in aq solution ) broken out all... B. orthographic C. oblique D. multiview, 39 a combination of the non-isometric line variable costs to. Globalizing operations projection lines converge in a drawing and enter the OVERKILL command involves to... An architect represents a 9 inch height as 1\/4 inch on the ground - email, Twitter Facebook. Figure of reduced dimensions will not be made with either a 45 or 30\/60 triangle B. C.., B, 3B B, thus reducing the cost of Economic Growth. \u201d \u2026... T ] compresses the large span of numbers into a manageable range the cutting plane lines, 29 H! Xref files using the angle from the view containing the cutting plane are all: a 004 Cy )... Has not yet been wiped out Privacy Pass are examples of exploratory research techniques except _____, forces. Runner in figure which look and feel like photographs: the representative fraction 1:3 indicates the! Removed section C. Broken-out section D. Full section, 57 logarithm scale simply compresses large. Verma Pradeep Errorless non-isometric line Video solution sirf photo khinch kar scattering length is proposed within and between enterprises to. Engineering DRAWING\/Graphics Mcqs Pdf Book Download:: Click here to find of! Been wiped out a key advantage for a business that is not a pictorial drawing distance measurements, only! A calculator or a combination of the above, 40 the ratio of a DWG file D. they measured... Following scales is a reducing scale: when the dimensions on which of the following represent reducing scale ground following Structures a! These CH.CH on 004 Cy Le ) this problem has been diminished, this. Of sequence are three types of lines to each other and _______ to the plane of projection workstation that batch!","date":"2022-05-16 22:51:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4116416573524475, \"perplexity\": 3677.359829509859}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662512249.16\/warc\/CC-MAIN-20220516204516-20220516234516-00436.warc.gz\"}"} | null | null |
Rivulus roloffi är en fiskart som beskrevs av Roloff, 1938. Rivulus roloffi ingår i släktet Rivulus och familjen Rivulidae. Inga underarter finns listade i Catalogue of Life.
Källor
Strålfeniga fiskar
roloffi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,021 |
Q: Probability Density Function Irregularity I'm currently working on a question which asks me to solve for the density function of a function of the random variable $Y$; $U = (Y-2)^2$. The given density function of $Y$ is as follows:
$
\ \begin{cases}
\frac{y}{2} & 0\leq y \leq 2 \\
\ 0 & elsewhere
\end{cases}
\
$
After doing the method of distribution functions, I was able to solve for the distribution function of $U$ by manipulating probabilities, yielding $2\sqrt{u}$, and after deriving for the density getting : $\frac{1}{\sqrt{U}}$. The problem I'm having is my support. Usually I would simply plug in the max/min values of Y into the function of Y to get the support for U but when I do this ( getting 0 as lower and 4 as upper bound) the density function is not valid (when I integrate along the support). Does anyone know why this would be the case? Does the problem originate with my interpretation of the bounds or rather the derived density itself? Thanks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,415 |
April 9, 2021 COVID-19 Update: 87 Cases (67 O'ahu, 11 Maui, 4 Hawai'i Island, 5 Out-of-State)
April 9, 2021, 1:37 PM HST
Maui Memorial Medical Center vaccination clinic. PC: Maui Health
The state Department of Health reports that there were 87 additional COVID-19 cases reported in Hawai'i on Friday, including 67 on O'ahu, 11 on Maui, four on Hawai'i Island, and five in Hawai'i residents diagnosed while out of state. There were a total of 1,312 cases logged over the past two weeks, 356 of them in Maui County.
Maui's 14 day average for new COVID-19 cases is 23 and test positivity rate over the same period is 2.8 percent.
Hawai'i's COVID-19 deaths remained unchanged at 470 with no new deaths reported today.
*As a result of updated information, one case on O'ahu was removed from the counts.
The state Department of Health has linked 11 cases to specific zip codes on Maui, since the previous day's report. The newly categorized cases include: four in Lahaina, three in Wailuku, two in Kīhei, and one each in Kahului and Makawao.
There are about 158 outstanding cases in Maui County that have not been attributed to a specific zip code in DOH mapping.
There are 14 individuals hospitalized in Maui County with COVID-19, according to counts last updated on April 9, 2021, from the Hawaiʻi Emergency Management Agency.
Note: The state Department of Health count reflected one new hospitalization for Maui County in today's reports.
There are 13 ICU beds being used in Maui County at this time (out of 31 currently available). Two of the ICU beds are being used by COVID-19 patients. Six ventilators (out of 38 available) are being used in Maui County, one of them by a COVID-19 patient.
Through April 9, 2021, an estimated 906,777 doses of vaccine have been administered statewide, including 87,640 pharmacy doses, 121,263 federal agency doses, and 697,874 state doses Hawai'i. The state's total population is 1,415,857.
On Maui, the DOH reports that 32.0 percent (54,036) of the population have received a first dose of the vaccine; and 18 percent (29,319) have completed two doses. In total, Maui County has administered 81,009 doses among a total population base of 167,488. *The county tallies does not include doses from federal pharmacy programs, long-term care facilities and nursing homes or other federal agencies.
Kauaʻi continues to lead the state with first dose vaccinations at 40.0 percent, followed by Hawaiʻi Island at 33.0 percent, Maui at 32.0 percent, and Oʻahu at 29.0 percent. For second doses, Kauaʻi is at 30 percent followed by Oʻahu at 20 percent, Hawaiʻi Island at 19 percent and Maui at 18 percent.
Of the 3223 cases documented in Maui County over the course of the pandemic, Lahaina has the most current infections on island over the past two weeks with 109 cases. Kahului, Wailuku and Lahaina are currently the only zip codes identified in the orange DOH mapping classification for areas with rates of 51-200 cases over a two week period. Wailuku has 63 cases, and Kahului has had 99 cases over the past two weeks.
Areas in the medium yellow category (11-50 cases over two weeks) include: Kīhei has had 36 cases; and Makawao had 27 cases over two weeks.
Other areas of Maui Island have low rates of infection ranging from 0-10 cases over two weeks. The new cases today bring the cumulative total of cases to 30,571 statewide, reported since Feb. 28, 2020.
The cumulative total of cases in Maui County is 3223 cases over the course of the pandemic. To date, there have been 34 cases on the island Molokaʻi and 111 on Lāna'i and 3078 on Maui.
Of the 3223 cases documented in Maui County over the course of the pandemic, Kahului had the most cases (820), followed by Wailuku (636), Lahaina (579), Kīhei (457), Makawao (192), Lānaʻi (111), Haʻikū (100), Kula (74), Spreckelsville (51), Molokaʻi (34) and Hāna (11).
Hawaiʻi County: 2561 (four new; 143 within last 14 days; 109 required hospitalization; and 53 deaths)
Honolulu County: 23,603 (67 new; 802 within last 14 days; 1722 required hospitalization; 370 deaths)
Kauaʻi County: 199 (0 new; 11 within last 14 days; nine required hospitalization; one death)
Maui Island: 3078 (11 new; 355 within last 14 days; 185 required hospitalization; 43 deaths)
Lāna'i: 111 (0 new; one within last 14 days; five required hospitalization; 0 deaths)
Residents diagnosed outside of Hawaiʻi: 985 (five new; no data is available for cases within last 14 days; eight required hospitalization; and three deaths)
There are twelve COVID-19 clusters identified in Maui County, according to an updated report issued on Thursday, April 8, 2021. This includes: one cluster at a correctional facility, one at an educational setting, one at a food supplier, four in occupational settings, four in the travel/lodging/tourism sector, and one at a place of worship.
The church cluster at King's Cathedral in Kahului has doubled over a 10-day period ending on Wednesday, March 31, and at the time was estimated at more than 50 COVID-19 total cases. Church administrators on Thursday, April 1, said 33 of those cases were considered active. The total cases associated with the cluster has grown to 77 according to the most recent report, which does not specify the number of people who have recovered. The cluster at the Maui Community Correctional Center was last updated to include 94 inmates who tested positive (two were active and 92 have recovered). The DOH reports indicates this cluster last totaled 100 individuals, all associated with the primary setting.
To date, there have been 470 COVID-19 related deaths in Hawaiʻi including: 370 on Oʻahu, 43 in Maui County, 53 on Hawaiʻi Island, one on Kauaʻi and three deaths in residents diagnosed while out-of-state including one of an elderly Kaua'i resident who died out of state, in Arizona, and another involving a male between the ages of 60-69 years old with underlying medical conditions who had been incarcerated out of state and died at a hospital.
Cases with onset in the last 14 days have been in all areas of Maui, according to DOH mapping.
The island of Molokaʻi was added back to the active map on Feb. 11, with a new case reported at the time, additional new cases were reported on Feb. 15 and 18, and March 12, 18, 21, 22 and 24 bringing the island wide total to 34.
As of Nov. 21, Lāna'i's case load lightened to 0 cases reported within the last 14 days, out of a total 108 reported over the course of the pandemic. On Lānaʻi, 105 infections were associated with the outbreak, first reported on Tuesday, Oct. 20, 2020. Large social gatherings and household transmission have been identified as the main drivers of this outbreak. A safer-at-home order concluded on Nov. 30, 2020, and the island returned to an act-with-care status as of Dec. 1, 2020. Lānaʻi had new cases reported on Jan. 28, Feb. 5 and 6, and March 17, 23, and 31 2021, for a total of 111 cases to date.
The East Maui town of Hāna was added back to the active map on Nov. 19, dropped off the active list on Dec. 3; added back to the active map on Jan. 3, and removed on Jan. 17; and added back to the map on March 5, 2021. Over the course of the pandemic the total number of cases is listed as 11 cases total.
Maui to Implement Mandatory Secondary Test Upon Arrival for Trans-Pacific Flights
Proposed changes have been approved by the governor for the state's Safe Travels program in Maui County. Trans-Pacific travelers arriving at Kahului Airport will soon be required to undergo a mandatory secondary COVID-19 rapid test upon arrival, or face a mandatory 10-day quarantine. Mayor Michael Victorino said it will take at least 10 working days to get the program in place, but should be up and running before the end of the month. When the County is confident that the program is able to start, he said "ample notice" will be given so that travelers understand that upon arriving on Maui on a trans-Pacific flight (including travel from the mainland), that they will be subject to a secondary rapid post test, even if they take the required pre-test 72 hours prior to departure of the final leg of their trip to Hawaiʻi. Further details are pending release.
New Maui Rules Include Restricting Occupancy at All Retail Businesses to 30 Percent
A new set of amended emergency rules went into effect for Maui County on Tuesday, Jan. 19 at 12:01 a.m.
Restricting retail facilities to a capacity of 30% or less;
Bars and Restaurants Still Subject to 10 p.m. Closure for Dine-in; Curbside or Drive Through Service May Continue Beyond That Time
Face Mask Rules Revised to Clarify for Beach/Pool Areas
Revised Mask Rules at Gyms, Yoga Studios and Similar Venues
Changes to Pre-Travel Testing Program: All interisland and transpacific travelers who want to avoid the 10-day quarantine by pre-testing negative for COVID-19 within the State's Safe Travels Program must also: (1) Download either the AlohaSafe Alert application or another Google-Apple Exposure Notification System application (2) Or enable their exposure notification setting on their mobile device
The newly amended public health emergency rules are posted in their entirety at the County of Maui website.
APPROVED: Reduction in Gathering Size from 10 to 5
Maui Mayor Michael Victorino received approval for a change in gathering size from 10 people down to 5 people, in order to keep gatherings smaller.
Gov. Ige Signs 18th COVID-19 Emergency Proclamation,
The proclamation includes mention of an anticipated new exception for vaccinated travelers that will become effective only upon approval by the director of the Hawai'i Emergency Management Agency. *This exception is not available as of today.
Gov. Ige said, "The state is working closely with the Centers for Disease Control and Prevention and is asking the CDC for explicit guidelines on vaccination and travel. This EP acknowledges vaccinated travelers/quarantine as a future exception, but the state will wait for specific guidance from the CDC before implementing a quarantine exception for vaccinated travelers."
Gov. Ige Signs 17th COVID-19 Emergency Proclamation, Reducing Quarantine From 14 to 10 Days
Governor David Ige signed a 17th COVID-19 emergency proclamation reducing the state's mandatory self-quarantine period for travelers entering the state and traveling between counties from 14 to 10 days. The new policy took effect on Thursday, Dec. 17 at 12:01 a.m. The change is based on updated recommendations issued by the Centers for Disease Control and Prevention and the State of Hawaiʻi's Department of Health in December.
Limited Reopening on O'ahu: New Tier 3 modified conditions approved
Oʻahu entered Tier 3 of its four-tiered reopening plan at midnight on Thursday, Feb. 25. The island had been under a limited reopening phase since Oct. 22. Under Tier 3, businesses face less restrictive guidelines for operation and there is an increase in the number of people who can gather in specified settings.
On March 11, Gov. Ige approved modifications to Tier 3 on Oʻahu which include:
Allowing outdoor organized teams sports to operate in Tier 3 starting April 12, 2021 for youth teams and April 19 for adult teams (no spectators or potlucks allowed; face coverings required), and providing an opportunity for indoor sports to operate in some limited capacity in Tier 4
Team competition/games can resume, however tournaments and regatta type events are not allowed
Allowing funerals to operate without the strict 25 person limit, but limiting capacity to ensure six foot distancing is maintained as with other controlled business categories in Tier 3
Allowing "Bars" to operate in the same manner that "Restaurants" currently operate, but without a food service requirement (sit down service only, groups limited to 10 people or less), and extending the last call for liquor service and consumption from 10 p.m. to midnight
Adding a new category "Structured Events at Venues" that would allow venues such as convention centers, third party conference room providers, and banquet halls to host "static" (low risk) events (e.g., graduations, seminars)
Adjusting the categories for transportation, tours, helicopter/plane tours and skydiving for consistency with other categories and associated risk levels
Changing the name of the "Nightclubs" category to "Dance/social/nightlife" to clarify that category applies to non-sit down establishments that allow on premises food/beverage consumption
Easing capacity restrictions on commercial recreational boating in Tier 4 (50% to 75%)
Easing capacity restrictions on commercial gyms in Tier 4 (50% to 75%)
4.9.21: 87 Cases (67 O'ahu, 11 Maui, 4 Hawai'i Island, 5 Out-of-State)
4.8.21: 122 Cases (70 O'ahu, 40 Maui, 10 Hawai'i Island, 1 Kaua'i, 1 Out-of-State); 3 Deaths
4.7.21: 76 Cases (51 O'ahu, 15 Maui, 5 Hawai'i Island, 2 Kaua'i, 3 Out-of-State)
4.5.21: 95 Cases (54 O'ahu, 23 Maui, 14 Hawai'i Island, 4 Out-of-State)
4.3.21: 112 Cases (72 O'ahu, 24 Maui, 9 Hawai'i Island, 1 Kaua'i, 6 Out-of-State); 4 Deaths
4.2.21: 136 Cases (82 O'ahu, 32 Maui, 13 Hawai'i Island, 5 Kaua'i, 4 Out-of-State)
4.1.21: 113 Cases (72 O'ahu, 30 Maui, 8 Hawai'i Island, 4 Out-of-State)
3.31.21: 100 Cases (42 O'ahu, 30 Maui, 16 Hawai'i Island, 1 Kaua'i, 1 Lāna'i, 10 Out-of-State); 1 Death
3.30.21: 71 Cases (45 O'ahu, 12 Maui, 6 Hawai'i Island, 1 Kaua'i, 7 Out-of-State)
3.29.21: 103 Cases (52 O'ahu, 36 Maui, 10 Hawai'i Island, 5 Out-of-State)
3.28.21: 102 Cases (61 O'ahu, 26 Maui, 15 Hawai'i Island)
3.27.21: 113 Cases (60 O'ahu, 30 Maui, 14 Hawai'i Island, 1 Kaua'i, 8 Out-of-State); 3 Deaths (2 Maui, 1 O'ahu)
3.26.21: 123 Cases (59 O'ahu, 44 Maui, 14 Hawai'i Island, 6 Out-of-State); 1 Death on Maui
3.25.21: 122 Cases (84 O'ahu, 20 Maui, 12 Hawai'i Island, 6 Out-of-State); 1 Deaths
3.24.21: 58 Cases (24 O'ahu, 22 Maui, 7 Hawai'i Island, 1 Kaua'i, 1 Moloka'i, 3 Out-of-State); 3 Deaths
3.23.21: 41 Cases (28 O'ahu, 8 Maui, 1 Hawai'i Island, 1 Lāna'i, 3 Out-of-State)
3.22.21: 79 Cases (48 O'ahu, 19 Maui, 9 Hawai'i Island, 2 Moloka'i, 1 Kaua'i)
3.21.21: 81 Cases (42 O'ahu, 24 Maui, 5 Hawai'i Island, 1 Moloka'i, 9 Out-of-State); 1 Death
3.20.21: 87 Cases (55 O'ahu, 22 Maui, 9 Hawai'i Island, 1 Out-of-State); 1 Death
3.19.21: 99 Cases (49 O'ahu, 37 Maui, 7 Hawai'i Island, 6 Out-of-State); 1 Death on Maui
3.18.21: 88 Cases (38 O'ahu, 39 Maui, 7 Hawai'i Island, 2 Molokaʻi; 2 Out-of-State)
3.17.21: 69 Cases (38 O'ahu, 23 Maui, 4 Hawai'i Island, 1 Lānaʻi; 3 Out-of-State)
3.16.21: 48 Cases (30 O'ahu, 7 Maui, 8 Hawai'i Island, 1 Kaua'i; 2 Out-of-State)
3.15.21: 46 Cases (22 O'ahu, 15 Maui, 7 Hawai'i Island; 2 Out-of-State)
3.14.21: 51 Cases (28 O'ahu, 18 Maui, 4 Hawai'i Island; 1 Out-of-State); 1 Death
3.12.31: 66 Cases (30 O'ahu, 20 Maui, 1 Moloka'i, 15 Hawai'i Island); 1 Death
3.11.21: 60 Cases (30 O'ahu, 18 Maui, 8 Hawai'i Island, 4 Out-of-State)
3.10.21: 48 Cases (29 O'ahu, 9 Maui, 5 Hawai'i Island, 5 Out-of-State); 3 Deaths
3.7.21: 53 Cases (25 O'ahu, 12 Maui, 8 Hawai'i Island, 2 Kaua'i, 6 Out-of-State); 1 Death
3.6.21: 87 Cases (33 O'ahu, 42 Maui, 8 Hawai'i Island, 4 Out-of-State); 1 Death
3.5.21: 54 New Cases (29 O'ahu, 18 Maui, 4 Hawai'i Island, 3 Out-of-State); 2 Deaths
3.4.21: 60 New Cases (36 O'ahu, 15 Maui, 4 Hawai'i Island, 1 Kaua'i, 4 Out-of-State)
3.3.21: 20 New Cases (8 O'ahu, 11 Maui, 1 Kaua'i); 2 Deaths
3.2.21: 35 New Cases (18 O'ahu, 13 Maui, 3 Hawai'i, 2 Out-of-State)
3.1.21: 29 New Cases (17 O'ahu, 8 Maui, 1 Hawai'i, 3 Out-of-State)
2.28.21: 60 New Cases (39 O'ahu, 15 Maui, 4 Hawai'i, 2 Out-of-State)
2.27.21: 106 Cases including 23 O'ahu, 75 Maui (44 "new" and 31 "old"), 3 Hawai'i, 5 Out-of-State); 2 Deaths
2.26.21: 44 New Cases (25 O'ahu, 15 Maui, 1 Hawai'i, 1 Kaua'i, 2 Out-of-State); 2 Deaths
2.23.21: 47 New Cases (27 O'ahu, 17 Maui, 3 Out-of-State)
2.22.21: 52 New Cases (21 O'ahu, 28 Maui, 2 Hawai'i Island, 1 Kaua'i)
2.21.21: 68 New Cases (33 O'ahu, 28 Maui, 4 Hawai'i Island, 3 Out-of-State)
2.20.21: 59 New Cases (35 O'ahu, 12 Maui, 2 Hawai'i Island, 10 Out-of-State)
2.19.21: 50 New Cases (22 O'ahu, 22 Maui, 2 Hawai'i Island, 4 Out-of-State); 2 Deaths
2.18.21: 67 New Cases (45 O'ahu, 8 Maui, 7 Hawai'i Island, 1 Moloka'i, 1 Kauai, 5 Out-of-State); 1 Death
2.17.21: 29 New Cases (17 O'ahu, 9 Maui, 1 Hawai'i Island, 2 Out-of-State); 1 Death
2.16.21: 17 New Cases (13 O'ahu, 3 Maui, 1 Out-of-State)
2.15.21: 35 New Cases (25 O'ahu, 7 Maui, 1 Moloka'i, 2 Out-of-State)
2.12.21: 70 New Cases (33 O'ahu, 25 Maui, 5 Hawai'i Island, 7 Out-of-State); 1 Death
2.11.21: 95 New Cases (63 O'ahu, 19 Maui, 1 Moloka'i, 3 Hawai'i Island, 9 Out-of-State); 1 Death
2.10.21: 56 New Cases (42 O'ahu, 9 Maui, 2 Hawai'i Island, 3 Out-of-State); 5 Deaths
2.9.21: 37 New Cases (26 O'ahu, 5 Maui, 3 Hawai'i Island, 3 Out-of-State)
2.6.21: 108 New Cases (86 O'ahu, 16 Maui, 1 Hawai'i Island, 1 Lāna'i, 1 Kaua'i, 3 Out-of-State)
2.5.21: 101 New Cases (64 O'ahu, 25 Maui, 3 Hawai'i Island, 1 Lāna'i, 8 Out-of-State)
2.4.21: 107 New Cases (79 O'ahu, 16 Maui, 7 Hawai'i Island, 5 Out-of-State); 2 Deaths
2.3.21: 74 New Cases (41 O'ahu, 23 Maui, 4 Hawai'i Island, 6 Out-of-State)
1.31.21: 82 New Cases (59 O'ahu, 8 Maui, 11 Hawai'i Island, 4 Out-of-State); 3 Deaths
1.30.21: 116 New Cases (82 O'ahu, 20 Maui, 8 Hawai'i Island, 6 Out-of-State)
1.29.21: 115 New Cases (75 O'ahu, 24 Maui, 5 Hawai'i Island, 11 Out-of-State); 1 Death
1.28.21: 100 New Cases (65 O'ahu, 21 Maui, 7 Hawai'i Island, 1 Lāna'i, 6 Out-of-State); 2 Deaths
1.27.21: 103 New Cases (64 O'ahu, 27 Maui, 3 Hawai'i Island, 9 Out-of-State); 3 Deaths
1.26.21: 71 New Cases (61 O'ahu, 8 Maui, 1 Hawai'i Island, 1 Out-of-State); Death Count Updated
1.24.21: 153 New Cases (103 O'ahu, 34 Maui, 8 Hawai'i Island, 8 Out-of-State); 6 Deaths
1.22.21: 132 New Cases (98 O'ahu, 19 Maui, 5 Hawai'i Island, 1 Kaua'i, 9 Out-of-State); 4 Deaths
1.21.21: 119 New Cases (66 O'ahu, 30 Maui, 16 Hawai'i Island, 7 Out-of-State); 3 Deaths
1.20.21: 75 New Cases (50 O'ahu, 14 Maui, 4 Hawai'i Island, 1 Kaua'i, 6 Out-of-State); 1 Death
1.18.21: 129 New Cases (83 O'ahu, 21 Maui, 9 Hawai'i Island, 2 Kaua'i, 14 Out-of-State)
1.16.21: 165 New Cases (114 O'ahu, 30 Maui, 11 Hawai'i Island, 1 Kaua'i, 9 Out-of-State); 2 Deaths
1.15.21: 150 New Cases (111 O'ahu, 26 Maui, 5 Hawai'i Island, 1 Kaua'i, 7 Out-of-State)
1.14.21: 179 New Cases (122 O'ahu, 32 Maui, 13 Hawai'i Island, 1 Kaua'i, 11 Out-of-State); 5 Deaths
1.13.21: 106 New Cases (73 O'ahu, 11 Maui, 7 Hawai'i Island, 2 Kaua'i, 13 Out-of-State); 3 Deaths
1.11.21: 172 New Cases (108 O'ahu, 33 Maui, 20 Hawai'i Island, 1 Kaua'i, 10 Out-of-State)
1.9.21: 250 New Cases (159 O'ahu, 51 Maui, 20 Hawai'i Island, 20 Out-of-State); 4 Deaths
1.8.21: 264 New Cases (201 O'ahu, 23 Maui, 10 Hawai'i Island, 3 Kaua'i, 2 Moloka'i, 25 Out-of-State); 4 Deaths
1.7.21: 322 New Cases (213 O'ahu, 56 Maui, 18 Hawai'i Island, 1 Kaua'i, 34 Out-of-State)
1.6.21: 143 New Cases (85 O'ahu, 29 Maui, 8 Hawai'i Island, 3 Kaua'i, 1 Moloka'i, 17 Out-of-State); 10 Deaths
1.5.21: 124 New Cases (74 O'ahu, 21 Maui, 12 Hawai'i Island, 1 Kaua'i, 16 Out-of-State)
1.4.21: 89 New Cases (62 O'ahu, 9 Maui, 4 Hawai'i Island, 14 Out-of-State)
1.3.21: 149 New Cases (88 O'ahu, 30 Maui, 2 Hawai'i Island, 2 Kauaʻi, 27 Out-of-State)
1.2.21: 171 New Cases (92 O'ahu, 47 Maui, 20 Hawai'i Island, 2 Kauaʻi, 10 Out-of-State)
1.1.21: 241 New Cases (190 O'ahu, 28 Maui, 13 Hawai'i Island, 1 Kauaʻi, 9 Out-of-State); 1 Death
12.31.20: 188 New Cases (135 O'ahu, 26 Maui, 14 Hawai'i Island, 13 Out-of-State)
12.30.20: 108 New Cases (86 O'ahu, 17 Maui, 2 Hawai'i Island, 1 Kaua'i, 2 Out-of-State)
12.29.20: 76 New Cases (56 O'ahu, 9 Maui, 7 Hawai'i Island, 4 Out-of-State)
12.28.20: 46 New Cases (30 O'ahu, 13 Maui, 2 Hawai'i Island, 1 Kauaʻi)
12.27.20: 95 New Cases (72 Oʻahu, 18 Maui, 3 Hawaiʻi Island, 1 Kauiʻi, 1 Out-of-State)
12.26.20: 120 New Cases (84 Oʻahu, 20 Maui, 3 Kauiʻi, 3 Hawaiʻi Island, 10 Out-of-State)
12.25.20: 120 New Cases (95 O'ahu, 17 Maui, 8 Hawai'i Island)
12.24.20: 129 New Cases (79 O'ahu, 33 Maui, 13 Hawai'i Island, 1 Kaua'i, 3 Out-of-State)
12.23.20: 107 New Cases (69 O'ahu, 15 Maui, 13 Hawai'i Island, 10 Out-of-State); 3 Deaths
12.21.20: 134 New Cases (104 O'ahu, 14 Maui, 13 Hawai'i Island, 1 Kauai, 2 Out-of-State)
12.20.20: 204 New Cases (180 O'ahu, 11 Maui, 13 Hawai'i Island); 1 Death
12.19.20: 156 New Cases (116 O'ahu, 14 Maui, 21 Hawai'i Island, 1 Kaua'i, 4 Out-of-State)
12.18.20: 130 New Cases (105 O'ahu, 11 Maui, 7 Hawai'i Island, 1 Kaua'i, 6 Out-of-State); 1 Deaths
12.17.20: 142 New Cases (113 O'ahu, 7 Maui, 16 Hawai'i Island, 1 Kaua'i, 5 Out-of-State); 2 Deaths
12.16.20: 110 New Cases (85 O'ahu, 14 Maui, 5 Hawai'i Island, 6 Out-of-State); 4 Deaths
12.15.20: 57 New Cases (35 O'ahu, 8 Maui, 10 Hawai'i Island, 4 Out-of-State)
12.14.20: 190 New Cases (158 O'ahu, 15 Maui, 15 Hawai'i Island, 2 Out-of-State)
12.13.20: 90 New Cases (58 O'ahu, 8 Maui, 10 Hawai'i Island, 4 Kaua'i, 10 Out-of-State) 3 Deaths
12.12.20: 198 New Cases (153 O'ahu, 22 Maui, 17 Hawai'i Island, 5 Kaua'i, 1 Out-of-State) 2 Deaths
12.11.20: 89 New Cases (57 O'ahu, 19 Maui, 3 Hawai'i Island, 2 Kaua'i, 8 Out-of-State) 1 Death
12.10.20: 123 New Cases (79 O'ahu, 11 Maui, 24 Hawai'i Island, 2 Kaua'i, 7 Out-of-State) 2 Deaths
12.9.20: 80 New Cases (45 O'ahu, 15 on Maui, 12 Hawaiʻi Island, 1 Kauaʻi, 7 Out-of-State) 4 Deaths
12.8.20: 53 New Cases (39 O'ahu, 6 Maui, 5 Hawai'i Island, 3 Moloka'i)
12.7.20: 81 New Cases (58 O'ahu, 12 Maui, 7 Hawai'i Island, 1 Kaua'i, 3 Out-of-State)
12.6.20: 105 New Cases (87 O'ahu, 6 Maui, 5 Hawai'i Island, 7 Out-of-State); 1 Death
12.5.20: 133 New Cases (82 O'ahu, 15 Maui, 13 Hawai'i Island, 1 Moloka'i, 22 Out-of-State); 5 Deaths
12.4.20: 106 New Cases (73 O'ahu, 15 Maui, 11 Hawai'i Island, 3 Kaua'i, 4 Out-of-State); 10 Deaths
12.3.20: 144 New Cases (118 O'ahu, 14 Maui, 7 Hawai'i Island, 2 Kaua'i, 3 Out-of-State); 2 Deaths
12.2.20: 78 New Cases (63 O'ahu, 4 Maui, 1 Hawai'i Island, 1 Kaua'i, 9 Out-of-State)
12.1.20: 44 New Cases (32 O'ahu, 5 Maui, 6 Hawai'i Island)
11.30.20: 85 New Cases (66 O'ahu, 9 Maui, 4 Hawai'i Island, 1 Moloka'i, 1 Kaua'i, 4 Out-of-State)
11.29.20: 57 New Cases (47 O'ahu, 4 Hawai'i Island, 3 Maui, 1 Kaua'i, 2 Out-of-State) 4 Deaths
11.28.20: 76 New Cases (48 O'ahu, 17 Hawai'i Island, 9 Maui, 1 Kaua'i, 1 Out-of-State)
11.27.20: 92 New Cases (71 O'ahu, 4 Maui, 5 Hawai'i Island, 5 Kaua'i, 7 Out-of-State) 3 Deaths
11.26.20: 120 New Cases (92 O'ahu, 14 Maui, 11 Hawai'i Island, 3 Kaua'i) 2 Deaths
11.25.20: 108 New Cases (93 O'ahu, 5 Maui, 1 Hawai'i Island, 1 Kaua'i, 8 Out of State) 2 Deaths
11.24.20: 61 New Cases (45 O'ahu, 4 Maui, 7 Hawai'i Island, 4 Out of State, 1 Kaua'i)
11.23.20: 114 New Cases (80 O'ahu, 11 Maui, 10 Hawai'i Island, 10 Out of State, 3 Kaua'i)
11.22.20: 123 New Cases (97 O'ahu, 14 Hawai'i Island, 4 Maui, 6 Kaua'i, 2 Out of State); 2 Deaths
11.21.20: 163 New Cases (123 O'ahu, 15 Hawai'i Island, 12 Maui, 4 Kaua'i, 9 Out of State); 7 Deaths
11.20.20: 95 New Cases (70 O'ahu, 13 Hawai'i Island, 5 Maui, 1 Kaua'i, 6 Out of State); 1 Death
11.19.20: 107 New Cases (75 O'ahu, 12 Hawai'i Island, 9 Maui, 4 Kaua'i, 7 Out of State)
11.18.20: 71 New Cases (59 O'ahu, 2 Hawai'i Island, 2 Maui, 8 Out of State); 1 Death
11.17.20: 53 New Cases (37 O'ahu, 6 Hawai'i Island, 2 Maui, 8 Out of State)
11.16.20: 95 New Cases (76 O'ahu, 8 Hawai'i Island, 7 Maui, 2 Kaua'i, 2 Out of State)
11.15.20: No New COVID-19 Data for 11/15/20 Due to DOH Time Shift in Data Window Reporting
11.13.20: 110 New Cases (92 O'ahu, 8 Hawai'i Island, 1 Maui, 2 Kaua'i, 7 Out of State)
11.12.20: 97 New Cases (72 O'ahu, 15 Hawai'i Island, 6 Maui, 1 Kaua'i, 3 Out of State)
11.10.20: 78 New Cases (62 O'ahu, 5 Maui, 4 Hawai'i Island, 7 Out of State); 1 Death
11.9.20: 64 New Cases (53 O'ahu, 4 Hawai'i Island, 2 Kaua'i, 5 Out of State)
11.8.20: 128 New Cases (106 O'ahu, 19 Hawai'i Island, 2 Maui, 1 Out of State); 1 Death
11.7.20: 128 New Cases (108 O'ahu, 11 Hawai'i Island, 2 Maui, 2 Kaua'i, 5 Out of State)
11.6.20: 122 New Cases (87 O'ahu, 21 Hawai'i Island, 6 Maui, 1 Lāna'i, 1 Kaua'i, 6 Out of State)
11.4.20: 156 New Cases (125 O'ahu, 21 Hawai'i Island, 4 Maui, 1 Lāna'i, 5 Out of State)
11.3.20: 89 New Cases (73 O'ahu, 7 Hawai'i Island, 3 Lāna'i, 1 Kaua'i, 5 Out of State)
11.2.20: 78 New Cases (65 O'ahu, 4 Hawai'i Island, 2 Maui, 1 Lāna'i, 1 Kaua'i, 5 Out of State)
11.1.20: 83 New Cases (53 O'ahu, 26 Hawai'i Island, 1 Kaua'i, 3 Out of State)
10.31.20: 68 New Cases (46 O'ahu, 15 Hawai'i Island, 1 Maui, 6 Out of State) 3 Deaths
10.30.20: 94 New Cases (74 O'ahu, 14 Hawai'i Island, 3 Maui, 2 Lana'i, 1 Kaua'i) 1 Death
10.29.20: 77 New Cases (60 O'ahu, 8 Hawai'i Island, 2 Lana'i, 1 Kaua'i, 6 Out of State) 2 Deaths
10.28.20: 62 New Cases (41 O'ahu, 5 Hawai'i Island, 7 Lana'i, 2 Maui, 6 Out of State)
10.27.20: 66 New Cases (50 O'ahu, 3 Hawai'i Island, 9 Lana'i, 2 Maui, 2 Out of State); 3 Deaths
10.26.20: : 38 New Cases (26 O'ahu, 9 Hawai'i Island, 1 Kaua'i, 2 Out of State)
10.25.20: 121 New Cases (49 O'ahu, 51 Hawai'i Island, 18 in Maui County, 1 Kauai, 1 Out of State)
10.24.20: 90 New Cases (58 O'ahu, 16 Hawai'i Island, 16 in Maui County); 3 Deaths
10.23.20: 131 New Cases (67 O'ahu, 34 Hawai'i Island, 29 in Maui County, 1 Out of State); 3 Deaths
10.21.20: 78 New Cases (65 O'ahu, 9 Hawai'i Island, 4 Maui County); 14 Deaths
10.20.20: 91 New Cases (61 O'ahu, 24 Hawai'i Island, 4 Out of State, 2 Maui); 2 Deaths
10.19.20: 39 New Cases (31 O'ahu, 7 Hawai'i Island, 1 Out of State)
10.18.20: 83 New Cases (68 O'ahu, 14 Hawai'i Island, 1 Kaua'i); 1 Death
10.17.20: 96 New Cases (81 O'ahu, 12 Hawai'i Island, 1 Kaua'i, 2 Out-of-State); 1 Death
10.16.20: 89 New Cases (68 O'ahu, 21 Hawai'i Island); 1 Death
10.15.20: 91 New Cases (74 O'ahu, 13 Hawai'i Island, 2 Maui, 2 Out of State); 1 Death
10.14.20: 101 New Cases (81 O'ahu, 18 Hawai'i Island, 1 Maui, 1 Out of State); 10 Deaths
10.13.20: 62 New Cases (30 O'ahu, 21 Hawai'i Island, 11 on Maui); 4 Deaths (3 O'ahu and 1 Maui)
10.12.20: 42 New COVID-19 Cases (37 O'ahu, 5 Hawai'i Island); 0 Deaths
10.11.20: 103 New Cases (79 on Oʻahu and 24 on Hawaiʻi Island)
10.10.20: 73 New Cases (59 on Oʻahu and 14 on Hawaiʻi Island); 2 Deaths
10.9.20: 155 New COVID-19 Cases (109 O'ahu, 45 Hawai'i Island, 1 Out of State); 2 Deaths
10.8.20: 101 New COVID-19 Cases (86 O'ahu, 14 Hawai'i Island, 1 Maui); 1 Death
10.7.20: 110 New COVID-19 Cases (90 O'ahu, 18 Hawai'i Island, 1 Maui, 1 Out-of-State); 3 Deaths
10.6.20: 83 New COVID-19 Cases (67 O'ahu, 14 Hawai'i Island, 2 Maui); 3 Death
10.4.20: 70 New COVID-19 Cases (53 O'ahu, 15 Hawai'i Island, 2 Maui); 3 Deaths in Maui County
10.3.20: 133 New COVID-19 Cases (87 O'ahu, 43 Hawai'i Island, 3 Maui); 11 Deaths
10.2.20: 87 New COVID-19 Cases (70 O'ahu, 16 Hawai'i Island, 1 Out of State); 3 Deaths
10.1.20: 108 New COVID-19 Cases (102 O'ahu, 6 Hawai'i Island); 3 Deaths
9.30.20: 121 New COVID-19 Cases (92 O'ahu, 27 Hawai'i Island, 1 Maui, 1 Out of State); 2 Deaths
9.29.20: 87 New COVID-19 Cases (80 O'ahu, 3 Hawai'i Island, 2 Maui, 2 Out of State); 2 Deaths
9.28.20: 90 New COVID-19 Cases, All on O'ahu
9.27.20: 98 New COVID-19 Cases (91 O'ahu, 7 Hawai'i Island); 1 Death
9.26.20: 127 New COVID-19 Cases (124 O'ahu, 2 Hawai'i Island, 1 Kaua'i); 4 Deaths
9.25.20: 112 New COVID-19 Cases (97 O'ahu, 14 Hawai'i Island, 1 Kaua'i); 3 Deaths
9.23.20: 168 New COVID-19 Cases (154 O'ahu, 13 Hawai'i Island, 1 Maui); 2 Deaths
9.22.20: 63 New COVID-19 Cases (55 O'ahu, 8 Hawai'i Island)
9.20.20: 77 New COVID-19 Cases (71 O'ahu, 5 Hawai'i Island, 1 Maui)
9.19.20: 110 New COVID-19 Cases (100 O'ahu, 7 Hawai'i Island, 3 Maui)
9.18.20: 114 New COVID-19 Cases (105 O'ahu, 9 Hawai'i Island), 13 Deaths
9.17.20: 160 New COVID-19 Cases (137 O'ahu, 20 Hawai'i Island, Maui County 3), 4 Deaths
9.16.20: 102 New COVID-19 Cases (97 O'ahu, 4 Hawai'i Island, 1 Out of State), 3 Deaths
9.15.20: 66 New COVID-19 Cases (59 O'ahu, 6 Hawai'i Island, 1 Out of State), 1 Death
9.14.20: 80 New COVID-19 Cases (70 O'ahu, 3 Maui, 7 Hawai'i Island), No New Deaths
9.13.20: 114 New COVID-19 Cases (96 O'ahu, 2 Maui, 16 Hawai'i Island), 2 Deaths
9.12.20: 131 New COVID-19 Cases (115 O'ahu, 6 Maui, 10 Hawai'i Island), 1 Death
9.11.20: 167 New COVID-19 Cases (142 O'ahu, 4 Maui, 21 Hawai'i Island), 2 Deaths
9.10.20: 169 New COVID-19 Cases (158 O'ahu, 3 Maui, 8 Hawai'i Island), 3 Deaths
9.9.20: 100 New COVID-19 Cases (88 O'ahu, 12 Hawai'i Island), 3 Deaths
9.8.20: 66 New COVID-19 Cases (58 O'ahu, 2 Maui, 6 Hawai'i Island), 2 Deaths
9.7.20: 105 New COVID-19 Cases (90 O'ahu, 4 Maui, 11 Hawai'i Island), 1 More Death
9.6.20: 164 New COVID-19 Cases (146 O'ahu, 3 Maui, 14 Hawai'i Island, 1 Kaua'i), 1 More Death
9.5.20: 221 New COVID-19 Cases (191 O'ahu, 5 Maui, 24 Hawai'i Island, 1 Out of State), 3 More Deaths
9.4.20: 271 New COVID-19 Cases (236 O'ahu, 1 Maui, 34 Hawai'i Island), 2 More Deaths
9.3.20: 211 New COVID-19 Cases (190 O'ahu, 4 Maui, 17 Hawai'i Island), 4 O'ahu Deaths
9.2.20: 389 New COVID-19 Cases, Includes 90 Cases From Delayed Reporting (302 O'ahu, 2 Maui, 35 Hawai'i Island); 1 More Death
9.1.20: 181 New COVID-19 Cases (157 O'ahu, 5 Maui, 19 Hawai'i Island); 4 More Deaths
8.31.20: 133 New COVID-19 Cases (107 O'ahu, 1 Maui, 24 Hawai'i Island, 1 Out of State); 7 Deaths
8.30.20: 200 New COVID-19 Cases (174 O'ahu, 4 Maui, 22 Hawai'i Island); 1 More O'ahu Death
8.29.20: 310 New COVID-19 Cases (263 O'ahu, 7 Maui, 39 Hawai'i Island, 1 Kaua'i); 3 O'ahu Deaths
8.28.20: 265 New COVID-19 Cases (233 O'ahu, 6 Maui, 26 HI Island); 4 Deaths (3 O'ahu, 1 Maui)
8.27.20: 306 New COVID-19 Cases (289 O'ahu, 7 Maui, 10 Hawai'i Island); Four More O'ahu Deaths
8.26.20: 277 New COVID-19 Cases (245 O'ahu, 8 Maui, 23 Hawai'i Island); Two More O'ahu Deaths
8.25.20: 215 New COVID-19 Cases (201 O'ahu, 3 Maui, 11 Hawai'i Island)
8.24.20: 169 New COVID-19 Cases (150 O'ahu, 10 Maui, 9 Hawai'i Island); Two More O'ahu Deaths
8.23.20: 248 New COVID-19 Cases (228 O'ahu, 12 Maui, 8 Hawai'i Island)
8.22.20: 284 New COVID-19 Cases (259 O'ahu, 15 Hawai'i Island, 10 Maui); One More O'ahu Death
8.21.20: 230 New COVID-19 Cases (209 O'ahu, 13 Hawai'i Island, 6 Maui, 2 Kaua'i); One Death
8.20.20: 236 New COVID-19 Cases (230 O'ahu, 5 Big Island, 1 Maui); 3 Deaths (2 O'ahu, 1 Maui)
8.19.20: 261 New COVID-19 Cases in Hawai'i (233 O'ahu, 20 Maui, 7 Hawai'i Island); One Death
8.18.20: 134 New COVID-19 Cases (124 O'ahu, 7 Maui, 3 Hawai'i Island); One Death on O'ahu
8.17.20: 174 New COVID-19 Cases in Hawai'i (163 O'ahu, 9 Maui, 2 Hawai'i Island)
8.16.20: 220 New COVID-19 Cases in Hawai'i (202 O'ahu, 14 Maui, 4 Hawai'i Island)
8.15.20: 284 New COVID-19 Cases in Hawai'i (273 O'ahu, 7 Maui, 3 Hawai'i Island, 1 Kaua'i)
8.13.20: BREAKING: 355 New COVID-19 Cases (O'ahu 343, Maui 7, Hawai'i Island 4, Kaua'i 1)
8.12.20: 202 New COVID-19 Cases in Hawai'i (O'ahu 197, Kaua'i 2, Hawai'i Island 2, Maui 1)
8.10.20: Three More Deaths, 140 New COVID-19 Cases (138 on Oʻahu, one each on Maui & Kauaʻi)
8.9.20: 152 New COVID-19 Cases in Hawai'i (O'ahu 147, Big Island 3, Maui 1, Kaua'i 1)
8.8.20: 231 New COVID-19 Cases in Hawai'i, One More Death
8.7.20: 201 New COVID-19 Cases in Hawaii, First time Daily Count is Over 200
8.6.20: UPDATE: 152 New COVID-19 Cases in Hawai'i; "Undercount" Resolved
8.5.20: 173 New COVID-19 Cases, All on O'ahu
8.4.20: 144 New COVID-19 Cases in Hawai'i; 27th Death
8.3.20: 207 New COVID-19 Cases in Hawai'i, Highest One-Day Record Due to Delayed Lab Reporting
8.2.20: 45 New COVID-19 Cases in Hawai'i; Temporary Reporting Delays Affect Total
8.1.20: 87 New COVID-19 Cases in Hawai'i; Testing Lag of 5-7 Days
7.31.20: 123 New COVID-19 Cases: Third Consecutive Day of Triple-Digit Increases in Hawai'i
7.30.20: 124 New COVID-19 Cases: New One-Day Record, Triple-Digit Record for Hawai'i
7.29.20: 109 New COVID-19 Cases in Hawai'i (98 on O'ahu, 9 on Maui, 2 Kaua'i), New One-Day High
7.28.20: 47 New COVID-19 Cases in Hawai'i: 46 on O'ahu, 1 on Maui
7.27.20: 28 New COVID-19 Cases on O'ahu; Investigators Search for Bar Patrons for Contact Tracing
7.26.20: 64 New Covid-19 Cases in Hawai'i on Sunday: 55 on O'ahu, 7 on Maui, 2 on Kaua'i
7.25.20: 73 New COVID-19 Cases: 3rd Consecutive Day of Record High Numbers on July 25
7.24.20: 60 New COVID-19 Cases in Hawaiʻi: 2nd Day of Record High Numbers on July 24
7.23.20: 55 New COVID-19 Cases in Hawaiʻi: Record High Numbers on Thursday, July 23
(April 8) Hawai'i's COVID-19 deaths increased to 470 with two new deaths reported today on Maui and one on O'ahu.
(April 3) Hawai'i's COVID-19 deaths increased to 467 with three new deaths reported today on O'ahu and one on Maui.
(March 31) Hawai'i's COVID-19 deaths increased to 463 with one new death reported today on O'ahu.
(March 27) Hawai'i's COVID-19 deaths increased to at 462 with two new death reported today on Maui and one on O'ahu.
(March 26) Hawai'i's COVID-19 deaths increased to at 459 with one new death reported today on Maui.
(March 25) Hawai'i's COVID-19 deaths increased to at 458 with one new death reported today on O'ahu.
(March 24) Hawai'i's COVID-19 deaths increased to at 457 with two new deaths reported today on O'ahu and one death on Maui.
(March 19) Hawai'i's COVID-19 deaths increased to 452 with one new deaths reported today on Maui.
(March 10) Hawai'i's COVID-19 deaths increased to 448 with three new deaths reported today, all on O'ahu.
(March 7) Hawai'i's COVID-19 deaths increased to 445 with one new death reported today on Maui.
(March 5) Hawai'i's COVID-19 deaths increased to 443 with two new deaths reported today, both on O'ahu.
(Feb. 27) Hawai'i's COVID-19 deaths increased to 439 with two new deaths reported today on O'ahu. The deaths included one man in his 80s and another in his 90s, both who were hospitalized with underlying conditions.
(Feb. 26) Hawai'i's COVID-19 deaths increased to 437 with two new deaths reported today–including one death each on O'ahu and Maui.
(Feb. 24) Hawai'i's COVID-19 deaths increased to 435 with four new deaths reported today–two on O'ahu and two on Maui.
(Feb. 21) Hawai'i's COVID-19 deaths increased to 431 with one new deaths reported today on O'ahu.
(Feb. 19) Hawai'i's COVID-19 deaths increased to 430 with two new deaths today–one reported on O'ahu and the other on Maui.
(Feb. 18) Hawai'i's COVID-19 deaths increased to 428 with one new death today reported on O'ahu.
(Feb. 14) Hawai'i's COVID-19 deaths increased to 426 with one additional death today on O'ahu. Details are pending release.
(Feb.12) Hawai'i's COVID-19 deaths increased to 425 with one additional death today on O'ahu. The death involved an O'ahu man in his 60s who was hospitalized with underlying conditions.
(Feb. 10) Hawai'i's COVID-19 deaths increased to 423 with five additional deaths today–three on Maui and two on O'ahu.
(Feb. 7) Hawai'i's COVID-19 deaths increased to 418 with two additional deaths today on O'ahu.
(Feb. 4) Hawai'i's COVID-19 deaths increased to 416 with two additional deaths today. on O'ahu.
(Feb. 3) Hawai'i's COVID-19 deaths increased to 414 with four additional deaths today. on O'ahu.
(Jan. 31) Hawai'i's COVID-19 deaths increased to 410 with three additional deaths today–two on O'ahu and one on Maui.
(Jan. 29) Hawai'i's COVID-19 deaths increased to 407 with one additional deaths today on O'ahu.
(Jan. 28) Hawai'i's COVID-19 deaths increased to 406 with three additional deaths today–both on O'ahu.
(Jan. 26) Hawai'i's COVID-19 deaths were adjusted upward to 401. This accounts for the 59 (revised by DOH) previously unreported COVID-19 related deaths that were confirmed on Monday. The deaths occurred in August through December, 2020 with 51 deaths reported on O'ahu, six on Hawai'i and three on Maui.
(Jan. 24) Hawai'i's COVID-19 deaths increased to 342 with six new deaths reported today–five on O'ahu, and one on Hawai'i Island.
(Jan. 23) Hawai'i's COVID-19 deaths increased to 336 with four new deaths reported today on O'ahu.
(Jan. 21) Hawai'i's COVID-19 deaths increased to 328 with three new deaths reported today on O'ahu.
(Jan. 20) Hawai'i's COVID-19 deaths increased to 325 with one new death reported today on Maui. The victim is a Maui woman, 70-79 yrs, who died in the hospital with underlying conditions.
(Jan. 19) Hawai'i's COVID-19 deaths increased to 324 with two new deaths reported today on O'ahu.
(Jan. 14) Hawai'i's COVID-19 deaths increased to 318 with four new deaths reported today on O'ahu and one on Maui. All had underlying conditions and were hospitalized at the time of their deaths. The Maui case involved a man in his 90s.
(Jan. 13) Hawai'i's COVID-19 deaths increased to 312 with two new deaths reported today on O'ahu and one on Maui. The O'ahu deaths included a female between the ages of 40-49 years and a male 50-59 years old. The Maui death involved a male between 60-69 years old. All three had underlying conditions are were hospitalized at the time of their passing.
(Jan. 10) Hawai'i's COVID-19 deaths increased to 309 with two new deaths reported today, one on O'ahu and one in a Hawaiʻi resident diagnosed outside of the state.
(Jan. 9) Hawai'i's COVID-19 deaths increasz`ed to 307 with four new deaths reported today, all on O'ahu.
(Jan. 8) Hawai'i's COVID-19 deaths increased to 303 with four new deaths reported today, all on O'ahu.
(Jan. 6) Hawai'i's COVID-19 deaths increased by 10 today to 299 with nine new deaths reported today on O'ahu and one on Maui. The state Department of Health reports that the deaths occurred between Nov. 12 and Dec. 19. The Maui death involved a man between the ages of 70 and 77, who was hospitalized with underlying conditions. The Maui death was the 18th since the start of the pandemic.
(Jan. 1) Hawai'i's COVID-19 deaths increased by one to 289 with a new death reported today on Oʻahu.
(Dec. 31) Hawai'i's COVID-19 deaths increased by three to 288 with new deaths reported today on O'ahu (2) and Hawai'i Island (1).
(Dec. 23) Hawai'i's COVID-19 deaths increased to 285 with three new deaths reported today on the island of O'ahu.
(Dec. 20) Hawai'i's COVID-19 deaths increased to 282 with one new death reported today on the island of O'ahu.
(Dec. 18) Hawai'i's COVID-19 deaths increased to 281 with one new death reported today on the island of O'ahu. The latest case involves a female between the ages of 60-69 years old, who had underlying conditions and was hospitalized at the time of her death.
(Dec. 17) Hawai'i's COVID-19 deaths increased to 280 with two new deaths reported today–one on the island of O'ahu, and the other on Hawai'i Island. The latest case involves a female between the ages of 60-69 years old, who had underlying conditions and was hospitalized at the time of her death.
(Dec. 16) Hawai'i's COVID-19 deaths increased to 278 with four new deaths reported today on the island of O'ahu.
(Dec. 13) Hawai'i's COVID-19 deaths increased to 274 with three new deaths reported today on O'ahu.
(Dec. 12) Hawai'i's COVID-19 deaths increased to 271 with two new deaths reported today on O'ahu.
(Dec. 11) Hawai'i's COVID-19 deaths increased to 269 with one new death reported today on O'ahu. The latest passing is a man, 50-59-years old, who died at home. He had underlying health conditions.
(Dec. 9) Hawai'i's COVID-19 deaths increased to 266 with four new deaths reported today on O'ahu.
(Dec. 6) Hawai'i's COVID-19 death toll increased to 262 with one new death reported today on O'ahu.
(Dec. 5) Hawai'i's COVID-19 death toll increased to 261 with two deaths on Hawai'i Island and three new deaths reported today on O'ahu.
(Dec. 4) Hawai'i's COVID-19 death toll increased to 256 with seven new deaths reported today on Hawai'i Island and three new deaths on O'ahu. Seven deaths at a Hawai'i island long-term care facility, which experienced a COVID-19 outbreak in October, have now been classified by DOH as coronavirus related, as a result of updated information. The deaths include three women and four men, all who had underlying health conditions and ranged in age from 60 to 80+ years old. Three additional deaths were reported from O'ahu today, also occurring more than two-weeks ago. All were men, ranging in age from 30 to 80+ years old. All had underlying health conditions and two of the men had been in the hospital.
(Dec. 3) Hawai'i's COVID-19 death toll increased to 246 with two new deaths reported today on O'ahu. A woman, 80-years-old or older, had been hospitalized with underlying conditions. A man, in the 70-79-year-old age group, had also been in the hospital and had underlying conditions.
(Nov. 29) Hawai'i's COVID-19 death toll increased by four to 244 with two new deaths reported today on O'ahu, and two new deaths on Hawaiʻi Island.
(Nov. 27) Hawai'i's COVID-19 death toll increased to 240 with three news deaths reported today on O'ahu. All were from Honolulu, had underlying conditions and had been hospitalized. Two were male, one between the age of 60-69 years old and the other between 70-79 years old. The other was female, between the age of 70-79 years old.
(Nov. 26) Hawai'i's COVID-19 death toll increased to 237 with two news deaths reported today on O'ahu.
(Oct. 14) Hawai'i's COVID-19 death toll increased to 183, with 10 new deaths. This includes five deaths on Maui,three on Hawai'i Island and two on O'ahu. Seven of the deaths occurred between Aug. 15 and Sept. 30, 2020 and are now reported as a result of updated information received on their cause of death. The other three deaths, one on Hawai'i Island and two on Oahu are recent, as of Oct. 1. The Maui deaths included three men (one between 60 and 69 years old and two above the age of 80) and two women over the age of 80. All had underlying health conditions. One of the women died at a nursing home and the other Maui patients were hospitalized.
(April 6)Maui reported its first COVID-19 related death on Monday, April 6, of an adult male over the age of 65 with underlying health conditions and exposure to travelers.
Private Paradise Villas
Property Care Coordinator
South Maui · 2 weeks ago
April 10, 2021 COVID-19 Update: 114 Cases (74 O'ahu,… April 10, 2021
April 16, 2021 COVID-19 Update: 98 Cases (67 O'ahu,… April 16, 2021
April 1, 2021 COVID-19 Update: 114 Cases (72 O'ahu,… April 1, 2021
This comments section is a public community forum for the purpose of free expression. Although Maui Now encourages respectful communication only, some content may be considered offensive. Please view at your own discretion. View Comments (15) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,050 |
Q: OneDrive direct download URLs for files and file versions I am putting the question in two lines
Does OneDrive supports direct download URLs for files? Are URLs returned by OneDrive are same for mobile and web ? Does OneDrive returns file versions via rest API calls ?
Read below for more details.
I am building a standalone app which uses various OneDrive REST apis for different operations, e.g. upload/download/update/browse a document/folder.
I have following needs which I am not able to accomplish based on readings from OneDrive rest api support page https://msdn.microsoft.com/en-us/library/office/dn659752.aspx.
*
*Direct Download URLs: I want to give user of my app a facility to directly downloading a file. I know currently 'link' property of file object gives a direct view URL which takes user to OneDrive account and from there user can download the file by clicking download button. I need a link for direct download of the file (such link when typed in a browser which has authenticated OneDrive session should directly prompt the download box.)
*Are direct view/download link different for browser and mobile cients?
*Versions: How do I get version numbers for files programmatically. I do not see version info in JSON objects returned via REST calls. [I have read in one of the post that versions are not supported ]
https://msdn.microsoft.com/EN-US/library/dn631834.aspx?f=255&MSPPError=-2147217396
This link shows a JSON representation of a file object.
A: *
*Direct Download URL-
See this link to download from URL
Getting BLOB data from XHR request
*Yes view/download different for browser and mobile, see below to access download box.
hope you are trying to access some file from your download folder!!
If you, you can use url like this-
file:///storage/sdcard0/download/file-name.mp4
This worked for me.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,798 |
\section{Impact of halo mass definition}\label{sec:mass_comp}
\begin{figure}
\begin{centering}
\includegraphics[scale = .41]{figures/auto_Galacticus_vs_mass.pdf}\caption[galaxy 2PCF for \textsc{Galacticus}\ model with different definitions of halo mass]{2PCF for the \textsc{Galacticus}\ model for galaxies with $M* > 10^{10}\, h^{-1} \, {\rm M_{\odot}}$. Each line represents a different halo mass definition used for the model.}
\label{fig:mass_comp_2pcf}
\par\end{centering}
\end{figure}
Halo mass is one of the properties that galaxy formation models use to determine the population and properties of galaxies. Because of this the models might produce different results if they use different definitions of halo mass. In order to make fair comparisons in our analysis we need to study the dependence on the halo mass definition.
In Fig. \ref{fig:mass_comp_2pcf} we show the 2-Point Correlation Function (2PCF) for different halo mass definitions using the \textsc{Galacticus}\ model for galaxies with $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. We only show this model because it used all the different mass definitions and also because we expect the other models to show similar behaviour. Although we only present one model, the different mass definitions reveal only very small changes to the clustering compared with the differences between the models. Thus, the clustering of the models is not affected by the halo mass definition significantly. This result is also independent of the stellar mass selection used. Hence, the results of our study do not depend on the masses used and we focus on few definitions. In this paper we use $M_{200m}$ (defined as the mass enclosed in a radius within the density is $200$ times the mean density) in all the models which used this mass to obtain a catalogue, and $M_{200c}$ (defined as the mass enclosed in a radius within the density is $200$ times the critical density) or $M_{FOF}$ (defined from the total number of particles belonging to the FOF group) for the other models, which do not use $M_{200m}$. The results and conclusions of the paper do not depend on the mass definitions used.
\section{Discussion and conclusions}\label{sec:conclusions}
In this article, we present a comparison of the clustering and halo occupation statistics of 12 different galaxy formation models. We use a dark matter only N-body simulation and run SAMs and HOD-based models with the same dark matter and merger trees input, and compare the results of mean halo occupation numbers, radial distributions of galaxies in haloes and 2-Point Correlation Functions (2PCFs). The goal of this paper is to study the clustering and distribution of galaxies in haloes, and to understand the roles of different galaxy types, in particular of orphan satellites (satellites which are not assigned to any dark matter subhalo). This work is part of a series of papers comparing galaxy formation models that started with K15.
The most important results of the study can be summarized as follows:
(1) The slope in the mean occupation number of orphan satellites as a function of halo mass is much shallower in HOD models than in SAMs, due to the different treatments of orphan satellites between both approaches. Orphan satellites in SAMs originate from the disruption of subhaloes, and this happens more often in massive haloes. However, in this study most of the HOD models populate satellites in subhaloes and only when there are more satellites than subhaloes these extra galaxies are considered as orphans. As massive haloes have many subhaloes, the number of orphan satellites in these HOD models is not as high as in SAMs.
(2) HOD models have a steeper radial distribution of orphan satellites in haloes than SAMs. This is because HOD models distribute orphan satellites following a NFW profile, independently of the substructure and evolution of the haloes. This allows HOD models to populate with more orphan satellites in the inner and denser regions of the haloes than SAMs. SAMs are constrained to where subhaloes have been disrupted, and this causes a lower density in the innermost regions, where subhaloes quickly merge into the central structure, as well as galaxies merging with other galaxies for some models. When comparing all the satellites, the different models have a scatter of $2$ times the measurement uncertainty (due to the limited volume used) in their radial distribution of galaxies for $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$. The scatter is smaller for higher thresholds, since orphan satellites become less relevant.
(3) Using all the galaxies above a certain mass threshold for the measurements of 2PCFs (see Fig. \ref{fig:2pcf_mstar_all}), we see a scatter of a factor of $2$ between the models. However, part of this scatter is due to the lower clustering found for the models that do not have orphan galaxies by construction, an indication of the importance of orphan galaxies on galaxy clustering. We find a good agreement between the models for central galaxies and at large scales for all the galaxy selections. Using a larger volume in simulations would allow us to measure linear bias, and this would be a valuable extension to this work.
(4) HOD models and SAMs have significant differences in their clustering of orphan satellites (see bottom panels of Fig. \ref{fig:2pcf_mstar_types}). Both SAMs and HODs show good agreement for models of the same kind, but SAMs have a higher 2PCF than HOD models. This is due to the differences on the halo occupation numbers of orphan satellites between both schemes. Although HOD models show a steeper radial distribution for orphan galaxies than SAMs, they statistically occupy less massive haloes. The clustering at small scales is strongly affected by the halo occupation of massive haloes, and because of this the orphan satellites in SAMs show a higher 2PCF at these scales than HOD models.
It is important to notice that the models used have not been re-calibrated for this particular simulation. The agreement between the models could be improved by calibrating the models in the simulation where the comparison has been done or even using the same observational constraints (Knebe et al., in prep.).
This study is limited by the resolution of the simulation. A higher resolution simulation would allow us to study smaller scales, and would also allow to detect subhaloes in inner regions of the haloes. This could have an impact on both the satellite distributions at small scales. In addition, a comparison with hydro-dynamic simulations would be useful to study the baryonic effects on both galaxy and dark matter clustering. It has been shown that baryons affect the dark matter distribution at small scales \citep{Tissera2010,Sawala2013,Cui2012,Cui2014,Cui2016}. Finally, another interesting extension would be the comparison of SAMs with new implementations that take into account observations of galaxy clustering to constrain their parameters \citep{vanDaalen2016}.
\section{Introduction}
In $\Lambda$CDM cosmology, gravitational evolution causes dark matter to cluster around peaks of the initial density field and to collapse into virialized objects (i.e., dark matter haloes). Structures form hierarchically, such that smaller haloes merge to form larger and more massive haloes. All galaxies are thought to form as a result of gas cooling at the center of the potential well of dark matter haloes. When a halo and its `central' galaxy are accreted by a larger halo, it becomes a subhalo and its galaxy becomes a `satellite' galaxy. However, due to tidal stripping and the gravitational interaction of subhaloes with their environment (other subhaloes, the gravitational potencial of the halo centre, etc.), sometimes can be distrupted and the galaxy, if it survives, becomes an `orphan' galaxy.
In addition to mergers, haloes also grow by smooth accretion and galaxies grow by \textit{in situ} star formation when fuel (i.e., cold gas) is available \citep{Cooray2002,Sheth2002,vandenBosch2002,Gill2004b,Delucia2004,Vandenbosch2005,Diemand2007,Giocoli2008}.
In this paradigm of hierarchical structure formation, there is a correlation between halo formation, their abundances and the surrounding large-scale structure where more massive haloes tend to reside \citep{Mo1996,Sheth2002}. Most galaxy formation models implicitly assume that the properties of a galaxy are determined primarily by the mass and formation history of the dark matter halo within which it formed \citep{White1978,White1991,Cole1991,Lacey1991,Baugh1999,Benson2001}. Thus, the correlation between halo properties and environment (matter density, substructure, etc.) induces a correlation between galaxy properties and environment.
There are multiple statistical quantities used to study large-scale structure, and here we focus on the two-point correlation function, the radial distribution and the mean occupation number of galaxies. Clustering studies have shown that a variety of galaxy properties (such as luminosity, color, stellar mass, star formation rate and morphology) are dependent on the environment and halo properties across a wide range of scales. Galaxy formation models in simulations are crucial to study the connection between galaxies and haloes, and hence it is important to understand the consistency or differences between different galaxy formation models.
Galaxy formation is a complex, nonlinear process, driven by the interplay of many different physical mechanisms \citep[e.g.][]{Benson2010}. The goal of galaxy formation models is to estimate the statistical properties of the galaxy population given some set of assumptions and thereby to better understand the physical processes involved. One fruitful approach has been to utilize Semi-Analytic Models (SAMs) of galaxy formation \citep[e.g.][]{Cole2000,Hatton2003,Cattaneo2006,Cora2006,Croton2006,Baugh2006,DeLucia2007,Monaco2007,LoFaro2009,Benson2012,Lee2013,Henriques2013,Baugh2013,GonzalezPerez2014,Gargiulo2015}, in which a statistical estimate of the distribution of dark matter haloes and their merger history---either coming from cosmological simulations or extended Press-Schechter/Lagrangian methods---is combined with simplified yet physically motivated prescriptions of processes such as star formation, gas cooling, feedback from supernovae and active galactic nuclei (AGN), etc. that allows one estimate the distribution of galaxy properties. New models are starting now to also use observations of galaxy clustering to constrain their parameters \citep{vanDaalen2016}.
An alternative approach to SAMs are (analytic) dark matter halo occupation models, which determine the halo occupation of galaxies based on the properties of their parent halo. Usually observations of clustering are used to constrain this occupation. This approach is used to study the link between galaxy formation and halo assembly (see \citealt{Cooray2002}; \citealt{Mo2010} for a review). Halo models of galaxy abundances and clustering generally consist of Halo Occupation Distribution (or conditional luminosity functions) \citep[HOD; e.g.][]{Seljak2000,Scoccimarro2001,Berlind2002,Cooray2002,Yang2003,Kravtsov2004,Cooray2006,Guo2016b} and (sub)halo abundance matching (HAM or SHAMs; \citealt{Vale2006}; \citealt{Conroy2006}; \citealt{Hearin2013}; \citealt{Reddick2013}; \citealt{Guo2016b}). Such models are useful for exploring the relations between galaxy formation and dark matter halo assembly in the context of the large-scale structure of the Universe.
Subhaloes closer to the halo centre tend to accrete earlier on \citep{Gao2004}, and therefore tidal stripping (a process which is in part numerical but also physical) has more time to act on these subhaloes. Hence, they are more frequently disrupted and this is the reason why subhaloes are anti-biased with respect to the dark matter \citep[see e.g. ][]{Ghigna2000,Diemand2004,Pujol2014b} in these regions.
In simulations, lack of mass resolution causes the disappearance of a subhalo, causing the galaxy to become an \emph{orphan}. Sometimes the halo finder will merge a subhalo with its parent halo, but the subhalo can reappear when its member particles bounce out of the halo. There are different ways to follow the positions of these orphan galaxies, and these can lead to different clustering of galaxies, especially on small scales \citep{Gao2004,Wang2006,Guo2011,Budzynski2012,Lee2014}.
In \cite{Gao2004},
they used high-resolution resimulations of galaxy clusters and analysed
the radial density profiles of both subhaloes and galaxies from a SAM. This study showed that by including orphan galaxies the radial
density distribution was very close to that of the dark matter, as inferred in
the observational data. They also argued that increasing the
resolution would not improve the situation. This result has been confirmed in
\cite{Guo2011}, who showed that orphans are still dominating the
central regions of galaxy clusters when increasing the resolution of the simulations. They showed that by tracking the position of the most bound particle at the time of disruption convergence between simulations of different resolution was achieved. In \cite{Wang2006}, an HOD approach was used, but using the number and positions of
galaxies from a SAM, and they showed that orphan galaxies are
needed to reproduce the clustering signal at small scales, also confirmed by recent studies \citep{Budzynski2012}. Finally, \cite{Kang2012}, \cite{Guo2013} and \cite{Henriques2013} showed that cosmology, within the current precision, has no impact in the clustering when compared to the differences from galaxy formation physics, even on large scales.
The trajectory and lifetime of orphan galaxies can be determined from different approaches in SAMs. On one side, some SAMs immediately merge galaxies with the central galaxy when the subhalo is lost, and then they have no orphan satellites by construction. Other SAMs define an analytical orbit for the orphan galaxies according to the position and velocity of the galaxies when they became orphan. The radius of the orbit is then continuously decreased until it merges with the central galaxy. Finally, other SAMs define the position and velocity of orphan galaxies directly from the dark matter particle that was the most gravitationally bound from the disrupted subhalo. And other SAMs \citep{Guo2011} use a combination of both the analytical orbits with the dark matter particle trajectories. All these different treatments of orphan galaxies can have consequencies on the abundance and distributions of such galaxies, especially at small scales.
SAM and HOD have important differences on the treatment of orphan galaxies. First of all, while SAMs make use of the merger trees to derive the initial trajectories of orphan galaxies, the HOD models define the galaxy distribution from the present distribution of haloes, without using information from their evolution. Moreover, classical HODs do not account for the presence of substructures - they are built on top of dark matter haloes and the population of satellites is just distributed according to an NFW model.
Given the variety of galaxy formation models that are used in simulations nowadays, it is important to study the differences that arise from the different treatment of galaxy formation physics in each model. In fact, many efforts have been done comparing different galaxy formation models and their physical prescriptions \citep{Somerville1999,Fontanot2009,Kimm2009,Contreras2013,Delucia2011,Fontanot2011,Fontanot2012,Kang2014,Somerville2015,Guo2016b}. This study focuses on the differences in the galaxy clustering for a large variety of models run on the same simulation and with the same merger tree.
In an attempt to put together a large representation of the models from the literature in an extensive comparison study, \cite{Knebe2015} (K15 hereafter) presented 14 models (12 SAMs and 2 HOD models) using the same simulation input (halo catalogues and merger trees) and analyzed the consistency between the models looking at the stellar mass function, the star formation, stellar-to-halo mass relations, stellar mass fractions or abundance of galaxies per halo.
The present paper is a complement of K15, where we study the consistency between several galaxy formation models on the clustering and the distribution of galaxies in haloes. We analyze the consequences of the differences between the models on the distribution of the galaxies inside and outside haloes. We do this by comparing the Two Point Correlation Function of galaxies, the halo occupation number and the radial distribution of galaxies in haloes. We also analyse orphan satellites separately in order to focus on the consequences from the different treatments of orphan satellites between HOD models and SAMs.
This paper is organized as follows. In the next section, Section~\ref{sec:simulation}, we describe the dark matter halo simulation and the orphan treatments of the galaxy formation models. Then we describe our methodology in Section~\ref{sec:methodology}. We present our results, including comparisons of halo occupation numbers, radial distribution of galaxies in haloes and galaxy clustering in Section~\ref{sec:results}. Finally, we end by summarizing and discussing our results in Section~\ref{sec:conclusions}.
\section{Methodology}\label{sec:methodology}
\begin{figure*}
\begin{centering}
\includegraphics[scale = .43]{figures/hod_1e9_all.pdf}
\includegraphics[scale = .43]{figures/hod_1e10_all.pdf}
\caption[Mean halo occupation number comparison of the different models]{Mean halo occupation number as a function of halo mass for the different models using all the galaxy types. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, while the right shows galaxies with $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. }
\label{fig:hod_comp}\par\end{centering}
\end{figure*}
To study the clustering between the models, we will use the 2PCF, which describes the excess of probability $dP$ over a random distribution of finding pairs of galaxies at a given separation $r$:
\begin{equation}
dP = n(1 + \xi(r))dV,
\end{equation}
where $\xi(r)$ represents the 2PCF at a separation $r$ and $n$ is the number density of galaxies. There are several estimators of the 2PCF \citep{Kerscher2000,Coil2013}, but for our study we use the estimator described by the following formula:
\begin{equation}
\xi(r) = \frac{DD(r)}{RR(r)} -1,
\end{equation}
where $\xi(r)$ is the 2PCF as a function of scale, $DD(r)$ is the number of data pairs separated a distance $r$ between them, and $RR(r)$ is the number of random pairs at the same distance. $DD(r)$ and $RR(r)$ are normalized by $n_D(n_D - 1)$ and $n_R(n_R - 1)$ respectively, where $n_D$ and $n_R$ are the numbers of data and random points used. This estimator is equivalent to \cite{Landy1993} when the random sample is large enough, as it is here, where we use $n_R = 10^6$. Due to the size of the simulations, we calculate $\xi(r)$ up to $R=6 \, h^{-1} \, {\rm Mpc}$, since the measurement becomes noisy for larger scales. Due to the resolution, the minimum scale for the study of $\xi(r)$ is $R = 300 \, h^{-1} \, {\rm kpc}$ \citep{Guo2011}.
To calculate the errors of the 2PCF we use the Jack-Knife method \citep{Norberg2009}. We divide the simulation box into $64$ cubic subvolumes, and we measure the 2PCF $64$ times excluding each time one of the subsamples. We obtain the error from these measurements using the unbiased standard deviation according to the following formula:
\begin{equation}
\Delta \xi(r) = \sqrt{ \left(\frac{N_{JK} - 1}{N_{JK}}\right) \sum^{N_{JK}}_{i = 1} (\xi_i(r) - \bar{\xi}(r))^2 },
\end{equation}
where $N_{JK}$ is the number of Jack-Knife subsamples used and $\xi_i(r)$ corresponds to the measurement of $\xi(r)$ excluding the $i$th subsample.
The errorbars, computed with jacknife, give an idea of the scatter that we would expect from different realizations of the same volume and number density. However, in this study we compare different models run on the same haloes, and then these errors do not reflect the uncertainties of the scatter between the models. Differences between models are systematic and could be significant even when they are below the error bars. Then, differences below the errorbar must be taken with care in this study since the same comparison applied to a larger volume could reduce the errorbars but not necessarily the difference between the models.
\section*{Acknowledgements}
The authors would like to express special thanks to the Instituto de F\'isica Te\'orica (IFT-UAM/CSIC in Madrid) for its hospitality and support, via the Centro de Excelencia Severo Ochoa Program under Grant no. SEV-2012-0249, during the three week workshop 'nIFTy Cosmology' where this work developed. We further acknowledge the financial support of the 2014 University of Western Australia Research Collaboration Award for 'Fast Approximate Synthetic Universes for the SKA', the ARC Centre of Excellence for All Sky Astrophysics (CAASTRO) grant number CE110001020, and the two ARC Discovery Projects DP130100117 and DP140100198. We also recognize support from the Universidad Autonoma de Madrid (UAM) for the workshop infrastructure.
Funding for this project was partially provided by the Spanish Ministerio de Ciencia e Innovaci\'{o}n (MICINN), Consolider-Ingenio CSD2007- 00060, European Commission Marie Curie Initial Training Network CosmoComp (PITNGA-2009-238356). We acknowledge support from the European Commission's Framework Programme 7, through the Marie Curie International Research Staff Exchange Scheme LACEGAL (PIRSES-GA-2010-269264).
AP acknowledges support from beca FI and 2009-SGR-1398 from Generalitat de Catalunya, project AYA2012-39620 and AYA2015-71825 from MICINN, and from a European Research Council Starting Grant (LENA-678282).
RAS acknowledges support from the NSF grant AST-1055081.
FJC acknowledges support from the Spanish Ministerio de Econom\'{i}a y Competitividad project AYA2012-39620.
SAC acknowledges grants from CONICET (PIP-220), Argentina.
DJC acknowledges receipt of a QEII Fellowship from the Australian Government.
PJE is supported by the SSimPL programme and the Sydney Institute for Astronomy (SIfA) via ARC grant, DP130100117.
WC and CP acknowledge support of ARC DP130100117.
FF acknowledges financial support from the grants PRIN INAF 2010 "From the dawn of galaxy formation" and PRIN MIUR 2012 "The Intergalactic Medium as a probe of the growth of cosmic structures".
JGB is supported by Spain through the MINECO grant FPA2015-68048, as well as the Consolider-Ingenio 2010 Programme of MICINN under grant PAU
CSD2009-00060 and the Severo Ochoa Programme SEV-2012-0249.
VGP acknowledges support from a European Research Council Starting Grant (DEGAS-259586). This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure.
The work of BMBH was supported by a Zwicky Prize fellowship and
by Advanced Grant 246797 "GALFORMOD" from the European Research Council.
MH acknowledges financial support from the European Research Council via an Advanced Grant under grant agreement no. 321323 NEOGAL.
AK is supported by the {\it Ministerio de Econom\'ia y Competitividad} and the {\it Fondo Europeo de Desarrollo Regional} (MINECO/FEDER, UE) in Spain through grants AYA2012-31101 and AYA2015-63810-P as well as the Consolider-Ingenio 2010 Programme of the {\it Spanish Ministerio de Ciencia e Innovaci\'on} (MICINN) under grant MultiDark CSD2009-00064. He also acknowledges support from the {\it Australian Research Council} (ARC) grant DP140100198.
PM has been supported by a FRA2012 grant of the University of Trieste, PRIN2010-2011 (J91J12000450001) from MIUR, and Consorzio per la Fisica di Trieste.
NDP P was supported by BASAL PFB-06 CATA, and Fondecyt 1150300. Part of the calculations presented here were run using the Geryon cluster at the Center for Astro-Engineering at U. Catolica, which received funding from QUIMAL 130008 and Fondequip AIC-57.
CP acknowledges support of the Australian Research Council (ARC) through Future Fellowship FT130100041 and Discovery Project DP140100198.
RSS thanks the Downsbrough family for their generous support.
PAT acknowledges support from the Science and Technology Facilities Council (grant number ST/L000652/1).
SKY acknowledges support from the National Research Foundation of Korea (Doyak 2014003730). Numerical simulations were performed using the KISTI supercomputer under the programme of KSC-2013-C3-015.
The authors of the paper contributed in the following ways: AP, RAS and EG lead the study and wrote the paper. AK, JGB and FRP organized the second week of the nIFTy workshop from where this study began. The authors listed in Table \ref{tab:models} performed their galaxy formation models using their code, in particular AB, FJC, AC, SC, DC, GDL, FF, VGP, BH, JL, PM, RAS, RS, CVM, and SY actively ran their models with the assistance of JH, MH, and CS. WC, DC, PJE, CP, and JO assisted with the analysis and data format issues. All authors had the opportunity to proof read and comment on the paper.
\section{Results}\label{sec:results}
\begin{figure*}
\begin{centering}
\includegraphics[scale = .43]{figures/hod_1e9_cent.pdf}
\includegraphics[scale = .43]{figures/hod_1e10_cent.pdf}
\includegraphics[scale = .43]{figures/hod_1e9_sat.pdf}
\includegraphics[scale = .43]{figures/hod_1e10_sat.pdf}
\includegraphics[scale = .43]{figures/hod_1e9_orph.pdf}
\includegraphics[scale = .43]{figures/hod_1e10_orph.pdf}
\caption[Mean halo occupation number comparison of the different models]{Mean halo occupation number as a function of halo mass for the different models and galaxy types. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, while the right shows $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. Top panels show central galaxies, middle panels satellite galaxies and bottom panels orphan satellites. }
\label{fig:hod_comp_types}\par\end{centering}
\end{figure*}
In this section we present the model comparison of the mean halo occupation number, the 2PCFs and the radial distribution of the galaxies in the haloes, with an emphasis on the orphan galaxy distribution. We use the galaxy catalogues at redshift $z=0$ and apply different stellar mass cuts in order to see the mass dependence of the convergence and differences of the models.
\subsection{Mean halo occupation number}\label{sec:hod}
In this section we study the mean number of galaxies populating haloes within a given range in mass. In the halo model paradigm it is usually assumed that haloes cluster according only to their masses. Hence, the distribution of galaxies provides a window into the clustering.
In both SAM and HOD models the galaxy populations are commonly characterized by central and satellite galaxies. According to these models, every halo can be occupied by at most one central galaxy, and only the haloes that contain a central galaxy can have a non-zero number of satellites:
\begin{equation}
\langle N|M,M_\ast\rangle \,\equiv\, \langle N_\mathrm{cen}|M,M_\ast\rangle ( 1 + \langle N_\mathrm{sat}|M,M_\ast\rangle ),
\end{equation}
with
\begin{equation}
\langle N_\mathrm{cen}|M,M_\ast\rangle <1,
\end{equation}
where $\langle N|M,M_\ast\rangle$ is the mean number of galaxies $N$ of stellar mass $M_*$ that populate haloes of mass $M$, and $N_{cen}$ and $N_{sat}$ are the number of central and satellite galaxies respectively.
According to all this, for $\langle N | M \rangle \lesssim 1$ the contribution of the HOD mainly comes from the central galaxies and for $\langle N | M \rangle > 1$ the contribution mainly comes from the satellite galaxies.
In Fig. \ref{fig:hod_comp} we show the comparison of the mean halo occupation numbers of the models. This measurement corresponds to the mean number of galaxies per halo as a function of halo mass. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, and right panels show galaxies with $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. This figure gives similar information as Figures $13$ and $14$ from K15, where the number of galaxies is normalized by the halo mass to explore the specific frequency of galaxies as a function of halo mass.
We can see a scatter where each model starts populating galaxies for low stellar masses, which is a consequence of the different implementations of cooling, reionization and stellar feedback. In particular, the minimum mass where all the haloes are populated (i.e. where $\langle N(M) \rangle = 1$) changes a factor of $3$ between the models for galaxies of a stellar mass threshold of $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$. This scatter decreases to a factor of $2$ when the $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$ cut is applied, except for \textsc{Galacticus}.
If we focus on the right panel, we see a strong difference in \textsc{Galacticus}\ for $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$, where these massive galaxies also populate very small haloes. This is due to the excess of galaxies at these masses for this model, that can be seen as a bump around $M_* \sim 2-3 \times 10^{10}\, h^{-1} \, {\rm M_{\odot}}$ in the stellar mass function from Fig. $2$ in K15. This comes from the fact that the galaxy formation model has been calibrated using another simulation to match observations. Changing the simulation without recalibrating the stellar mass function has a significant impact on \textsc{Galacticus}\ (see Fig. 6 of K15). We also like to mention that the fact that \textsc{SkibbaHOD}\ and \textsc{SkibbaSUBs}\ are not identical due to some stochastic components of the models. We also note that \textsc{Sage}\ and \textsc{GalICS 2.0}\ show the lowest occupation number at high masses in the left panel. This is expected since these models do not have orphan satellites by construction.
In order to study the contributions of the different galaxy types, we show in Fig. \ref{fig:hod_comp_types} the mean occupation number of galaxies split into galaxy types. Top panels show central galaxies, middle panels show satellite galaxies (orphan and non-orphan) and bottom panels show orphan satellites. The same stellar mass cuts as in Fig. \ref{fig:hod_comp} has been applied here for left and right panels.
Given the fact that all the haloes are populated by a central galaxy, the mean occupation number for a given stellar mass cut is directly related to the fraction of central galaxies that are more massive than the given stellar mass cut. Models whose central galaxies are less massive will present lower occupation numbers when a stellar mass cut is applied.
If we focus on the top left panel where the central galaxies are shown for a stellar mass cut of $10^9\, h^{-1} \, {\rm M_{\odot}}$, we see that \textsc{Sag}, \textsc{Galacticus}, \textsc{GalICS 2.0}\ and \textsc{DLB07}\ are the models that show higher occupation numbers, while \textsc{Lgalaxies}\ is the model that has the lowest values. This is consistent with Table $3$ of K15, where we see that \textsc{Sag}, \textsc{Galacticus}, \textsc{GalICS 2.0}\ and \textsc{DLB07}\ are precisely the models that present more central galaxies above $10^9\, h^{-1} \, {\rm M_{\odot}}$, and \textsc{Lgalaxies}\ presents the lowest number.
If we focus on the middle panels, where satellite galaxies are plotted, we see that the models show a large scatter at small masses, but the number of galaxies per halo increases with mass with a similar slope for larger masses. This agreement in the slope of the relation means that the galaxy formation models distribute satellite galaxies in haloes in a similar way. We note the different behaviour shown by \textsc{Morgana}, which shows a significantly lower occupation number for small haloes. This is due to the decoupled modelling of satellite galaxies with respect to substructures, than can leave some naked substructures without any satellite galaxy in it. The reason of this treatment of satellite galaxies is that the model was originally designed to be run in \textsc{Pinocchio} simulations, where only haloes (but not subhaloes) are obtained from the output of the simulations.
\begin{figure*}
\begin{centering}
\includegraphics[scale = .43]{figures/dens_prof_1e9_all.pdf}
\includegraphics[scale = .43]{figures/dens_prof_1e10_all.pdf}
\includegraphics[scale = .43]{figures/dens_prof_1e9_orph.pdf}
\includegraphics[scale = .43]{figures/dens_prof_1e10_orph.pdf}
\caption[Radial distribution comparison of the different models]{Comparison of the radial distributions of the different models. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, while in the right with $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. In the top panels we show all the satellite galaxies, while bottom panels show only orphan satellites. Each panel includes the residual with respect to the median of the distribution in each $R/R_{200}$ bin. }
\label{fig:dens_prof}\par\end{centering}
\end{figure*}
The differences between HOD models and SAMs are stronger on the bottom panels, where we show the mean halo occupation number for orphan satellites. We can see that the slope in the mean halo occupation numbers of HOD models is much shallower than SAMs for $M_* > 10^{12}\, h^{-1} \, {\rm M_{\odot}}$, showing a difference between HOD models and SAMs of one order of manitude higher at $M_* \approx 10^{14}\, h^{-1} \, {\rm M_{\odot}}$ than at $M_* \approx 10^{12}\, h^{-1} \, {\rm M_{\odot}}$. This is because massive haloes have many substructures and hence the HOD models occupy them with as many non-orphan satellites as possible. If the total occupation number is not high enough, then the number of orphan satellites in these haloes is low. On the contrary, orphan satellites in SAMs originate from the disruption of subhaloes, and this usually happens in high density environments with strong gravitational interactions. Consequently, in massive haloes, many subhaloes can interact with the environment and suffer tidal stripping. Hence, we expect that the orphan occupation increases quickly with halo mass, as we can see from these panels.
Finally, note also that \textsc{SkibbaHOD}\ has a higher amplitude than \textsc{SkibbaSUBs}, and the slope is similar to some SAMs at high enough halo masses. This is due to the fact that \textsc{SkibbaHOD}\ populates the haloes only with orphan satellites by construction, for which the total number is significantly higher than \textsc{SkibbaSUBs}. Also see that most of the SAMs show a good agreement in this mean halo occupation number of orphan satellites for $M^* > 10^9\, h^{-1} \, {\rm M_{\odot}}$, specially for \textsc{DLB07}, \textsc{Galacticus}, \textsc{Galform}\ and \textsc{Sag}, and there is a good agreement between \textsc{Lgalaxies}\ and \textsc{ySAM}\ too. However, the differences become more significant for $M^* > 10^{10}\, h^{-1} \, {\rm M_{\odot}}$. \textsc{Morgana}\ shows the lowest occupation number due to the shorter merger times implemented in this model.
\subsection{Radial distributions}\label{sec:dens_prof}
In this section we compare the radial distributions measured in all the models. We do the measurement from the following equation:
\begin{equation}
n(R/R_{200}) = \frac{N_g}{(4 \pi /3) [(R + \Delta R)^3 - R^3]},
\end{equation}
where $n(R/R_{200})$ is the number density of galaxies in the radial annulus $R$ to $R + \Delta R$, with $R$ referring to the radial distance to the halo centre, and $N_g$ is the number of galaxies between $R$ and $R + \Delta R$. $R_{200}$ is the radius that encloses $200$ times the critical density. So, this basically describes the density of galaxies as a function of the radial distance of the halo centre.
Fig. \ref{fig:dens_prof} shows the radial distribution of the different models, applying the same stellar mass thresholds of $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$ (left) and $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$ (right) as in the previous figure. Top panels show all the satellites, while bottom panels show only orphan satellites. Each panel shows the residual as $n(R)/\bar{n}(R) - 1$, where $\bar{n}(R)$ is the median of the distribution at each $R$ bin.
\begin{figure*}
\begin{centering}
\includegraphics[scale = .43]{figures/auto_1e9_all.pdf}
\includegraphics[scale = .43]{figures/auto_1e10_all.pdf}
\caption[2PCF comparison of the different models]{Comparison of the 2PCF measurements of the different galaxy formation models using a stellar mass threshold. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, while right panels correspond to $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. Both panels use all the galaxies of the models. }
\label{fig:2pcf_mstar_all}
\par\end{centering}
\end{figure*}
\begin{figure*}
\begin{centering}
\includegraphics[scale = .43]{figures/auto_1e9_cent.pdf}
\includegraphics[scale = .43]{figures/auto_1e10_cent.pdf}
\includegraphics[scale = .43]{figures/auto_1e9_sat.pdf}
\includegraphics[scale = .43]{figures/auto_1e10_sat.pdf}
\includegraphics[scale = .43]{figures/auto_1e9_orph.pdf}
\includegraphics[scale = .43]{figures/auto_1e10_orph.pdf}
\caption[2PCF comparison of the different models]{Comparison of the 2PCF measurements of the different galaxy formation models using a stellar mass threshold. Left panels show galaxies with $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$, while in the right shows galaxies with $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. In the top panels we show the central galaxies, middle panels all the satellite galaxies, and bottom panels orphan satellites. }
\label{fig:2pcf_mstar_types}
\par\end{centering}
\end{figure*}
In the top panels we only show those models that have implemented a treatment for orphan satellites (or have not done it by construction). The differences between most of the models are lower than $2\sigma$ for $M_* > 10^9 \, h^{-1} \, {\rm M_{\odot}}$ and lower than $1\sigma$ for $M_* > 10^{10} \, h^{-1} \, {\rm M_{\odot}}$. We see that the scatter is smaller than a factor of $2$ for all scales smaller than $0.2 R/R_{200}$ and for all the models except \textsc{SkibbaHOD}, \textsc{GalICS 2.0}\ and \textsc{Sage}. \textsc{SkibbaHOD}\ shows the highest radial distribution, a factor of $2$ higher than the median. This is because all the satellites are orphans, and hence all the galaxies follow a NFW profile instead of following substructures. The fact that most of the models agree with the radial distribution of \textsc{SkibbaHOD}\ except for the smallest scales is consistent with previous studies \citep{Gao2004,Kang2014,Pujol2014b,vanDaalen2016}.
On the other hand, \textsc{Sage}\ and \textsc{GalICS 2.0}\ show much flatter radial distributions than the rest of the models. This is because these models have no orphans, and hence only non-orphan galaxies contribute to these distributions. These two models then show the contribution of non-orphan satellite galaxies to the radial distributions, showing agreement with the rest of the models only at the largest scales.
We see that the HOD models (\textsc{Mice}, \textsc{SkibbaHOD}\ and \textsc{SkibbaSUBs}) present a steeper slope of the radial distributions than SAMs in all the cases. Given the similarities between the top and bottom panels, we see that these differences basically come from the different treatments of orphan satellites, which dominate the smallest scales (satellites only contribute to the large scales of the panels). This indicates that the orphan satellites have a very important role in the distribution of galaxies in haloes, consistent with the conclusions at \cite{Gao2004}. While HOD models distribute orphan satellites without any information from substructure or evolution (in \textsc{SkibbaHOD}\ this is the case for all satellites), orphan satellites from SAMs are a consequence of subhalo disruption, and hence the positions of orphan satellites are correlated with substructure.
Moreover, orphan satellites in SAMs are limited in the densest regions, close to the halo centre, where orphans merge quickly with the central galaxy. This exclusion effect in the inner parts of the halo is one reason of why SAMs show a lower orphan density at the smallest scales.
\subsection{Two point galaxy correlation functions}\label{sec:2pt}
We now compare the 2PCF between the different models. We have applied the same stellar mass thresholds used previously, and we also study the different galaxy types separately. Again, we show the residuals with respect to the median of the measurements in order to see the scatter between the models. Since the stellar mass functions are different between the models, the number density of galaxies for the same stellar mass cut can be different. We have also studied the differences using number density cuts instead of stellar mass cuts and we obtain the same results, meaning that the difference between the models is not due to the differences between their number densities.
In Fig. \ref{fig:2pcf_mstar_all} we show the 2PCFs of galaxies for all the models that have computed the orphan positions (and those which did not do it by construction), using all the galaxies. For both stellar mass cuts we find a good agreement, with most of the models consistent within the error bars (although we must be carefull when interpreting the errorbars, as discussed in \S\ref{sec:methodology}). This is an encouraging result, since it highlights a consistency between the models even when most of them did not use any observations of clustering to constrain their parameters. However, we note that \textsc{GalICS 2.0}\ and \textsc{Sage}\ show a lower clustering at small scales, showing a factor of $2$ lower at the smallest scales. Again, this is a consequence of the fact that these models do not have orphan satellites. These models then show the impact that excluding orphan satellites can have on the galaxy clustering predictions.
We split the galaxy samples according to their type in Fig. \ref{fig:2pcf_mstar_types}. We show in the top and middle panels the 2PCF of galaxies for central and satellite galaxies respectively. In the cases of central galaxies the models have very good agreement, while some differences appear for satellite galaxies. The scatter between models in the middle panels is larger for smaller stellar masses, since galaxies are more dominated by orphan galaxies. In particular, all the satellites in \textsc{SkibbaHOD}\ are orphans, causing a large difference with the rest of the models at the smallest scales due to the orphan radial distributions discussed previously. We also note a high clustering signal for \textsc{SkibbaSUBs}, and a lower signal for \textsc{GalICS 2.0}\ (which has no orphans) and \textsc{Sag}. For most of the models, the scatter between the models is in general lower than a $25\%$ for the satellite galaxies and lower than $20\%$ for the central galaxies.
In the bottom panels of Fig. \ref{fig:2pcf_mstar_types} we focus on the models that have orphan satellites in order to study their distribution. For both stellar mass thresholds we see a strong and significant difference between the HOD models and SAMs. The HOD models show a lower clustering amplitude, and they all agree between them, while SAMs agree between them but with a higher amplitude and different shape than the HODs.
The clustering of 2PCF on small scales depends on two main factors, the halo occupation number and the density profile of galaxies in haloes, especially in massive haloes. Although the HOD models show a slightly steeper radial distribution of orphan satellites with respect to SAMs, they also show a much flatter halo occupation number distribution, as indicated in the lower panel of Fig. \ref{fig:hod_comp_types}. This implies that orphan satellites in HODs populate less massive haloes than in SAMs. This is the main reason for the lower 2PCF of orphan satellites in HOD models compared to SAMs, since small scale clustering is strongly affected by the occupation numbers in massive haloes.
This difference between the clustering of orphan satellites of SAMs and HOD models is large, reaching an order of magnitude at scales of $\approx 1 \, h^{-1} \, {\rm Mpc}$.
The impact of orphan satellites in the agreement between models will depend in general on the orphan fraction of the galaxy samples, and we have seen that it also depends on the galaxy formation model. In this analysis the models that computed the orbits and positions of the galaxies are those with the lowest orphan fractions, meaning that the impact of orphan satellites on the clustering of the rest of the models might be stronger.
\section{Simulation data}\label{sec:simulation}
For this study we use a dark matter halo catalogue generated from a \textsc{Gadget-3} N-body simulation (\citealt{Springel2005b}) of a $62.5 \, h^{-1} \, {\rm Mpc}$ side box. We use $270^3$ particles with a particle mass resolution of $9.31 \times 10^8 \, h^{-1} \, {\rm M_{\odot}}$, producing an output of $62$ snapshots. From each snapshot we generate a halo catalogue using the \textsc{SUBFIND} (\citealt{Springel2001}) code, that generates haloes and subhaloes from dark matter overdensities. We used the code \textsc{MergerTree} to generate the merger trees of the haloes \footnote{MergerTree forms part of the \textsc{AHF} package (\citealt{Knollmann2009})}.
From the simulation we obtained several mass definitions for the haloes that can be used for the galaxy formation models. The mass definitions used are detailed in Appendix \ref{sec:mass_comp}.
Some properties of the galaxy formation models can be sensitive to the mass definition and to the galaxy formation models. We discuss the mass definition criteria in Section \ref{sec:methodology}.
We use several galaxy formation models together with this dark matter only simulation for the comparison analysis. Some of them are SAMs of galaxy formation, while others are based on the HOD model. We refer to K15 for a detailed description of these models and some comparisons between them. In this section we enumerate the models, their acronyms and references in Table \ref{tab:models}, and briefly describe the treatment and merging of the orphan satellites of each, since this is one of the most relevant aspects for this comparison analysis.
All the models were originally calibrated to reproduce a given set of observations. However, each model uses different observational data and simulated cosmologies to calibrate its parameters, as stated in the corresponding papers describing the models. It is worth stressing that our strategy forces all models to the same underlying merger tree, therefore we do not expect the original calibrations to be optimal. We have seen in K15 that this leads to model-to-model variations larger than if they were all calibrated for this particular simulation. Nonetheless, we are interested in the general agreement between the different galaxy formation models.
\begin{table*}
\begin{center}
\begin{tabular}{c c c c l}
\hline
Model & Type & Orphans & Orphan positions & Reference \\
\hline
DLB07 & SAM & YES & NO & \citealt{DeLucia2007} \\
Galacticus & SAM & YES & NO & \citealt{Benson2012} \\
Galform & SAM & YES & NO & \citealt{GonzalezPerez2014} \\
GalICS 2.0 & SAM & NO & - & \citealt{Hatton2003,Cattaneo2006,Cattaneo2017} \\
LGALAXIES & SAM & YES & YES & \citealt{Henriques2013} \\
MICE & HOD & YES & YES & \citealt{Carretero2015} \\
MORGANA & SAM & YES & NO & \citealt{Monaco2007,LoFaro2009} \\
SAG & SAM & YES & YES & \citealt{Cora2006,Gargiulo2015} \\
SAGE & SAM & NO & - & \citealt{Croton2016} \\
SkibbaSUBs & HOD & YES & YES & \citealt{Skibba2006, Skibba2009} \\
SkibbaHOD & HOD & YES & YES & \citealt{Skibba2006, Skibba2009} \\
ySAM & SAM & YES & YES & \citealt{Lee2013} \\
\hline
\end{tabular}
\caption{This table shows the list of galaxy formation models used in this paper. The first column shows the acronyms used for each of the models. The second column specifies whether the model is a SAM or HOD. The third column specifies if the model has implemented a treatment of orphan satellites. The fourth column specifies if the model has calculated the positions or orbits of the orphan satellites for this work. Finally, a list of the references is shown in the last column. }
\label{tab:models}
\end{center}
\end{table*}
\subsection{Treatment of orphan satellites}\label{sec:orphan_treatments}
The treatment of the orphan satellites (satellites with no associated dark matter subhalo) has a direct impact on galaxy clustering. In this section we give a brief overview on how models deal with the orphan population (if any) and we refer the interested reader to K15 for more details on the modeling of other physical processes.
\subsubsection{\textsc{DLB07}, \textsc{Galacticus}\ and \textsc{Galform}\ (SAM)}
In these models, when a subhalo disappears (it is stripped below the resolution of the parent simulation), a merger time is assigned to its galaxies according to some variations of the Chandrasekar formula and galaxies are merged when this time is over. These galaxies are assumed to continue orbiting within their parent halo until dynamical friction causes it to merge with the central galaxy. Positions and velocities of orphan satellites are assumed to be traced by those of the most bound particles of substructures at the last time they were identified. This information was not provided for the simulation used in this study. Therefore, the positions of orphan satellites in these models cannot be used for the clustering analysis presented here.
\subsubsection{\textsc{GalICS 2.0}\ (SAM)}
In this model, the effects of the merging timescale are degenerate with those of supernova and AGN feedback and the shock-heating scale. Then, the same effects in the stellar mass function due to the contribution of orphan satellites can also be obtained without them by lowering the efficiency of supernova feedback or the shock heating mass. This model contains a free parameter that sets the dynamical friction efficiency, producing orphan satellites only when this parameter is larger than $0$ (otherwise galaxies always merge when haloes and subhaloes merge). As a good fit to observations (not incuding clustering observations) was obtained without the need of orphan satellites, we decided to use the simplest solution and set the dynamical friction efficiency parameter to $0$. Because of this, \textsc{GalICS 2.0}\ has no orphans.
\subsubsection{\textsc{Lgalaxies}\ (SAM)}
In the original form of \textsc{Lgalaxies}\ the positions of orphans are followed by tracking the most-bound particle of their host dark matter halo just before it was tidally disrupted. The orphan satellite is then placed not at the current position of the particle with which it is identified, but at a position whose (vector) offset from the central galaxy is reduced from that of the particle by a factor of $(1-\delta t/t_{friction})$ where $\delta t$ is the time since the dynamical friction clock was started. This time dependence is based on a simple model for a satellite with ``isothermal" density structure spiralling to the centre of an isothermal host on a circular orbit \citep{Guo2013,Henriques2013}.
Since the dynamical information of most-bound particles is not available for the current simulation, for this work \textsc{Lgalaxies}\ simply decays the positions of orphans from their value at the time they become orphans. Instead of $(1-\delta t/t_{friction})$, a factor of $2 \times
\sqrt{1-\delta t/t_{friction}}$ is used in order to obtain satellite profiles and small scale clustering that roughly resemble those from the default model.
\subsubsection{\textsc{Mice}\ (HOD)}
The galaxy population in haloes is determined from the halo mass, independently of their substructure. In the original implementation of the model, the luminosity function and the colour-magnitude diagrams are determined from observations \citep{Blanton2005b}. Then, the galaxies are split into centrals and satellites. Using a modified NFW profile for the satellite distribution inside haloes, the occupation of galaxies as a function of halo mass is calibrated in order to reproduce the 2-Point Correlation Function (2PCF) of galaxies from observations. The modification of the NFW profile corresponds to a slightly steeper distribution that improves the clustering consistency with observations \citep{Zehavi2011}.
For this particular project, once the number of satellite galaxies in a halo is set, each satellite galaxy is assigned to a different subhalo. When there are more satellites than subhaloes, the excess of satellites are considered orphans and populate the halo according to a modified NFW profile. This is not the approach used in the original implementation of the model, where all the satellite galaxies are distributed with the modified NFW profile. In this case, we will be able to study the clustering of galaxies that follow subhaloes (the non-orphan satellites) and those that are consistent with the original implementation of the HOD model (the orphan satellites).
Although originally this model was implemented and calibrated in the MICE simulation \citep{Crocce2015,Fosalba2015,Fosalba2015b} to make it consistent with clustering observations, in this simulation we used the same parameters obtained from the calibration in the MICE simulation, and hence the model is not necessarily reproducing the clustering observations in this study. In order to reproduce clustering observations, we would need to recalibrate the parameters of the model in this simulation and cosmology (the original simulation used the parameters $\Omega_m = 0.25$, $\Omega_\Lambda = 0.75$ and $\sigma_8 = 0.8$, while this simulation has $\Omega_m = 0.272$, $\Omega_\Lambda = 0.728$ and $\sigma_8 = 0.807$, to mention some parameters).
\subsubsection{\textsc{Morgana}\ (SAM)}
This model has been originally designed to work with merger trees
generated by the Lagrangian code \textsc{Pinocchio} \citep{Monaco2002}, and hence some adjustments have been needed in order to use it
interface with \textsc{SUBFIND} based merger trees.
As \textsc{Morgana}\ does not explicitely follow the evolution of substructures,
only central galaxies are linked to a dark matter structure. Whenever a dark matter halo
becomes a substructure, its galaxies become satellites and each of them
receive a merging time (computed from the \cite{Taffoni2003}
prescriptions), which is defined independently from substructure
evolution. As these merger times are estimated statistically, the
merger of a satellite galaxy with the central object is decoupled from
its parent substructure survival, i.e. satellite galaxies may merge
before their host substructure is lost (giving rise to a population
of substructures whose galaxies have already disappeared; the other models assign a residual merger time when the subhalos is lost, assuring that no merger takes place before the subhalo disappears) or, vice versa,
after it. In both cases they are considered as orphans, and placed at
the centre of the host main halo (meaning that we do not track the trajectories of the orphans). As this assignment of the position
of the orphans is not physical when describing small scale clustering,
we exclude them for the analysis of this model. The other non-central galaxies are considered non-orphan satellites for this analysis (differently than in K15, where all satellite galaxies of this model are considered orphans), and their position is then defined from their subhaloes.
Moreover, merging times computed from the \cite{Taffoni2003}
prescriptions are typically shorter than those estimated from N-Body
simulations \citep{Delucia2010}. Both effects have important
implications on the satellite number density, showing a lower halo occupation number as we discuss later.
\subsubsection{\textsc{Sag}\ (SAM)}
When the subhaloes are no longer identified due to the mass loss form the merging with a larger structure, their galaxies become orphans. The trajectory of orphan satellites is calculated to be a circular orbit with a velocity determined by the virial velocity of the host subhalo and initially located at a halocentric distance given by the virial radius of the subhalo. The decaying radial distance is estimated from the dynamical friction, with position and velocity components randomly generated. The orphan satellites finally merge with the central galaxy of the substructure in which they reside according to the dynamical friction time-scale \citep{Binney1987b}. Because of this time-scale, orphan satellites can be found inside the biggest substructure of the halo (where the central galaxy resides) or inside the substructure of another (satellite) galaxy.
\subsubsection{\textsc{Sage}\ (SAM)}
When a halo/central galaxy system is captured by something larger to become a subhalo/satellite galaxy, the \emph{expected} average merger time of the system is calculated using the \cite{Binney1987b} dynamical friction formula. The subhalo/satellite is then tracked with time until its subhalo-to-baryonic mass ratio falls below a critical threshold, taken as 1.0. At this point the current survival time as a subhalo/satellite is compared to the expected merger time calculated at infall. If the subhalo/satellite has survived longer than average we say it is more resistant to disruption and the satellite is merged with the central in the usual way. Otherwise the satellite is disrupted and its stars are added to a new intra-``cluster'' mass reservoir. As a consequence, SAGE does not produce an orphan galaxy population, since the decision about (and implementation of) the ultimate fate of a satellite is always made before (or when) its subhalo is lost in the merger tree.
This model is an update of \cite{Croton2006}, and the suppression of orphan galaxies and the satellite treatment are some of the changes. Satellite galaxies in \cite{Croton2006} were found to be too red, mainly because of the instantaneous hot gas stripping that was causing a premature supression of star formation. In \textsc{Sage}\ satellite galaxies are treated more like central galaxies, in the sense that hot gas stripping now happens in proportion to the subhalo mass stripping. The lack of orphans makes the model to be resolution dependent, since the population of satellite galaxies depends on the resolution and detection of subhaloes in the simulations. However, most modern cosmological simulations have sufficiently high resolution to recover the galaxy population down to the limit of that typically probed by current surveys \citep{Croton2016}.
\subsubsection{\textsc{SkibbaSUBs}\ (HOD)}
As in \textsc{Mice}, the occupation of galaxies in haloes are determined as a function of the halo mass, independently of the substructure, and calibrated to recover luminosity function, colour magnitude diagrams and the clustering 2PCF \citep{Blanton2003,Zehavi2005}.
Satellite galaxies are distributed in subhaloes, and the exceeding galaxies are considered orphan satellites and populate the haloes according to the NFW profile. For an additional comparison in this study, we also constructed a catalogue where all the satellite galaxies follow the NFW profile (independently of the substruture of the haloes), that we call \textsc{SkibbaHOD}. The difference between \textsc{SkibbaSUBs}\ and \textsc{SkibbaHOD}\ is that the satellite galaxies have a different density profile. In \textsc{SkibbaSUBs}\ the satellite galaxies follow the subhaloes (except the orphan satellites) while in \textsc{SkibbaHOD}\ all satellite galaxies follow an NFW profile. This is useful to understand the importance of the different treatments of the satellite distribution and how they affect clustering. In this study \textsc{SkibbaHOD}\ will be treated as a reference for an NFW-based model.
As in \textsc{Mice}, the parameters have been calibrated in another simulation and cosmology, and hence the model does not necessarily reproduce clustering observations even they were fit to do it in the original simulation.
\subsubsection{\textsc{ySAM}\ (SAM)}
All subhaloes are tracked even after the halo finder loses them in the central dense region of a main halo, and populated with a resident galaxy. In these cases, galaxies only merge when they are closer to the centre of the halo than $0.1R_{vir}$. If a substructure disappears before reaching the central region of its host halo, the galaxy is considered orphan and \textsc{ySAM}\ calculates its mass \citep{Battin1987} and orbit \citep{Binney1987b} analytically until approaching the very central regions. This has a large impact on the lifetime of subhaloes and galaxy merging timescale \citep{Yi2013}.
\subsection{Orphan fraction}
In Fig. \ref{fig:orph_frac} we show the orphan fraction as a function of stellar mass for the galaxy formation models that have orphan satellites. This figure is similar to Fig.12 in K15 where the same fraction has been plotted, but as a function of halo mass. The orphan fraction is defined as:
\begin{equation}
f_{\mbox{orph}} (M_*) = \frac{N_{\mbox{orph}}(M_*)}{N(M_*)},
\end{equation}
where $N_{\mbox{orph}}(M_*)$ is the number of orphan satellites in the catalogue with stellar mass $M_*$ and $N(M_*)$ is the total number of galaxies with the same stellar mass. We see that the fraction decreases with mass.
This trend is expected since small subhaloes are more easily affected by tidal stripping, so orphan satellites tend to originate from small subhaloes. As the mass of the orphan satellites is strongly related to the mass of the subhaloes at the time of accretion, their masses then tend to be small.
Because of this, orphan satellites are more important at small masses than at large masses, and this implies that the role of orphan satellites on galaxy clustering will mostly be important for low mass thresholds \citep{Gao2004,Wang2006,Guo2011,Budzynski2012,Lee2014}. The large orphan fractions of \textsc{DLB07}, \textsc{Galacticus}\ and \textsc{Sag}\ at $M_* > 3\times 10^{11}\, h^{-1} \, {\rm M_{\odot}}$ are not significant due to the low number of galaxies with these masses, that makes very few orphan satellites (less than 5) represent a large fraction.
We can see that the fraction of orphan satellites depends strongly on the different galaxy formation model, with the scatter between the models being large. The models that show the lowest orphan fractions (for high masses) are \textsc{ySAM}\ and \textsc{Morgana}, as expected from their treatment of orphan satellites. On one hand, \textsc{ySAM}\ tracks the galaxies in the substructures even after its mass has decreased below the resolution level. These galaxies are still considered non-orphan galaxies, and because of this it is more difficult for a galaxy to become orphan in this model. On the other hand, in \textsc{Morgana}\ the galaxy merger times are shorter than those estimated from N-body simulations, which means that orphan satellites merge more quickly with central galaxies. Therefore there are fewer orphans for this model. Interestingly, \textsc{Galacticus}\ and \textsc{Galform}\ show the highest orphan fractions. Several studies \citep{Contreras2013,Campbell2015,Simha2016} show that the analytical equation used in \textsc{Galform}\ allows galaxies to orbit around the central galaxy for longer than other approximations, also causing a more centrally concentrated distribution of satellite galaxies. Note that the models that track the positions of the orphan satellites, and hence the models that we will use to measure the clustering of orphan satellites, are the ones that present the lowest orphan fraction at small masses. This means that the results of the distribution of orphan satellites that we show in this study might have a stronger impact on galaxy clustering for the rest of the models.
\begin{figure}
\begin{centering}
\includegraphics[scale = .58]{figures/frac_orph_all.pdf}
\caption[Orphan fraction as a function of stellar mass]{Comparison of the orphan fraction (with respect to all the galaxies) of the different galaxy formation models as a function of the stellar mass. Each line corresponds to a different model. \textsc{Sage}\ and \textsc{GalICS 2.0}\
did not consider orphans in this work.}
\label{fig:orph_frac}\par\end{centering}
\end{figure} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,168 |
\section{Introduction}
With the increasing adoption of Internet of Things (IoT) \cite{da2014internet} technology, sensor data from various situations ranging from wearables to industrial machines is becoming increasingly available. Deep neural networks (DNNs) have been successfully applied on such data for forecasting, classification, anomaly detection, diagnostics, and prognostics \cite{p:lstm-ad,lai2018modeling,khan2018review}, \cite{wang2019deep},
with several applications in healthcare, utilities, etc.
Further, as shown in \cite{TIMENET, fawaz2018data}, a deep learning model trained on time-series across domains shows improved performance
on other domains. The potential for transfer across tasks is expected to be even more likely when tasks are similar, e.g., different instances of the same underlying dynamical system, e.g. humans in the case of activity recognition, or similar machines (engines/turbines, etc.) in the case of equipment health monitoring.
In the extreme case, we might even like to consider data from all such instances as one dataset.
In the existing approaches, we face two important challenges in learning across a sequence of such multi-sensor tasks: (i) The number of sensors
used in each instance often differ, e.g., different people may be using different subsets of equipment (e.g., wrist-bands, phones, sensor-equipped shoes, etc.), and at different times. Similarly, machines installed in different factories, vehicles, or OEMs may be installed with a smaller or larger set of sensors often (e.g., temperature, vibration, pressure, etc.). However, most deep learning models typically assume fixed dimensional inputs and can handle variable temporal dimensions, i.e. variable length of the time series, but fail to adapt and generalize to variability in the number of available dimensions in multivariate
time series, where one or more dimensions are completely missing. Further, it may often be the case that some patterns (e.g., activities) cannot be detected using fewer sensors, and so tasks may also differ in the number of classes being encountered.
(ii) While it is becoming increasingly common for artificial intelligence (AI) software providers to retain the right to continuously improve their machine learning models as new data is encountered, legal restrictions usually prohibit the storage and use of customer data beyond a certain limit, e.g. for privacy reasons
in the case of human activities, and even in the case of industrial data, the data used to train AI software legally belongs to the customer
and cannot be stored and used by the software vendor forever. Since tasks are encountered sequentially over time, a software vendor cannot use standard training techniques requiring multiple passes over all datasets to improve their models over time.
Motivated by the above, we consider a setting where multiple tasks with scarce and partially-observed sensor data, i.e., {\bf variable input dimensions} and {\bf classes} arrive sequentially. We would like to continuously improve a deep learning model as tasks are encountered, {\bf without being able to access data from previous tasks}, so as to {\it improve performance on subsequent tasks as well as on previous tasks}. (The latter requirement is actually important in practice as it is the rationale used by AI software vendors to claim the right to improve their models using customers' data, i.e., that this will enable future versions of the software to automatically perform better on their original task.) Note that this setting is related to the {\it online} continual learning problem \cite{kirkpatrick2017overcoming}, as also class-incremental learning \cite{kundu2020universal} where (in the case of classification) tasks may have varying subsets of classes. However, such approaches do not typically consider the variability in input dimensions, which is the main focus of this work.
Our key contributions can be summarized as follows:
\begin{itemize}[leftmargin=*]
\item We motivate and formulate a novel and practically important problem of {\bf continual learning} from partially-observed multi-sensor (multivariate time-series) data, with a focus on the need to adapt to {\bf varying input dimensions (VID)} and classes across tasks with {\bf data-access prohibited across tasks}.
\item We propose a novel modularized neural network architecture to handle variable input dimensions, with two main modules:
i. a {\bf core dynamics module} comprising an RNN that models the underlying dynamics of the system,
ii. a {\bf conditioning module using a graph neural network} (GNN \cite{battaglia2018relational}) that adjusts the activations of the core dynamics module for each time-series based on the combination of sensors available, effectively exhibiting different behavior depending on the available sensors.
\item To leverage partially-observed sensor data, we build upon the idea of generative replay \cite{shin2017continual}, where task-specific generators generate variable dimensional time-series for subsequent processing by the task-solving core dynamics and conditioning modules.
\item We present an extensive empirical evaluation on diverse tasks from three real-world datasets to prove the advantage of our approach in contrast to vanilla architectures in terms of the ability to adapt to variable input dimensions and changing classes.
\end{itemize}
\begin{figure*}[h]
\centering
\includegraphics[scale=0.17,trim={3cm 22.25cm 0cm 14.5cm}, clip]{images/Approach.pdf}
\caption{Scenarios considered (A and B). \label{fig:approach}}
\end{figure*}
\section{Problem Definition\label{sec:prob}}
Consider $m$ tasks $\mathcal{T} = (\mathcal{T}_{1}, \cdots, \mathcal{T}_{m})$
arriving sequentially.
A task $\mathcal{T}_{i}$ ($i = 1, 2, \cdots, m)$, consists of $N$ instances of labeled multi-sensor data $\mathcal{D}_i = \{(\mathbf{x}_{i,r},y_{i,r})\}_{r=1}^N$, each time-series $\mathbf{x}_{i,r} \in \mathcal{X}_i$ with an associated target $y_{i,r} \in \mathcal{Y}_i$. The target space $\mathcal{Y}_i$ corresponds to class labels in case of classification tasks, or real-valued numbers (or vector) in case of regression tasks.
The goal in each task is to learn a \textit{solver model} ${M}_{i}$ with trainable parameters $\boldsymbol{ \theta}_i$ such that $M_i\!\!:\mathcal{X}_i \!\rightarrow \mathcal{Y}_i$, with estimated target $\hat{y}_{i, r} = {M}_{i}(\mathbf{x}_{i, r}; \boldsymbol{\theta}_{i})$.
Any task $\mathcal{T}_i$ is not allowed access to data from previous tasks $\mathcal{T}_j$ ($j={1,\ldots,i-1}$), but can leverage relevant generated samples obtained using generative models of these previous tasks.
We consider the following partially-observed multi-sensor tasks under the above-mentioned basic setup:
i. tasks have a \textit{variable input space}, as illustrated in Fig. \ref{fig:approach}A, i.e. the input dimensions or available sensors change across tasks, i.e. for $r$-th time-series $\mathbf{x}_{i,r} = \{\mathbf{x}_{i,r}^t\}_{t=1}^{T_i}$ of length $T_i$ with $\mathbf{x}_{i,r}^t \in \mathbb{R}^{d_i}$, $ d_i \in \{1,2,\ldots,d\}$ is the number of available dimensions or sensors out of a maximum of $d$ possible dimensions. We refer to the set of available dimensions in $\mathcal{T}_i$ as $\mathcal{S}_i \subseteq \mathcal{S} = \{s_1,s_2,\ldots,s_d\}$, where $\mathcal{S}$ is the set of all the possible sensors.
ii. \textit{variable target space}: where classes of interest can vary across tasks, as illustrated in Fig. \ref{fig:approach}B; this is applicable for classification tasks only (but not for regression tasks).
The variable target space can correspond to either i. \textit{incrementally increasing classes}, where new classes are added for each new task while all old classes persist, or ii. a subset of old classes are present in the new task along with some new classes, i.e. the \textit{partially changing classes}.
\section{Related Work}
\subsection{Handling variable-dimensional time-series}
Several approaches in literature deal with variations in input data along temporal or spatial dimensions, i.e. variable length of time-series, or variable image sizes, e.g. via recurrent neural networks (RNNs) \cite{cho2014learning} for the time dimension, and variants of pooling operations for image applications \cite{he2015spatial}.
Several approaches for handling missing values in multi-sensor data via neural networks have been proposed.
For instance, \cite{che2018recurrent} study missing value problem in multi-sensor data by proposing a variant of the gated recurrent units (GRUs) \cite{cho2014learning} using knowledge of which dimensions of the input are missing and for how long.
However, handling VID is a more difficult problem compared to the missing value problem in time-series.
Approaches dealing with missing values in time-series are not directly applicable in the VID setting where one or more dimensions of the time-series are completely missing (i.e. the missing \%age for the dimension/sensor is 100\%). Since these approaches would typically rely on one or more past values to extrapolate, and do not consider scenarios where a dimension is completely missing from first to last time step.
The generalization ability of GNNs across varying nodes has been studied in, e.g. GRU-CM \cite{gupta2020handling} and NerveNet \cite{wang2018nervenet} use the inherent ability of GNNs to generalize well on unseen combinations of test instances that are different from training instances. In GNMR \cite{narwariya2020graph}, GNNs leverage the knowledge of readily available graph structure to process multi-sensor time series data for the Remaining Useful Life (RUL) estimation in equipment health monitoring. However, these approaches neither take into account data-access restrictions across tasks nor variability in target spaces.
Modular Universal Reparameterization (MUiR) \cite{meyerson2019modular} attempts to learn core neural modules which can be transferred across varying input and output dimensions.
It relies on learning the core module by solving several sets of architecture-task problems with varying input and output dimensions.
However, MUiR does not study multi-sensor data setting, and relies on solving several tasks to learn the core neural module. Instead, our approach relies on a conditioning vector obtained via GNNs to allow adaptability to VID, and addresses additional constraints on simultaneous task-access as required by MUiR.
Recently, a neuro-evolutionary approach has been proposed in \cite{elsaid2020neuro} which studies the problem of structure-adaptive for time-series prediction. It relies on a series of mutation operations and crossover operations over the neural units. Instead of the computationally expensive neuro-evolutionary approaches, we take a different perspective on the problem of adapting to varying dimensionality where graph neural networks (GNNs) are used.
The recently proposed CondConv \cite{yang2019condconv} is similar in spirit to our work, i.e. it attempts to dynamically adapt the parameters of the neural network conditioned on the current input. While it focuses on adapting to each input with same dimensionality, our work focuses on adapting to inputs of varying dimensionality.
Though significantly different in implementation and the end-objective, our approach also draws inspiration from such works, including \cite{rosenbaum2019routing,andreas2016neural}, where the parameters of the core neural network are dynamically adjusted as per the input.
\subsection{Continual Learning}
We review three recent approaches for continual learning. i. Experience replay: This approach maintains a storage buffer for rehearsing all previous tasks, e.g. \cite{rolnick2018experience,isele2018selective} and uses them along with new task data to avoid catastrophic forgetting. Our work is orthogonal to this approach, as real data-access prohibited across tasks. ii. Generative replay: Instead of storing real previous tasks data, a generative model can be used to mimic the real tasks distribution, e.g. \cite{shin2017continual,velik2014brain}. The generated samples of all previous tasks are used along with current tasks to adapt on old tasks. iii. Task-specific model: Instead of storing or replaying previous tasks data, different model components such as expanding model for each new task \cite{yoon2017lifelong} or using supermask to select task-specific subnetworks, while keeping the weights of model \cite{wortsman2020supermasks} fixed to avoids catastrophic forgetting.
However, these approaches focus on variable classes across tasks, and do not usually consider the VID setting. In fact, as we show empirically in Section \ref{sec:exp}, these approaches struggle in the VID setting.
\section{Approach\label{sec:approach}}
Our approach consists of learning a series of generator and solver models, one for each task, as shown in Fig. \ref{fig:approach}A and \ref{fig:relevant_samples}.
For any task $\mathcal{T}_i$, a solver model ${M}_i$ is trained in a supervised manner using labeled data, a generator model ${G}_i$ is trained in an unsupervised manner as a generative model for the data $\mathcal{D}_i$ from $\mathcal{T}_i$.
We denote the original and generated data for $\mathcal{T}_i$ as $\mathcal{D}_i$ and $\hat{\mathcal{D}}_i$, respectively.
While ${G}_i$ generates unlabeled time-series data, the corresponding labels are estimated using the learned ${M}_i$.
This labeled generated data $\hat{\mathcal{D}}_i$ serves as additional training data for subsequent data-scarce tasks $\mathcal{T}_j$ ($j=i+1,\ldots,m$) instead of $\mathcal{D}_i$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.2,trim={35cm 25.25cm 31cm 12.0cm}, clip]{images/relevant_samples.pdf}
\caption{Using relevant generator and solver for augmenting training data for task ($\mathcal{T}_i$). \label{fig:relevant_samples}}
\end{figure}
\begin{figure*}[h!]
\centering
\includegraphics[scale=0.215,trim={22cm 18.25cm 25cm 8cm}, clip]{images/Solver_CM.pdf}
\caption{Flow-diagram of the Solver-CM model using GNN-based Conditioning Module. The available sensors in a time series and corresponding active embeddings and nodes are shown in color while the missing sensor/dimension is shown in grey color (dashed). The active nodes corresponding to the available sensors exchange messages with each other via a graph neural network, and generate a conditioning vector that is used by the core dynamics module (a recurrent neural network) processing the time series to estimate the target. \label{fig:solver_CM}}
\end{figure*}
\subsection{Generator Models}
Our approach using generated data for subsequent tasks is similar in spirit to existing work in generative replay (GR) for continual learning, e.g. \cite{shin2017continual,achille2018life}.
However, instead of maintaining a common generator (CG) across tasks, we maintain a separate independent generator (IG) ${G}_i$ for each task. We refer prior work such as \cite{shin2017continual} relying on CG as \textbf{GR-CG}, and our proposed approach relying on IG as \textbf{GR-IG}.
This design choice of having an GR-IG per task is crucial to handle VID across tasks: when dimensions across tasks vary, a GR-CG with fixed input dimensions $d$ cannot easily adapt to the missing $d-d_i$ dimensions in $\mathcal{T}_i$ as there is no training signal for those dimensions. Even if the GR-CG can model the $\mathcal{S}_i$ sensors of the current task $\mathcal{T}_i$ well, but it fails to adapt to a new combination of sensors in $\mathcal{S}_{i+1}$ from the task $\mathcal{T}_{i+1}$.
In contrast, while training the GR-IG ${G}_i$, the input dimensions correspond only to the original relevant sensors $\mathcal{S}_i$ for that task, and the burden of handling VID is passed on to the solver model ${M}_i$ whose novel modular structure relying on GNNs is specifically designed to be able to handle VID data.
We analyze and validate the advantage of GR-IG over GR-CG empirically in Section \ref{sec:exp}.
In this work, we use variational autoencoders (VAEs) \cite{kingma2013auto,achille2018life} as generative models for the data distribution $p(\mathcal{D})$. We use VAEs for ease of training. However, the proposed approach is agnostic to the choice of generator, and other generative models such as generative adversarial networks (GANs, as used in \cite{shin2017continual}), are equally applicable. The encoder and decoder of VAEs are RNNs with GRUs that allow for capturing the temporal dependencies in multi-sensor data. The loss function is the same as in standard VAEs (refer \cite{kingma2013auto} and Appendix for more details).
\subsection{Solver Model}
The training data $\tilde{\mathcal{D}}_i$ for the solver model ${M}_{i}$ consists of original data $\mathcal{D}_i$ from $\mathcal{T}_i$ as well as relevant\footnote{We also consider the scenario of using all classes instead of relevant classes samples from the previous generated tasks data, but found using the relevant classes samples performs better, especially in partially changing classes scenario.} generated data $\bigcup_{j=1}^{i-1}\hat{\mathcal{D}_{j}}$ from the previous tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$. This inevitably results in VID data if different tasks have different dimensions.
However, the training data across tasks can have VID as well as variable target space. VID is handled via solver model, whereas variable target space is handled by using only those generated samples from the previous tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$ for which the estimated class labels correspond to classes in $\mathcal{T}_i$, as illustrated in Fig. \ref{fig:approach}B.
Any point $\mathbf{x}^t$ at a timestep $t$ in time-series $\mathbf{x} \in \tilde{\mathcal{D}}_i = \bigcup_{j=1}^{i-1}\hat{\mathcal{D}_{j}} \cup \mathcal{D}_i$, is mapped to a vector in $\mathbb{R}^d$ with a fixed consistent indexing on the sensors in $\mathcal{S}$, irrespective of the number of available sensors in $\mathbf{x}$. The unavailable sensors are mean-imputed, i.e. the mean value of the sensor across other instances where it is available is used.
The solver model consists of the two modules, as illustrated in Fig. \ref{fig:solver_CM}:
i. \textit{core dynamics module} (CDM), which is a standard GRU network that learns the dynamics of the system and ingests a fixed-dimensional time-series where the missing dimensions are imputed with a constant (mean) value,
ii. \textit{conditioning module} (CM), which generates a ``conditioning vector'' as a function of the set of available sensors irrespective of the readings those sensors take for a particular time-series instance as shown in Appendix Fig. \ref{fig:Conditioning module}.
This conditioning vector is passed as an additional input to the CDM allowing it to adjust its internal computations and activations according to the combination of available sensors.
The conditioning vector is in-turn obtained from ``sensor embedding vectors'' via a GNN.
Note that the CDM and the CM along with the sensor embeddings are trained in an end-to-end fashion via stochastic gradient descent (SGD). We next provide architecture details of the two modules.
\subsubsection{Conditioning Module (CM)}
The CM based on GNNs is one of the crucial modules in the overall architecture as it allows to handle missing dimensions (variables) in multivariate time series. For this, we consider a fully-connected bi-directional graph where each node corresponds to a sensor, such that each active node (corresponding to available sensors in a time series) in the graph is connected to every other active node.
Any node in the graph is updated by using aggregated messages from its neighboring nodes. The conditioning vector is obtained as a summary of the messages exchanged between the active nodes. This conditioning vector guides the core dynamics module (CDM) about the current ``context'', i.e. available sensors, so that the computations of the CDM can be adjusted according to the available sensors.
More specifically, corresponding to the set of sensors $\mathcal{S}$, consider a graph $G(\mathcal{V},\mathcal{E})$ with nodes or vertices $\mathcal{V}$ and edges $\mathcal{E}$, with one node $v_s \in \mathcal{V}$ for every $s \in \mathcal{S}$ such that $|\mathcal{V}|=|\mathcal{S}|$.
The set of neighboring nodes for any node $v_s$ is denoted by $\mathcal{N}_{G}(v_s)$.
Each sensor $s \in \mathcal{S}$ is associated with a learnable embedding vector $\mathbf{v}_s \in \mathbb{R}^{d'}$, where $d'$ is a hyperparameter.
While training a model for $\mathcal{T}_i$, the possible dimensions or sensor combinations can be any of $\mathcal{S}_1, \mathcal{S}_2,\ldots,\mathcal{S}_i$, corresponding to the sensor combinations present in $\hat{\mathcal{D}}_1, \hat{\mathcal{D}}_2, \ldots,\hat{\mathcal{D}}_{i-1}$ or $\mathcal{D}_i$. For a given set or combination of sensors $\mathcal{S}_j$ ($j=1,\ldots, i$), only the corresponding nodes $\mathcal{V}_j \subseteq \mathcal{V}$ are considered to be \textit{active}, and contribute to obtaining the combination-specific conditioning vector $\mathbf{v}_{\mathcal{S}_j}$.
For the active nodes, the graph is assumed to be \textit{fully-connected} such that each active node in the graph is connected to every other active node.
Any edge is active only if both the participating nodes are active.
The GNN corresponding to this graph consists of a node-specific feed-forward network $f_n$ and an edge-specific feed-forward network $f_e$; $f_n$ and $f_e$ are shared across the nodes and edges in the graph, respectively.
For any active node $v_a \in \mathcal{V}_i$, the node vector $\mathbf{v}_a$ is updated using the GNN as follows:
\begin{align}
\mathbf{v}_{al} &= f_{e}([\mathbf{v}_{a},\mathbf{v}_{l}]; \boldsymbol{\theta}_{i}^{e}) \label{eq1}, \quad \forall v_l \in \mathcal{N}_{G}(v_a),\\
\tilde{\mathbf{v}}_{a} &= f_{n}([\mathbf{v}_{a},\sum_{\forall l} \mathbf{v}_{al}]; \boldsymbol{\theta}_{i}^{n}) \label{eq2},
\end{align}
where $f_{e}$ and $f_{n}$ both consist of feedforward layers with Leaky ReLUs \cite{maas2013rectifier} and dropout \cite{srivastava2014dropout}, and with learnable parameters $ \boldsymbol{\theta}_{i}^{e}$ and $ \boldsymbol{\theta}_{i}^{n}$, respectively.
While $f_{e}$ computes the message from node $v_l$ to $v_a$, $f_{n}$ updates the node vector $\mathbf{v}_a$ to $\tilde{\mathbf{v}}_a$ using messages from its neighboring nodes (Eqn. \ref{eq2}).
Finally, the conditioning vector $\mathbf{v}_{\mathcal{S}_j}$ specific to the set of sensors $\mathcal{S}_j$ is obtained from the updated node vectors as
\begin{equation}\label{eq3}
\mathbf{v}_{\mathcal{S}_j} = \mathtt{max}(\{\tilde{\mathbf{v}}_a\}_{v_a \in \mathcal{V}_j}),
\end{equation}
where $\mathtt{max}$ returns the dimension-wise maximum value\footnote{We also tried averaging instead of max operation to summarize the vectors of available sensors, but found max to work better.} across the updated node vectors.
It is noteworthy that the summation over the messages across nodes in Eqn. \ref{eq2} and the $\mathtt{max}$ operation in Eqn. \ref{eq3} essentially provide the desired ability to process VID.
\subsubsection{Core Dynamics Module (CDM)}
As mentioned earlier, any time-series $\mathbf{x}_{i, r} \in \tilde{\mathcal{D}}_i$ is first converted to the $d$-dimensional time-series $\tilde{\mathbf{x}}_{i, r}$ with mean-imputation for the unavailable sensors.
This time-series along with its conditioning vector, say $\mathbf{v}_{\mathcal{S}_j}$ (assuming this time-series comes from generated data $\hat{\mathcal{D}}_j$) for the task $\mathcal{T}_{i}$ are processed by the CDM as follows:
\begin{align}
\mathbf{z}^t_{i, r} &= GRU([\tilde{\mathbf{x}}_{i, r}^t, \mathbf{v}_{\mathcal{S}_j}],\mathbf{z}^{t-1}_{i, r}; \boldsymbol{\theta}_{i}^{GRU}), \quad t:1,\ldots,T_i \label{eq6}\\
\hat{y}_{i, r} &=f_{o}(\mathbf{z}^{T_i}_{i, r};\boldsymbol{\theta}_{i}^{o})\label{eq7},
\end{align}
where $GRU$ is a multi-layered RNN having $\boldsymbol{\theta}_{i}^{GRU}$ learnable parameters that gives feature vector $\mathbf{z}^{T_i}$ at the last timestep $T_i$, and $\mathbf{v}_{\mathcal{S}_j}$ is concatenated with $d$-dimensional sensor values at each time-step.
At last, the estimate $\hat{y}_{i, r}$ for $y_{i, r}$ is obtained via $f_{o}$ consisting of ReLU layer(s) followed by softmax or sigmoid layer corresponding to classification or regression, respectively, with $\hat{y}_{i, r} \in [0,1]^K$ for classification, and $\hat{y}_{i, r} \in \mathbb{R}$ for regression. For the RUL estimation regression task, we use min-max normalized targets such that they lie in $[0,1]$.
All the steps described above from the Eqn. \ref{eq1} to Eqn. \ref{eq7} can be consolidated as follows:
$\hat{y}_{i,r} = {M}_{i}(\mathbf{x}_{i,r}; \boldsymbol{\theta}_{i})$,
where $\boldsymbol{\theta}_{i}$ includes all the parameters $(\boldsymbol{\theta}_{i}^{e}, \boldsymbol{\theta}_{i}^{n}, \boldsymbol{\theta}_{i}^{GRU}$ and $ \boldsymbol{ \theta}_{i}^{o})$ of $M_{i}$.
\section{Training Objective}
Denoting the training objectives for classification (standard cross-entropy loss) or regression (squared-error loss) as $\mathcal{L}(y_{i, r}, \hat{y}_{i, r}; \boldsymbol{\theta}_{i})$, we have:
\begin{dmath}
\mathcal{L}_i= \frac{q}{N}\sum_{r = 1}^{N}\mathcal{L}\big(y_{i, r}, \hat{y}_{i, r}; \boldsymbol{\theta}_{i}\big) + \Big(\frac{1 - q}{(i-1)N}\Big)\sum_{j = 1}^{i-1}\sum_{r=1}^{N}\mathcal{L}\Big(y_{j,r}, {M}_{i}(\mathbf{\hat{x}}_{j,r}; \boldsymbol{\theta}_i); \boldsymbol{\theta}_{i}\Big),
\end{dmath}
where, $0 < q \leq 1$ is the importance given to data from $\mathcal{T}_i$ and $N$ is the number of instances. Higher values of $q$ imply more weightage to samples from $D_i$. In this work, we use equal weightage to all samples, i.e. $q=0.5$. Note that $y_{j,r}$ is the label corresponding to class with maximum probability as given by the solver model ${M}_{j}$ for the generated sample $\mathbf{\hat{x}}_{j,r}$.
\section{Evaluation\label{sec:exp}}
\subsection{Experimental Setup}
We consider five tasks arriving sequentially with each sequential task containing a small amount of labeled training data (only 5$\%$ of training instances in case of DSADS \cite{altun2010human} and HAR \cite{anguita2012human}, and 10 engines data in Turbofan \cite{saxena2008damage} - FD001 ). The training data for each task undergoes an 80-20 split into a train and validation set, where the validation set is used for early stopping.
Apart from evaluating for fixed input dimensions (FID) and fixed classes across tasks, we consider: i. variable input dimensions (VID) across the tasks for all three datasets, and ii. variable target space across the tasks for the classification datasets.
We randomly remove 40\% of sensors in each task for VID scenario.
For \textit{incrementally increasing classes} scenario, one class is added for each new task while all the old classes persist in DSADS and HAR. For the \textit{partially changing classes} scenario, two and one old class(es) are replaced with two and one new class(es) for each new task in case of DSADS and HAR, respectively.
For DSADS and HAR datasets, we use sensor-wise z-normalization for each input dimension/sensor, whereas sensor-wise min-max normalization is used for the Turbofan dataset. We used classification error rates and root mean squared error (RMSE) as the performance metrics for the classification and regression, respectively. For each scenario, we consider five different sets\footnote{In variable target space, each set has different combination of classes.} of five sequentially arriving tasks. In Table \ref{tab:results}, we report the average value across five sets. Refer Appendix for further details on datasets and hyperparameters.
\subsection{Approaches Considered}
For comparison, we consider the following approaches:
\begin{table*}[h]
\caption{Performance comparison of various approaches on sequential tasks. *Note: UB with standard solver for VID scenario does not serve as an upper bound despite access to all past data due to difficulty in handling VID. Bold implies approach is statistically significance against others with $p < 0.05$. \textit{Our proposed approach GR-IG (ours) has significant gain compared to other approaches on most of the tasks, in both the scenarios FID and VID.}}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|c|c|cccccc||cccccc|ccccc|}
\hline
&&\multicolumn{7}{c||}{\textbf{Fixed Input Dimensions (FID) }}& \multicolumn{11}{c|}{\textbf{Variable Input Dimensions (VID)}} \\
\cline{3-20}
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Dataset}\\\\\\\end{tabular}}&
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Scenario}\\\\\\\end{tabular}}&
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Task}\\\textbf{ID}\\\\\end{tabular}} & \multicolumn{6}{c||}{\textbf{Standard Solver}} & \multicolumn{6}{c|}{\textbf{Standard Solver}} & \multicolumn{5}{c|}{\textbf{Solver-CM }} \\
\cline{4-20}
&&&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{TS}\\\end{tabular}}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB}\\\end{tabular}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{TS}\\\end{tabular}}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB$^*$}\\\end{tabular}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB}\\\end{tabular}}\\
\cline{7-8} \cline{13-14} \cline{18-19}
&&&\multicolumn{1}{l|}{} &\multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & &\multicolumn{1}{l|}{} &\multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & & \multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & \\
\hline
\parbox[t]{2mm}{\multirow{15}{*}{\rotatebox[origin=c]{90}{\textbf{DSADS}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fixed\\ Classes\\\end{tabular}}
&1 & 16.3 & 16.3 & 16.2 & 16.3 & 16.3 & \textit{16.3} & 27.8 & 27.8 & 29.2 & 27.8 & 27.8 & \textit{27.8} & 27.8 & 29.2 & 27.8 & 27.8 & \textit{27.8} \\
&&2&11.9 & 12.0 & 11.9 & 10.9 & \textbf{10.5} & \textit{9.0} & 29.4 & 30.4 & 27.3 & 29.9 & 28.6 & \textit{25.6} & 26.6 & 25.7 & 25.2 & \textbf{20.7} & \textit{13.9} \\
&&3&17.9 & 14.7 & 12.7 & 12.4 & \textbf{9.7} & \textit{6.5} & 29.6 & 26.2 & 28.3 & 36.5 & 25.1 & \textit{22.7} & 23.4 & 26.8 & 36.1 & \textbf{18.7} & \textit{11.3} \\
&&4&18.9 & 13.2 & 13.6 & 10.2 & \textbf{8.0} & \textit{4.5} & 26.1 & 26.9 & 25.1 & 58.3 & 23.6 & \textit{20.9} & 23.3 & 24.1 & 52.2 & \textbf{16.2} & \textit{7.7} \\
&&5&15.7 & 12.3 & 12.1 & 8.7 & \textbf{7.4} & \textit{4.0} & 26.1 & 24.1 & 25.3 & 54.7 & 22.4 & \textit{19.1} & 22.3 & 22.6 & 48.6 & \textbf{13.9} & \textit{7.0} \\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Incrementally\\ Increasing\\ Classes\\\end{tabular}}
&1 & 13.4 & 13.4 & 13.8 & 13.4 & 13.4 & \textit{13.4} & 19.1 & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} \\
&&2&15.6 & 14.9 & 13.8 & 13.2 & \textbf{11.1} & \textit{9.7} & 21.4 & 26.9 & 22.9 & 24.4 & 22.8 & \textit{18.8} & 25.6 & 20.6 & 21.2 & \textbf{15.3} & \textit{12.7} \\
&&3&12.8 & 11.9 & 10.9 & 11.3 & \textbf{9.2} & \textit{6.9} & 15.7 & 34 & 19.5 & 39.4 & 26.3 & \textit{25.0} & 24.3 & 18.6 & 33.5 & \textbf{13.5} & \textit{10.8} \\
&&4&11.7 & 8.4 & 9.4 & 7.5 & \textbf{6.0} & \textit{4.3} & 21.2 & 35.7 & 26.1 & 45.3 & 30.7 & \textit{23.7} & 28.9 & 23.6 & 38.3 & \textbf{13.9} & \textit{9.5} \\
&&5&9.2 & 8.3 & 8.4 & \textbf{6.3} & \textbf{6.1} & \textit{3.8} & 13.4 & 39.6 & 24.8 & 47.6 & 36.3 & \textit{27.9} & 27.8 & 12.3 & 43.1 & \textbf{11.8} & \textit{6.7}\\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Partially\\ Changing\\ Classes\\ \end{tabular}}
&1 & 13.4 & 13.4 & 13.8 & 13.4 & 13.4 & \textit{13.4} & 19.1 & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} \\
&&2&12.3 & 11.7 & 11.2 & 11.3 & \textbf{9.5} & \textit{8.0} & 16.5 & 29.9 & 20.5 & 26.7 & 23.1 & \textit{17.7} & 26.1 & 19.7 & 27.9 & \textbf{14.9} & \textit{13.1} \\
&&3&13.5 & 12.4 & 15.0 & 12.4 & \textbf{9.8} & \textit{7.4} & 18.5 & 25.4 & 19.4 & 32.4 & 25.4 & \textit{18.8} & 22.3 & 17.1 & 29.4 & \textbf{14.8} & \textit{13.4} \\
&&4&14.3 & 13.4 & 15.1 & 12.4 & \textbf{10.1} & \textit{7.2} & 19.3 & 23.7 & 23.3 & 41.2 & 22.0 & \textit{19.5} & 20.9 & 20.4 & 36.7 & \textbf{17.7} & \textit{13.9} \\
&&5&15.4 & 13.3 & 12.7 & 10.6 & \textbf{8.8} & \textit{6.5} & 18.6 & 23.4 & 19.7 & 30.5 & 21.7 & \textit{18.6} & 20.6 & 19.0 & 32.1 & \textbf{15.2} & \textit{11.6} \\
\hline \hline
\parbox[t]{2mm}{\multirow{15}{*}{\rotatebox[origin=c]{90}{\textbf{HAR}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fixed \\ Classes\\\end{tabular}}
&1 & 15.7 & 15.7 & 16.1 & 15.7 & 15.7 & \textit{15.7} & 30.2 & 30.2 & 30.9 & 30.2 & 30.2 & \textit{30.2} & 30.2 & 30.9 & 30.2 & 30.2 & \textit{30.2} \\
&&2&17.1 & 16.3 & 16.2 & 15.8 & \textbf{15.4} & \textit{14.4} & 41.5 & 46.9 & 41.2 & 46.9 & 40.1 & \textit{37.4} & 44.9 & 38.8 & 37.3 & \textbf{30.8} & \textit{27.7} \\
&&3&18.2 & 16.1 & 15.8 & 15.1 & \textbf{14.4} & \textit{12.8} & 34.4 & 42.7 & 33.2 & 52.1 & 34.0 & \textit{32.4} & 40.6 & 32.1 & 40.9 & \textbf{30.2} & \textit{26.2} \\
&&4&17.4 & 15.5 & 15.2 & \textbf{13.4} & 13.8 & \textit{11.9} & 34.6 & 45.7 & 32.2 & 72.2 & 31.4 & \textit{28.4} & 43.0 & \textbf{28.4} & 68.2 & \textbf{28.2} & \textit{23.6} \\
&&5&17.6 & 15.7 & 15.2 & 13.5 & \textbf{13.1} & \textit{11.4} & 35.0 & 31.3 & 32.1 & 72.9 & 29.0 & \textit{26.2} & 30.4 & 32.5 & 71.5 & \textbf{23.7} & \textit{20.1} \\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Incrementally\\ Increasing\\ Classes\\\end{tabular}}
&1 & 17.6 & 17.6 & 16.9 & 17.6 & 17.6 & \textit{17.6} & 23.7 & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} & 23.7 & 23.1 & 23,7 & 23.7 & \textit{23.7} \\
&&2&18.0 & 13.7 & 14.3 & 14.1 & \textbf{11.9} & \textit{10.7} & 20.3 & 19.6 & 23.2 & 23.3 & 18.6 & \textit{18.5} & 18.1 & 22.4 & 19.2 & \textbf{13.5} & \textit{11.0} \\
&&3&18.6 & 12.1 & 13.4 & 15.1 & \textbf{10.4} & \textit{8.5} & 23.8 & 24.5 & 25.7 & 36.3 & 24.6 & \textit{20.1} & 22.4 & 24.5 & 35.0 & \textbf{16.2} & \textit{14.0} \\
&&4&17.5 & 17.6 & 15.1 & 14.6 & \textbf{12.6} & \textit{10.2} & 27.6 & 31.2 & 28.7 & 41.7 & 35.1 & \textit{26.6} & 27.8 & 27.9 & 41.3 & \textbf{24.0} & \textit{19.6} \\
&&5&21.7 & 19.2 & 18.2 & 17.0 & \textbf{14.2} & \textit{11.6} & 28.4 & 30.3 & 31.3 & 50.7 & 36.9 & \textit{32.4} & 29.4 & 30.7 & 47.4 & \textbf{22.9} & \textit{19.6}\\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Partially \\Changing \\ Classes\\ \end{tabular}}
&1 & 17.6 & 17.6 & 16.9 & 17.6 & 17.6 & \textit{17.6} & 23.7 & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} \\
&&2&17.4 & 16.0 & 15.0 & 14.5 & \textbf{13.6} & \textit{12.1} & 28.1 & 31.1 & 27.5 & 35.4 & 31.2 & \textit{26.7} & 29.9 & 26.9 & 28.2 & \textbf{25.8} & \textit{23.0} \\
&&3&17.1 & 14.9 & 15.8 & 14.2 & \textbf{13.2} & \textit{11.9} & 31.1 & 32.9 & 29.9 & 42.0 & 37.9 & \textit{29.4} & 32.1 & 29.1 & 39.2 & \textbf{27.6} & \textit{24.7} \\
&&4&13.5 & 12.8 & 13.4 & \textbf{10.5} & \textbf{10.8} & \textit{8.0} & 22.2 & 27.4 & 20.4 & 41.8 & 31.8 & \textit{22.1} & 25.8 & 20.5 & 36.9 & \textbf{18.8} & \textit{15.1} \\
&&5&16.9 & 14.9 & 16.1 & 12.2 & \textbf{9.9} & \textit{7.8} & 23.3 & 31.2 & 23.0 & 38.4 & 30.5 & \textit{25.5} & 28.2 & 23.3 & 34.4 & \textbf{19.6} & \textit{15.2}\\
\hline \hline
\parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{90}{\textbf{Turbofan}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Not\\Applicable\\(Regression Task)\end{tabular}}
&1 & 16.7 & 16.7 & 17.4 & 16.7 & 16.7 & \textit{16.7} & 17.3 & 17.3 & 18.2 & 17.3 & 17.3 & \textit{17.3} & 17.3 & 18.2 & 17.3 & 17.3 & \textit{17.3} \\
&&2&17.7 & 17.3 & 16.9 & 16.9 & \textbf{16.2} & \textit{15.7} & 18.7 & 18.2 & 19.0 & 18.9 & 18.4 & \textit{17.4} & 17.9 & 18.7 & 17.7 & \textbf{16.8} & \textit{15.9} \\
&&3&18.9 & 17.6 & 17.1 & \textbf{15.8} & 16.3 & \textit{15.0} & 19.3 & 18.7 & 22.9 & 19.3 & 17.6 & \textit{16.2} & 18.1 & 18.8 & 18.3 & \textbf{17.1} & \textit{15.8} \\
&&4&18.2 & 17.0 & 17.5 & \textbf{15.6} & \textbf{15.7} & \textit{14.7} & 16.5 & 17.4 & 17.9 & 17.9 & 17.5 & \textit{15.3} & 17.1 & \textbf{16.2} & 16.8 & \textbf{15.7} & \textit{15.1} \\
&&5&16.2 & 16.7 & 16.8 & 15.5 & \textbf{15.2} & \textit{13.9} & 17.5 & 17.6 & 18.2 & 18.3 & 17.2 & \textit{16.1} & 16.3 & 16.6 & 17.6 & \textbf{16.0} & \textit{14.9}\\
\hline
\end{tabular}}
\label{tab:results}
\end{table*}
\begin{figure*}[h!]
\captionsetup[subfigure]{font=small,labelfont=small, justification=centering}
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[scale=0.39,trim={0.23cm 0.8cm 0.25cm 0},clip]{images/img/dsads_fixed_all_classes.pdf}
\caption{DSADS \\(Fixed\\ Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/dsads_fixed_openset_classes.pdf}
\caption{DSADS \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/har_fixed_all_classes.pdf}
\caption{HAR \\(Fixed \\Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/har_fixed_openset_classes.pdf}
\caption{HAR \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/turbofan_fixed.pdf}
\caption{Turbofan \\(Regression \\Task)}
\end{subfigure}
\vspace{2mm}
\centerline{\newline(A) FID Scenario}
\vspace{3mm}
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[scale=0.39,trim={0.23cm 0.8cm 0.25cm 0},clip]{images/img/dsads_var_all_classes.pdf}
\caption{DSADS \\(Fixed\\ Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/dsads_var_openset_classes.pdf}
\caption{DSADS \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/har_var_all_classes.pdf}
\caption{HAR \\(Fixed \\Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/har_var_openset_classes.pdf}
\caption{HAR \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/turbofan_var.pdf}
\caption{Turbofan \\(Regression \\Task)}
\end{subfigure}
\vspace{2mm}
\centerline{\newline(B) VID Scenario}
\vspace{3mm}
\caption{Percentage gain of respective solver model (Standard Solver in FID scenario and Solver-CM in VID scenario) over the TS standard solver model. Respective solver model $M_i$ when used along with GR-IG have +ve gain on previous tasks $\mathcal{T}_{j}$ ($j=1,\ldots, i-1$) indicating the ability of the solver model with GR-IG to retain the knowledge of past tasks. TS standard solver model and solver model with GR-CG struggle to perform well on previous tasks, especially in the VID scenario. Our approach GR-IG is better than SMT indicating the advantage of generated previous tasks data from generative model and CM over SMT, which only uses the current task data to learn task-specific supermask. \label{fig:continual}}
\end{figure*}
\begin{enumerate}
\item \textbf{Task-Specific (TS)} is the vanilla approach or baseline which uses only the data from current task, i.e. $\mathcal{D}_i$, for learning the task-specific model $M_i$. This approach serves as a lower bound for evaluation.
\item \textbf{Fine-tune (FT)}: This approach also uses the data from the current task only (as in TS) but the parameters of the current solver model $M_i$ are initialized using those\footnote{For the final classification layer in case of incrementally increasing and partially changing classes, relevant parameters are used.} of the previous solver model $M_{i-1}$.
\item \textbf{Common Generator (GR-CG)} \cite{shin2017continual}: This approach maintains a single common generator (CG) across all previous tasks to generate samples from all of them (refer Section \ref{sec:approach}), while the solver model can be randomly initialized vanilla GRU model or the proposed conditioning module based solver model.
\item \textbf{Supermasks with Transfer (SMT)} \cite{wortsman2020supermasks}: This approach maintains the same core solver model but learns task-specific supermasks. The task-specific supermask allows the model to retain knowledge to solve each task in different units of the same core base neural network. The supermask weights for the new task are initialised from the running mean of all the previous tasks supermask weights, resulting in transfer of knowledge. We found this initialization strategy to work better than the vanilla `Supermasks without Transfer' (SM) approach (in concurrence with the observations made in \cite{wortsman2020supermasks}), and report results for SMT alone.
\item \textbf{Upper Bound (UB)}: This approach assumes access to all relevant original data from previous tasks. This approach is equivalent to having an ideal generator and solver for all previous tasks. Though access to original data from previous tasks is assumed, the available dimensions across tasks vary for the VID scenario while all dimensions are available for the FID scenario.
\end{enumerate}
Our proposed approach using independent generators for each previous task is referred to as \textbf{GR-IG}.
We refer to the GRU-based solver model using the GNN-based conditioning module as \textbf{Solver-CM}, while the vanilla GRU-based solver as \textbf{Standard Solver}.
\subsection{Results and Observations}
We make following key observations from Tables \ref{tab:results} and \ref{tab:Heterogeneousresults}:
\begin{itemize}[leftmargin=*]
\item As more tasks arrive, the gap between TS and UB widens, indicating scope for positive-transfer from old tasks to new tasks; this is expected as the underlying dynamical system is the same across tasks. We observe the same empirically: solvers using generative models (GR-CG and GR-IG) significantly improve upon TS in the FID scenario. For the partial changing classes scenario, GR-IG improves upon the baseline while GR-CG struggles, indicating the ability of GR-IG to provide better quality generated data. This is further helped by estimated labels from task-specific solvers, while GR-CG does not have this advantage.
\item \textit{GR-IG performs better than GR-CG in most cases} with FID as well as VID, proving the advantage of GR-IG over GR-CG. The samples generated via GR-CG are the combination of all previous tasks with diversity across classes (and input dimensions in VID scenario), which cannot be captured by one VAE.
\item SMT performs better than TS on FID scenario with standard solver whereas, on VID scenario SMT with Standard Solver performs worse than TS mostly in incrementally increasing classes and partial changing classes. The gains observed when using Solver-CM instead of Standard Solver for VID scenario confirms the advantage of GNN-based conditioning module for dimension-adaptability.
Our proposed approach always performs better than SMT in both FID and VID scenarios, showing the advantage of using the independent generator (GR-IG) for each task.
\item For scenarios with partially-observed sensors, i.e. VID scenario, GR-CG performs worse than TS. Unlike the FID scenario, even GR-IG and UB with access to all data from previous tasks fail to improve upon TS when using standard solver.
Furthermore, vanilla transfer via FT models is ineffective in VID scenario.
These observations confirm the \textit{non-trivial nature of the VID problem}, and inability of GR-CG to adapt to variable dimensions.
\textit{Only Solver-CM with GNN-based Conditioning Module for dimension-adaptability and GR-IG for generating past data consistently performs better than the TS on VID scenarios}. Combination of Solver-CM and GR-IG is the best performing model in most scenarios across datasets and tasks.
\item \textit{Continual learning}:
We also evaluate how well Solver-CM in VID scenario and standard solver in FID scenario with GR-IG performs on tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$. As shown in Fig. \ref{fig:continual}, in both FID and VID scenarios, we observe that while the TS model on one task struggles to generalize on other tasks, our approach performs better than SMT and GR-CG and is comparable to the UB in all scenarios. This shows that the \textit{samples generated from GR-IG are better than GR-CG in both FID and VID scenarios and also solvers with CM (GNN-based conditioning module) are able to adapt variable dimension in VID scenario}, which in-turn helps to solve the previous tasks better by virtue of capabilities similar to generative replay \textit{GR-CG} as in \cite{shin2017continual}.
\begin{table}[ht]
\centering
\caption{Performance comparison for generalization to unseen sensor combinations at test time. \textit{Solver-MH}: the conditioning module (CM) is removed and the conditioning vector is replaced with a multi-hot vector indicating the active sensors. \textit{Solver-SE}: the GNN is removed and replaced with a simple max-pooling over the sensor embeddings. Standard Solver-A: the upper bound that assumes access to all sensors in the train and test set. \textit{In both the settings, No Fine-tuning and Fine-tuning, Solver-CM generalizes better on unseen sensor combination than other solver models.} }
\scalebox{0.82}{
\begin{tabular}{|c|cccc|cccc|c|}
\hline
&\multicolumn{4}{c}{\textbf{No Fine-tuning}} &
\multicolumn{4}{|c|}{\textbf{Fine-tuning}} & \\
\cline{2-9}
&\multicolumn{8}{c|}{\textbf{Solver-}}& \\
\cline{2-9}
{\begin{tabular}[c]{@{}c@{}}\textbf{Dataset}\\\\\end{tabular}}
&{\begin{tabular}[c]{@{}c@{}}\textbf{Stan-}\\\textbf{dard}\end{tabular}} &{\begin{tabular}[c]{@{}c@{}}\textbf{
MH}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{CM}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{Stan-}\\\textbf{dard}\end{tabular}} &{\begin{tabular}[c]{@{}c@{}}\textbf{MH}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{CM}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{Standard}\\\textbf{Solver-A}\\\textbf{(UB)}\end{tabular}} \\
\hline
DSADS
& 8.3 &8.9&7.8& \textbf{5.9} & 4.7 &6.1& 3.4 & \textbf{3.3} & 1.5 \\
\hline
HAR
& 12.8& 13.5 &11.8& \textbf{11.4} & 12.1 & 12.4&10.7 & \textbf{10.5} & 6.3 \\
\hline
Turbofan
& 15.5 & 16.4&14.9&\textbf{14.6} & 16.2 &15.6&14.7 & \textbf{14.2} &13.6 \\
\hline
\end{tabular}
}
\label{tab:Heterogeneousresults}
\end{table}
\item \textit{Ablation Study of Solver-CM}: We consider VID multi-sensor tasks with 100\% training instances instead of the scarce training instances in the sequential-tasks setup discussed above. The training time-series correspond to certain combinations of sensors, while the test set contains sensor combinations unseen during training.
Any time-series instance in train or test set has $\approx 0.4$ fraction sensors missing. For all datasets, 40\% of the instances are used for training, 10\% for validation, 10\% for fine-tuning (ignored in no fine-tuning setting), and remaining 40\% for testing. From Table \ref{tab:Heterogeneousresults} we observe that Solver-SE is better than Solver-MH, indicating the advantage of using the compressed information of active sensors in the form of sensor embeddings over a multi-hot vector. But still, Solver-SE fails to generalize on unseen sensors combinations. We use GNN and sensor embeddings in Solver-CM to improve further, where GNN allows to exchange messages between the active sensors. These messages passing amongst the active sensors inherently allow for combinatorial generalization on unseen sensor combinations without fine-tuning. For fine-tuning, we consider small amounts of data for the same sensor combinations as the test set, which further improves the results.
\end{itemize}
In summary, while GR-CG class of methods like \cite{shin2017continual,velik2014brain} would usually suffice to deal with lack of access to past data by providing generated data, and SMT \cite{wortsman2020supermasks} would allow to adapt across tasks with varying classes via supermasks or similar methods, neither class of methods suffice for the variable input dimensions scenario as indicated by their inferior performance in comparison to GR-IG with CM-based solver. Furthermore, conditioning module based on GNNs is critical to the success of the proposed approach.
\section{Conclusion and Future work}
In this work, we have motivated a practical problem of enabling knowledge transfer from previous tasks to new tasks under restricted data-sharing across sequentially arriving multi-sensor tasks with varying input dimensions. We have proposed a deep learning architecture to deal with
multivariate time series classification and regression tasks that works across the partially-observed sensor (variable input dimensions) tasks as well as changing classes across tasks.
Our results on three publicly available datasets prove the efficacy of the proposed approach to leverage generated data from previous tasks to improve performance on (partially-observed multi-sensor) new tasks, without any explicit original data exchange.
Though we have evaluated the proposed approach for activity recognition and remaining useful life estimation applications, we believe the proposed approach is generic and would be useful in other multivariate time series applications.
Furthermore, combining the benefits of generative replay as we have presented, with meta-learning (e.g., on lines of \cite{riemer2018learning})
to further improve transfer of the solver models used in our approach is an interesting future direction to explore
especially in context of variable-dimensional input spaces. Extensions of this work in privacy-preserving scenarios is another interesting future direction.
\bibliographystyle{abbrv}
\section{Introduction}
With the increasing adoption of Internet of Things (IoT) \cite{da2014internet} technology, sensor data from various situations ranging from wearables to industrial machines is becoming increasingly available. Deep neural networks (DNNs) have been successfully applied on such data for forecasting, classification, anomaly detection, diagnostics, and prognostics \cite{p:lstm-ad,lai2018modeling,khan2018review}, \cite{wang2019deep},
with several applications in healthcare, utilities, etc.
Further, as shown in \cite{TIMENET, fawaz2018data}, a deep learning model trained on time-series across domains shows improved performance
on other domains. The potential for transfer across tasks is expected to be even more likely when tasks are similar, e.g., different instances of the same underlying dynamical system, e.g. humans in the case of activity recognition, or similar machines (engines/turbines, etc.) in the case of equipment health monitoring.
In the extreme case, we might even like to consider data from all such instances as one dataset.
In the existing approaches, we face two important challenges in learning across a sequence of such multi-sensor tasks: (i) The number of sensors
used in each instance often differ, e.g., different people may be using different subsets of equipment (e.g., wrist-bands, phones, sensor-equipped shoes, etc.), and at different times. Similarly, machines installed in different factories, vehicles, or OEMs may be installed with a smaller or larger set of sensors often (e.g., temperature, vibration, pressure, etc.). However, most deep learning models typically assume fixed dimensional inputs and can handle variable temporal dimensions, i.e. variable length of the time series, but fail to adapt and generalize to variability in the number of available dimensions in multivariate
time series, where one or more dimensions are completely missing. Further, it may often be the case that some patterns (e.g., activities) cannot be detected using fewer sensors, and so tasks may also differ in the number of classes being encountered.
(ii) While it is becoming increasingly common for artificial intelligence (AI) software providers to retain the right to continuously improve their machine learning models as new data is encountered, legal restrictions usually prohibit the storage and use of customer data beyond a certain limit, e.g. for privacy reasons
in the case of human activities, and even in the case of industrial data, the data used to train AI software legally belongs to the customer
and cannot be stored and used by the software vendor forever. Since tasks are encountered sequentially over time, a software vendor cannot use standard training techniques requiring multiple passes over all datasets to improve their models over time.
Motivated by the above, we consider a setting where multiple tasks with scarce and partially-observed sensor data, i.e., {\bf variable input dimensions} and {\bf classes} arrive sequentially. We would like to continuously improve a deep learning model as tasks are encountered, {\bf without being able to access data from previous tasks}, so as to {\it improve performance on subsequent tasks as well as on previous tasks}. (The latter requirement is actually important in practice as it is the rationale used by AI software vendors to claim the right to improve their models using customers' data, i.e., that this will enable future versions of the software to automatically perform better on their original task.) Note that this setting is related to the {\it online} continual learning problem \cite{kirkpatrick2017overcoming}, as also class-incremental learning \cite{kundu2020universal} where (in the case of classification) tasks may have varying subsets of classes. However, such approaches do not typically consider the variability in input dimensions, which is the main focus of this work.
Our key contributions can be summarized as follows:
\begin{itemize}[leftmargin=*]
\item We motivate and formulate a novel and practically important problem of {\bf continual learning} from partially-observed multi-sensor (multivariate time-series) data, with a focus on the need to adapt to {\bf varying input dimensions (VID)} and classes across tasks with {\bf data-access prohibited across tasks}.
\item We propose a novel modularized neural network architecture to handle variable input dimensions, with two main modules:
i. a {\bf core dynamics module} comprising an RNN that models the underlying dynamics of the system,
ii. a {\bf conditioning module using a graph neural network} (GNN \cite{battaglia2018relational}) that adjusts the activations of the core dynamics module for each time-series based on the combination of sensors available, effectively exhibiting different behavior depending on the available sensors.
\item To leverage partially-observed sensor data, we build upon the idea of generative replay \cite{shin2017continual}, where task-specific generators generate variable dimensional time-series for subsequent processing by the task-solving core dynamics and conditioning modules.
\item We present an extensive empirical evaluation on diverse tasks from three real-world datasets to prove the advantage of our approach in contrast to vanilla architectures in terms of the ability to adapt to variable input dimensions and changing classes.
\end{itemize}
\begin{figure*}[h]
\centering
\includegraphics[scale=0.17,trim={3cm 22.25cm 0cm 14.5cm}, clip]{images/Approach.pdf}
\caption{Scenarios considered (A and B). \label{fig:approach}}
\end{figure*}
\section{Problem Definition\label{sec:prob}}
Consider $m$ tasks $\mathcal{T} = (\mathcal{T}_{1}, \cdots, \mathcal{T}_{m})$
arriving sequentially.
A task $\mathcal{T}_{i}$ ($i = 1, 2, \cdots, m)$, consists of $N$ instances of labeled multi-sensor data $\mathcal{D}_i = \{(\mathbf{x}_{i,r},y_{i,r})\}_{r=1}^N$, each time-series $\mathbf{x}_{i,r} \in \mathcal{X}_i$ with an associated target $y_{i,r} \in \mathcal{Y}_i$. The target space $\mathcal{Y}_i$ corresponds to class labels in case of classification tasks, or real-valued numbers (or vector) in case of regression tasks.
The goal in each task is to learn a \textit{solver model} ${M}_{i}$ with trainable parameters $\boldsymbol{ \theta}_i$ such that $M_i\!\!:\mathcal{X}_i \!\rightarrow \mathcal{Y}_i$, with estimated target $\hat{y}_{i, r} = {M}_{i}(\mathbf{x}_{i, r}; \boldsymbol{\theta}_{i})$.
Any task $\mathcal{T}_i$ is not allowed access to data from previous tasks $\mathcal{T}_j$ ($j={1,\ldots,i-1}$), but can leverage relevant generated samples obtained using generative models of these previous tasks.
We consider the following partially-observed multi-sensor tasks under the above-mentioned basic setup:
i. tasks have a \textit{variable input space}, as illustrated in Fig. \ref{fig:approach}A, i.e. the input dimensions or available sensors change across tasks, i.e. for $r$-th time-series $\mathbf{x}_{i,r} = \{\mathbf{x}_{i,r}^t\}_{t=1}^{T_i}$ of length $T_i$ with $\mathbf{x}_{i,r}^t \in \mathbb{R}^{d_i}$, $ d_i \in \{1,2,\ldots,d\}$ is the number of available dimensions or sensors out of a maximum of $d$ possible dimensions. We refer to the set of available dimensions in $\mathcal{T}_i$ as $\mathcal{S}_i \subseteq \mathcal{S} = \{s_1,s_2,\ldots,s_d\}$, where $\mathcal{S}$ is the set of all the possible sensors.
ii. \textit{variable target space}: where classes of interest can vary across tasks, as illustrated in Fig. \ref{fig:approach}B; this is applicable for classification tasks only (but not for regression tasks).
The variable target space can correspond to either i. \textit{incrementally increasing classes}, where new classes are added for each new task while all old classes persist, or ii. a subset of old classes are present in the new task along with some new classes, i.e. the \textit{partially changing classes}.
\section{Related Work}
\subsection{Handling variable-dimensional time-series}
Several approaches in literature deal with variations in input data along temporal or spatial dimensions, i.e. variable length of time-series, or variable image sizes, e.g. via recurrent neural networks (RNNs) \cite{cho2014learning} for the time dimension, and variants of pooling operations for image applications \cite{he2015spatial}.
Several approaches for handling missing values in multi-sensor data via neural networks have been proposed.
For instance, \cite{che2018recurrent} study missing value problem in multi-sensor data by proposing a variant of the gated recurrent units (GRUs) \cite{cho2014learning} using knowledge of which dimensions of the input are missing and for how long.
However, handling VID is a more difficult problem compared to the missing value problem in time-series.
Approaches dealing with missing values in time-series are not directly applicable in the VID setting where one or more dimensions of the time-series are completely missing (i.e. the missing \%age for the dimension/sensor is 100\%). Since these approaches would typically rely on one or more past values to extrapolate, and do not consider scenarios where a dimension is completely missing from first to last time step.
The generalization ability of GNNs across varying nodes has been studied in, e.g. GRU-CM \cite{gupta2020handling} and NerveNet \cite{wang2018nervenet} use the inherent ability of GNNs to generalize well on unseen combinations of test instances that are different from training instances. In GNMR \cite{narwariya2020graph}, GNNs leverage the knowledge of readily available graph structure to process multi-sensor time series data for the Remaining Useful Life (RUL) estimation in equipment health monitoring. However, these approaches neither take into account data-access restrictions across tasks nor variability in target spaces.
Modular Universal Reparameterization (MUiR) \cite{meyerson2019modular} attempts to learn core neural modules which can be transferred across varying input and output dimensions.
It relies on learning the core module by solving several sets of architecture-task problems with varying input and output dimensions.
However, MUiR does not study multi-sensor data setting, and relies on solving several tasks to learn the core neural module. Instead, our approach relies on a conditioning vector obtained via GNNs to allow adaptability to VID, and addresses additional constraints on simultaneous task-access as required by MUiR.
Recently, a neuro-evolutionary approach has been proposed in \cite{elsaid2020neuro} which studies the problem of structure-adaptive for time-series prediction. It relies on a series of mutation operations and crossover operations over the neural units. Instead of the computationally expensive neuro-evolutionary approaches, we take a different perspective on the problem of adapting to varying dimensionality where graph neural networks (GNNs) are used.
The recently proposed CondConv \cite{yang2019condconv} is similar in spirit to our work, i.e. it attempts to dynamically adapt the parameters of the neural network conditioned on the current input. While it focuses on adapting to each input with same dimensionality, our work focuses on adapting to inputs of varying dimensionality.
Though significantly different in implementation and the end-objective, our approach also draws inspiration from such works, including \cite{rosenbaum2019routing,andreas2016neural}, where the parameters of the core neural network are dynamically adjusted as per the input.
\subsection{Continual Learning}
We review three recent approaches for continual learning. i. Experience replay: This approach maintains a storage buffer for rehearsing all previous tasks, e.g. \cite{rolnick2018experience,isele2018selective} and uses them along with new task data to avoid catastrophic forgetting. Our work is orthogonal to this approach, as real data-access prohibited across tasks. ii. Generative replay: Instead of storing real previous tasks data, a generative model can be used to mimic the real tasks distribution, e.g. \cite{shin2017continual,velik2014brain}. The generated samples of all previous tasks are used along with current tasks to adapt on old tasks. iii. Task-specific model: Instead of storing or replaying previous tasks data, different model components such as expanding model for each new task \cite{yoon2017lifelong} or using supermask to select task-specific subnetworks, while keeping the weights of model \cite{wortsman2020supermasks} fixed to avoids catastrophic forgetting.
However, these approaches focus on variable classes across tasks, and do not usually consider the VID setting. In fact, as we show empirically in Section \ref{sec:exp}, these approaches struggle in the VID setting.
\section{Approach\label{sec:approach}}
Our approach consists of learning a series of generator and solver models, one for each task, as shown in Fig. \ref{fig:approach}A and \ref{fig:relevant_samples}.
For any task $\mathcal{T}_i$, a solver model ${M}_i$ is trained in a supervised manner using labeled data, a generator model ${G}_i$ is trained in an unsupervised manner as a generative model for the data $\mathcal{D}_i$ from $\mathcal{T}_i$.
We denote the original and generated data for $\mathcal{T}_i$ as $\mathcal{D}_i$ and $\hat{\mathcal{D}}_i$, respectively.
While ${G}_i$ generates unlabeled time-series data, the corresponding labels are estimated using the learned ${M}_i$.
This labeled generated data $\hat{\mathcal{D}}_i$ serves as additional training data for subsequent data-scarce tasks $\mathcal{T}_j$ ($j=i+1,\ldots,m$) instead of $\mathcal{D}_i$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.2,trim={35cm 25.25cm 31cm 12.0cm}, clip]{images/relevant_samples.pdf}
\caption{Using relevant generator and solver for augmenting training data for task ($\mathcal{T}_i$). \label{fig:relevant_samples}}
\end{figure}
\begin{figure*}[h!]
\centering
\includegraphics[scale=0.215,trim={22cm 18.25cm 25cm 8cm}, clip]{images/Solver_CM.pdf}
\caption{Flow-diagram of the Solver-CM model using GNN-based Conditioning Module. The available sensors in a time series and corresponding active embeddings and nodes are shown in color while the missing sensor/dimension is shown in grey color (dashed). The active nodes corresponding to the available sensors exchange messages with each other via a graph neural network, and generate a conditioning vector that is used by the core dynamics module (a recurrent neural network) processing the time series to estimate the target. \label{fig:solver_CM}}
\end{figure*}
\subsection{Generator Models}
Our approach using generated data for subsequent tasks is similar in spirit to existing work in generative replay (GR) for continual learning, e.g. \cite{shin2017continual,achille2018life}.
However, instead of maintaining a common generator (CG) across tasks, we maintain a separate independent generator (IG) ${G}_i$ for each task. We refer prior work such as \cite{shin2017continual} relying on CG as \textbf{GR-CG}, and our proposed approach relying on IG as \textbf{GR-IG}.
This design choice of having an GR-IG per task is crucial to handle VID across tasks: when dimensions across tasks vary, a GR-CG with fixed input dimensions $d$ cannot easily adapt to the missing $d-d_i$ dimensions in $\mathcal{T}_i$ as there is no training signal for those dimensions. Even if the GR-CG can model the $\mathcal{S}_i$ sensors of the current task $\mathcal{T}_i$ well, but it fails to adapt to a new combination of sensors in $\mathcal{S}_{i+1}$ from the task $\mathcal{T}_{i+1}$.
In contrast, while training the GR-IG ${G}_i$, the input dimensions correspond only to the original relevant sensors $\mathcal{S}_i$ for that task, and the burden of handling VID is passed on to the solver model ${M}_i$ whose novel modular structure relying on GNNs is specifically designed to be able to handle VID data.
We analyze and validate the advantage of GR-IG over GR-CG empirically in Section \ref{sec:exp}.
In this work, we use variational autoencoders (VAEs) \cite{kingma2013auto,achille2018life} as generative models for the data distribution $p(\mathcal{D})$. We use VAEs for ease of training. However, the proposed approach is agnostic to the choice of generator, and other generative models such as generative adversarial networks (GANs, as used in \cite{shin2017continual}), are equally applicable. The encoder and decoder of VAEs are RNNs with GRUs that allow for capturing the temporal dependencies in multi-sensor data. The loss function is the same as in standard VAEs (refer \cite{kingma2013auto} and Appendix for more details).
\subsection{Solver Model}
The training data $\tilde{\mathcal{D}}_i$ for the solver model ${M}_{i}$ consists of original data $\mathcal{D}_i$ from $\mathcal{T}_i$ as well as relevant\footnote{We also consider the scenario of using all classes instead of relevant classes samples from the previous generated tasks data, but found using the relevant classes samples performs better, especially in partially changing classes scenario.} generated data $\bigcup_{j=1}^{i-1}\hat{\mathcal{D}_{j}}$ from the previous tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$. This inevitably results in VID data if different tasks have different dimensions.
However, the training data across tasks can have VID as well as variable target space. VID is handled via solver model, whereas variable target space is handled by using only those generated samples from the previous tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$ for which the estimated class labels correspond to classes in $\mathcal{T}_i$, as illustrated in Fig. \ref{fig:approach}B.
Any point $\mathbf{x}^t$ at a timestep $t$ in time-series $\mathbf{x} \in \tilde{\mathcal{D}}_i = \bigcup_{j=1}^{i-1}\hat{\mathcal{D}_{j}} \cup \mathcal{D}_i$, is mapped to a vector in $\mathbb{R}^d$ with a fixed consistent indexing on the sensors in $\mathcal{S}$, irrespective of the number of available sensors in $\mathbf{x}$. The unavailable sensors are mean-imputed, i.e. the mean value of the sensor across other instances where it is available is used.
The solver model consists of the two modules, as illustrated in Fig. \ref{fig:solver_CM}:
i. \textit{core dynamics module} (CDM), which is a standard GRU network that learns the dynamics of the system and ingests a fixed-dimensional time-series where the missing dimensions are imputed with a constant (mean) value,
ii. \textit{conditioning module} (CM), which generates a ``conditioning vector'' as a function of the set of available sensors irrespective of the readings those sensors take for a particular time-series instance as shown in Appendix Fig. \ref{fig:Conditioning module}.
This conditioning vector is passed as an additional input to the CDM allowing it to adjust its internal computations and activations according to the combination of available sensors.
The conditioning vector is in-turn obtained from ``sensor embedding vectors'' via a GNN.
Note that the CDM and the CM along with the sensor embeddings are trained in an end-to-end fashion via stochastic gradient descent (SGD). We next provide architecture details of the two modules.
\subsubsection{Conditioning Module (CM)}
The CM based on GNNs is one of the crucial modules in the overall architecture as it allows to handle missing dimensions (variables) in multivariate time series. For this, we consider a fully-connected bi-directional graph where each node corresponds to a sensor, such that each active node (corresponding to available sensors in a time series) in the graph is connected to every other active node.
Any node in the graph is updated by using aggregated messages from its neighboring nodes. The conditioning vector is obtained as a summary of the messages exchanged between the active nodes. This conditioning vector guides the core dynamics module (CDM) about the current ``context'', i.e. available sensors, so that the computations of the CDM can be adjusted according to the available sensors.
More specifically, corresponding to the set of sensors $\mathcal{S}$, consider a graph $G(\mathcal{V},\mathcal{E})$ with nodes or vertices $\mathcal{V}$ and edges $\mathcal{E}$, with one node $v_s \in \mathcal{V}$ for every $s \in \mathcal{S}$ such that $|\mathcal{V}|=|\mathcal{S}|$.
The set of neighboring nodes for any node $v_s$ is denoted by $\mathcal{N}_{G}(v_s)$.
Each sensor $s \in \mathcal{S}$ is associated with a learnable embedding vector $\mathbf{v}_s \in \mathbb{R}^{d'}$, where $d'$ is a hyperparameter.
While training a model for $\mathcal{T}_i$, the possible dimensions or sensor combinations can be any of $\mathcal{S}_1, \mathcal{S}_2,\ldots,\mathcal{S}_i$, corresponding to the sensor combinations present in $\hat{\mathcal{D}}_1, \hat{\mathcal{D}}_2, \ldots,\hat{\mathcal{D}}_{i-1}$ or $\mathcal{D}_i$. For a given set or combination of sensors $\mathcal{S}_j$ ($j=1,\ldots, i$), only the corresponding nodes $\mathcal{V}_j \subseteq \mathcal{V}$ are considered to be \textit{active}, and contribute to obtaining the combination-specific conditioning vector $\mathbf{v}_{\mathcal{S}_j}$.
For the active nodes, the graph is assumed to be \textit{fully-connected} such that each active node in the graph is connected to every other active node.
Any edge is active only if both the participating nodes are active.
The GNN corresponding to this graph consists of a node-specific feed-forward network $f_n$ and an edge-specific feed-forward network $f_e$; $f_n$ and $f_e$ are shared across the nodes and edges in the graph, respectively.
For any active node $v_a \in \mathcal{V}_i$, the node vector $\mathbf{v}_a$ is updated using the GNN as follows:
\begin{align}
\mathbf{v}_{al} &= f_{e}([\mathbf{v}_{a},\mathbf{v}_{l}]; \boldsymbol{\theta}_{i}^{e}) \label{eq1}, \quad \forall v_l \in \mathcal{N}_{G}(v_a),\\
\tilde{\mathbf{v}}_{a} &= f_{n}([\mathbf{v}_{a},\sum_{\forall l} \mathbf{v}_{al}]; \boldsymbol{\theta}_{i}^{n}) \label{eq2},
\end{align}
where $f_{e}$ and $f_{n}$ both consist of feedforward layers with Leaky ReLUs \cite{maas2013rectifier} and dropout \cite{srivastava2014dropout}, and with learnable parameters $ \boldsymbol{\theta}_{i}^{e}$ and $ \boldsymbol{\theta}_{i}^{n}$, respectively.
While $f_{e}$ computes the message from node $v_l$ to $v_a$, $f_{n}$ updates the node vector $\mathbf{v}_a$ to $\tilde{\mathbf{v}}_a$ using messages from its neighboring nodes (Eqn. \ref{eq2}).
Finally, the conditioning vector $\mathbf{v}_{\mathcal{S}_j}$ specific to the set of sensors $\mathcal{S}_j$ is obtained from the updated node vectors as
\begin{equation}\label{eq3}
\mathbf{v}_{\mathcal{S}_j} = \mathtt{max}(\{\tilde{\mathbf{v}}_a\}_{v_a \in \mathcal{V}_j}),
\end{equation}
where $\mathtt{max}$ returns the dimension-wise maximum value\footnote{We also tried averaging instead of max operation to summarize the vectors of available sensors, but found max to work better.} across the updated node vectors.
It is noteworthy that the summation over the messages across nodes in Eqn. \ref{eq2} and the $\mathtt{max}$ operation in Eqn. \ref{eq3} essentially provide the desired ability to process VID.
\subsubsection{Core Dynamics Module (CDM)}
As mentioned earlier, any time-series $\mathbf{x}_{i, r} \in \tilde{\mathcal{D}}_i$ is first converted to the $d$-dimensional time-series $\tilde{\mathbf{x}}_{i, r}$ with mean-imputation for the unavailable sensors.
This time-series along with its conditioning vector, say $\mathbf{v}_{\mathcal{S}_j}$ (assuming this time-series comes from generated data $\hat{\mathcal{D}}_j$) for the task $\mathcal{T}_{i}$ are processed by the CDM as follows:
\begin{align}
\mathbf{z}^t_{i, r} &= GRU([\tilde{\mathbf{x}}_{i, r}^t, \mathbf{v}_{\mathcal{S}_j}],\mathbf{z}^{t-1}_{i, r}; \boldsymbol{\theta}_{i}^{GRU}), \quad t:1,\ldots,T_i \label{eq6}\\
\hat{y}_{i, r} &=f_{o}(\mathbf{z}^{T_i}_{i, r};\boldsymbol{\theta}_{i}^{o})\label{eq7},
\end{align}
where $GRU$ is a multi-layered RNN having $\boldsymbol{\theta}_{i}^{GRU}$ learnable parameters that gives feature vector $\mathbf{z}^{T_i}$ at the last timestep $T_i$, and $\mathbf{v}_{\mathcal{S}_j}$ is concatenated with $d$-dimensional sensor values at each time-step.
At last, the estimate $\hat{y}_{i, r}$ for $y_{i, r}$ is obtained via $f_{o}$ consisting of ReLU layer(s) followed by softmax or sigmoid layer corresponding to classification or regression, respectively, with $\hat{y}_{i, r} \in [0,1]^K$ for classification, and $\hat{y}_{i, r} \in \mathbb{R}$ for regression. For the RUL estimation regression task, we use min-max normalized targets such that they lie in $[0,1]$.
All the steps described above from the Eqn. \ref{eq1} to Eqn. \ref{eq7} can be consolidated as follows:
$\hat{y}_{i,r} = {M}_{i}(\mathbf{x}_{i,r}; \boldsymbol{\theta}_{i})$,
where $\boldsymbol{\theta}_{i}$ includes all the parameters $(\boldsymbol{\theta}_{i}^{e}, \boldsymbol{\theta}_{i}^{n}, \boldsymbol{\theta}_{i}^{GRU}$ and $ \boldsymbol{ \theta}_{i}^{o})$ of $M_{i}$.
\section{Training Objective}
Denoting the training objectives for classification (standard cross-entropy loss) or regression (squared-error loss) as $\mathcal{L}(y_{i, r}, \hat{y}_{i, r}; \boldsymbol{\theta}_{i})$, we have:
\begin{dmath}
\mathcal{L}_i= \frac{q}{N}\sum_{r = 1}^{N}\mathcal{L}\big(y_{i, r}, \hat{y}_{i, r}; \boldsymbol{\theta}_{i}\big) + \Big(\frac{1 - q}{(i-1)N}\Big)\sum_{j = 1}^{i-1}\sum_{r=1}^{N}\mathcal{L}\Big(y_{j,r}, {M}_{i}(\mathbf{\hat{x}}_{j,r}; \boldsymbol{\theta}_i); \boldsymbol{\theta}_{i}\Big),
\end{dmath}
where, $0 < q \leq 1$ is the importance given to data from $\mathcal{T}_i$ and $N$ is the number of instances. Higher values of $q$ imply more weightage to samples from $D_i$. In this work, we use equal weightage to all samples, i.e. $q=0.5$. Note that $y_{j,r}$ is the label corresponding to class with maximum probability as given by the solver model ${M}_{j}$ for the generated sample $\mathbf{\hat{x}}_{j,r}$.
\section{Evaluation\label{sec:exp}}
\subsection{Experimental Setup}
We consider five tasks arriving sequentially with each sequential task containing a small amount of labeled training data (only 5$\%$ of training instances in case of DSADS \cite{altun2010human} and HAR \cite{anguita2012human}, and 10 engines data in Turbofan \cite{saxena2008damage} - FD001 ). The training data for each task undergoes an 80-20 split into a train and validation set, where the validation set is used for early stopping.
Apart from evaluating for fixed input dimensions (FID) and fixed classes across tasks, we consider: i. variable input dimensions (VID) across the tasks for all three datasets, and ii. variable target space across the tasks for the classification datasets.
We randomly remove 40\% of sensors in each task for VID scenario.
For \textit{incrementally increasing classes} scenario, one class is added for each new task while all the old classes persist in DSADS and HAR. For the \textit{partially changing classes} scenario, two and one old class(es) are replaced with two and one new class(es) for each new task in case of DSADS and HAR, respectively.
For DSADS and HAR datasets, we use sensor-wise z-normalization for each input dimension/sensor, whereas sensor-wise min-max normalization is used for the Turbofan dataset. We used classification error rates and root mean squared error (RMSE) as the performance metrics for the classification and regression, respectively. For each scenario, we consider five different sets\footnote{In variable target space, each set has different combination of classes.} of five sequentially arriving tasks. In Table \ref{tab:results}, we report the average value across five sets. Refer Appendix for further details on datasets and hyperparameters.
\subsection{Approaches Considered}
For comparison, we consider the following approaches:
\begin{table*}[h]
\caption{Performance comparison of various approaches on sequential tasks. *Note: UB with standard solver for VID scenario does not serve as an upper bound despite access to all past data due to difficulty in handling VID. Bold implies approach is statistically significance against others with $p < 0.05$. \textit{Our proposed approach GR-IG (ours) has significant gain compared to other approaches on most of the tasks, in both the scenarios FID and VID.}}
\centering
\scalebox{0.8}{
\begin{tabular}{|c|c|c|cccccc||cccccc|ccccc|}
\hline
&&\multicolumn{7}{c||}{\textbf{Fixed Input Dimensions (FID) }}& \multicolumn{11}{c|}{\textbf{Variable Input Dimensions (VID)}} \\
\cline{3-20}
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Dataset}\\\\\\\end{tabular}}&
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Scenario}\\\\\\\end{tabular}}&
\multirow{4}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Task}\\\textbf{ID}\\\\\end{tabular}} & \multicolumn{6}{c||}{\textbf{Standard Solver}} & \multicolumn{6}{c|}{\textbf{Standard Solver}} & \multicolumn{5}{c|}{\textbf{Solver-CM }} \\
\cline{4-20}
&&&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{TS}\\\end{tabular}}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB}\\\end{tabular}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{TS}\\\end{tabular}}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB$^*$}\\\end{tabular}}&\multicolumn{1}{l|}{\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{FT}\\\end{tabular}}}&\multirow{2}{*}{\begin{tabular}[c|]{@{}c@{}}\textbf{SMT}\\\end{tabular}}&\multicolumn{2}{|c|}{\textbf{GR}} &\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}\cellcolor[gray]{0.7}\textbf{UB}\\\end{tabular}}\\
\cline{7-8} \cline{13-14} \cline{18-19}
&&&\multicolumn{1}{l|}{} &\multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & &\multicolumn{1}{l|}{} &\multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & & \multicolumn{1}{l|}{} &\ \textbf{\cite{wortsman2020supermasks}}& \multicolumn{1}{|l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{CG}\\\cite{shin2017continual}\end{tabular}}} & \multicolumn{1}{l|}{ \textbf{\begin{tabular}[c]{@{}c@{}}\textbf{IG}\\\textbf{(ours)}\end{tabular}}} & \\
\hline
\parbox[t]{2mm}{\multirow{15}{*}{\rotatebox[origin=c]{90}{\textbf{DSADS}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fixed\\ Classes\\\end{tabular}}
&1 & 16.3 & 16.3 & 16.2 & 16.3 & 16.3 & \textit{16.3} & 27.8 & 27.8 & 29.2 & 27.8 & 27.8 & \textit{27.8} & 27.8 & 29.2 & 27.8 & 27.8 & \textit{27.8} \\
&&2&11.9 & 12.0 & 11.9 & 10.9 & \textbf{10.5} & \textit{9.0} & 29.4 & 30.4 & 27.3 & 29.9 & 28.6 & \textit{25.6} & 26.6 & 25.7 & 25.2 & \textbf{20.7} & \textit{13.9} \\
&&3&17.9 & 14.7 & 12.7 & 12.4 & \textbf{9.7} & \textit{6.5} & 29.6 & 26.2 & 28.3 & 36.5 & 25.1 & \textit{22.7} & 23.4 & 26.8 & 36.1 & \textbf{18.7} & \textit{11.3} \\
&&4&18.9 & 13.2 & 13.6 & 10.2 & \textbf{8.0} & \textit{4.5} & 26.1 & 26.9 & 25.1 & 58.3 & 23.6 & \textit{20.9} & 23.3 & 24.1 & 52.2 & \textbf{16.2} & \textit{7.7} \\
&&5&15.7 & 12.3 & 12.1 & 8.7 & \textbf{7.4} & \textit{4.0} & 26.1 & 24.1 & 25.3 & 54.7 & 22.4 & \textit{19.1} & 22.3 & 22.6 & 48.6 & \textbf{13.9} & \textit{7.0} \\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Incrementally\\ Increasing\\ Classes\\\end{tabular}}
&1 & 13.4 & 13.4 & 13.8 & 13.4 & 13.4 & \textit{13.4} & 19.1 & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} \\
&&2&15.6 & 14.9 & 13.8 & 13.2 & \textbf{11.1} & \textit{9.7} & 21.4 & 26.9 & 22.9 & 24.4 & 22.8 & \textit{18.8} & 25.6 & 20.6 & 21.2 & \textbf{15.3} & \textit{12.7} \\
&&3&12.8 & 11.9 & 10.9 & 11.3 & \textbf{9.2} & \textit{6.9} & 15.7 & 34 & 19.5 & 39.4 & 26.3 & \textit{25.0} & 24.3 & 18.6 & 33.5 & \textbf{13.5} & \textit{10.8} \\
&&4&11.7 & 8.4 & 9.4 & 7.5 & \textbf{6.0} & \textit{4.3} & 21.2 & 35.7 & 26.1 & 45.3 & 30.7 & \textit{23.7} & 28.9 & 23.6 & 38.3 & \textbf{13.9} & \textit{9.5} \\
&&5&9.2 & 8.3 & 8.4 & \textbf{6.3} & \textbf{6.1} & \textit{3.8} & 13.4 & 39.6 & 24.8 & 47.6 & 36.3 & \textit{27.9} & 27.8 & 12.3 & 43.1 & \textbf{11.8} & \textit{6.7}\\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Partially\\ Changing\\ Classes\\ \end{tabular}}
&1 & 13.4 & 13.4 & 13.8 & 13.4 & 13.4 & \textit{13.4} & 19.1 & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} & 19.1 & 17.9 & 19.1 & 19.1 & \textit{19.1} \\
&&2&12.3 & 11.7 & 11.2 & 11.3 & \textbf{9.5} & \textit{8.0} & 16.5 & 29.9 & 20.5 & 26.7 & 23.1 & \textit{17.7} & 26.1 & 19.7 & 27.9 & \textbf{14.9} & \textit{13.1} \\
&&3&13.5 & 12.4 & 15.0 & 12.4 & \textbf{9.8} & \textit{7.4} & 18.5 & 25.4 & 19.4 & 32.4 & 25.4 & \textit{18.8} & 22.3 & 17.1 & 29.4 & \textbf{14.8} & \textit{13.4} \\
&&4&14.3 & 13.4 & 15.1 & 12.4 & \textbf{10.1} & \textit{7.2} & 19.3 & 23.7 & 23.3 & 41.2 & 22.0 & \textit{19.5} & 20.9 & 20.4 & 36.7 & \textbf{17.7} & \textit{13.9} \\
&&5&15.4 & 13.3 & 12.7 & 10.6 & \textbf{8.8} & \textit{6.5} & 18.6 & 23.4 & 19.7 & 30.5 & 21.7 & \textit{18.6} & 20.6 & 19.0 & 32.1 & \textbf{15.2} & \textit{11.6} \\
\hline \hline
\parbox[t]{2mm}{\multirow{15}{*}{\rotatebox[origin=c]{90}{\textbf{HAR}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Fixed \\ Classes\\\end{tabular}}
&1 & 15.7 & 15.7 & 16.1 & 15.7 & 15.7 & \textit{15.7} & 30.2 & 30.2 & 30.9 & 30.2 & 30.2 & \textit{30.2} & 30.2 & 30.9 & 30.2 & 30.2 & \textit{30.2} \\
&&2&17.1 & 16.3 & 16.2 & 15.8 & \textbf{15.4} & \textit{14.4} & 41.5 & 46.9 & 41.2 & 46.9 & 40.1 & \textit{37.4} & 44.9 & 38.8 & 37.3 & \textbf{30.8} & \textit{27.7} \\
&&3&18.2 & 16.1 & 15.8 & 15.1 & \textbf{14.4} & \textit{12.8} & 34.4 & 42.7 & 33.2 & 52.1 & 34.0 & \textit{32.4} & 40.6 & 32.1 & 40.9 & \textbf{30.2} & \textit{26.2} \\
&&4&17.4 & 15.5 & 15.2 & \textbf{13.4} & 13.8 & \textit{11.9} & 34.6 & 45.7 & 32.2 & 72.2 & 31.4 & \textit{28.4} & 43.0 & \textbf{28.4} & 68.2 & \textbf{28.2} & \textit{23.6} \\
&&5&17.6 & 15.7 & 15.2 & 13.5 & \textbf{13.1} & \textit{11.4} & 35.0 & 31.3 & 32.1 & 72.9 & 29.0 & \textit{26.2} & 30.4 & 32.5 & 71.5 & \textbf{23.7} & \textit{20.1} \\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Incrementally\\ Increasing\\ Classes\\\end{tabular}}
&1 & 17.6 & 17.6 & 16.9 & 17.6 & 17.6 & \textit{17.6} & 23.7 & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} & 23.7 & 23.1 & 23,7 & 23.7 & \textit{23.7} \\
&&2&18.0 & 13.7 & 14.3 & 14.1 & \textbf{11.9} & \textit{10.7} & 20.3 & 19.6 & 23.2 & 23.3 & 18.6 & \textit{18.5} & 18.1 & 22.4 & 19.2 & \textbf{13.5} & \textit{11.0} \\
&&3&18.6 & 12.1 & 13.4 & 15.1 & \textbf{10.4} & \textit{8.5} & 23.8 & 24.5 & 25.7 & 36.3 & 24.6 & \textit{20.1} & 22.4 & 24.5 & 35.0 & \textbf{16.2} & \textit{14.0} \\
&&4&17.5 & 17.6 & 15.1 & 14.6 & \textbf{12.6} & \textit{10.2} & 27.6 & 31.2 & 28.7 & 41.7 & 35.1 & \textit{26.6} & 27.8 & 27.9 & 41.3 & \textbf{24.0} & \textit{19.6} \\
&&5&21.7 & 19.2 & 18.2 & 17.0 & \textbf{14.2} & \textit{11.6} & 28.4 & 30.3 & 31.3 & 50.7 & 36.9 & \textit{32.4} & 29.4 & 30.7 & 47.4 & \textbf{22.9} & \textit{19.6}\\
\cline{2-20}
&\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Partially \\Changing \\ Classes\\ \end{tabular}}
&1 & 17.6 & 17.6 & 16.9 & 17.6 & 17.6 & \textit{17.6} & 23.7 & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} & 23.7 & 23.1 & 23.7 & 23.7 & \textit{23.7} \\
&&2&17.4 & 16.0 & 15.0 & 14.5 & \textbf{13.6} & \textit{12.1} & 28.1 & 31.1 & 27.5 & 35.4 & 31.2 & \textit{26.7} & 29.9 & 26.9 & 28.2 & \textbf{25.8} & \textit{23.0} \\
&&3&17.1 & 14.9 & 15.8 & 14.2 & \textbf{13.2} & \textit{11.9} & 31.1 & 32.9 & 29.9 & 42.0 & 37.9 & \textit{29.4} & 32.1 & 29.1 & 39.2 & \textbf{27.6} & \textit{24.7} \\
&&4&13.5 & 12.8 & 13.4 & \textbf{10.5} & \textbf{10.8} & \textit{8.0} & 22.2 & 27.4 & 20.4 & 41.8 & 31.8 & \textit{22.1} & 25.8 & 20.5 & 36.9 & \textbf{18.8} & \textit{15.1} \\
&&5&16.9 & 14.9 & 16.1 & 12.2 & \textbf{9.9} & \textit{7.8} & 23.3 & 31.2 & 23.0 & 38.4 & 30.5 & \textit{25.5} & 28.2 & 23.3 & 34.4 & \textbf{19.6} & \textit{15.2}\\
\hline \hline
\parbox[t]{2mm}{\multirow{5}{*}{\rotatebox[origin=c]{90}{\textbf{Turbofan}}}} &
\multirow{5}{*}{\begin{tabular}[c]{@{}c@{}}Not\\Applicable\\(Regression Task)\end{tabular}}
&1 & 16.7 & 16.7 & 17.4 & 16.7 & 16.7 & \textit{16.7} & 17.3 & 17.3 & 18.2 & 17.3 & 17.3 & \textit{17.3} & 17.3 & 18.2 & 17.3 & 17.3 & \textit{17.3} \\
&&2&17.7 & 17.3 & 16.9 & 16.9 & \textbf{16.2} & \textit{15.7} & 18.7 & 18.2 & 19.0 & 18.9 & 18.4 & \textit{17.4} & 17.9 & 18.7 & 17.7 & \textbf{16.8} & \textit{15.9} \\
&&3&18.9 & 17.6 & 17.1 & \textbf{15.8} & 16.3 & \textit{15.0} & 19.3 & 18.7 & 22.9 & 19.3 & 17.6 & \textit{16.2} & 18.1 & 18.8 & 18.3 & \textbf{17.1} & \textit{15.8} \\
&&4&18.2 & 17.0 & 17.5 & \textbf{15.6} & \textbf{15.7} & \textit{14.7} & 16.5 & 17.4 & 17.9 & 17.9 & 17.5 & \textit{15.3} & 17.1 & \textbf{16.2} & 16.8 & \textbf{15.7} & \textit{15.1} \\
&&5&16.2 & 16.7 & 16.8 & 15.5 & \textbf{15.2} & \textit{13.9} & 17.5 & 17.6 & 18.2 & 18.3 & 17.2 & \textit{16.1} & 16.3 & 16.6 & 17.6 & \textbf{16.0} & \textit{14.9}\\
\hline
\end{tabular}}
\label{tab:results}
\end{table*}
\begin{figure*}[h!]
\captionsetup[subfigure]{font=small,labelfont=small, justification=centering}
\centering
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[scale=0.39,trim={0.23cm 0.8cm 0.25cm 0},clip]{images/img/dsads_fixed_all_classes.pdf}
\caption{DSADS \\(Fixed\\ Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/dsads_fixed_openset_classes.pdf}
\caption{DSADS \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/har_fixed_all_classes.pdf}
\caption{HAR \\(Fixed \\Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/har_fixed_openset_classes.pdf}
\caption{HAR \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.15cm 0.8cm 0.25cm 0},clip]{images/img/turbofan_fixed.pdf}
\caption{Turbofan \\(Regression \\Task)}
\end{subfigure}
\vspace{2mm}
\centerline{\newline(A) FID Scenario}
\vspace{3mm}
\begin{subfigure}{0.23\textwidth}
\centering
\includegraphics[scale=0.39,trim={0.23cm 0.8cm 0.25cm 0},clip]{images/img/dsads_var_all_classes.pdf}
\caption{DSADS \\(Fixed\\ Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/dsads_var_openset_classes.pdf}
\caption{DSADS \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/har_var_all_classes.pdf}
\caption{HAR \\(Fixed \\Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/har_var_openset_classes.pdf}
\caption{HAR \\(Incrementally \\Increasing Classes)}
\end{subfigure}
\begin{subfigure}{0.185\textwidth}
\centering
\includegraphics[scale=0.39,trim={2.45cm 0.8cm 0.25cm 0},clip]{images/img/turbofan_var.pdf}
\caption{Turbofan \\(Regression \\Task)}
\end{subfigure}
\vspace{2mm}
\centerline{\newline(B) VID Scenario}
\vspace{3mm}
\caption{Percentage gain of respective solver model (Standard Solver in FID scenario and Solver-CM in VID scenario) over the TS standard solver model. Respective solver model $M_i$ when used along with GR-IG have +ve gain on previous tasks $\mathcal{T}_{j}$ ($j=1,\ldots, i-1$) indicating the ability of the solver model with GR-IG to retain the knowledge of past tasks. TS standard solver model and solver model with GR-CG struggle to perform well on previous tasks, especially in the VID scenario. Our approach GR-IG is better than SMT indicating the advantage of generated previous tasks data from generative model and CM over SMT, which only uses the current task data to learn task-specific supermask. \label{fig:continual}}
\end{figure*}
\begin{enumerate}
\item \textbf{Task-Specific (TS)} is the vanilla approach or baseline which uses only the data from current task, i.e. $\mathcal{D}_i$, for learning the task-specific model $M_i$. This approach serves as a lower bound for evaluation.
\item \textbf{Fine-tune (FT)}: This approach also uses the data from the current task only (as in TS) but the parameters of the current solver model $M_i$ are initialized using those\footnote{For the final classification layer in case of incrementally increasing and partially changing classes, relevant parameters are used.} of the previous solver model $M_{i-1}$.
\item \textbf{Common Generator (GR-CG)} \cite{shin2017continual}: This approach maintains a single common generator (CG) across all previous tasks to generate samples from all of them (refer Section \ref{sec:approach}), while the solver model can be randomly initialized vanilla GRU model or the proposed conditioning module based solver model.
\item \textbf{Supermasks with Transfer (SMT)} \cite{wortsman2020supermasks}: This approach maintains the same core solver model but learns task-specific supermasks. The task-specific supermask allows the model to retain knowledge to solve each task in different units of the same core base neural network. The supermask weights for the new task are initialised from the running mean of all the previous tasks supermask weights, resulting in transfer of knowledge. We found this initialization strategy to work better than the vanilla `Supermasks without Transfer' (SM) approach (in concurrence with the observations made in \cite{wortsman2020supermasks}), and report results for SMT alone.
\item \textbf{Upper Bound (UB)}: This approach assumes access to all relevant original data from previous tasks. This approach is equivalent to having an ideal generator and solver for all previous tasks. Though access to original data from previous tasks is assumed, the available dimensions across tasks vary for the VID scenario while all dimensions are available for the FID scenario.
\end{enumerate}
Our proposed approach using independent generators for each previous task is referred to as \textbf{GR-IG}.
We refer to the GRU-based solver model using the GNN-based conditioning module as \textbf{Solver-CM}, while the vanilla GRU-based solver as \textbf{Standard Solver}.
\subsection{Results and Observations}
We make following key observations from Tables \ref{tab:results} and \ref{tab:Heterogeneousresults}:
\begin{itemize}[leftmargin=*]
\item As more tasks arrive, the gap between TS and UB widens, indicating scope for positive-transfer from old tasks to new tasks; this is expected as the underlying dynamical system is the same across tasks. We observe the same empirically: solvers using generative models (GR-CG and GR-IG) significantly improve upon TS in the FID scenario. For the partial changing classes scenario, GR-IG improves upon the baseline while GR-CG struggles, indicating the ability of GR-IG to provide better quality generated data. This is further helped by estimated labels from task-specific solvers, while GR-CG does not have this advantage.
\item \textit{GR-IG performs better than GR-CG in most cases} with FID as well as VID, proving the advantage of GR-IG over GR-CG. The samples generated via GR-CG are the combination of all previous tasks with diversity across classes (and input dimensions in VID scenario), which cannot be captured by one VAE.
\item SMT performs better than TS on FID scenario with standard solver whereas, on VID scenario SMT with Standard Solver performs worse than TS mostly in incrementally increasing classes and partial changing classes. The gains observed when using Solver-CM instead of Standard Solver for VID scenario confirms the advantage of GNN-based conditioning module for dimension-adaptability.
Our proposed approach always performs better than SMT in both FID and VID scenarios, showing the advantage of using the independent generator (GR-IG) for each task.
\item For scenarios with partially-observed sensors, i.e. VID scenario, GR-CG performs worse than TS. Unlike the FID scenario, even GR-IG and UB with access to all data from previous tasks fail to improve upon TS when using standard solver.
Furthermore, vanilla transfer via FT models is ineffective in VID scenario.
These observations confirm the \textit{non-trivial nature of the VID problem}, and inability of GR-CG to adapt to variable dimensions.
\textit{Only Solver-CM with GNN-based Conditioning Module for dimension-adaptability and GR-IG for generating past data consistently performs better than the TS on VID scenarios}. Combination of Solver-CM and GR-IG is the best performing model in most scenarios across datasets and tasks.
\item \textit{Continual learning}:
We also evaluate how well Solver-CM in VID scenario and standard solver in FID scenario with GR-IG performs on tasks $\mathcal{T}_1,\ldots,\mathcal{T}_{i-1}$. As shown in Fig. \ref{fig:continual}, in both FID and VID scenarios, we observe that while the TS model on one task struggles to generalize on other tasks, our approach performs better than SMT and GR-CG and is comparable to the UB in all scenarios. This shows that the \textit{samples generated from GR-IG are better than GR-CG in both FID and VID scenarios and also solvers with CM (GNN-based conditioning module) are able to adapt variable dimension in VID scenario}, which in-turn helps to solve the previous tasks better by virtue of capabilities similar to generative replay \textit{GR-CG} as in \cite{shin2017continual}.
\begin{table}[ht]
\centering
\caption{Performance comparison for generalization to unseen sensor combinations at test time. \textit{Solver-MH}: the conditioning module (CM) is removed and the conditioning vector is replaced with a multi-hot vector indicating the active sensors. \textit{Solver-SE}: the GNN is removed and replaced with a simple max-pooling over the sensor embeddings. Standard Solver-A: the upper bound that assumes access to all sensors in the train and test set. \textit{In both the settings, No Fine-tuning and Fine-tuning, Solver-CM generalizes better on unseen sensor combination than other solver models.} }
\scalebox{0.82}{
\begin{tabular}{|c|cccc|cccc|c|}
\hline
&\multicolumn{4}{c}{\textbf{No Fine-tuning}} &
\multicolumn{4}{|c|}{\textbf{Fine-tuning}} & \\
\cline{2-9}
&\multicolumn{8}{c|}{\textbf{Solver-}}& \\
\cline{2-9}
{\begin{tabular}[c]{@{}c@{}}\textbf{Dataset}\\\\\end{tabular}}
&{\begin{tabular}[c]{@{}c@{}}\textbf{Stan-}\\\textbf{dard}\end{tabular}} &{\begin{tabular}[c]{@{}c@{}}\textbf{
MH}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{CM}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{Stan-}\\\textbf{dard}\end{tabular}} &{\begin{tabular}[c]{@{}c@{}}\textbf{MH}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}}& {\begin{tabular}[c]{@{}c@{}}\textbf{CM}\end{tabular}} & {\begin{tabular}[c]{@{}c@{}}\textbf{Standard}\\\textbf{Solver-A}\\\textbf{(UB)}\end{tabular}} \\
\hline
DSADS
& 8.3 &8.9&7.8& \textbf{5.9} & 4.7 &6.1& 3.4 & \textbf{3.3} & 1.5 \\
\hline
HAR
& 12.8& 13.5 &11.8& \textbf{11.4} & 12.1 & 12.4&10.7 & \textbf{10.5} & 6.3 \\
\hline
Turbofan
& 15.5 & 16.4&14.9&\textbf{14.6} & 16.2 &15.6&14.7 & \textbf{14.2} &13.6 \\
\hline
\end{tabular}
}
\label{tab:Heterogeneousresults}
\end{table}
\item \textit{Ablation Study of Solver-CM}: We consider VID multi-sensor tasks with 100\% training instances instead of the scarce training instances in the sequential-tasks setup discussed above. The training time-series correspond to certain combinations of sensors, while the test set contains sensor combinations unseen during training.
Any time-series instance in train or test set has $\approx 0.4$ fraction sensors missing. For all datasets, 40\% of the instances are used for training, 10\% for validation, 10\% for fine-tuning (ignored in no fine-tuning setting), and remaining 40\% for testing. From Table \ref{tab:Heterogeneousresults} we observe that Solver-SE is better than Solver-MH, indicating the advantage of using the compressed information of active sensors in the form of sensor embeddings over a multi-hot vector. But still, Solver-SE fails to generalize on unseen sensors combinations. We use GNN and sensor embeddings in Solver-CM to improve further, where GNN allows to exchange messages between the active sensors. These messages passing amongst the active sensors inherently allow for combinatorial generalization on unseen sensor combinations without fine-tuning. For fine-tuning, we consider small amounts of data for the same sensor combinations as the test set, which further improves the results.
\end{itemize}
In summary, while GR-CG class of methods like \cite{shin2017continual,velik2014brain} would usually suffice to deal with lack of access to past data by providing generated data, and SMT \cite{wortsman2020supermasks} would allow to adapt across tasks with varying classes via supermasks or similar methods, neither class of methods suffice for the variable input dimensions scenario as indicated by their inferior performance in comparison to GR-IG with CM-based solver. Furthermore, conditioning module based on GNNs is critical to the success of the proposed approach.
\section{Conclusion and Future work}
In this work, we have motivated a practical problem of enabling knowledge transfer from previous tasks to new tasks under restricted data-sharing across sequentially arriving multi-sensor tasks with varying input dimensions. We have proposed a deep learning architecture to deal with
multivariate time series classification and regression tasks that works across the partially-observed sensor (variable input dimensions) tasks as well as changing classes across tasks.
Our results on three publicly available datasets prove the efficacy of the proposed approach to leverage generated data from previous tasks to improve performance on (partially-observed multi-sensor) new tasks, without any explicit original data exchange.
Though we have evaluated the proposed approach for activity recognition and remaining useful life estimation applications, we believe the proposed approach is generic and would be useful in other multivariate time series applications.
Furthermore, combining the benefits of generative replay as we have presented, with meta-learning (e.g., on lines of \cite{riemer2018learning})
to further improve transfer of the solver models used in our approach is an interesting future direction to explore
especially in context of variable-dimensional input spaces. Extensions of this work in privacy-preserving scenarios is another interesting future direction.
\bibliographystyle{abbrv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,966 |
As the higher education landscape shifts with emerging tech like AI to improve the learning experience, educators need to ensure the technology they choose—from projectors to VR devices—supports their students' needs. Read on to learn what the future of EDU may look like for distributed, international classrooms.
A recent interview with a top Udacity leader suggested that the company is moving away from massive open online courses (MOOCs) and toward a different model of learning.
Salwa Muhammad is vice president of admissions and Udacity Connect at Udacity, an online education platform. Her focus of interest is on informal and experiential learning outside of the classroom and in online spaces. Prior to her work at Udacity, she was the program director of international internships and careers at Wellesley College. In this interview, Muhammad discusses her thoughts on the role that technology plays in international higher education and the future of the field.
Q: What do you think the future holds for higher education, and how will international higher education, specifically, be affected?
A: Udacity is an online education company focused on lifelong learning. Given the low completion rates of free online courses (i.e., MOOCS) across the board, Udacity began offering its paid Nanodegree programs in late 2014, providing microcredentials that connect learning to jobs. The average Nanodegree program takes 6-9 months (approximately 10-15 hours per week) to complete, in subjects including self-driving cars, artificial intelligence, virtual reality, robotics, machine learning, data science, digital marketing, and mobile and web development. In addition to Nanodegree coursework, students receive personalized services such as one-on-one coaching, moderated forums, project reviews, and career and job-placement support. More than 53,000 students are enrolled in Udacity Nanodegree programs and thousands of graduates have secured jobs. Udacity Nanodegree programs are built with, and recognized by, industry leaders including Google, Facebook, IBM Watson, Nvidia, Mercedes Benz, and AT&T.
The future of higher education is promising, provided that educators align with employers and help to connect learning to jobs. U.S. and international higher education institutions alike will benefit from the changing landscape if their services better meet students' needs. Removing the geographic barriers of higher education by making it available online will help usher in a new era of global education, global students, and global citizens. But there must be an alignment of the skills taught with the skills needed. And learning must be lifelong, not confined to a four-year undergraduate program.
Q: As technology steadily becomes a bigger part of international higher education, do you believe that technology will not only change the pedagogy but will also shape what is and is not taught?
Technology should scaffold, not define, learning. Educators build the foundation and structure of learning, and then choose the technologies that best support their curriculum design and delivery. A pencil and paper can be useful for conveying certain concepts, while a programmable mini-supercomputer can be the right tool to teach a very different set of skills. This leads to mindful use of technology in the classroom.
At Udacity, we utilize technology as a distribution platform and put our classroom online. More students can access our curriculum because of technology, and it also affords them flexibility in when, where, and how they learn. Technology is either central or tangential to everything we teach. Technology, as a teaching tool, is only as useful as how easy it is to adopt and how many students find it helpful.
On the other hand, technology is changing that landscape and is the biggest catalyst for change in the workforce. Technology has helped each generation get different and better jobs. We focus on teaching students how to use the most useful technology to do their jobs. While we use technology to be better teachers, we also adapt to our students' needs in parallel with the roles that are in high demand with employers.
Q: As someone who has worked, studied, and lived in such a diversity of places-from Bangladesh to Saudi Arabia to Uganda-in your experience, what's the biggest misconception that people abroad have about U.S. higher education? And, conversely, what's the biggest misconception that U.S. colleges and universities have about students coming from abroad?
One big misconception about U.S. higher education that many people have is that there is a linear path to the best jobs through the highest-ranked schools. While the schools are publicly ranked (and by multiple different sources), the students are not, and having a good grade-point average from a top school doesn't guarantee them a job in their field at the end. To get their dream jobs, graduates have to be well-rounded with professional networks and experience and realize that most career paths don't go in a straight line.
U.S. universities are prone to developing misconceptions about international students by simply labeling everyone as "international," which is far too simplistic and can even be harmful. It's important not to generalize students based on the continents or countries they come from, the languages they speak, or their religions.
"International" encompasses so many different aspects of a student's culture, background, and aspirations that are important to take into account. We have the tendency to find a rule book or manual to help navigate studying abroad. However, if we flip roles and ask ourselves how we would write a guidebook for studying in the United States, we are very aware of the diversity within the United States and wouldn't necessarily have rules. The most important thing we can do is to put ourselves in someone else's shoes and understand that it is impossible to know all the intricacies of a culture, other than our own. But respect, interest, and mutual understanding go both ways.
International students are unwittingly given the huge responsibility in U.S. colleges and universities to represent their countries, cultures, and, sometimes, even entire continents and educate everyone else. Your sense of identity is heightened and you have to pick a box. Am I Muslim enough to represent the religion? Do I identify as being Bangladeshi? Can I really be considered as American when I travel abroad now after spending most of my life here in the United States? I wrestle with these questions even now, but I had to make a choice very quickly when I first arrived in the United States as an international student. I had to learn what the stigmas were and how to navigate and advocate for myself. I remember reasoning with myself that this was the diversity tax of being allowed to be an international student in a foreign country.
We can help to ease this burden by not expecting international students to fit into our need of figuring out cultures within parameters we are comfortable with. We can examine existing practices of having student orientation based on ethnicity or assess the purpose of having cultural centers and academic centers at higher education institutions. We need to better understand whether these student services are aligned with how our international students want to be represented and supported, rather than retrofit these students into our own preconceived notions of their roles and responsibilities based on why and how we value diverse cultures.
Removing the geographic barriers of higher education by making it available online will help usher in a new era of global education, global students, and global citizens. But there must be an alignment of the skills taught with the skills needed. And learning must be lifelong, not confined to a four-year undergraduate program.
Copyright NAFSA: Association of International Educators, Jan./Feb. 2018. This interview has been condensed for brevity. This interview has been condensed for brevity.This article was written by Lisa Schock from International Educator and was legally licensed through the NewsCred publisher network. Please direct all licensing questions to legal@newscred.com. | {
"redpajama_set_name": "RedPajamaC4"
} | 316 |
Q: Python Regex to parse email URLs but excluding the public email I am parsing a file that have entries like:
xxx-yy.biz. 39405 A 156.154.66.33
mail.global.com. 3464 A 115.113.9.64
xyx xyx xyx
webmail.xyz.com. 1463 A 115.113.9.64
gmail.com. 3464 A 115.113.9.22
I am trying to extact the URL and its IP address with string "mail" in it:
for line in (dnsfile):
match = re.search(r'(.*mail.*?)\s+(.*)\s+A\s+(.*)', line)
and match.group(1) and match.group(2) is giving me URL and IP.
I want to extent this search so that I don't want to parse public emails like: gmail, hotmail, yahoo,mail. More general : exclude a list of words in this
search.
A: You can use a negative look ahead, but you need to add the start and end anchors so you need re.DOTALL flags too (make the anchors to match from start and end of each line), you can create your negative look-ahead with joining the list of words with | :
re.search(r'^(?!{})(.*mail.*?)\s+(.*)\s+A\s+(.*)$'.format('|'.join(list_of_domin)),line,re.DOTALL)
See demo https://regex101.com/r/bF5xQ3/1
A: If it's not a requirement to have it as part of the regexp you could do a simple array search
nothanks = ['gmail.com', 'hotmail.com']
for line in (dnsfile):
match = re.search(r'(.*mail.*?)\.\s+(.*)\s+A\s+(.*)', line)
if match:
if not match.group(1) in nothanks:
print match.group(1)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,502 |
\section{Introduction}
With the rapid growth in popularity of Cloud services, object storage systems (e.g.,
Amazon S$3$, IBM COS or OpenStack Swift~\cite{swift}) have gained momentum.
These storage systems offer consolidated storage at scale, with high degrees of availability
and bandwidth at low cost. Proof of that is the recent trend of serverless computing.
Due to the high difficulty of function-to-function communication\footnote{Some
works have shown that cloud functions can communicate directly using NAT (Network Address Translation) traversal techniques.
However, direct communication between functions is not supported by cloud providers.},
many serverless systems use
object storage for passing data between functions~\cite{Sampe2018, pu2019shuffling, muller20, primula, sonic},
which has revived the interest in this type of object storage.
~Although very useful for cloud applications, object storage systems offer
a small number of options to keep sensitive data safe. Few (or no) efforts
have been realized on security issues such as data confidentiality, data integrity, or access control, to mention a few.
For instance, online storage services such as Amazon S$3$, or IBM COS, only provide
server-side encryption for protecting objects at rest~\cite{s3, cos}. Similar words can be
said for the access control of individual objects\footnote{In general terms, cloud object stores enforce access at the container level
rather than at the object level.}, which is currently either realized via simple object ACLs (Access Control Lists) as in S$3$~\cite{s3},
or not possible at all as in OpenStack Swift~\cite{biswas15}.
This poor interface is insufficient for many applications. For instance,
it does not enable in-place queries on encrypted data, transparent and secure data sharing,
and access control based on the content of an object. But also, it is
very difficult to make it evolve to meet the changing needs of applications
and withstand the test of time. In practice, most of these object storage systems leave no
other choice to modifying the system internals to incorporate new security mechanisms
at the object level. This requires a deep knowledge of the system, extreme care
when modifying critical software that took years of code-hardening
to trust, and significant cost and time (see, for instance,~\cite{li19}, where it is described the ``daunting'' task
of deploying new erasure coding solutions in object stores such as OpenStack Swift~\cite{swift} and Ceph~\cite{weil06}).
Rather than relying on object storage systems to change,~we advocate in this paper
to ``work around'' the traditionally rigid object storage APIs by embracing a \emph{software-defined}
approach. Similarly to software-defined networking (SDN), we argue that
the separation of the ``control logic'' from the ``data protection logic''
can give the needed flexibility and ease of use to enable users,
programmers and sysadmins to custom-fit access control
and object protection. To give an example, pretend that
upon certain conditions, parts of an encrypted object need to be re-encrypted to share~it with the mobile
users of the application. Further, these conditions may depend on the contents of the object itself (e.g., on sensitive data such as sexual
orientation), which must always remain confidential from the server.
Simply put, what we pursue is to offer users the ability to succinctly express this behavior at the object storage level,
and enforce~it by calling the corresponding re-encryption modules.
~Nevertheless, a software-defined solution to enhance object storage data protection
requires solving several issues at~two levels:
\begin{itemize}
\item At the control plane, by enabling the composition of per-object protection services. These compositions should be
expressed concisely and in a manner agnostic to the data protection code and the remaining storage stack.
\item At the data plane, by making it truly programmable. To put it baldly, the data plane should not only allow to plug-in new protection
logic in the critical I/O path, but to run it safely. In addition, it should enable the re-usability of the protection capabilities,
so that users can compose new data protection controls.
\end{itemize}
In this research work, we present \textsc{Egeon}, a novel software-defined data protection framework for object storage.
\textsc{Egeon}\hphantom{} exports a scripting API to define privacy policies for protecting objects. These policies enable data owners to declaratively
set complex data protection services through the composition of user-defined transformations run in a serverless fashion.
These functions represent the elementary processing units in \textsc{Egeon},
and may be re-used and linked together to implement complex privacy policies. The major feature of these transformations is that
they are executed ``inline'' by \textsc{Egeon}~upon a standard~\texttt{Get} request.
In this way, users can trivially display multiple views from the same object, and
modify these views by updating the policies. If some functionality to
implement a view is missing, \textsc{Egeon}~provides a
simple API to deploy new transformations and customize the access to data objects.
In this sense, one of the primary contributions of \textsc{Egeon}~is the ability to provide privacy-compliant transformed views of
the underlying data on the fly. Perception of privacy can vary broadly
across applications. As an example, a dataset created by a hospital may include personally identifiable information (PII) that is not needed when it is
processed by a data analytics engine. Nonetheless, if the same dataset is accessed by medical personnel, a richer view of it should
be given. Thus, a practical system needs to support a range of privacy preferences.
By adopting a software-defined storage architecture, \textsc{Egeon}\hphantom{}
allows users to express their privacy preferences as policies~in the control plane and produce views
conforming to the policies by running transformations in the data plane. In this work,
we focus on (cryptographic) transformations that process data as streams, that is, as
data is being retrieved from object storage nodes. Consequently, the first bytes of transformed views are
received as soon as possible, which permits \textsc{Egeon}~to scale to arbitrary object sizes without important
penalty on end-to-end latency.
The \textsc{Egeon}~prototype we present in this research has been implemented atop
OpenStack Swift~\cite{swift}. We took Swift~because it is open source and a production quality
system. Its sizable developer community ensures that our new properties are built on
code that is robust and that will be soon evaluated. It must be noticed
that the design concepts underpinning \textsc{Egeon}~are generic and
could be easily ported to other storage substrates.
For example, object
classes allow to extend Ceph by loading custom code into Object Storage Daemons (OSDs),
which can be run from within a \texttt{librados} application~\cite{librados}. Thus, with some effort, it would be possible to leverage object classes to
implement transformed views of data.
~~Our performance evaluation of \textsc{Egeon}~shows promise in developing data
protection services with low overhead directly into the object storage. For instance, the overhead~of
a \texttt{NOOP} policy, where a storage function simpy echoes the input data, is of around $9$ ms. Also,
depending on the amount~of protected data filtered out in the transformed views,~end-to-end
latency can be even lower with \textsc{Egeon}~due to the savings in network communication (up to $72.1$x for
a $4$G mobile use case).
\begin{figure}
\includegraphics[width=0.5\textwidth]{Egeon.png}
\caption{\textsc{Egeon}'s architecture.}
\label{fig:architecture}
\vspace{-20pt}
\end{figure}
\section{Design}
\textsc{Egeon}~is a software-defined data protection framework that augments
object storage with composable security services to enforce users' privacy
preferences over protected data. These services are built up as
pipelines of serverless functions. Fig.~\ref{fig:architecture} illustrates an overview of \textsc{Egeon}'s
architecture. Our objective is to enable authorized users or applications to access protected
data without violating the privacy policies of data owners.~We
have designed \textsc{Egeon}~to make it easy the leverage of state-of-the-art privacy
solutions (such as content-level access control, homomorphic encryption,
encrypted keyword search, ...) while preserving the normal data flow in the consuming applications.
Concretely, we achieve this by introducing a logical separation between the \emph{privacy plane}, where
data owners set their privacy preferences, and the \emph{data plane}, where the creation of privacy-compliant transformed views
happens. This separation allows for heterogeneous policies atop the same
data without having to modify the system internals to enforce advanced policies at
the object level.
\textsc{Egeon}'s architecture consists of the following components:
\medskip
\noindent\textbf{Privacy Plane}.~~The privacy plane in \textsc{Egeon}~corresponds to the control plane of a software-defined
architecture~\cite{crystal, FGCScrystal}, but specialized for data protection. In practice, this means that \textsc{Egeon}~provides
its own script language to assist data owner~in composing data protection services from elementary serverless functions
in the \emph{data plane}. The textual, JSON-based language supports conditions and compositions to build up inline privacy transformers (e.g.,
see Listing~\ref{lst:1}). Moreover, \textsc{Egeon}~offers an API to allow data owners to manage the life-cycle of their~data protection policies.
For performance reasons, once uploaded, the policies are automatically compiled into Java bytecode and stored in the
\emph{Metadata Service}. For fast access, this service has been built on top of Redis~\cite{redis}, an in-memory, low-latency key value store.
\medskip
\noindent\textbf{Data Plane}. In \textsc{Egeon}, the data plane has a critical role. The data plane is responsible for
generating the privacy-compliant data views. And hence, it must be extensible to accommodate new
functionality that enables privacy transformations on data. Particularly, in this realization of \textsc{Egeon},
we have focused~on inline privacy transformations as data is retrieved from storage nodes HDDs. As mandated by
the policies in the privacy plane, privacy transformations are constructed from pipelines of user-defined functions
executed as serverless functions by \textsc{Egeon}. Namely, a user integrating a new transformation only needs to
contribute the logic. Resource allocation and execution of the chain
of transformations is automatically handled by \textsc{Egeon}, bringing a true serverless experience to users.
To minimize execution overhead, since many cryptographic operations are CPU-intensive, \textsc{Egeon}~abides by
the principles of reactive programming and runs a transformation only when is strictly needed, instead of
continuously on the data streams. More concretely, \textsc{Egeon}~extends the observer pattern~\cite{observer} and execute a certain
transformation in the pipeline when an event occurs~\cite{event}. Consequently, \textsc{Egeon}~better utilizes the available resources in
the storage servers by balancing the load across the chain of transformations. To better understand this, pretend
that a user wants to compute the average salary of employees in a department $\mathtt{X}$. Now suppose that all the employee
records have been saved in a single JSON document with all the salary values homomorphically encrypted. Thanks to
\textsc{Egeon}~reactive core, the transformation to average the salary will only be run
when the event ``$\mathtt{employee\,of\,department\,X}$'' comes through the data stream, thereby saving CPU resources.
\subsection{Threat Model}
Specifically, \textsc{Egeon}~enforces user's privacy preferences via function composition.
That is, users are ensured that their data is transformed as it goes through
a pipeline of transformation functions before it is released to applications.
In the meantime, the original data remains end-to-end encrypted.
We assume an \emph{honest-but-curious}~\cite{cryptdb} storage servers, i.e.,
the server performs the computations correctly but will analyze all
observed data to learn as much information as possible. We also presume the existence of an identity service (IDS) such as OpenStack Keystone for user authentication.
We leverage~this service for authentication of the storage functions. An IDS is~a standard requirement in multi-user systems
and can even be a trustworthy external entity.
~Consumers of shared data are semi-trusted, in that they~do not
collude with the servers to leak the data or keys. This~is~a reasonable assumption for groups of data consumers that
are acquainted with each other. Further,
\textsc{Egeon}~assumes that~the applications behave correctly and do not hand out user keys
to malicious parties. Finally, we assume state-of-the-art security
mechanisms to be in place for user devices, and that all parties
communicate over secure channels.
In this setting, \textsc{Egeon}\hphantom{} enforces \emph{data confidentiality}, making sure that
the adversary learns nothing about the data streams, except what can be learned
from the transformed views.
\smallskip
\noindent\textbf{Robustness.}~While \textsc{Egeon}~is able to handle various types~of failures
in practice, provable robustness against misconfigured, or even malicious
privacy policies, and data producers is out of scope for this
research. A malicious user sending corrupted tokens cannot
compromise privacy but could alter the output~of a
transformed view.
\subsection{Privacy Plane}
\label{sec:privacy}
~In the privacy plane, \textsc{Egeon}~provides the capabilities for data owner to set their privacy preferences ---i.e., user-centric privacy---,
and what transformations will be required to apply to enforce a privacy policy. These transformations are specified by their unique
name, and their existence is verified when the privacy policy is to be compiled. A privacy policy applies to~a single object. In this
paper, we do not consider the question of how to set privacy policies for group of objects and how~they should look like. This
question has been left for future work.
In \textsc{Egeon}, targeted data objects in the privacy policies are specified by its full resource path. Following OpenStack Swift specs~\cite{swift}, the access path to an
object is structured into three parts: ${\small \mathtt{/account/container/object}}$. As an example, for the ${\small \mathtt{rose.jpg}}$ object in the ${\small \mathtt{images}}$
container in the $1234$ account, the resource path is: ${\small\mathtt{/1234/images/rose.jpg}}$.
Data owners can translate their preferences over an object to a set of transformations by mapping them in a JSON-based schema language.
In addition to some meta-information (e.g., policy identifier), this schema permits data owners to formalize conditions at the policy level
using a rich set of operators such as ${\small ``\mathtt{StringLike}"}$, $\small ``\mathtt{NumericLessThan}"$, etc. Importantly,~the language
enables data owners to build date expressions using operators like $\small{``\mathtt{DateNotEquals}"}$, which makes it possible~to
express temporal restrictions. For instance, pretend that a data owner wishes to prevent that a document can be accessed~on weekends.
She could indicate this through the date expression: ${\footnotesize \mathtt{``DateNotEquals": \left\{``Day": [``Sat",``Sun"]\right\}}}$.
More interestingly, this schema allows composing complex data protection transformations from elementary UDFs. This can be achieved
by adding each individual UDF as a step~in~the transformation pipeline defined in the JSON object ${\small ``\mathtt{Action}"}$.
This object contains two name-value pairs: ${\small ``\mathtt{StartAt}"}$, which indicates the first transformation in the pipeline, and
${\small ``\mathtt{Steps}"}$, which is another JSON object that specifies the transformation UDFs along with their input parameters.
Since transformations run only when the corresponding events come through the data stream, this schema allows data owners to
specify the observed events for each transformation to execute. To do so, there exists a field named ${\small ``\mathtt{EventType}"}$
to signal the event to be observed by a particular transformation UDF. A typical ${\small ``\mathtt{EventType}"}$ block looks like
this:
\begin{lstlisting}[language=jsonnonumber]
{
"Type": "<type_of_event>",
"Input": [<parameter_block>, ...],
}
\end{lstlisting}
where the ${\small ``\mathtt{Type}"}$ field specifies the type of an event (e.g., an XPath~\cite{xpath} event) and
the array ${\small ``\mathtt{Input}"}$ lists the parameters that are required for this type of event. For instance, if
a data owner wanted to apply a transformation UDF over all $\small \mathtt{salary}$ elements of an XML document,
she could do so by setting an XPath event as follows:
\begin{lstlisting}[language=jsonnonumber]
{
"Type": "XPathEvent",
"Input": [{"Predicate": "//salary"}],
}
\end{lstlisting}
Similarly, a transformation UDF block is defined as follows:
\begin{lstlisting}[language=jsonnonumber]
{
"Id": "<identifier_of_UDF>",
"EventType": "<event_block>",
"Input": [<parameter_block>, ...],
"Next": (<identifier_of_UDF>|"End")
}
\end{lstlisting}
The ${\small ``\mathtt{Id}"}$ field is a string which uniquely identifies the UDF,
while the ${\small ``\mathtt{Input}"}$ field permits to specify the parameters
for the transformation UDF (e.g., the targeted security level of a cryptosystem). Finally, the
${\small ``\mathtt{Next}"}$ field indicates the next step to follow in the pipeline,
or ${\small ``\mathtt{End}"}$ to indicate the end of the chain of transformations.
An example of a real policy can be found in Listing~1. This policy provides
transformed views over a JSON file containing employee records of the format:
\begin{lstlisting}[language=jsonnonumber]
{
"employee": {
"name": "Alice",
"identification": {
"SSN": "32456677"
},
"salary": 50000
}
...
}
\end{lstlisting}
As a first transformation in the chain, this policy uses
content-level access control~\cite{biswas15} (CLAC). Very succinctly, CLAC works as follows. It assigns targeted JSON elements
an object label (${\small \mathtt{olabel}}$) and each user~a user-label ($\mathtt{ulabel}$). Then, it allows to define rules in the form~of $(\mathtt{ulabel}, \mathtt{olabel})$,
which means that the JSON elements labeled with any of the $\mathtt{olabel}$s are allowed to be read
by the users labeled with the corresponding $\mathtt{ulabel}$s. To indicate what JSON items to protect,
CLAC uses JSONPath in this case.
In this policy, we assume two types of users: treasurers with user label $``\mathtt{treasurer}"$
and regular users with label ${\small ``\mathtt{user}"}$. We protect the salaries with the object
label $``\mathtt{sensitive}"$ and specify the single rule $(``\mathtt{treasurer}", ``\mathtt{sensitive}")$,
which means that only treasurers will have access to the salaries. The field
$``\mathtt{salary}"$ is identified using the JSONPath expression:
$`\mathtt{\$.employee.salary}$' (Listing~1, line $14$).
The second transformation applies homomorphic encryption on the field
$``\mathtt{salary}"$ to prevent the servers from learning the
employee salaries~\cite{pilatus}, while computing the average salary of employees (Listing~1, lines $20$-$30$).
As a final transformation in the chain, the policy uses proxy re-encryption~\cite{pilatus} to convert the
homomorphically encrypted average salary to a ciphertext under the receiver's key. Thus, the
transformed view can be decrypted by the receiver, without the data owner having to share her
private key nor performing any encryption for the receiver on her personal device.
To wrap up, this policy will generate two transformed views of the same data.
For regular users, it will only be executed the first step (lines $10$-$19$): the CLAC transformation.
Due to lack of permissions, the CLAC module will eliminate the encrypted $``\mathtt{salary}"$ field,
and output a transformed view with the rest~of information. For a treasurer, it will output the same
view as~any regular user (without individual salaries), but enriched with the average salary encrypted under
her public key thanks to proxy re-encryption (Listing~1, lines $31$-$37$).
We want to note that the different transformations UDF only execute when the corresponding JSONPath
events come along. To wit, proxy re-encryption will only be run one time, after~the JSON field named $``\mathtt{average\_salary}"$
is added at the end of the response by the second transformation in the chain.
\begin{lstlisting}[caption={A sample policy to process employee records.}\label{lst:1},language=json,firstnumber=1,captionpos=b]
{
"Id": "employee.policy",
"Object": "v1/{account}/{container}/employees.json",
"Condition": {
"DateNotEquals": { "Day": ["Sat","Sun"] }
},
"Action": {
"StartAt": "Step1",
"Steps": {
"Step1": {
"Id": "CLAC",
"EventType": {
"Type": "JSONPathMarkerEvent",
"Input": [{"Predicate":"$.employee.salary", "olabel": "sensitive" }]
},
"Input": [{"ulabel": "treasurer",
"olabel": "sensitive" }],
"Next": "Step2"
},
"Step2": {
"Id": "SUM",
"EventType": {
"Type": "JSONPathEvent",
"Input": [{"Predicate":"$.employee.salary"}]
},
"Input": [{"average": true},
{"keyOwner": "meta://Alice/keys/hom",
{"output":"$.average_salary"}],
"Next": "Step3"
},
"Step3": {
"Id": "PRE",
"EventType": {
"Type": "JSONPathEvent",
"Input": [{"Predicate":"$.average_salary"}]
},
"Next": "End"
}}}
\end{lstlisting}
As a final word, this
example clearly displays how \textsc{Egeon}\hphantom{} is capable of performing real-time
privacy transformations~atop the same data object for a variety of application scenarios,~thus
enhancing the rigid interface of object storage systems.
\subsection{Data Plane}
\label{sec:data}
~~The focus of \textsc{Egeon}~is on inline privacy transformations, where
data streams are ``observables'' and user-defined privacy transformations
are ``observers'' subscribed to the data streams.
As as soon as an event is observed, it will be delivered to the subscribed observers.
If there are no events on the data stream then the original data stream is pushed
back to the user. In~this sense, a privacy transformation is nothing but a function taking an
observable as input and returning another observable as its output. This design has
the advantage that transformations can be chained together to generate complex
data views compliant with the policies in the privacy plane.
\smallskip
\noindent\textbf{Runtime.} Currently, \textsc{Egeon}'s runtime is Java-based. Thus,~the chain of
transformations is run within a Java Virtual Machine (JVM) wrapped
within a Docker container to guarantee a high level of isolation between
two different data transformations.
\smallskip
\noindent\textbf{Policy Enforcement.}
~~Upon a new \texttt{Get} request,
\textsc{Egeon}~starts up a thread inside the JVM to perform three
tasks. We refer~to this thread as the ``master thread''. The three tasks in order
of execution are:
\begin{enumerate}
\item \emph{Policy loading}, where the master thread loads the policy into memory and evaluates its conditions clauses.
\item \emph{Observable setting}, where the master thread opens a~data stream to the target object, namely, the observable, if
the policy conditions are fulfilled. To this aim, it creates an instance of the appropriate subclass of the abstract class
$\mathtt{StreamBuilder}$ that the \textsc{Egeon}'s engine leverages to start parsing the object.
Subclasses are required since~the specific
logic to parse the data stream and generate the events depends on the type of file.
Specifically, \textsc{Egeon}\vphantom{} picks up
the proper $\mathtt{StreamBuilder}$ subclass based on the object extension
(e.g., $``\mathtt{.json}"$ for~JSON documents) using \emph{factory} methods.
\item \emph{Transformations setting}, where the master thread makes an instance of each
transformation UDF in the pipeline and chains them together. The $\mathtt{StreamBuilder}$
subclass generates events as the data is being parsed, and notifies the subscribed
transformation UDFs in the same order as dictated in the privacy policy. To wit,
in Listing~\ref{lst:1}, upon the JSONPath event $`\mathtt{\$.employee.salary}$',
\textsc{Egeon}~will execute first the CLAC transformation, followed by the SUM transformation.
\end{enumerate}
In \textsc{Egeon}, we assume that an observable can only handle~one event at a time. We adopted
this design to minimize compute resources at the storage layer, so that both
the data stream and all its transformations operate in the same thread, in our case, the master
thread. Nevertheless, this can be easily changed by switching to a different thread and integrating some
additional logic for coordinating the threads.
Since each transformation UDF acts as an observer, another responsibility of
the master thread is to subscribe each privacy transformation to the events
specified in the policy. To this aim, it invokes the method~${\tiny \mathtt{install(Event\,event, UDF\,observer)}}$ in
the ${\small \mathtt{StreamBuilder}}$ class, where the parameter ${\small \mathtt{observer}}$ is the
transformation UDF bound to the ${\small \mathtt{event}}$.
\smallskip
\noindent\textbf{Extensibility.} At the time of this writing, \textsc{Egeon}~implements three types of data sources: XML, JSON and CSV
documents, a number of events including XPath and JsonPath expressions, CSV field and records, etc., and multiple transformation
UDFs (see \S\ref{sec:trans} for further details). However, \textsc{Egeon}~is extensible, and new events, observables and observers can be incorporated by
extending the abstract classes ${\small \mathtt{Event}}$, ${\small \mathtt{StreamBuilder}}$ and
${\small \mathtt{UDF}}$, respectively.
Due to space constraints, we only show here an example of a transformation ${\small \mathtt{UDF}}$ to
perform summations on ciphertexts~\cite{pilatus} to see how easy
it is to code a transformation UDF (Listing~2).
\begin{lstlisting}[caption={A transformation UDF to perform summations on ciphertexts.},language=Java, captionpos=b, numbers=left,
numberstyle=\footnotesize]
package com.urv.egeon.function;
import com.urv.egeon.runtime.api.crypto.Homomorphic;
import com.urv.egeon.runtime.api.parser.UDF.ContextUDF;
import com.urv.egeon.runtime.api.parser.UDF.UDF;
import com.urv.egeon.runtime.api.parser.event.Event;
public class Sum extends UDF {
private Homomorphic accum = new Homomorphic();
@Override
public Event update(Event e, ContextUDF ctx) {
try {
this.accum.add((String) e.getValue());
} catch (Exception ex) {
// first execution of the UDF
this.accum.setKeys((String) ctx.getParameter("keyOwner"));
this.accum.setCipher(this.accum.fromSerial((String) e.getValue()));
} finally {
return e;
}
}
@Override
public Object complete(ContextUDF ctx) { ... }
}
\end{lstlisting}
As shown in Listing~2, a UDF has two methods: the method $\mathtt{update}$,
which is invoked every time a new subscribed event is emitted, and the method $\mathtt{complete}$,
which is called when~the data stream is finalized (empty). The $\mathtt{update}$ method has two arguments
of type $\mathtt{Event}$ and $\mathtt{ContextUDF}$. The first argument encapsulates the details of the emitted event.
For instance, for the homomorphic summation UDF of Listing~2, this may mean a JSONPath event, alongside
the value of the selected field by the JSONPath expression (e.g., an encrypted salary value), and accessible via
the method $\mathtt{getValue}$ (Listing~2, line $14$). The $\mathtt{ContextUDF}$ encapsulates the access to the
request metadata, such as HTTP headers and cryptographic keys and tokens sent out by the client,
the object's metadata, and specific parameters required for the UDF to work, such as the data owner's public key to
operate on the encrypted values (Listing~2, line $17$). All this information is automatically made available by
\textsc{Egeon}~to the transformation UDF. This includes the input parameters~set in the privacy policy (e.g., Listing~1, lines $26$-$28$).
~~It is worth to mention here that the input arguments whose values have~the format of
$``\mathtt{meta://key}"$ are automatically downloaded from the Metadata
Service using the key $``\mathtt{key}"$. In this way, a data owner can change the input parameters (e.g., her
homomorphic public key) without having to re-compile the policy. An example of this can be found in
Listing~1, line $27$, for the argument $``\mathtt{keyOwner}"$, whose value is retrieved from the Metadata
Service behind the scenes, and made accessible to the summation UDF via the context
(Listing~2, line $17$).
~~The purpose of the method $\mathtt{complete}$ is to provide a hook for
developers to perform final computations and append their result at the
end of the data stream. For instance, this could be useful to compute
the average over encrypted data and append the result as new item
at the end of a JSON document.
\subsection{Data Transformation UDFs}
\label{sec:trans}
\noindent\textsc{Egeon}~comes up with a library of reusable functions~(UDFs) for
inline data protection. While some of these functions apply cryptographic
transformations, others act on raw data, e.g., by filtering out protected
parts of a JSON object to non-privileged users~\cite{biswas15}. All functions are composable
to provide complex transformed data views. These
are the following:
\smallskip
\noindent\textbf{Homomorphic encryption (HOM).}~~HOM
is a cryptosystem (typically, IND-CPA secure) that allows the server to perform
computations directly on encrypted data, the final result being
decrypted by the user devices. For general operations, HOM is prohibitively slow.~~However, it is
efficient for summation.~To support summation, along with proxy re-encryption (PRE),
we adopted the homomorphic cryptosystem of~\cite{pilatus}. We chose this scheme,
because it allows secure data sharing and is tailored~to mobile platforms
and constrained IoT devices. Concretely,~we implemented two UDFs:
\begin{itemize}
\item Summation (\texttt{SUM}): This UDF supports the summation of ciphertexts, such
that the result is equal to the addition~of the plaintext values:$\,\mathtt{Enc}(m_1)\cdot\mathtt{Enc}(m_2)=\mathtt{Enc}(m_1 + m_2)$.
\item Re-Encryption (\texttt{PRE}). Succinctly, \texttt{PRE} allows the storage
servers to convert ciphertexts under the data owner's key to ciphertexts under authorized users keys without
leaking the plaintext. Therefore, the data owner can securely share data with other users, i.e., without sharing her
private key nor performing any encryption for them on her personal device.
To do so, the data owner $d$ solely needs
to issue a re-encryption token for a user $u$ based on his public key $pk_u$ as the $\mathtt{Token}_{d\rightarrow u}$.
With this token, the \texttt{PRE} UDF~can automatically re-encrypt data on behalf of the data owner without her
intervention. Further, the re-encryption tokens are unidirectional and non-transitive
\end{itemize}
For both UDFs, we assume integers of $\leq 32$ bits with $128$-bits of security. The implementation
makes use of the optimal Ate pairing~\cite{ate} over Barreto-Naehrigopera elliptic curve~\cite{bn}, and
also applies the Chine Remainder Theorem (CRT) to optimize
decryption~\cite{pilatus}. For all this cryptographic processing,~we~use the RELIC toolkit~\cite{relic}.
\smallskip
\noindent\textbf{Keyword search (SEARCH).} To allow keyword searches (as the SQL``\texttt{ILIKE}'' keyword),
we make use of a cryptographic scheme for keyword searches on encrypted text~\cite{peks}. As above, we have
chosen this scheme because it is multi-user. To put~it baldly, before storing an object, the data owner first
selects~the users with whom she wants to share her data and then encrypts it with their public keys. To
search for keywords in the shared object, a user makes a trapdoor $\mathtt{Trap}_W$ for the keyword~set~$W$
using his private key, and then sends it to the server. The server runs the \texttt{SEARCH} UDF, which takes the
public key of the user, the trapdoor $\mathtt{Trap}_W$ and the encrypted text, and returns ``yes'' if contains
$W$ or ``no'' otherwise. As expected, this scheme~is proved secure against chosen keyword attacks~(IND-CKA).
~One major advantage of this scheme is its short ciphertext size. Concretely, for $n$ users, it requires
$(n + \ell + 1) \cdot L1$ bits, where $\ell$ is the number of keywords and $L1$ is the bit length~of
the underlying finite field $\mathbb{F}_q$, which is much smaller than the deployment of $n$ separate
instances of a public key searchable encryption scheme, one for each user. For the implementation of this
scheme, we use the ``SS$512$'' elliptic curve, a symmetric curve with a $512$-bit base field, which provides
a security level of $80$-bits, from the jPBC library~\cite{JPBC}.
\smallskip
\noindent\textbf{Content-level access control (CLAC).} As an example of a non-cryptographic function,
we decided to implement content-level access control~\cite{biswas15}. The central idea of \texttt{CLAC}
is to enforce access control at the content level to restrict who reads which part
of a document. To give a concrete example, consider that a hospital stores its patient records
as big JSON object. These records should be accessed
differently by different personnel. For example, a ``doctor'' could
see health information from her patients, while a ``receptionist'' should only
view basic profile information about the patients. With \texttt{CLAC}, a data owner
can define content-level policies to censor access to parts of a data object. Remember that
in object storage systems such as Swift or Amazon S$3$, once an object is made accessible to someone,
she retrieves the full content of the object. So, there is no way to hide out sensitive information to that user.
Our implementation of \texttt{CLAC} overcomes this limitation. As introduced in \S\ref{sec:privacy},
\texttt{CLAC} borrows the LaBAC model (Label Based Access Control Model)~\cite{biswas15}. Either
be an XML element, a JSON element, or a CSV column, an object label (${\small \mathtt{olabel}}$)~is
assigned on the targeted item. Similarly, each authorized user is given a user label ($\mathtt{ulabel}$). Then,
the LaBAC model works by specifying tuples of rules in the form~of $(\mathtt{ulabel}, \mathtt{olabel})$,
which tell that only the users labeled with $\mathtt{ulabel}$ can access the items labeled with that $\mathtt{olabel}$.
For instance, if only users with the user label $``\mathtt{manager}"$ were authorized to access items labeled with $``\mathtt{restricted}"$,
a data owner should have to set the rule $(``\mathtt{manager}", ``\mathtt{restricted}")$ to effectively formalize access control.
As in the LaBAC model, our implementation admits hierarchies of both $\mathtt{ulabel}$s and $\mathtt{olabel}$s to
rank users and objects.
~One interesting side effect of our reactive data plane is that our \texttt{CLAC} implementation is generic,
and not tied to a specific file type. What changes is the mechanism to identify the items, which depends
on the object type. That is, for XML, an XPath expression should be used to indicate that a certain
element has $``\mathtt{restricted}"$ access, while for a JSON document, a JSON predicate should be used in its stead.
We have abstracted this coupling by the definition of specific marker events as shown in Listing~1, lines $22$-$25$.
\section{Implementation}
\label{sec:impl}
We have constructed \textsc{Egeon}~by extending Zion~\cite{zion}, a
data-driven serverless computing middleware for object storage. In
particular, we have implemented \textsc{Egeon}~on top of OpenStack Swift~\cite{swift},
a highly-scalable object store.
OpenStack Swift is split into several components. The main components
are the object and proxy servers. While the object servers are responsible for the storage and management of the objects,
the proxy servers expose the RESTful Swift API (e.g., $\mathtt{GET\,/v1/account/container/object}$ to get object content)
and stream objects to and from the clients upon request.
To be as non-intrusive as possible, the only modification we perform in the default Swift architecture is the
deployment of~a custom Swift middleware to intercept $\mathtt{GET}$, or read, requests~at the proxy servers~\cite{middleware}.
This middleware also provides a simple API to manage the life-cycle of privacy policies. Essentially, it communicates
with the Metadata Service to store and retrieve the privacy policies for protected data objects. Recall that the Metadata
Service leverages Redis~\cite{redis} to yield sub-millisecond access latency to metadata.
To optimize request matching even further, we have collocated the Redis instance with the proxy server.
~~As Zion, \textsc{Egeon}~uses containers to sandbox the execution of the chain of privacy transformations.
However, contrary to Zion, which is a general-purpose serverless platform, \textsc{Egeon}\hphantom{} employs
a single, optimized serverless function to produce the privacy-compliant views of the
underlying data. This function deserializes the compiled privacy policy, loads it into memory,
evaluates the conditions from the clauses, and if it ``applies'',~it runs the the pipeline of UDF transformations.
This design has two main benefits. On the one hand, \textsc{Egeon}~runtime starts up faster. On the other hand,
UDF transformations enjoy of a great level of isolation. Simply put, they have neither direct network access nor access to the local Linux file
system, among other namespaced resources, which protects the whole system from malicious transformations.
Also, to enhance response time for policies that are accessed frequently, \textsc{Egeon}~runtime deploys a cache for policies
using the Google Guava caching library~\cite{guava} configured with a least-recently-used (LRU) eviction policy.
~~Resource allocation is managed by Zion. \textsc{Egeon}~does not contribute any optimization at this level.
When a read request comes along, our Swift middleware contacts the Zion service, which manages
the containers in the object servers, and starts up a new one if necessary.
\section{Evaluation}
~In this section, we evaluate key aspects of \textsc{Egeon}'s~design and our prototype implementation.
First, we begin with a series of microbenchmarks to judge aspects such as system overhead,
throughput and the performance of the cryptographic operators in isolation. Finally, we
assess \textsc{Egeon}'s flexibility to compose complex data views.
\smallskip
\noindent\textbf{System setup.} All the experiments have been conducted in a cluster of
$8$ machines: $2$ Dell PowerEdge R320 machines with $12$GB RAM, which operate as Swift Proxy servers,
and $6$ Dell PowerEdge R320 machines with $8$GB RAM and $4$-core CPUs, which act as
object servers. The version of OpenStack Swift~is Stein $5.2.0$. All the machines run
Ubuntu Server $20.04$LTS~and are interconnected through $1$GbE links. The client machine for the experiments is
equipped with a Intel Core i5-4440 CPU with 4 cores and $8$GB RAM.
\smallskip
\noindent\textbf{Competing systems}. In some tests, we have compared \textsc{Egeon}\hphantom{} against plain Zion~\cite{zion}
and Vertigo~\cite{vertigo}, all deployed in the same Swift cluster as above. Concretely, we have used Zion~as a baseline
to assess the overhead added by \textsc{Egeon}'s software-defined architecture to the original Zion design, while we have chosen Vertigo as an
example of a general-purpose, software-defined object storage system. As \textsc{Egeon}, Vertigo allows users to create pipelines
of storage functions, each implemented as an OpenStack Storlet~\cite{storlet}. Hence, it is a good representative to act as
a proving ground for the performance of \textsc{Egeon}~against a similar software-defined architecture. It must be noticed that
the control plane in Vertigo is programmatic, while in \textsc{Egeon}\hphantom{} is declarative through the use of JSON-based privacy policies.
\subsection{Microbenchmarks}
We have run several microbenchmarks:
\medskip
\noindent\textbf{Cryptographic operations.}
To better understand the sources of overhead incurred by \textsc{Egeon}, we
examined the throughput of the individual transformation UDFs, since
different privacy policies may result in various transformation mixes.
For each type of cryptographic transformation, we measured the number
of operations per second that the \textsc{Egeon}~runtime can perform on a
object server in the data plane, as well as the latency.~The meaning of each operation depends on
the specific type of~the cryptographic transformation UDF.
For \texttt{HOM}, this refers to~the
summation of two encrypted $32$-bit integers. For \texttt{PRE}, it refers to the proxy re-encryption of
a single encrypted $32$-bit integer, while for \texttt{SEARCH}, it represents the
search of a keyword in~an encrypted document with the same keyword. The results of this experiment are given in
Table~\ref{tab:OPsSecond}. As expected, we can observe that the latency of the cryptographic operations is
in the order of a few milliseconds, which is acceptable for many use cases. Due to the added latency of the cryptographic
transformations, a reactive data plane such as that available in \textsc{Egeon}, which~is driven exclusively
by the events appearing in the data streams, can be of great help to define privacy policies that minimize~the
number of cryptographic operations (for instance, by skipping unneeded data in the first steps of the
transformation chain).
\medskip
\noindent\textbf{Overhead.} To provide a full picture of the overheads incurred by \textsc{Egeon}, we
measured the latency introduced by \textsc{Egeon}~to the I/O path with respect to Zion and vanilla Swift.
To measure the overhead, we utilized the Time to First Byte (TTFB),~which captures how long the
client needed to wait before receiving~its first byte of the response payload from the Swift servers. To
make measurements more precise, we colocated the client with one of the Swift proxy servers. As a client,
we used \texttt{pycurl} to generate the \texttt{Get} requests for three different data file sizes.
For each object size, we performed $1$K requests. For Zion,~each request caused the invocation of a \texttt{NOOP} function
that simply echoes the input stream to the output (see Listing~1 in \cite{zion}~for further details). For \textsc{Egeon}, we
set up a \texttt{NOOP} policy which includes a single \texttt{NOOP} UDF in the transformation chain. This UDF
does nothing:
\smallskip
\begin{lstlisting}[caption={A no-operation (NOOP) UDF.},language=Java, captionpos=b, numbers=left,
numberstyle=\footnotesize]
package com.urv.egeon.function;
import com.urv.egeon.runtime.api.parser.UDF.ContextUDF;
import com.urv.egeon.runtime.api.parser.UDF.UDF;
import com.urv.egeon.runtime.api.parser.event.Event;
public class Noop extends UDF {
@Override
public Event update(Event e, ContextUDF ctx) { return e; }
@Override
public Object destroy(ContextUDF ctx) { return null; }
}
\end{lstlisting}
The results are given in Fig.~\ref{fig:2}. As seen in this figure, Zion and \textsc{Egeon}~are on par, which
demonstrates that \textsc{Egeon}\vphantom{} software-defined architecture adds little overhead to Zion.
With respect to vanilla Swift, both systems add around $9$ms of extra latency, which can be
considered very small. To wit, serverless function invocation in major cloud providers usually take
between $25$ to $320$ms in warm state~\cite{peeking}.
Since Zion does not support the pipelining of functions, we compared \textsc{Egeon}~against Vertigo.
Recall that Vertigo enables users to chain several Storlets together, where each Storlet~can
implement some reusable storage function such as decryption, compression, etc. We repeated
the same experiment as above, but evaluating chains of increasing length. For \textsc{Egeon}, we
set up chains of \texttt{NOOP} UDFs, while for Vertigo, we did the same, but for pipelines of
\texttt{NOOP} Storlets. The results are depicted~in Fig.~\ref{fig:3}. We can see that while the
overhead keeps constant in \textsc{Egeon}, Vertigo shows a linear increase in latency. The reason
for such a difference is the reactive core of \textsc{Egeon},~which~does nothing if the
transformations are not subscribed to any event. Vertigo, however, wires each
Storlet with their neighbors in~the chain, which takes some time, albeit each of them just
copies the input to the output. This strongly reinforces the idea that for a software-defined
data protection system to be useful, it~is not a good idea to route the data streams through
a pipeline~of functions, but rather to act on them when it is strictly needed. Indeed, Vertigo's
overhead is more than one magnitude higher than \textsc{Egeon}'s overhead as shown in Fig.~\ref{fig:3}.
\begin{table}[t!]
\caption{Performance of cryptographic primitives available in \textsc{Egeon}.}
\label{tab:OPsSecond}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\textbf{Transformation UDF} & \textbf{Throughput} (ops/sec) & \textbf{Latency} (ms)\\ \hline\hline
Homomorphic Addition (\texttt{SUM}) & $616$ & $1.62$ \\
Proxy Re-Encryption (\texttt{PRE}) & 137 & $7.29$ \\
Keyword Search (\texttt{SEARCH}) & 166 & $6.02$ \\ \hline
\end{tabular}
\end{center}
\vspace{-20pt}
\end{table}
\begin{figure*}
\centering
\subfloat[][TTFB for a $10$KB object.]{
\includegraphics[width=0.325\textwidth]{Fig2a}
}
\subfloat[][TTFB for a $100$KB object.]{
\includegraphics[width=0.325\textwidth]{Fig2b}
}
\subfloat[][TTFB for a $1$MB object.]{
\includegraphics[width=0.325\textwidth]{Fig2c}
}
\caption{Time to First Byte (TTFB) for different object sizes.}
\label{fig:2}
\vspace{-10pt}
\end{figure*}
\begin{figure}
\centering
\subfloat[][Overhead of \textsc{Egeon}.]{
\includegraphics[width=0.245\textwidth]{Fig3a}
\label{fig:3a}
}
\subfloat[][Overhead of Vertigo.]{
\includegraphics[width=0.245\textwidth]{Fig3b}
\label{fig:3b}
}
\caption{Overhead of chain setup of \textsc{Egeon}~versus Vertigo.}
\vspace{-5pt}
\label{fig:3}
\end{figure}
\medskip
\noindent\textbf{Throughput.} As a final microbenchmark, we quantified the impact of \textsc{Egeon}~on
the system throughput. As above, \textsc{Egeon}\vphantom{} was compared to Zion and vanilla Swift to
give real sense of its performance. For this experiment, we utilized the \texttt{getput} benchmarking tool suite~\cite{getput} for Swift.
And in particular, the \texttt{gpsuite} to conduct parallel tests with multiple clients. More concretely, we run \texttt{gpsuite} in one of
the Swift proxy servers for $10$ seconds and for different object sizes. We considered a replication factor of $3$, and
instrumented \texttt{gpsuite} to stress $3$ out of the $6$ object servers in the data plane. As in the overhead test, a \texttt{NOOP}
function call per request was made for Zion and a \texttt{NOOP} policy plus \texttt{NOOP} UDF for \textsc{Egeon}. Table~\ref{tab:IOPS}
reports the maximum throughput in operations per second attained by each system. Similarly to what was observed for the overhead,
\textsc{Egeon}~and Zion perform in similar terms. More interestingly, as the object size increases, the gap between both \textsc{Egeon}~and Zion and Swift
grows. We investigated this issue and we found that this happens due to a higher CPU interference caused by the JVM used to run
the \textsc{Egeon}~logic and the \texttt{NOOP} code~in Zion, respectively.
\begin{figure*}
\centering
\subfloat[][Throughput (ops/sec) for a $1$MB object.]{
\includegraphics[width=0.3\textwidth]{Fig4a}
\label{fig:4a}
}\quad\quad\quad
\subfloat[][Slowdown of \textsc{Egeon}~over vanilla Swift.]{
\includegraphics[width=0.325\textwidth]{Fig4b}
\label{fig:4b}
}
\caption{Throughput and latency of \textsc{Egeon}~under a multi-client setting.}
\vspace{-15pt}
\end{figure*}
Also, Fig.~\ref{fig:4a} plots the throughput for an increasing number of emulated clients for a $1$MB object, which
exhibits the same behavior as before. That is, \textsc{Egeon}~and Zion showing a similar performance, while Swift
delivering a much higher throughput due to the absence of any computation in the I/O path. Finally, Fig.~\ref{fig:4b} illustrates
\textsc{Egeon}'s slowdown relative to vanilla Swift, calculated as $\mathrm{Slowdown = \frac{\textsc{Egeon}\,download\,time}{Swift\,download\,time}}$,
as a function of the object size ($x$-axis) and the number of concurrent clients ($y$-axis). As can be seen in the figure, the slowdown
factor~does not increase steeply. Rather, it increases gradually in both axes, never doubling the latency. This indicates that pushing
down data protection logic to the storage may be acceptable for many applications.
\begin{table}[t!]
\centering
\caption{Maximum throughput (ops/sec) for different object sizes.}
\label{tab:IOPS}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{System} & \multicolumn{3}{c|}{\textbf{Object size}} \\ \cline{2-4}
& $100$KB & $1$MB & $10$MB \\ \hline\hline
Swift & 157,81 & 111,43 & 31,62\\
Zion & 140,72 & 79,69 & 21,17 \\
Egeon & 140,56 &79,59 & 21,26 \\ \hline
\end{tabular}
\vspace{-15pt}
\end{table}
\subsection{Applications}
To evaluate the composability of \textsc{Egeon}, we have designed two privacy policies that capture the
different complexities of real-world applications. These applications are the following:
\smallskip
\noindent\textbf{Covid-$19$ use case.} In this use case, we demonstrate the same policy
of Listing~1, but applied to healthcare. We use the JSON file reported by the US government that
summarizes the patient impact on healthcare facilities caused by Covid-$19$~\cite{covid19} (April 2021).
Specifically, we exchange the user label $\mathtt{``treasurer"}$ by $\mathtt{``state\,coordinator"}$, and label the field
that reports the sum of patients hospitalized in a pediatric inpatient bed in~$7$-day periods [FAQ-$10$.b)] as
$\mathtt{``sensitive"}$, as it reveals which hospitals~may be collapsed. The rest of information is ignored.
As a result, the transformed view for state officials only bears a single homomorphically encrypted value
that aggregates the sum of all healthcare facilities. As in Listing~1, the policy links $3$ UDFs:
\texttt{CLAC}$\rightarrow$\texttt{HOM}$\rightarrow$\texttt{PRE}. To play out with the file size, we split
this dataset into three smaller files based on increasingly smaller time periods: year, month and week.
\smallskip
\noindent\textbf{Adult dataset~\cite{adult} use case.} This dataset in CSV format from
the UCI Machine Learning Repository~\cite{adult} has $48842$ records and $14$ attributes.
Some of these attributes can leak sensitive information such as race and occupation. For this
use case,~we have decided to protect the attribute $\#7$: occupation, with the \texttt{SEARCH}
scheme to allow for type-of-employment searches on encrypted text. To prove
composability, we have used \texttt{CLAC}~to protect three attributes out of the
four attributes chosen for this experiment. We have used one object label: $\mathtt{``sensitive}"$, and one user label:
$\mathtt{``HR\,manager"}$, so that only a human resources manager can retrieve the three protected columns. Further,
we have encrypted other fields to increase the file size to $134$MB, to later split it into $3$ smaller files based on
the attribute $\#6$: marital-status. The policy is as follows (some fields have been omitted for brevity):
\begin{lstlisting}[language=jsonnonumbersmall]
{
"Object": "v1/{account}/{container}/adult.csv",
"Action": {
"StartAt": "Step1",
"Steps": {
"Step1": {
"Id": "CLAC",
"EventType": {
"Type": "ColumnMarkerEvent",
"Input": [{"columns": [2,6,7], "olabel": "sensitive" }]
},
"Input": [{"ulabel": "HR manager", "olabel": "sensitive" }],
"Next": "Step2"
},
"Step2": {
"Id": "SEARCH",
"EventType": {"Type": "ColumnEvent", "Input": [{"column":7}]},
"Next": "End"
}}}
\end{lstlisting}
\begin{table}[h]
\caption{Network Speeds. }
\label{tab:NetworkSpeeds}
\centering
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Network} & 4G & Fiber & LAN \\ \hline\hline
\textbf{Median speed (Mbps)} & $28.9$~\cite{4GUS} & $55.98$~\cite{fiberUS} & $887$ (SpeedTest) \\ \hline
\end{tabular}
\end{table}
\smallskip
\noindent\textbf{Experiment.} In this test, we measure the time to download~the raw files directly from Swift
against the time to download the transformed views generated by \textsc{Egeon}. The goal is to decide if it is worth
to push the privacy transformations into the object store instead of running them on the user devices and VMs,~so that
software-defined data protection is within reach. To do so, we have capped the ingoing
bandwidth of our client machine to emulate different network speeds and customary scenarios: Fiber
network speeds to emulate home and business users, $4$G network bandwidth to emulate mobile users, and finally, LAN
to simulate a scenario where the client and Swift servers reside on the same local area network (e.g., a university
intranet). The exact network speeds are listed in Table~\ref{tab:NetworkSpeeds}. We performed $1$K
executions per object size and network speed.
\begin{figure}
\centering
\subfloat[][Covid-$19$ use case.]{
\includegraphics[width=0.245\textwidth]{Fig5}
\label{fig:5}
}
\subfloat[][Adult dataset use case.]{
\includegraphics[width=0.245\textwidth]{Fig6}
\label{fig:6}
}
\caption{Performance of \textsc{Egeon}~in two composite policies.}
\vspace{-15pt}
\end{figure}
The results are plotted in Fig.~\ref{fig:5} for the Covid-$19$ use case, and in Fig.~\ref{fig:6} for the Adult dataset.
Error bars display the standard deviations of results, which are indeed very narrow.
Non-surprisingly, we can see that \textsc{Egeon}~lowers the download time
significantly for the slow $4$G and fiber connections, which means that pushing down privacy transformations
to storage~is a good deal better than the naive approach of encrypting data on the client side and retrieve
the whole file as alleged~by~cloud providers such as AWS for S$3$. The savings in some scenarios can be
dramatic such as in the Covid-$19$ use case, where just~a few bytes (e.g., aggregates such as \texttt{SUM}, \texttt{COUNT} and \texttt{AVG}, etc.)
are consumed by the application, reaching $72.1$X speedup for $4$G mobile terminals. For the LAN setting, the benefits are not so
clear, and for the Adult dataset, \textsc{Egeon}~shows~a~slowdown factor of $3.3$X in the worst case due to the heavy
computations associated with keyword search---actually, the test primitive of \texttt{SEARCH} requires three pairing operations~\cite{peks}.
Either way,~we believe that \textsc{Egeon}'s fine-grained data protection capabilities outweigh the slight loss of performance.
\section{Related Work}
\noindent\textbf{Software-defined storage systems}. A first category of related work comprises software-defined approaches for
storage, and in particular, for object storage systems. The common feature of these approaches is that they break
the vertical alignment of conventional storage infrastructures by reorganizing the I/O stack to decouple the control and data flows into two
planes~of functionality---control and data. A number of proposals have followed this approach, including IOFlow~\cite{ioflow}, sRoutes~\cite{sroute},
Retro \cite{retro}, Vertigo~\cite{vertigo} and Crystal~\cite{crystal, FGCScrystal}.
Among them, only Vertigo and Crystal have been tailored~to object storage. As \textsc{Egeon}, both systems have been deployed atop
OpenStack Swift. But unlike \textsc{Egeon}, their data plane~is based on OpenStack Storlets~\cite{storlet}. A Storlet is a piece of Java logic~that
is injected into the data plane to run custom storage services over incoming I/O requests. As in \textsc{Egeon}, this
design increases the modularity and programmability of the data plane stages, fostering reutilization. However,
Storlet-made pipelines are not reactive, wasting resources when we are only interested in specific elements
of the data stream. Moreover, the Storlet-enabled data plane of Vertigo and Crystal emphasizes
control-flow over data-flow, making it hard to explicitly represent the (cryptograhic) transformations of
objects. Per contra, \textsc{Egeon}'s data plane is driven solely by the events showing up in the data streams, which
makes it easy to reuse the same transformations over and over again on different data. Only the events must~be
re-defined in the policies.
Finally, it is worth to note that to the best of our knowledge, we are not aware of another software-defined
storage~system that automatically enforces privacy policies along the I/O path as \textsc{Egeon}. Software-defined security
has remained within the boundaries of software-defined networking (see, for instance, Fresco~\cite{fresco}).
The only exception is the recent vision paper~\cite{Zsolt21} on software-defined data protection. Like \textsc{Egeon},
\cite{Zsolt21} argues that the key ideas of software-defined storage can be translated to the data protection domain.
However, the approach of \cite{Zsolt21}~is radically different. Instead of adding privacy controls to the I/O stack
of a disk-based object storage system, \cite{Zsolt21} assumes all in-storage processing to occur on FPGAs and ``smart storage'' devices,
for we see \cite{Zsolt21} as an orthogonal work to us.
\medskip
\noindent\textbf{Privacy Policy Enforcement}. There exist many systems that enforce
privacy policies automatically. Most of these systems resort to Information Flow Control (IFC) as a means to
control how information flows through the system. See, for instance, Riverbed~\cite{riverbed}, which
uses IFC to enforce user policies on how a web service should release sensitive user data. In contrast to these
systems, \textsc{Egeon}~follows a software-defined approach to leverage the storage resources and enforce the privacy
policies where data is. Similar to our transformation chains, Zeph~\cite{zeph} proposes to enforce privacy controls cryptographically
but over encrypted stream processing pipelines. Specifically for storage, Guardat~\cite{guardat}, at the block level, and Pesos~\cite{pesos}, at
the object level, enable users to specify security policies, for instance,~to
stipulate that accesses to a file require a record be added to an append-only log file.
Nonetheless, these systems do not permit the composition of
advanced privacy controls as \textsc{Egeon}, and thus, fall short to empower users with
strong data controllers.
\section{Conclusions}
As increasingly much more sensitive data is being collected to
gain valuable insights, the need to natively integrate privacy
controls into the storage systems is growing in importance. In
particular, the poor interface of object storage systems,
which lacks of sophisticated data protection mechanisms, along with
the inherent difficulties to refactor them, have motivated us~to
design and implement \textsc{Egeon}. To put in a nutshell, \textsc{Egeon}~is a novel software-defined data protection framework for object storage.
It allows data owners to define privacy policies on~how their data can be shared, which permit the composition of
data transformations to build sophisticated data protection controls. In this way, data owners can specify multiple views from the same data piece,
and modify these views by only updating the policies (e.g., by modifying the chain of transformations that produce a particular view), leaving the
system internals intact. The \textsc{Egeon}\hphantom{} prototype has been coded atop
OpenStack Swift. And our evaluation results demonstrate that \textsc{Egeon}\hphantom{} adds little
overhead to the system, yet empowering users with~the needed controls to ensure strong data protection.
\section*{Acknowledgment}
This research has been partly supported
by EU under grant agreement No. 825184 and the Spanish
Government through project PID2019-106774RB-C22. Marc
Sanchez-Artigas is a Serra-H\'{u}nter Fellow.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,920 |
Aporocidaris eltaniana is een zee-egel uit de familie Ctenocidaridae.
De wetenschappelijke naam van de soort werd in 2000 gepubliceerd door Mooi, David, Fell & Choné.
Cidaroidea | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,803 |
'use strict';
// NEEDS REFACTORING!
var map, marker, panorama; // created in video loadedmetadata handler
var video = document.querySelector('video');
var mapElement = document.getElementById('map');
var timeOfDay = document.getElementById('timeOfDay');
var VIDEO_OFFSET = 1331362993; // time of first point, at start of video
// in effect, if the browser supports addTextTrack(), this demo will work...
// - older non-working versions use addTrack()
if (typeof video.addTextTrack === 'undefined') {
document.querySelectorAll('.trackNotSupported').forEach(showElement);
document.querySelectorAll('.trackSupported').forEach(hideElement);
} else { // track implemented!
// need to get wait for video to load before getting duration, etc.,
// in order to create track
setupMap();
}
function setupMap() {
var track = video.addTextTrack('metadata', 'GBike track', 'en');
track.mode = 'hidden';
var points = [];
fetch('data/gbike.xml').then(function(response) {
return response.text();
}).then(function(text) {
var domParser = new window.DOMParser();
var document = domParser.parseFromString(text, 'text/xml');
var pElements = document.querySelectorAll('p');
pElements.forEach(function(p) {
points.push({
'lat': p.getAttribute('a'),
'lng': p.getAttribute('b'),
't': p.getAttribute('t')
});
});
setupPointsAndPath(track, points);
});
}
function setupPointsAndPath(track, points) {
var startLatLng = new google.maps.LatLng(points[0].lat, points[0].lng);
var options = {
center: startLatLng,
zoom: 18,
mapTypeId: google.maps.MapTypeId.SATELLITE
};
map = new google.maps.Map(mapElement, options);
var path = [];
for (var i = 0; i !== points.length; ++i) { // not very efficient...
var point = points[i];
path.push(new google.maps.LatLng(point.lat, point.lng));
var startTime = point.t - VIDEO_OFFSET;
var endTime = i === points.length - 1 ?
video.duration : points[i + 1].t - VIDEO_OFFSET;
var cue = new window.VTTCue(startTime, endTime, JSON.stringify(point));
track.addCue(cue);
}
track.oncuechange = function() {
// 'this' is a textTrack, and there is only one active cue in this example
cue = this.activeCues[0];
if (typeof cue === 'undefined') {
return;
}
point = JSON.parse(cue.text);
var newLatLng = new google.maps.LatLng(point.lat, point.lng);
if (!map.getBounds().contains(newLatLng)) {
map.setCenter(newLatLng);
}
var oldLatLng = marker.getPosition();
var heading = getHeading(oldLatLng, newLatLng);
var pov = {
'heading': heading,
'pitch': panorama.getPov().pitch,
'zoom': panorama.getPov().zoom
};
panorama.setPov(pov);
panorama.setPosition(newLatLng);
marker.setPosition(newLatLng);
timeOfDay.textContent =
new Date(point.t * 1000).toLocaleTimeString();
};
var panoramaOptions = {
position: startLatLng,
pov: {
heading: 332, // hack :)
pitch: 10,
zoom: 1
}
};
panorama = new google.maps.StreetViewPanorama(
document.getElementById('panorama'), panoramaOptions);
map.setStreetView(panorama);
var polyline = new google.maps.Polyline({
path: path,
strokeColor: '#ff0000',
strokeOpacity: 0.2,
strokeWeight: 5
});
polyline.setMap(map);
marker = new google.maps.Marker({
position: new google.maps.LatLng(points[0].lat, points[0].lng),
map: map,
title: 'gbike!'
});
google.maps.event.addListener(map, 'click', function(event) {
var min = 360; // good enough...
var nearest;
var clickLat = event.latLng.lat();
var clickLng = event.latLng.lng();
for (i = 0; i !== points.length; ++i) {
var pointLat = points[i].lat;
var pointLng = points[i].lng;
var distance = Math.sqrt(Math.pow(clickLat - pointLat, 2) +
Math.pow(clickLng - pointLng, 2));
if (distance < min) {
min = distance;
nearest = i;
}
}
point = points[nearest];
var latLng = new google.maps.LatLng(point.lat, point.lng);
if (!map.getBounds().contains(latLng)) {
map.setCenter(latLng);
}
marker.setPosition(latLng);
video.currentTime = point.t - VIDEO_OFFSET + 1;
});
}
function degreesToRadians(deg) {
return deg * (Math.PI / 180);
}
// Adapted from http://econym.org.uk/gmap/example_dist.htm
// Returns the bearing in degrees between two points.
// North = 0, East = 90, South = 180, West = 270.
function getHeading(from, to) {
// See T. Vincenty, Survey Review, 23, No 176, p 88-93,1975.
// Convert to radians.
var lat1 = degreesToRadians(from.lat());
var lon1 = degreesToRadians(from.lng());
var lat2 = degreesToRadians(to.lat());
var lon2 = degreesToRadians(to.lng());
// Compute the angle.
var angle = -Math.atan2(Math.sin(lon1 - lon2) *
Math.cos(lat2), Math.cos(lat1) * Math.sin(lat2) -
Math.sin(lat1) * Math.cos(lat2) * Math.cos(lon1 - lon2));
if (angle < 0.0) {
angle += Math.PI * 2.0;
}
// And convert result to degrees.
var degreesPerRadian = 180.0 / Math.PI;
angle = angle * degreesPerRadian;
angle = angle.toFixed(1);
return parseInt(angle);
}
function hideElement(element) {
element.style.display = 'none';
}
function showElement(element) {
element.style.display = 'block';
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 872 |
ACCEPTED
#### According to
Index Fungorum
#### Published in
Persoonia 6(3): 325 (1971)
#### Original name
Conocybe fibrillosipes Watling
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,604 |
These are used for their anti-inflammatory effects in asthma. They are very effective in asthma, but are of limited use in COPD.
Their main role is with the decreased formation of cytokines. Particularly, Th2 cytokines as these recruit and activate eosinophils, and are also released by eosinophils.
They also reduce the production of the vasodilators PGE2 and PGE1, by inhibiting the production of COX-2.
Ultimately, these two effects reduce the influx of eosinophils to the lung. Long term use of steroids will also ultimately reduce the responsiveness of the lung to bronchoconstrictors.
There is also decreased production of IL-3 – which regulates mast cell production – and thus long term there are fewer mast cells residing in the lungs, and the early phase reaction is also inhibited (particularly useful in allergen and exercise activated disease).
There is also upregulation of β2 receptors – which helps enhance the effect of β2 agonists.
Beclomethasone – this is one of the most widely used inhaled corticosteroids. It is poorly absorbed by the gut (so reduces systemic availability when drug is swallowed). It is also inactivated very slowly once it enters the systemic circulation, and thus this is a draw back.
Budesonide –undergoes massive fist pass metabolism if it gets into the systemic circulation, and is preffered to beclomethasone when large doses of drug are needed.
Fluticasone – is very poorly absorbed from the gut, and is used in similar situations to budesonide.
The full effect of these drugs is only seen 1-2 weeks of daily inhalation, however, the initial effects occur just 6-12 hours after first use.
Generally the drugs only ever exhibit systemic effects when inhaled in very high doses.
Hoarseness of the voice – as the drugs can affect the laryngeal muscles. | {
"redpajama_set_name": "RedPajamaC4"
} | 500 |
In the End è un singolo della rock band statunitense Black Veil Brides, tratto dal loro terzo album Wretched and Divine. La canzone è stata distribuita il 31 ottobre 2012 negli Stati Uniti, come primo singolo tratto dall'album. Si tratta del sesto singolo del gruppo.
Video musicale
Il video del brano è stato pubblicato su YouTube il 12 dicembre 2012. Il video mostra la band suonare in diversi ambienti, ovvero un deserto, una chiesa e una fabbrica abbandonata.
Formazione
Andy Biersack - voce
Jake Pitts - chitarra solista
Jinxx - chitarra ritmica, violino, violoncello, cori
Ashley Purdy - basso, cori
Christian "CC" Coma - batteria, percussioni, cori
Note
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 265 |
\section{Introduction}
Gini mean difference (GMD) was introduced by Corrado Gini in 1914 as an alternative measure of variability. Since then, GMD and its derivatives such as Gini index have been widely used in a variety of research fields especially in finance, economics and social welfare (Yitzhaki and Schechtman, 2013). Rather than the assumption on the finite second moment, the GMD only requires existence of the finite mean of the distribution (Yitzhaki, 2003). Hence GMD is more robust than the variance and it is often used for heavy-tailed asymmetric distributions, although it is less robust than some scale measures without any moment conditions. On the other hand, GMD is highly efficient. The relative efficiencies (RE) of the sample GMD with respect to sample standard deviation are about 0.98 under the normal distributions, 1.21 under the Laplace distribution and 1.86 under the $t(5)$ distribution (Nair, 1936; Gerstenberger and Vogel, 2015). With a little loss on efficiency, GMD gains robustness against departures from normal distributions.
In this paper, we extend GMD to the multivariate case. We propose the Gini covariance matrix (GCM) as (2 times) the covariance of $\bi X$ with its spatial rank $\bi r(\bi X)$, which is a direct generalization from a covariance representation of the univariate GMD. While the covariance matrix (Cov) is the covariance of $\bi X$ with itself and the rank covariance matrix (RCM) (Visuri {\em et al.}, 2000) is the covariance of $\bi r(\bi X)$ with $\bi r(\bi X)$, intuitively GCM is a new scatter measure between Cov and RCM. With no surprise, the efficiency and robustness of sample GCM are between those of sample Cov and RCM. In terms of balance between efficiency and robustness, sample Gini covariance matrix provides us an extra method for multivariate statistical inference including multivariate analysis of variance, principle component analysis, factor analysis, and canonical correlation analysis.
As any estimator based on spatial signs and ranks, GCM is only orthogonally equivariant. In order to gain fully affine equivariant property, we utilize a transformation-retransformation (TR) technique (Charkraborty and Chaudhuri, 1996; Serfling, 2010) to obtain an affine equivariant version of GCM. The well-known scatter Tyler M-functional (Tyler, 1987) is a TR version of the spatial sign covariance matrix. D\"{u}mbgen (1998) considered symmetrized TR spatial sign covariance matrix on the difference of two independent vectors $\bi X_1-\bi X_2$. D\"{u}mbgen {\em et al.} (2015) provided a general treatment on M-functionals of scatter based on symmetrizations of arbitrary order. Our TR Gini covariance matrix turns out to be a pairwise symmetrized scatter M-functional. Compared to the regular M-functional, the symmetrized one has several advantages as emphasized in Sirki\"{a} {\em et al.} (2007). The distribution of pairwise differences is symmetric at $\bi 0$, hence avoids imposing some arbitrary definition of location for non-symmetric distributions. For elliptical distributions, there is no need to estimate location simultaneously for scatter M-estimators.
Hence they avoid restrictive regularity conditions for joint existence of location and scatter estimators and may take fewer iterations to converge than their counterparts. Further, a symmetrized scatter matrix has the so-called block independence property: it is a block diagonal matrix if the block components of the random vector are independent. Such a property holds naturally for the regular covariance matrix but may not for general M-functionals and some robust alternatives (Nordhausen and Tyler, 2015). Both versions of GCM have the block independence property and hence can be applied to independent component analysis (Hyv\"{a}rinen {\em et al.}, 2001) or invariant coordinate selection (Tyler {\em et al.}, 2009). The price to pay for those advantages of the pairwise difference approach is an increase of computation burden and a loss of some robustness. If a procedure has computation complexity $O(n)$, its symmetrized one may require $O(n^2)$, although D\"{u}mbgen {\em et al.} (2016) have presented new algorithms for symmetrized M-estimators to reduce the computation time substantially. For large $n$, they approximate symmetrized estimators by considering the surrogate ones rather than all pairwise differences. Their algorithms can easily be adopted for our estimators. The decrease of robustness in symmetrized procedures seems to be understandable since one single outlier affects $n-1$ pairwise differences. One must take into consideration of efficiency, robustness and computation when choosing a proper procedure for applications at hand.
Koshevoy {\em et al.} (2003) considered other multivariate extensions of mean deviation and Gini mean difference using the geometric volumes of zonotopes and lift-zonotopes. Their covariance matrices share many similar properties as our proposed ones, but ours enjoy simplicity and computational ease. Although the approach differs from ours, it is worthwhile to mention that Serfling and Xiao (2007) also generalize GMD to multivariate case through L-moment approach.
The remainder of the paper is organized as follows. In Section 2, we first review the Gini mean difference and the spatial rank function, then introduce the Gini covariance matrix and its affine equivariant version. Section 3 explores the influence functions of the two Gini covariance matrices. Section 4 presents estimation of the two Gini covariance matrices and asymptotical properties of the estimators. Asymptotic and finite sample efficiencies of the proposed TR Gini covariance estimator have been studied and compared with other estimators. The paper ends with some final comments in Section 5. All proofs are reserved to Appendix Section.
\section{Two Gini Covariance Matrices}
\subsection{Gini Mean Difference}
Gini mean difference (GMD) was introduced as an alternative measure of variability to the usual standard deviation. For a random variable $X$ from a univariate distribution $F$, the GMD of $X$ (or $F$) is
\begin{equation}\label{eqn:gmd}
\sigma_g=\sigma_g(X)=\sigma_g(F)=\mathbb{E} |X_1-X_2|,
\end{equation}
where $X_1$ and $X_2$ are independent random variables from $F$. In contrast, the variance of $X$ (or $F$) is
$$\sigma^2_v(F)=\mbox{var}(X)= \frac{1}{2}\mathbb{E} (X_1-X_2)^2.$$
Rather than the assumption of finite second moment, the GMD only requires existence of a finite mean of $F$. Hence the Gini mean difference is often used for heavy-tailed asymmetric distributions, especially in social welfare and the fields of decision-making under financial risk.
Among many representations such as Lorenz curve or $L$-functional formulations (Yitzhaki and Schechtman, 2013), we are interested in covariance formulations. One of them is
\begin{equation*}\label{eqn:gmdf}
\sigma_g(F)=4\mbox{Cov}(X, F(X)).
\end{equation*}
While the variance is the covariance of $X$ with itself, the GMD is (4 times) the covariance of $X$ with $F(X)$. In this spirit, two Gini-type alternatives to the usual covariance for measuring the dependence of random variable $X$ and another random variable $Y$ with distribution function $H$ are
\begin{equation*}
\mbox{Cov}_g(X,Y)=4\mbox{Cov}(X, H(Y)), \;\; \mbox{Cov}_g(Y,X)=4\mbox{Cov}(Y, F(X)).
\end{equation*}
Such extensions are natural and useful (Yitzhaki, 2003; Carcea and Serfling, 2015). However, a major drawback is the asymmetry between $X$ and $Y$, i.e., $\mbox{Cov}_g(X,Y) \neq\mbox{Cov}_g(Y,X)$, in general. An even worse part is that $\mbox{Cov}_g(X,Y)$ and $\mbox{Cov}_g(X,Y)$ may have different signs in some cases (Yitzhaki, 2003), which brings substantial difficulty in interpretation. The asymmetry stems from the usage of $F(X)$ or $H(Y)$, which can be thought as a standardized marginal rank. A symmetry one calls for a `joint' rank of $X$ and $Y$.
The other covariance type formulation for GMD is
\begin{equation*}\label{eqn:gmd2f}
\sigma_g(F)=2\mbox{Cov}(X, 2F(X)-1),
\end{equation*}
allowing an insightful interpretation: $\sigma_g(X)$ is twice of the covariance of $X$ and the {\em centered} rank function $r(X)=2F(X)-1$. $r(X)$ is centered because $\mathbb{E} r(X)=0$ if $F$ is continuous. So
\begin{equation}\label{eqn:gmdr}
\sigma_g(F)=2\mbox{Cov}(X, r(X))=2 \mathbb{E} (Xr(X)).
\end{equation}
A nice generalization of the centered rank in high dimensions provides a joint rank, and along with the representation of GMD in \eqref{eqn:gmdr} yields a natural extension of GMD for a multivariate distribution $F$.
\subsection{Spatial Rank Function} \label{sec:sr}
Let $\bi X$ be a $d$-variate random vector from a continuous distribution $F$ with a finite first moment and the expected Euclidean distance from $\bi x$ to $\bi X$ be $D(\bi x, F)=\mathbb{E}_F\|\bi x-\bi X\|$. Then the gradient of $D$ is denoted as the centered {\em spatial rank function} (M\"{o}tt\"{o}nen {\em et al.}, 1997), that is,
\begin{equation}\label{eqn:sr}
\bi r(\bi x)=\nabla_{\bi x} D(\bi x, F)=\mathbb{E} \frac{\bi x-\bi {X}}{\|\bi x-\bi X\|}=\mathbb{E} \{\bi s(\bi{x}-\bi{X})\},
\end{equation}
where $\bi s(\bi{x})=\bi{x}/\|\bi{x}\|$ $(\bi s(\bi{0})=\bi{0})$ is the
{\em spatial sign function} in $\mathbb{R}^d$. The spatial rank function is the {\em expected direction} from $\bi X$ to $\bi x$. We call it centered because a random rank is centered at $\bi 0$, that is, $\mathbb{E} \bi r(\bi X)=\bi 0$. The solution of $\bi x$ in $\bi r(\bi x)=\bi 0$ is called the spatial median of $F$, which minimizes $D$. In the univariate case, the derivative of $D(x,F)=\mathbb{E} |x-X|$ with respect to $x$ leads to the univariate centered rank function $r(x)=\mathbb{E} \mbox{sign}(x-X)=2F(x)-1\in [-1,1]$ if $F$ is continuous. Clearly, the median of $F$ has a center rank 0.
The spatial rank function has many nice properties. The rank function $\bi r(\bi x)$ characterizes the distribution $F$ (up to a location shift) (Koltchinskii, 1997; Oja, 2010), which means that if we know the rank function, we know the distribution (up to a location shift). Under weak assumptions on $F$, $\bi r(\bi x)$ is a one-to-one mapping from $\bi x$ to a vector inside the unit ball with the magnitude $\|\bi r(\bi x)\| \in [0,1]$ and the center of the unit ball is the spatial median of $F$.
Marginal ranks are less interesting because they are neither rotation nor scale equivariant. Also they lack the efficiency at the normal model since they loss dependence information (Visuri {\em et al.}, 2000). Based on the expected geometric volume of the simplex formed by $\bi x$ and $d$ random vectors from $F$, i.e, $D(\bi x, F)=\mathbb{E}_F V(\bi x, \bi X_1, ..., \bi X_d)$, Oja rank and sign functions are defined analogously (Oja, 1983). The major concern of this joint rank function is the computation of its sample version, especially for high dimensions. However, the sample spatial rank is simple and easy to compute, which makes it advantageous and feasible in practice. For this reason, we will use the spatial rank function to define our multivariate Gini covariance matrix.
\subsection{Gini Covariance Matrix}
\begin{definition}
For a $d$-variate random vector $\bi X$ from a continuous distribution $F$ possessing a finite first moment, the Gini covariance matrix of $\bi X$ or $F$ (GCM) is defined as
\begin{equation*}
\bi \Sigma_g=\bi\Sigma_g(F)=2\mathbb{E}[ \bi X\bi r^T(\bi X)],
\end{equation*}
where $\bi r(\bi x)$ is the spatial rank function defined in (\ref{eqn:sr}).
\end{definition}
As we can see, the definition of the Gini covariance matrix is a direct generalization from \eqref{eqn:gmdr}. Equivalently, let $\bi X_1$ and $\bi X_2$ be independent random vectors from $F$ and $\bi s(\bi x)$ be the spatial sign function, then we have
\begin{align}
&\bi \Sigma_g=2\mathbb{E}\bi X_1\mathbb{E} [\bi s^T(\bi X_1-\bi X_2)|\bi X_1]=2\mathbb{E}[ \bi X_1\bi s^T(\bi X_1-\bi X_2)] \nonumber\\
&=\mathbb{E}\frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|}. \label{eqn:gcm}
\end{align}
The third equality in \eqref{eqn:gcm} is a result of
$$\mathbb{E}\frac{\bi X_1(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|}=-\mathbb{E}\frac{\bi X_2(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|}.$$
From \eqref{eqn:gcm}, it is easy to prove that $\bi\Sigma_g$ is positive definite since $F$ is continuous.
Equation \eqref{eqn:gcm} additionally recovers the $L_1$ metric representation of Gini mean difference \eqref{eqn:gmd} when $d=1$. It also demonstrates that it is a pairwise difference approach defined without reference to a location parameter.
From \eqref{eqn:gcm}, we can also write $\bi\Sigma_g$ as $\mathbb{E}(\bi X_1-\bi X_2)\bi s^T(\bi X_1-\bi X_2)$, which is the expected matrix of the product of a pairwise difference and its directional sign function. If we only use directional information of $F$, we obtain
$$
\mathbb{E} \bi s(\bi X_1-\bi X_2)\bi s^T(\bi X_1-\bi X_2)=\mathbb{E}\frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|^2}.
$$
The resulting matrix is known as the symmetrized spatial sign covariance matrix (SSCM), which has been studied by Visuri {\em et al.} (2000), Croux {\em et al.} (2002) and Taskinen {\em et al.} (2012).
The spatial rank covariance matrix (RCM) is defined as the covariance matrix of spatial rank. That is,
\begin{align}\label{eqn:rcm}
\mathbb{E}\bi r(\bi X)\bi r ^T(\bi X)=\mathbb{E} \bi s(\bi X_1-\bi X_2)\bi s^T(\bi X_1-\bi X_3)=\mathbb{E}\frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_3)^T}{\|\bi X_1-\bi X_2\|\|\bi X_1-\bi X_3\|}.
\end{align}
RCM and its modified version have been studied by Visuri {\em et al.} (2000) and Yu {\em et al.} (2015). Since RCM uses three independent random vectors in its definition, the sample RCM is more efficient than the sample SSCM.
Before we explore properties of the Gini covariance matrix, it is worthwhile to note that another useful extension of GMD from the covariance representation based on the spatial rank function is $2\mathbb{E}\bi X^T\bi r(\bi X)$. This generalization coincides with the multivariate Gini mean difference defined in Koshevoy and Mosler (1997). That is,
$2\mathbb{E}\bi X^T\bi r(\bi X)=\mathbb{E} \|\bi X_1-\bi X_2\|$. Clearly, it is also an immediate extension from the $L_1$ metric representation of \eqref{eqn:gmd}.
\subsection{Properties of Gini covariance matrix}
We study properties of the Gini covariance matrix under elliptical distributions.
\begin{definition}
A $d$-variate absolutely continuous random vector $\bi X$ has an elliptical distribution if its density function is of the form
\begin{equation}
f(\bi x| \bi \mu, \bi\Sigma)
=|\bi\Sigma|^{-1/2} g\{(\bi x-\bi {\mu})^T\bi\Sigma^{-1}(\bi x-\bi{\mu})\},\label{eqn:elliptical}
\end{equation}
for some positive definite symmetric matrix $\bi \Sigma$ and nonnegative function $g$ with $\int_0^\infty t^{d/2-1}g(t)dt < \infty$.
\end{definition}
If $t^{(d-k)/2-1}g(t)$ is integrable, then the $k^{th}$ moment of $\bi X$ exits. The parameter $\bi{\mu}$ is the symmetric center and it equals the first moment if it exists. The scatter parameter $\bi\Sigma$ is proportional to the covariance matrix when it exists. It should be noted that elliptical distributions can also be defined through the characteristic functions without assuming densities.
In addition, the variates $R=\|\bi \Sigma^{-1/2}(\bi X-\bi \mu)\|$ and $\bi U=\{\bi \Sigma^{-1/2}(\bi X-\bi \mu)\}/R$ are independent with $\bi U$ being uniformly distributed on the unit sphere and $R$ having density
\begin{equation}\label{eqn:pdfr}
f_r(r)=\frac{2\pi ^{d/2}}{\Gamma(d/2)} r^{d-1} g(r^2),
\end{equation}
where $\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}dt$ is the gamma function. The independence of $R$ and $\bi U$ follows from Lemma 1 of Stamatis {\em et al.} (1981). Note that if the covariance matrix of $\bi X$ exists, it equals $(\mathbb{E} R^2/d) \bi \Sigma$. More details on the elliptical distribution family refer to Fang and Anderson (1990).
The family of elliptical distributions is denoted as ${\cal E}(\bi\mu, \bi\Sigma, g)$. If $\bi \mu=\bi 0$ and $\bi \Sigma=\bi I_{d}$ (the $d\times d$ identity matrix), we call the distribution spherically symmetric and denote it as $F_0(g)$.
The family of elliptical distributions contains a quite rich collection of models. Perhaps the most widely used one is the Gaussian distribution, in which
\begin{equation*}\label{eqn:normal}
g(t)=(2\pi)^{-d/2}e^{-t/2}.
\end{equation*}
Other than that, $t$ distributions are commonly used in modeling data with heavy-tailed regions. In the case of the $t$ distributions,
$$g(t)=\frac{\Gamma[(\nu+d)/2]}{\Gamma(\nu/2)(\nu\pi)^{d/2}}(1+t/\nu)^{-(d+\nu)/2},$$
where $\nu$ is the degree of freedom parameter. $\nu$ determines the fatness of the tail regions. For $\nu=1$, it is called $d$-variate Cauchy distribution, which has very heavy tails where even the first moment does not exist. When $\nu \rightarrow \infty$, it yields the Gaussian distribution.
A quite flexible elliptical family is called Kotz type distributions (Kotz, 1975; Nadarajah, 2003), in which the density is of the form \eqref{eqn:elliptical} with
\begin{equation*}
g(t)=c(d,\alpha, \beta, \gamma) t^{\alpha-1}e^{-\gamma t^{\beta}}. \label{eqn:kotztype}
\end{equation*}
The parameters are $\beta, \gamma >0$, $\alpha>1-d/2$ and $c(d,\alpha, \beta, \gamma)$ is the normalization constant. Clearly, when $\beta=1$, $\alpha=1$ and $\gamma=1/2$, the distribution reduces to the Gaussian distribution. The heaviness (or lightness) of tail regions of distributions mainly depends on $\beta$.
In particular, we take the special case of $\beta=1/2$, $\alpha=1$ and $\gamma=1$ for demonstration in later sections, that is,
\begin{equation}
g(t)=\frac{\Gamma(d/2)}{2 \pi^{d/2}\Gamma(d)}e^{-\sqrt{t}}. \label{eqn:kotz}
\end{equation}
We call it the Kotz distribution. For $d=1$, the Kotz distribution reduces to the Laplace distribution. It can be viewed as a multivariate generalization of Laplace distribution. Arslan (2010) also considered this distribution and extended it to asymmetry distributions by introducing a skewness parameter.
The following theorem states the relationship of the Gini covariance matrix and the scatter matrix $\bi \Sigma$ in elliptical distributions.
\begin{theorem}\label{thm:eigv}
If $\bi X$ is elliptically distributed from $F$ with the first moment and the scatter parameter $\bi \Sigma$ having the spectral decomposition $V \Lambda V^T$, then $\bi \Sigma_g=V\Lambda_gV^T$ with
$$\Lambda_g=\mbox{diag}(\lambda_{g,1},...,\lambda_{g,d})=c(F) \mathbb{E}\left[\frac{\Lambda^{1/2}\bi U\bi U^T\Lambda^{1/2}}{\sqrt{\bi U^T\Lambda \bi U}}\right], $$
and hence $$ \lambda_{g,i}=c(F)\mathbb{E}\left[\frac{\lambda_i u_i^2}{\sqrt{\sum_{j=1}^d\lambda_j u_j^2}}\right],$$
where $\bi U=(U_1,...,U_d)^T$ is uniformly distributed on the unit sphere, $\lambda_i$'s are eigenvalues of $\bi \Sigma$ and $c(F)$ is a constant depending on distribution $F$.
\end{theorem}
The proof of Theorem \ref{thm:eigv} goes along the lines of the proof of Theorem 1 in Taskinen {\em et al.} (2012) and is given (together with all other proofs) in the Appendix. The main consequence is that the same orthogonal matrix $V$ diagonalizes both $\bi \Sigma$ and $\bi \Sigma_g$. In other words, the Gini covariance matrix $\bi \Sigma_g$ has the same eigenvectors as $\bi \Sigma$. Consequently, the Gini covariance matrix can be used for principal component analysis.
\begin{remark}\label{rem:identity}
In the case of an elliptical distribution $F$ in $ {\cal E}(\bi \mu,\bi \Sigma,g)$ having $\bi \Sigma = \bi I_{d}$, it holds that $\lambda_{g,i}=c(F)\mathbb{E} (U_i^2/\| \bi U \|)=c(F)\mathbb{E} U_i^2=c(F)/d$ for all $i=1,...,d$ and hence $\bi \Sigma_g=
\frac{ c(F)}{d}\bi I_{d}$. In other words, for spherical distributions $F_0(g)$ (even $\bi \mu \neq \bi 0$), their Gini covariance matrix is the identity matrix multiplied by a factor. Dividing by this factor makes GCM estimator Fisher consistent to the scatter parameter at $F_0(g)$. An estimator $T(F_n)$ is Fisher consistent to $\theta$ if $T(\lim_{n\rightarrow \infty} F_n) =\theta$ where $F_n$ is the empirical distribution of sample $\bi X_1,...,\bi X_n$ from $F$.
\end{remark}.
\begin{remark}\label{rem:cf}
For any elliptical distribution $F$ in ${\cal E}(\bi \mu, \bi \Sigma, g)$ and the associated spherical distribution $F_0(g)$, the constant $c(F) =c(F_0)= \mathbb{E}_{F_0} \|\bi X_1-\bi X_2\|$, where $\bi X_1$ and $\bi X_2$ are independent random vectors from $F_0(g)$. Let $c_1(F_0) =\mathbb{E}_{F_0} \|\bi X\|$. For Gaussian distributions, $c(F_0)=\sqrt{2}c_1(F_0)$. However, such a relationship may not hold for other elliptical distributions.
\end{remark}
\begin{remark}\label{rem:normal}
If $F$ is a multivariate normal distribution ${\cal N}_d(\bi \mu,\bi \Sigma)$,
$c(F)=\sqrt{2}\mathbb{E} (D^{1/2}),$ where $D=(\bi X-\bi \mu)^T\bi\Sigma^{-1}(\bi X-\bi \mu)$ has a $\chi^2$ distribution with $d$ degrees of freedom. Hence
$c(F)={2\Gamma[(d+1)/2]}/{\Gamma(d/2)}.$
For a univariate normal distribution ${\cal N}(\mu,\sigma^2)$, the Gini covariance is reduced to the Gini mean difference that equals $2\sigma/\sqrt{\pi}$.
\end{remark}
Spatial signs and spatial ranks are orthogonally
equivariant in the sense that for any $d\times d$ orthogonal matrix $O$ ($O^T=O^{-1}$), $d$-dimensional vector $\bi b$ and nonzero scalar $c$,
letting $\bi X^*=c O\bi X+\bi b$ with the distribution $F^*$,
$$ \bi s(\bi X^*)=\mbox{sign}(c)O \bi s(\bi X), \;\mbox{ and } \;\;
\bi r(\bi X^* , F^*)=\mbox{sign}(c)O\bi r(\bi X, F).$$
Therefore, we have the orthogonal equivariance property of GCM as follows.
\begin{align} \label{eqn:Ortho}
\bi \Sigma_g(cO\bi X+\bi b) =\bi \Sigma_g(F^*)= |c|O\bi \Sigma_g(\bi X)O^T.
\end{align}
Orthogonal equivariance property of the Gini covariance matrix $\bi \Sigma_g$ holds under any distribution with a finite first moment. Orthogonal equivariance ensures that under rotation, translation and homogeneous scale change, the quantities are transformed accordingly. However, it does not allow heterogeneous scale changes. The equality does not hold for a general $d\times d$ nonsingular matrix $A$. Hence the Gini covariance matrix is not fully affine equivariant.
\subsection{The Affine Equivariant Version of GCM}
In order to achieve full affine equivariance, we use the transformation - retransformation (TR) technique, which serves as standardization of multivariate data. More details can be found in Charkraborty and Chaudhuri (1996) and Serfling (2010). The affine equivariant counterpart of the Gini covariance matrix is denoted as $\bi\Sigma_G$. The idea is that if $\bi X_1$ and $\bi X_2$ are independent random vectors from $F$ and they are transformed or standardized to be $\bi Z_i=\bi \Sigma_G^{-1/2}(\bi X_i-\bi\mu)$ for $i=1,2$, then $\bi Z_1$ and $\bi Z_2$ are independently distributed from the spherical distribution $F_0$ with the scatter matrix $\bi I_{d}$. By Remark \ref{rem:identity}, we thus have
\begin{align}
\bi \Sigma_g(F_0)=\mathbb{E}\frac{(\bi Z_1-\bi Z_2)(\bi Z_1-\bi Z_2)^T}{\|\bi Z_1-\bi Z_2\|} =\frac{c(F)}{d} \bi I_d. \label{eqn:trgcm1}
\end{align}
Since $\bi Z_1-\bi Z_2=\bi \Sigma_G^{-1/2}(\bi X_1-\bi X_2)$, the middle term of (\ref{eqn:trgcm1}) is $$\mathbb{E} \frac{\bi \Sigma_G^{-1/2}(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T\bi \Sigma_G^{-1/2}}{\sqrt{(\bi X_1-\bi X_2)^T\bi \Sigma_G^{-1}(\bi X_1-\bi X_2)}}. $$
Thus, the TR version of the GCM is defined as follows.
\begin{definition}\label{def:afgcm}
For a $d$-variate elliptical distribution $F$ with existing first moment, its TR version of the Gini covariance matrix, denoted as $\bi \Sigma_G$, is defined as the solution of
\begin{equation}\label{eqn:trgcm2}
\bi \Sigma_G= \frac{d}{c(F)}\mathbb{E} \frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T}{\sqrt{(\bi X_1-\bi X_2)^T\bi \Sigma_G^{-1}(\bi X_1-\bi X_2)}},
\end{equation}
where $\bi X_1, \bi X_2 \stackrel{iid}{ \sim} F$ and $c(F)$ is given in Remark \ref{rem:cf}.
\end{definition}
\begin{theorem}\label{thm:afeq}
The matrix valued functional $\bi \Sigma_G(\cdot)$ is a scatter matrix in the sense that for any nonsingular $d\times d$ matrix $A$ and $d$-vector $\bi b$, $\bi \Sigma_G(F^*)=A\bi \Sigma_G(F)A^T$, where $\bi X$ is distributed from $F$ and $F^*$ is the distribution of $A\bi X+\bi b$.
\end{theorem}
\begin{remark}
For $d=1$, the affine equivariant Gini mean difference $\sigma_G$ satisfies $\sigma_G^{1/2}= \mathbb{E}_F|X_1-X_2|/c(F)=\sigma$. Hence, $\sigma_G$ is Fisher consistent to the squared scale parameter for distributions in the location-scale family. Also $\sigma_G$ only assumes the existence of first moment compared to the second moment needed for the variance $\sigma_v^2=\mathbb{E}(X-\mu)^2=1/2\mathbb{E}(X_1-X_2)^2$.
\end{remark}
Note that the affine equivariant version of GCM $\bi \Sigma_G$ is a symmetrized M-functional. Sirki\"{a} {\em et al.} (2007) studied a general symmetrized M-functional $\bi \Sigma_M$ that solves
$$ \mathbb{E} [w_1(R_{12}(\bi \Sigma_M)) \bi U_{12}(\bi \Sigma_M)\bi U_{12}^T(\bi \Sigma_M)-w_2(R_{12}(\bi \Sigma_M)) \bi I_d] = \bi 0,$$
where $w_1$ and $w_2$ are real-valued functions on $[0,\infty)$, $\bi X_1, \bi X_2\stackrel{iid}{ \sim} F$, $\bi Z_{12}(\bi \Sigma_M)=\bi \Sigma_M^{-1/2}(\bi X_1-\bi X_2)$, $R_{12}(\bi \Sigma_M)=\|\bi Z_{12}(\bi \Sigma_M)\|$ and $\bi U_{12}(\bi \Sigma_M)=R_{12}(\bi \Sigma_M)^{-1}\bi Z_{12}(\bi \Sigma_M)$. Clearly, the weight functions for $\bi \Sigma_G$ are $w_1(t)=t$ and $w_2(t)= c(F)/d$. For the case of $w_1(t)=t^2$ and $w_2(t)=2$, the covariance matrix is obtained. If $w_1(t)=d$, $w_2(t) =1$ and an additional condition that the trace of the matrix is $d$, the symmetrized Tyler M-functional called D{\" u}mbgen's M-functional is obtained (D{\" u}mbgen, 1998). D{\" u}mbgen's M-functional is also a TR-version of symmetrized spatial sign matrix. The very recent paper (D{\" u}mbgen {\em et al}., 2015) considers a framework to generalize M-functionals based on symmmetrizations of arbitrary order. Our TR Gini covariance matrix can be treated as an example of their Case 1 with the symmetrization of order 2.
As defined using pairwise differences, the symmetrized M-functional is obtained without reference to the location parameter $\bi \mu$. Maronna (1976) considered simultaneous location and scatter M-functionals. In this paper, since we focus on symmetrized scatter M-functionals, we assume $\bi \mu$ known for scatter M-functionals. Letting $\bi Z(M)= M^{-1/2}(\bi X-\bi \mu)$, $R( M)=\|\bi Z(M)\|$ and $\bi U(M)=R(M)^{-1}\bi Z(M)$, we obtain a regular scatter M-functional that solves
$$ \mathbb{E} [w_1(R(M)) \bi U(M)\bi U^T(M)-w_2(R(M)) \bi I_d] = \bi 0.$$
For $w_1(t)=t^2, w_2(t)=1$, the covariance matrix is obtained. Tyler M-functional is the case of $w_1(t)=d$, $w_2(t) =1$ with an additional condition that the trace of the matrix $M$ is $d$. The case of $w_1(t)=t$ and $w_2(t)=1$ is called Kotz functional, denoted as $\bi \Sigma_K$. The rational for such a name is because it equals the scatter parameter of the Kotz distribution (\ref{eqn:kotz}).
Note that a $L_1$-type M-functional considered by Roelant and Van Aelst (2007) and Arslan (2010) is the simultaneous location and scatter Kotz M-functional. Our TR Gini covariance matrix can be viewed as the symmetrized Kotz functional. In other words, TR Gini covariance is a multivariate extension of $\mathbb{E}|X_1-X_2|$, while $\bi \Sigma_K$ is for $\mathbb{E} |X-\mu|$. It is worth to mention that Koshevoy {\em et al.} (2003) considered extensions of mean deviation and mean difference using volumes of zonotope and lift-zonotope. For $F\in {\cal E}(\bi \mu,\bi \Sigma, g) $, we have that $\bi \Sigma_K(F) =\{\mathbb{E} R /d\}^2 \bi \Sigma$ with $R=\|\bi \Sigma^{-1/2}(\bi X-\bi \mu)\|$ and the zonoid covariance matrix (ZCM) $\bi \Sigma_Z(F)$ is equal to $c(d) \bi \Sigma_K(F)$ with the factor $c(d)$ depending on the dimension $d$ but independent on the distribution $F$.
Instead of taking the TR technique, Visuri {\em et al.} (2000) re-estimated each eigenvalue of the spatial rank covariance (RCM) defined in \eqref{eqn:rcm} to make it affine equivariant. Yu {\em et al.} (2015) used the median of absolute deviation (MAD) to estimate scale of each univariate projected data on each of eigenvector directions. Thus the resulting RCM is affine equivariant and robust. However, it may trade off too much efficiency for robustness. The simulation in later section confirms its relatively low finite sample efficiency comparing to symmetrized M-estimators. Also, as cautioned in Nordhausen and Tyler (2015), those robust alternatives may
not have the block independence property.
\subsection{Block Independence Property}
One important property of symmetrized scatter functionals is the independence property, or more generally, the block independence property. A scatter functional with the block independence property means that it is a block diagonal matrix if the block components of the random vector are independent. Such a property holds naturally for the regular covariance matrix, but it may not hold for general M-functionals and some other robust scatter functionals, as noted in Nordhausen and Tyler (2015). They proved that any symmetrized scatter functionals have the block independent property. In fact, such a result holds for any symmetrized orthogonally equivariant covariance matrix since the proof of their theorem only uses the conditions of symmetry and orthogonal equivariance. As a result, our two versions of GCM have the block independence property as stated in the following corollary.
\begin{corollary}\label{thm:IP}
Let $\bi X^T =(\bi X_1^T, \bi X_2^T, ...,\bi X_k^T)$ have $k$ independent blocks with dimensions $d_1,...,d_k$ ($d_1+d_2+...+d_k=d$). Then $\bi \Sigma_g(\bi X)$ and $\bi \Sigma_G(\bi X)$ are block diagonal matrices with block dimensions $d_1,...,d_k$.
\end{corollary}
The block independence property is beneficial in many applications, for example, in independent component analysis (Oja {\em et al.}, 2006), in independent subspace analysis (Nordhausen and Oja, 2011), or in invariant coordinate selection (Tyler {\em et al.}, 2009).
In the next section, we study the robustness properties of the two Gini covariance matrices along with the Kotz functional through the influence function approach.
\section{Influence function}
The influence function (IF) introduced by Hampel (1974) is a standard
heuristic tool for measuring the effect of infinitesimal perturbations on a
functional $T$. For a distribution $F$ on $\mathbb{R}^d$ and a covariance functional $T: F \mapsto T(F) \in \mathcal{M}^+$ with $\mathcal{M}^+$ being the set of $d \times d$ positive definite matrices, the IF of $T$ at $F$ may be expressed as
\[
\mbox{IF}(\bi{x};T,F)
=\displaystyle \lim_{\varepsilon \rightarrow 0}
\frac{T((1-\varepsilon)F+\varepsilon\delta_{\bi{x}})-T(F)}{\varepsilon},
\;\;\;\bi{x}\in\mathbb{R}^d,
\]
where $\delta_{\bi{x}}$ denotes the point mass distribution at $\bi{x}$. Not only is the IF a local robustness measure of $T(F)$, it is also useful in deriving asymptotic efficiency of the corresponding estimator $T(F_n)$, where $F_n$ is the empirical distribution.
\begin{proposition} \label{thm:ifgc}
The influence function of the Gini covariance matrix $\bi \Sigma_g$ is
$$
IF(\bi x;\bi \Sigma_g,F)=2\mathbb{E} \frac{(\bi X_1-\bi x)(\bi X_1-\bi x)^T}{\|\bi X_1-\bi x\|}-2\bi \Sigma_g.
$$
\end{proposition}
\begin{remark}
For $d=1$, we obtain the influence function for the Gini mean difference, that is, $IF(x;\sigma_g,F)= 2\mathbb{E} |X_1-x|-2\sigma_g$, which is approximately linear for large $|x|$ in contrast to the quadratic form in $IF(x; \sigma_v^2,F)=\mathbb{E}(X_1-x)^2-\sigma_v^2=(x-\mu)^2$, the influence function of the regular variance.
\end{remark}
The influence function of the affine equivariant GCM is more complicated than that of GCM. Hampel {\em et al.} (1986) showed that, for an affine equivariant scatter functional $M(\cdot)$, the influence function of $M$ at a spherical distribution $F_0(g)$ in $\mathbb{R}^d$ is given by
\begin{equation}\label{eqn:if}
IF(\bi x; M, F_0) = \alpha_{M}(\|\bi x\|)\frac{\bi x \bi x^T}{\|\bi x\|^2}-\beta_{ M}(\|\bi x\|)\bi I_{d},
\end{equation}
where $\alpha_{M}$ and $\beta_{M}$ are two real valued functions depending on $F_0(g)$. Then the influence function of $ M$ at an elliptical distribution $F(\bi \mu, \bi \Sigma, g)$ is
$$IF(\bi x; M, F)=\bi \Sigma^{1/2}IF(\bi \Sigma^{-1/2}(\bi x-\bi \mu); M, F_0)\bi \Sigma^{1/2}.$$
The following corollary states the influence function of the TR version of GCM, which is obtained as a special case of Theorem 2 in Sirki\"{a} {\em et al.} (2007) with $w_1(t)=t$ and $w_2(t)=c(F_0)/d$.
\begin{corollary}\label{thm:ifafgc}
The influence function of the affine equivariant version of the Gini covariance matrix $\bi \Sigma_G$ at a spherical distribution $F_0$ is of the form \eqref{eqn:if} with
\begin{eqnarray*}
\alpha_{\Sigma_G}(\|\bi x\|)&=&\frac{2d(d+2)}{(d+1)c(F_0)}\mathbb{E}\left[ (\left\|\bi X_1-\|\bi x\|\bi e_1\right\|)-\frac{d (\bi X_1)_2^2}{\|\bi X_1-\|\bi x\|\bi e_1\|}\right],\\
\beta_{\Sigma_G}(\|\bi x\|)&=&4-\frac{2d}{(d+1)c(F_0)}\mathbb{E}\left[(\|\bi X_1-\|\bi x\|\bi e_1\|)+\frac{(d+2) (\bi X_1)_2^2}{\|\bi X_1-\|\bi x\|\bi e_1\|}\right],
\end{eqnarray*}
where $(\bi X_1)_2$ denotes the second coordinate of $\bi X_1$, $\bi e_1=(1,0,...,0)^T$, and $c(F_0)=\mathbb{E}_{F_0} \|\bi X_1-\bi X_2\|$.
\end{corollary}
\begin{remark}
For $d=1$, the influence function for $\sigma_G$ is $IF(x; \sigma_G, F_0)=\alpha_{\sigma_G}(|x|)-\beta_{\sigma_G}(|x|)=4\mathbb{E}|X_1-|x||/c(F_0)-4$, where $c(F_0)=\mathbb{E}_{F_0}|X_1-X_2|$. Again, it is approximately linear in large values of $|x|$.
\end{remark}
\begin{figure}[tbh]
\centering
\begin{tabular}{ll}
\includegraphics[width=0.49\linewidth]{Alpha.pdf}
&
\includegraphics[width=0.49\linewidth]{Beta.pdf}
\end{tabular}
\caption {Functions $\alpha_{M}(r)$ (left panel) and $\beta_{M}(r)$ (right panel) for covariance matrix, Tyler M functional, D\"{u}mbgen functional, Kotz functional and TR Gini covariance matrix under the bivariate standard normal distribution. \label{fig:IF}}
\end{figure}
Applying the result of M-functional from Huber and Ronchetti (2009) (pp. 220-222) to the Kotz functional, we have the following corollary.
\begin{corollary}\label{thm:ifkotz}
The influence function of Kotz functional $\bi \Sigma_K$ at a spherical distribution $F_0$ is of the form \eqref{eqn:if} with
\[
\begin{array}{lcl}
\displaystyle\alpha_{\Sigma_K}(\|\bi x\|)= \frac{d(d+2)}{(d+1)c_1(F_0)}\|\bi x\|,&&
\displaystyle\beta_{\Sigma_K}(\|\bi x\|)=\frac{d}{c_1(F_0)}\left[2-\frac{\|\bi x \|}{d+1}\right],\\
\end{array}
\]
where $c_1(F_0) =\mathbb{E}_{F_0}\|\bi X_1\|$ with $\bi X_1$ from $F_0$.
\end{corollary}
\begin{remark}
For a spherical distribution $F_0(g)$, $c_1(F_0) = \mathbb{E} r$ where $r$ has the distribution of \eqref{eqn:pdfr}.
\end{remark}
\begin{remark}\label{rem:t}
If $F_0$ is a spherical $t$ distribution ${\cal T}_d(\nu)$ with $\nu>1$, $$c_1(F_0)= \frac{\nu^{1/2}\Gamma[(\nu-1)/2]}{\sqrt{2}\Gamma(\nu/2)}\frac{\sqrt{2}\Gamma[(d+1)/2]}{\Gamma(d/2)}.$$
Note that when $\nu\rightarrow \infty$, using Stirling formula $\Gamma(\nu)\approx \sqrt{2\pi}e^{-\nu}\nu^{\nu-1/2}$, we have $\frac{\nu^{1/2}\Gamma[(\nu-1)/2]}{\sqrt{2}\Gamma(\nu/2)}\rightarrow 1$, which corresponds to the normal case in that $c_1(F_0)=c(F_0)/\sqrt{2}$ as in Remark \ref{rem:normal}.
\end{remark}
\begin{remark}\label{rem:kotz}
If $F_0$ is the spherical Kotz distribution \eqref{eqn:kotz}, then $c_1(F_0)=d$, the mean of Gamma$(d,1)$.
\end{remark}
Figure \ref{fig:IF} displays functions $\alpha_{M}(r)$ and $\beta_{M}(r)$ for covariance matrix, Tyler M functional, D\"{u}mbgen functional, Kotz functional and TR Gini covariance matrix under the bivariate standard normal distribution. From \eqref{eqn:if}, the function $\alpha$ is the influence of $\bi x$ on an off-diagonal element of ${ M}$, that is, $IF(\bi x; {M}_{ij},F_0)=\alpha_{M}(\|\bi x\|)u_iu_j$, where $u_i$ and $u_j$ are the $i^{th}$ and $j^{th}$ component of $\bi u=\bi x/\|\bi x\|$. The influences of diagonal elements of $M$ appear in both $\alpha$ and $\beta$ functions. In other words, $IF(\bi x; {M}_{ii}, F_0) =\alpha_{M}(\|\bi x\|)u_i^2-\beta_{M}(\|\bi x\|)$. This means that for boundedness of the influence at off-diagonal elements, a necessary and sufficient condition is that the $\alpha$ is bounded, while for diagonal elements, one needs boundedness on both $\alpha$ and $\beta$. As we can see from Figure \ref{fig:IF}, the $\alpha$ and $\beta$ functions of Tyler's and D\"{u}mbgen's M-functionals are bounded. The $\alpha$ function of the covariance matrix is quadratic in the radius $r=\|\bi x\|$, though its $\beta$ function is constant to be bounded. Both functions of the TR Gini covariance matrix are approximately linear for large $r$ and those of the Kotz functionals are linear. This suggests that the TR Gini covariance matrix and Kotz matrix give more protection to moderate outliers than the covariance matrix but they are not robust in the strict sense. The Kotz functional and its symmetrized version TR Gini covariance matrix are $L_1$ methods. They are more robust than $L_2$ methods, and also very efficient (as we will see in the next section). Such properties are also shared by the zonoid scatter matrix (Koshevoy {\em et al.}, 2003), Oja sign and rank covariance matrix (Ollila {\em et al.}, 2003; Ollila {\em et al.}, 2004). They all have influence functions linear or approximately linear in $r$.
Note that the influence function of the affine equivariant version of spatial rank covariance matrix (MRCM) considered by Visuri {\em et al.} (2000) can not be written as the form of (\ref{eqn:if}) because of the construction way of MRCM with nonlinear transformations. See Yu {\em et al.} (2015) for more details.
\section{Estimation}
\subsection{Sample Gini Covariance Matrix}
Suppose that ${\cal X}=\{\bi X_1,..., \bi X_n\}$ is a random sample from a continuous distribution $F$ in $\mathbb{R}^d$ and its empirical distribution is $F_n$. Then the sample counterpart of the Gini covariance matrix is obtained by replacing $F$ with the empirical distribution $F_n$ in \eqref{eqn:gcm}. That is,
\begin{align}
\hat{\bi\Sigma}_g=\bi\Sigma_g(F_n)=\frac{2}{n}\sum_{i=1}^n \bi X_i\bi r(\bi X_i)^T
={n \choose 2}^{-1}\sum_{ i<j}\frac{(\bi X_i-\bi X_j)(\bi X_i-\bi X_j)^T}{\|\bi X_i-\bi X_j\|}.\label{eqn:sgcm}
\end{align}
Clearly, the sample Gini covariance matrix $\bi \Sigma_g(F_n)$ is a matrix-valued $U$-statistic $U_n$ to estimate $\bi \Sigma_g(F)$ with the kernel $h(\bi x_1,\bi x_2) =(\bi x_1-\bi x_2)(\bi x_1-\bi x_2)^T/\|\bi x_1-\bi x_2\|$. A straightforward generalization of univariate results on non-degenerated $U$-statistics given in Serfling (1980) establishes $\sqrt{n}$-consistency of $\bi \Sigma_g(F_n)$. This means that for $F$ having a finite second moment,
\begin{align}\label{eqn:rngcm}
\sqrt{n}(\hat{\bi \Sigma}_g-\bi \Sigma_g)=\sqrt{n} (U_n-\bi \Sigma_g)=\sqrt{n}\left[\frac{1}{n}\sum_{i=1}^n IF(\bi X_i; \bi \Sigma_g,F)\right]+\bi R_n,
\end{align}
where the remainder term satisfies $\bi R_n \overset{p} {\to} \bi 0$. We have the following proposition.
\begin{proposition}\label{thm:gcmnormal}
Let $\bi X_1,...,\bi X_n$ be a random sample from $d$-variate distribution $F$ with a finite second moment. Then $\bi \Sigma_g(F_n)$ is an unbiased, $\sqrt{n}$-consistent estimator of $\bi \Sigma_g(F)$. Furthermore,
$$
\sqrt{n}\, vec (\hat{\bi \Sigma}_g-\bi \Sigma_g) \rightarrow {\cal N}_{d^2}(\bi 0, 4\mathbb{E}[\bi \phi_g(\bi X)\bi \phi_g(\bi X)^T]),
$$
where $\bi \phi_g(\bi x) = vec (\mathbb{E} h(\bi x, \bi X_1) - \bi \Sigma_g)$ with $h(\bi x,\bi x_1) =(\bi x-\bi x_1)(\bi x-\bi x_1)^T/\|\bi x-\bi x_1\|$ and $vec(A)$ stacks columns of $A$ to form a long column vector.
\end{proposition}
Note that $2\bi \phi_g(\bi x)=vec( IF(\bi x;\bi \Sigma_g,F))$. The assumption of a finite second moment guarantees existence of the covariance of the limiting distribution.
\subsection{Sample TR Gini Covariance Matrix}
Replacing $F$ with $F_n$ in \eqref{eqn:trgcm1}, the sample affine equivariant Gini covariance matrix $\hat{\bi \Sigma}_G$ is defined and it is the solution of
\begin{equation}\label{eqn:safgcm}
\frac{2}{n(n-1)}\sum_{1\leq i <j \leq n}\frac{\hat{\bi \Sigma}_G^{-1/2}(\bi X_i-\bi X_j)(\bi X_i-\bi X_j)^T\hat{\bi \Sigma}_G^{-1/2}}{\sqrt{(\bi X_i-\bi X_j)^T\hat{\bi\Sigma}_G^{-1}(\bi X_i-\bi X_j)}}-\frac{c(F)}{d}\bi I_d =\bi 0.
\end{equation}
The existence and uniqueness of the solution of \eqref{eqn:safgcm} can be established by checking the conditions of scatter M-estimators (Maronna,1976; Huber and Ronchetti, 2009). Those conditions for existence (E) and uniqueness (U) are also used for symmetrized M-estimators in Sirki\"{a} {\em et al.} (2007) and listed below
\begin{description}
\item [E1] $w_1(r)/r^2$ is decreasing, and positive when $r>0$.
\item[E2] $w_2(r)$ is increasing, and positive when $r\geq 0$.
\item [E3] $ w_1(r)$ and $w_2(r)$ are bounded and continuous.
\item[E4] $w_1(0)/w_2(0) < d$.
\item[E5] For any hyperplane $H$, let $P(H)$ be the fraction of pairwise difference belonging to that hyperplane. $P(H)<1-d w_2(\infty)/w_1(\infty)$ and $P(H)\leq 1/d$.
\item [U1] $w_1(r)/r^2$ decreasing.
\item [U2] $w_1(r)$ is continuous and increasing, and positive when $r>0$.
\item [U3] $w_2(r)$ is continuous and decreasing, non-negative, and positive when $0\leq r
<r_0$ for some $r_0$.
\item[U4] For all hyperplane $H$, $P(H)<1/2$.
\end{description}
Our affine equivariant version of Gini covariance estimator is the case with $w_1(t)=t$ and $w_2(t)=c(F)/d$. It satisfies all except Assumption E3 in which $w_1$ is bounded. However, if we replace E3 with E3',
\begin{description}
\item [E3'] The distribution of $F$ has a finite first moment,
\end{description}
then Lemma 8.3 in Huber and Ronchetti (2009) is still satisfied, hence our estimator does exist and exists uniquely. The assumption E3' is in agreement with the condition of D\"{u}mbgen {\em et al.} (2015) for the Case 1 in which $\psi(t)=\sqrt{t}$.
Intuitively, we can find the solution of the equation of \eqref{eqn:safgcm} by a common iterative algorithm:
\begin{equation}\label{eqn:alg}
\hat{\bi\Sigma}_G^{(t+1)} \longleftarrow \frac{2}{n(n-1)}\frac{d}{c(F)}\sum_{1\leq i <j \leq n}\frac{(\bi x_i-\bi x_j)(\bi x_i-\bi x_j)^T}{\sqrt{(\bi x_i-\bi x_j)^T (\hat{\bi\Sigma}_G^{(t)})^{-1}(\bi x_i-\bi x_j)}}.
\end{equation}
The initial value can take $\hat{\bi\Sigma}_G^{(0)}=\bi I_d$. The iteration stops when $\|\hat{\bi\Sigma}_G^{(t+1)}-\hat{\bi\Sigma}_G^{(t)}\|<\varepsilon$ for a pre-specified number $\varepsilon>0$, where $\|\cdot\|$ can take any matrix norm. Note that we need to know the distribution $F$ since $c(F)$ is included in (\ref{eqn:alg}). In this case the estimator is Fisher consistent to $\bi\Sigma$. Usually one makes the estimator Fisher consistent at the normal model. That is, one takes $c(F) = 2\Gamma[(d+1)/2]/\Gamma(d/2)$ as stated in Remark \ref{rem:normal}. If one is interested in estimation of correlation matrix or shape matrix (shape matrix is defined later at Section \ref{sec:AE}), there is no need to specify the distribution. One can delete the factor $d/c(F)$ in the equation of \eqref{eqn:safgcm} and obtain its solution for estimation of scatter matrix up to a factor.
The above algorithm is called the fixed-point algorithm and its convergence from any start points has been rigorously proved (Tyler, 1987). However, it can be rather slow for high dimensions and large sample sizes. The very recent paper by D\"{u}mbgen, Nordhausen and Schuhmacher (2016) provide much faster new algorithms by utilizing a Taylor expansion of second order of the target functional. For large $n$, they approximate symmetrized estimators by considering the surrogate ones rather than all pairwise differences. Based on their idea, an algorithm for our Gini estimator can be developed and added to their R package ``fastM" (D\"{u}mbgen {\em et al.}, 2014).
If we assume that the location parameter $\bi \mu$ is known, then the MLE of $\bi \Sigma$ in the Kotz distribution \eqref{eqn:kotz} is found to be a scatter M-estimator, which is the solution of $\hat{\bi \Sigma}$ in the equation below:
\begin{align}
\frac{1}{n}\sum_{i=1}^n\frac{(\bi X_i-{\bi \mu})(\bi X_i-{\bi \mu})^T} {\sqrt{(\bi X_i-{\bi\mu})^T\hat {\bi \Sigma}^{-1}(\bi X_i-{\bi \mu})}}= \hat{\bi \Sigma}.\label{eqn:Kotz_Sigma}
\end{align}
The solution of $\hat{\bi \Sigma}$ in \eqref{eqn:Kotz_Sigma} is denoted as $\hat {\bi \Sigma}_K$, which is ${\bi \Sigma}_K(F_n)$. Assuming a known location parameter is for avoiding some restrictive regularity conditions for the simultaneous M-estimators. The simultaneous one is treated in Roelant and Van Aelst (2007) and Arslan (2010). Our TR version GCM estimator is the symmetrized scatter MLE of the Kotz distribution without the need of reference to the location parameter, and hence avoids the above situation.
D\"{u}mbgen {\em et al.} (2015) provided a general treatment and asymptotics for M-estimation of multivariate scatter. The Kotz and TR Gini estimators are examples of their Case 1. Hence by using their Theorem 6.11, $\sqrt{n}$-consistency of $\hat {\bi \Sigma}_G$ and $\hat {\bi \Sigma}_K$ under a spherical distribution $F_0$ is established as follows.
\begin{proposition} \label{thm:consistent_trgcm}
Let $\bi X_1,...,\bi X_n$ be a random sample from a spherical distribution $F_0$ in $\mathbb R^d$. Under the assumption of finite second moment of $F_0$, $\hat{\bi \Sigma}_G$ is $\sqrt{n}$-consistent estimator of $\bi \Sigma_G(F_0)=\bi I_d$ and $\hat{\bi \Sigma}_K$ is $\sqrt{n}$-consistent estimator of $\bi \Sigma_K(F_0)=[c_1(F_0)]^2/d^2 \bi I_d$, where $c_1(F_0)=\mathbb{E}_{F_0}[\|\bi X\|]$.
\end{proposition}
\begin{remark}
If $F_0$ is the spherically distributed Kotz distribution, $\bi \Sigma_K(F_0)=\bi I_d$ and both $\hat{\bi \Sigma}_G$ and $\hat{\bi \Sigma}_K$ are consistent scatter estimators.
\end{remark}
Once we obtain the $\sqrt{n}$-consistency of $\hat{\bi \Sigma}_G$, we are able to use Theorem 4 of Sirki\"{a} {\em et al.} (2007), in which they assume $\sqrt{n}$-consistency of symmetrized M-estimators to establish asymptotic normality. In the following we give the result for our estimator.
\begin{corollary}\label{thm:normafgcm}
Let $\bi X_1,...,\bi X_n$ be a random sample from a spherical distribution $F_0$ in $\mathbb R^d$. If the covariance matrix (second moments) of $F_0$ exists, then
$$
\sqrt{n}\; vec(\hat{\bi \Sigma}_G-\bi I_{d})\rightarrow N_{d^2}(\bi 0, \mathbb{E}[ vec(IF(\bi X;\bi \Sigma_G,F_0))vec(IF(\bi X;\bi \Sigma_G,F_0))^T]).
$$
\end{corollary}
According to \eqref{eqn:if} and Corollary \ref{thm:ifafgc}, the covariance matrix of the limit distribution
$\mathbb{E}[ vec(IF(\bi X;\bi \Sigma_G,F_0))vec(IF(\bi X;\bi \Sigma_G,F_0))^T]$ can be written as
$$
ASV(\hat{\bi \Sigma}_{G_{12}};F_0)(\bi I_{d^2}+\bi 1_{d,d})+ASC(\hat{\bi \Sigma}_{G_{11}},\hat{\bi \Sigma}_{G_{22}};F_0)vec(\bi I_{d})vec(\bi I_{d})^T,
$$
where $\bi 1_{d,d}$ is $d^2\times d^2$ matrix with $(i,j)$-block being equal to a $d\times d$ matrix that has 1 at entry $(j,i)$ and 0 elsewhere. $ASV(\hat{\bi \Sigma}_{G_{12}};F_0)$ denotes the asymptotic variance of an off-diagonal element and $ASC(\hat{\bi \Sigma}_{G_{11}},\hat{\bi \Sigma}_{G_{22}};F_0)$ denotes the covariance of any two diagonal elements. With Corollaries \ref{thm:normafgcm} and \ref{thm:ifafgc}, we have
\begin{align}
&ASV(\hat{\bi \Sigma}_{G_{12}};F_0)
\nonumber\\
&= \frac{4d(d+2)}{(d+1)^2c^2(F_0)}\mathbb{E}\left[\mathbb{E}(\|\bi X_1-\|\bi X_2\|\bi e_1\| -\frac{d(\bi X_1)_2^2}{\|\bi X_1-\|\bi X_2\|\bi e_1\|})|\bi X_2\right]^2; \label{eqn:asvG}\\
&ASV(\hat{\bi \Sigma}_{G_{11}};F_0) \nonumber\\
&=\frac{2(d-1)}{d}ASV(\hat{\bi \Sigma}_{G_{12}};F_0)+16\left[\frac{\mathbb{E}\left[\mathbb{E}(\|\bi X_1-\|\bi X_2\|\bi e_1\|)|\bi X_2\right]^2}{c^2(F_0)}-1\right]; \nonumber\\
&ASC(\hat{\bi \Sigma}_{G_{11}},\hat{\bi \Sigma}_{G_{22}};F_0)=ASV(\hat{\bi \Sigma}_{G_{11}};F_0)-2ASV(\hat{\bi \Sigma}_{G_{12}};F_0).\nonumber
\end{align}
Using the affine equivariance property of $\hat{\bi \Sigma}_G$ and Kronecker product $\otimes$, the limiting distribution of $\sqrt{n}\;vec(\hat{\bi \Sigma}_G-\bi \Sigma)$ at the elliptical distribution $F$ is multivariate normal with zero mean and covariance matrix
\begin{equation}\label{eqn:ASVF}
ASV(\hat{\bi \Sigma}_{G_{12}};F_0)(\bi I_{d^2}+\bi 1_{d,d})(\bi \Sigma \otimes \bi \Sigma)+
ASC(\hat{\bi \Sigma}_{G_{11}},\hat{\bi \Sigma}_{G_{22}};F_0)vec(\bi \Sigma)vec(\bi \Sigma)^T.
\end{equation}
Checking the conditions (N1-N4) of MLE proposed by Huber (1967), we are able to establish the normality of Kotz estimator $\hat{\bi \Sigma}_K$ assuming a known location parameter.
\begin{proposition}\label{thm:normkotz}
Let $\bi X_1,...,\bi X_n$ be a random sample from spherical distribution $F_0(g)$ in $\mathbb R^d$. If the second moment of $F_0(g)$ exists and the first moment is known, then
$$
\sqrt{n}\; vec(\hat{\bi \Sigma}_K-\bi \Sigma_K)\rightarrow N_{d^2}(\bi 0, \mathbb{E}[ vec(IF(\bi X;\bi \Sigma_K,F_0))vec(IF(\bi X;\bi \Sigma_K,F_0))^T]).
$$
\end{proposition}
With the results of Corollary \ref{thm:ifkotz} and Proposition \ref{thm:normkotz}, we have
\begin{equation}\label{eqn:asvkotz}
\displaystyle ASV(\hat{\bi \Sigma}_{K_{12}};F_0)= \frac{d(d+2) \mathbb{E}_{F_0}[ \|\bi X\|^2]}{(d+1)^2 [c_1(F_0)]^2},
\end{equation}
in which $\mathbb{E}_{F_0}[ \|\bi X\|^2]=\mathbb{E} R^2$ with $R$ having the distribution of (\ref{eqn:pdfr}).
\subsection{Asymptotic Efficiency}\label{sec:AE}
Although our TR Gini covariance estimator is Fisher consistent to the scatter matrix since it is corrected by $c(F_0)/d$, we consider its shape estimator in order to compare its limiting efficiency with that of the Tyler and D\"{u}mbgen M-estimators. The shape matrix associated with the scatter functional $\bi \Sigma$ is
$$ W(F)=\frac{d}{Tr(\bi \Sigma(F))}\bi \Sigma(F).$$
Note that there are also other definitions for a shape matrix. For example, Paindaveine (2008) uses the determinant. Here we use the shape matrix based on the matrix trace because it allows us to compare asymptotic efficiency more easily. Tyler and D\"{u}mbgen estimators estimate the shape matrix. At elliptical distributions, all shape estimators estimate the same population quantity and hence are comparable without any correction factors. Theorem 5 of Sirki\"{a} {\em et al.} (2007) states that
a single number characterizes the limiting distribution of the shape estimators at $F_0$ and that number is the variance of off-diagonal elements of $\hat{\bi \Sigma}$ or $\hat{W}$, $\tau$. In general, the asymptotic relative efficiency (ARE) of an estimator $T_1$ with respect to another estimator $T_2$ is defined as the ratio of $ASV(T_2)$ and $ASV(T_1)$. Hence for shape estimators, the ARE of $\hat{W}_1$ with respect to $\hat{W}_2$ is $\tau( \hat{W}_2)/\tau( \hat{W}_1)$.
\begin{table}[thb]
\centering
\begin{tabular}{llcccccr}\toprule\toprule
&&${\cal T}_d(5)$&${\cal T}_d(6)$&${\cal T}_d(8)$&${\cal T}_d(15)$&${\cal T}_d(\infty)$&$Kotz(d)$\\ \midrule
\multirow{4}{*}{$d=2$} & Tyler& 1.50&1.00&0.75&0.59&0.50&0.83\\
& D\"umbgen & 2.36&1.57&1.26&1.01&0.91&1.22 \\
& Kotz&2.25 &1.56&1.22&1.00&0.88&1.25\\
& TR Gini&2.09&1.48&1.24&1.05&0.98&1.21\\
& Zonoid&2.00 &1.45& 1.18 &1.03 &0.96&1.11\\ \midrule
\multirow{4}{*}{$d=3$}& Tyler& 1.80&1.20&0.90& 0.71& 0.60& 0.90\\
& D\"umbgen & 2.38& 1.66& 1.27& 1.04& 0.92&1.18 \\
& Kotz&2.31 &1.60&1.25&1.03&0.91&1.20\\
&TR Gini&2.14&1.53&1.25&1.06&0.99&1.17\\
& Zonoid& 1.96 &1.43 &1.18& 1.04&0.97&1.07\\ \midrule
\multirow{4}{*}{$d=4$}&Tyler& 2.00&1.33&1.00&0.79&0.67&0.93\\
&D\"umbgen & 2.39& 1.69&1.30& 1.06&0.93 &1.15\\
& Kotz&2.34 &1.63&1.27&1.05&0.92&1.17\\
& TR Gini&2.21&1.56&1.26&1.09&0.99&1.15\\
& Zonoid& 1.93 &1.41& 1.17 &1.04&0.98&1.05 \\ \midrule
\multirow{4}{*}{$d=5$}&Tyler& 2.14&1.43&1.07&0.84&0.71&0.95\\
&D\"umbgen & 2.50& 1.71&1.31& 1.07& 0.94&1.13\\
& Kotz&2.37 &1.65&1.29&1.06&0.93&1.14\\
&TR Gini&2.28&1.57&1.26&1.09&0.99&1.11\\
& Zonoid& 1.91& 1.40 &1.17& 1.04&0.99&1.04 \\ \bottomrule\bottomrule
\end{tabular}
\caption{Asymptotic relative efficiencies of the shape estimators based on the Tyler M-estimator, D\"umbgen, Kotz M-estimator, TR Gini covariance estimator and Zonoid covariance estimator relative to the regular shape estimator at different distributions $F_0$ at different $d$-dimension.}
\label{tbl:ARE}
\end{table}
Listed in Table \ref{tbl:ARE} are the limiting efficiencies of shape estimators with respect to the shape estimator based on the regular sample covariance matrix (i.e. the regular shape estimator). The efficiencies are considered under spherical Kotz($d$) distribution and ${\cal T}_d(\nu)$ distributions at different dimensions $d$ with different degrees of freedom $\nu$, with $\nu=\infty$ referring to the normal case. The variance of the off-diagonal element of the regular shape estimator at $F_0$ equal to $1+\kappa(F_0)$, where $\kappa(F_0)$ is the kurtosis of $F_0$. That is, $\tau$ of the regular shape estimator is $(\nu-2)/(\nu-4)$ in the ${\cal T}_d(\nu)$-distributions for $\nu>4$ and $(d+3)/(d+1)$ in the $Kotz(d)$ distribution (Wang, 2009; Zografos, 2008). In the normal case, $\tau=1$ corresponds to that of the ${\cal T}_d(\nu)$-distribution case when $\nu\rightarrow \infty$. $\tau$ of the Tyler estimator is always $(d+2)/d$ for any distribution in $\mathbb{R}^d$. From (\ref{eqn:asvkotz}), the asymptotic variance of off-diagonal elements of the Kotz shape estimator under $F_0$ is equal to $d(d+2)\mathbb{E}[\|\bi X\|^2]/((d+1)^2[\mathbb{E}\|\bi X\|]^2)$ with $\bi X$ from $F_0$. For example, ASV of the Kotz shape estimator under the $Kotz(d)$ distribution is $(d+2)/(d+1)$. The variances of off-diagonal elements of the TR Gini shape estimator are given by (\ref{eqn:asvG}), and computed through a combination of numerical integration and Monte Carlo simulation. More specifically, for $d=2$, the inner expectation of (\ref{eqn:asvG}) is computed by a double integration and the outer expectation is estimated by an empirical mean on a sample of size $10^8$. For $d >2$, all calculations are through simulations on samples with size $10^8$. The asymptotic variance of off-diagonal elements of the zonoid shape estimator under $F_0$ is $d(4\mathbb{E}[\|\bi X\|^2]-3[\mathbb{E}\|\bi X\|]^2)/((d+2)[\mathbb{E}\|\bi X\|]^2)$ (Koshevoy {\em et al.}, 2003). For example, under the Kotz distributions, the ASV of the zonoid shape estimator is $(d+4)/(d+2)$.
From Table \ref{tbl:ARE}, it can be seen that the ARE of each shape estimator decreases as $\nu$ increases in ${\cal T}_d(\nu)$ distributions, and the ARE of Tyler, D\"umbgen, Kotz and TR Gini shape estimators increases as dimension $d$ increases. In the normal cases, TR Gini estimator has a 98\% ARE for $d=2$ and 99\% for $d\geq 3$. With very little loss in efficiency in the normal case, the TR Gini estimator gains efficiency in the heavy tailed distributions. For example, its ARE is greater than 2 relative to the regular shape estimator in the ${\cal T}_d(5)$ distribution. The Tyler estimator has the lowest ARE among all estimators except the Zonoid estimator for all distributions considered. In particular, the symmetrized D\"umbgen estimator is more efficient than its counterpart, the Tyler estimator, in all distributions. However, such a result does not hold for all symmetrized estimators. TR Gini shape estimator is more efficient than Kotz estimator in ${\cal T}_d(15)$ and ${\cal T}_d(\infty)$, but less efficient in the Kotz and ${\cal T}_d(\nu)$ distributions with $\nu=5,6$. It is worthwhile to point out that Gerstenberger and Vogel (2015) studied efficiency of Gini mean difference. Their results complement ours for Kotz and TR Gini estimator when $d=1$. The ARE's
of the zonoid shape estimator under ${\cal T}_d(\infty)$ are 0.96, 0.97, 0.98 and 0.99, respectively for $d=2,3,4,5$. Under ${\cal T}_2(\nu)$, their ARE's are 2.00, 1.45, 1.18 and 1.03, respectively for $\nu=5,6,8,15$. Those numbers are similar to (slightly smaller than) the ARE's of our TR Gini shape estimator, which is not surprising since both are multivariate extensions of the mean deviation or mean difference and both have linear or approximately linear influence functions. They are highly efficient at the normal and fairly robust at the heavy-tailed cases. For ${\cal T}_d(5)$, ${\cal T}_d(6)$, ${\cal T}_d(8)$ and Kotz distributions, the efficiency of the Zonoid shape estimator decreases with $d$, which is different from other estimators. At ${\cal T}_5(5)$, the Zonoid shape estimator is least efficient among M-estimators and symmetrized M-estimators, but it is much efficient than the regular shape estimator.
\subsection{Finite Sample Efficiency}
We conduct a small simulation to study finite sample efficiencies of the shape estimators with respect to the regular shape estimator. $M=10000$ samples of two different sample sizes ($n=50,200$) at two different dimensions ($d=2,5$) are drawn from spherical ${\cal T}$-distributions with 5, 8 and $\infty$ degrees of freedoms and from spherical Kotz distribution. We use R Package ``mnormt" (Azzalini and Genz, 2016) to generate samples from multivariate ${\cal T}$-distributions and normal distribution. We generate a random vector $\bi X$ from spherical Kotz distribution by $\bi X=R \bi U$, in which $R$ is distributed from the Gamma distribution with the shape parameter being $d$ and the scale parameter being 1 and $\bi U=\bi Z/\|\bi Z\|$ with $\bi Z$ being a vector formed by $d$ iid standard normal variables. If a random sample from Kotz$(\bi \mu,\bi \Sigma)$ is required, then by taking $\bi \Sigma$'s Cholesky decomposition $L$, we have $\bi Y =L\bi X +\bi \mu $ from Kotz$(\bi \mu,\bi \Sigma)$.
\begin{table}[ht]
\centering
\small
\begin{tabular}{@{}ll|ccccccccccr@{}}\toprule\toprule
&&\multicolumn{2}{c}{${\cal T}_d(5)$}&&\multicolumn{2}{c}{${\cal T}_d(8)$}&&\multicolumn{2}{c}{${\cal T}_d(\infty)$}&&\multicolumn{2}{r}{$Kotz(d)$}\\
\cmidrule(l){3-4}\cmidrule(l){6-7}\cmidrule(l){9-10}\cmidrule(l){12-13}
&$n \backslash d$ &$2$ & $5$ && $2$ & $5$ && $2$ & $5$&&$2$ & $5$ \\
\midrule
Tyler &50 &0.81& 1.12&&0.61& 0.71&&0.45&0.49 && 0.71&0.75\\
& 200 &1.14&1.60&& 0.82& 1.01&&0.59& 0.69& & 0.79& 0.90\\
&$\infty$ & 1.50& 2.14&& 0.75& 1.07&& 0.50& 0.71& & 0.83& 0.95\\ \midrule
D\"umbgen&50&1.27& 1.73&& 1.02& 1.15&& 0.83& 0.89 && 1.04&0.94\\
&200 &1.35& 1.88&& 1.03& 1.23&& 0.81& 0.91&& 1.17& 1.09\\
&$\infty$ & 2.36& 2.50&& 1.26& 1.31&& 0.91& 0.89& & 1.22& 1.13\\ \midrule
Kotz&50&1.41& 1.72&& 1.15& 1.19&& 0.91& 0.96 && 1.23&1.13\\
&200&1.54& 1.87&& 1.22& 1.27&& 0.95& 0.94& & 1.24& 1.14\\
&$\infty$ & 2.25& 2.37&& 1.22& 1.29&& 0.88& 0.93& & 1.25& 1.14\\ \midrule
TR Gini&50&1.31& 1.60&& 1.14& 1.18&& 0.98& 0.99& & 1.15&1.09\\
& 200&1.36& 1.67&& 1.16& 1.21&& 0.99& 0.99& & 1.18& 1.10\\
&$\infty$ & 2.09& 2.28&& 1.24& 1.26&& 0.98& 0.99& & 1.21& 1.11\\ \midrule
MRCM & 50 &0.95& 1.17&&0.72&0.71&&0.52&0.49&&0.78&0.75 \\
& 200&1.29&1.50&&0.92&0.92&&0.63&0.60&&0.84&0.81 \\
$(Q_n)$ &200&1.65&1.83&&1.11&1.19&&0.83&0.87&&1.07&1.04\\
\bottomrule\bottomrule
\end{tabular}
\caption{
Finite sample relative efficiencies of the shape estimators with respective to the regular shape matrix at different distributions $F_0$. }
\label{tbl:FRE}
\end{table}
In the simulation, all M-estimators and symmetrized M-estimators are calculated by the fixed-point algorithm. Tyler and Kotz shape estimators use the true location values in the computation. {\em tyler.shape} and {\em duembgen.shape} functions in R package ``ICSNP" (Nordhausen{\em et al.}, 2015) are used for computing Tyler and D\"umbgen estimators. Also {\em spatial.rank} function of ``ICSNP" is used for TR Gini shape estimator. The convergence criterion uses Frobenius matrix norm with $\varepsilon$ being the default value $10^{-6}$ and the maximum number of iterations setting to be 100. We also include the affine equivariant spatial rank shape estimator (MRCM) for comparison. It uses the median of absolute deviation (MAD) as univariate scale estimator. An alternative to MAD, $Q_n$, is also included to see efficiency improvements of MRCM. The Zonoid shape estimator is not included in the finite sample efficiency comparison study due to its high computation complexity $O(n^{d+1})$.
For each estimator, the mean squared errors of off-diagonal elements are computed. That is, $$MSE(\hat{\bi \Sigma}_{ij}) =\frac{1}{M}\sum_{m=1}^{M}(\hat{\bi \Sigma}_{ij}^{(m)}-\bi \Sigma_{ij})^2$$ for $i\neq j$. Obviously, here we have $\bi \Sigma_{ij}=\bi I_{ij}=0$. Since the off-diagonal elements have equal variances and are uncorrelated, the average of their MSEs is computed. The finite sample relative efficiencies listed in Table \ref{tbl:FRE} are ratios of the mean MSE of the regular shape matrix to that of each estimator. The asymptotic relative efficiencies ($n=\infty$) from Table \ref{tbl:ARE} are also listed in Table \ref{tbl:FRE} for convenient reference.
The results of finite sample study show that Kotz and TR Gini estimators have a relatively fast convergence to their limiting efficiencies. Even for $n=50$ of the normal and Kotz cases, their finite sample efficiencies are already close to the asymptotic ones. For the Tyler estimator, the convergence is slower, and the loss in efficiency is larger for finite sample sizes comparing to that of others.
In the case of the ${\cal T}(5)$ distribution, the convergence to the limiting efficiency is much slower than that of the other cases. Low efficiency of MRCM can be explained by low efficiency of the univariate scale estimator MAD. Improvement can be done by using other robust alternatives which are more efficient, as suggested by Rousseeuw and Croux (1993). They recommended $Q_n$ which is given by the 0.25 quantile of the pairwise distances multiplying some correction factor. For the normal distribution under the size $n=200$, if $Q_n$ is used, the RE of MRCM increases to 0.83 for $d=2$ and 0.87 for $d=5$. Similar improvements are observed for other distributions also.
\section{Conclusion}
We have extended the univariate Gini mean difference to the multivariate case and proposed two versions of Gini covariance matrix (GCM). New covariance matrices are based on pairwise differences. Thus the location center needs not be estimated nor known. Their properties have been explored. They possess the block independence property, which allow them beneficial in many applications. Their influence functions have been derived. It was found that the influence functions of GCM are approximately linear, which is unbounded. In a strict sense, they are not highly robust. However, they are highly efficient under normal distributions. They have greater than 98\% asymptotic relative efficiency with respect to sample covariance matrix. On the other hand, they are more robust than the covariance matrix which has influence function of a quadratic form. GCM will give more protection to moderate outliers than the covariance matrix. Similar properties are also shared by the Oja sign or rank covariance matrix and the zoniod or lift-zoniod covariance matrix, but our proposed ones enjoy computational ease. Hence the proposed affine equivariant GCM provides us an option for estimating scatter matrix with a consideration to balance well among efficiency, robustness and computation.
\section{Appendix}
\noindent {\bf Proof of Theorem \ref{thm:eigv}}. We first show that $\bi X_1-\bi X_2$ is elliptically distributed with center $\bf 0$ and scatter parameter $2\bf \Sigma$ by its characteristic function as follows.
$$ \mathbb{E} e^{i\bi t^T(\bi X_1-\bi X_2)}= \mathbb{E} e^{i\bi t^T\bi X_1}\mathbb{E} e^{-i\bi t^T\bi X_2}= e^{i\bi t^T\bi \mu}e^{-i\bi t^T\bi \mu}\psi^2(\bi t^T\bi \Sigma \bi t):=\psi^*(2\bi t^T\bi \Sigma \bi t),$$
where $\psi^*(s) = \psi^2(s/2)$. Note that except for normal distributions, $\bi X_1-\bi X_2$ has a different generating function $g^*$ from $g$, the one for $\bi X$.
Let $\bi Z_i=V^T(\bi X_i-\bi\mu)$ for $i=1,2$, then $\bi Z_1-\bi Z_2=V^T(\bi X_1-\bi X_2)$ follows a centered elliptical distribution with diagonal scatter matrix $2\Lambda$. We can write $(2\Lambda)^{-1/2}(\bi Z_1-\bi Z_2)= R\bi U$ with $R=\|(2\Lambda)^{-1/2}(\bi Z_1-\bi Z_2)\|$ and $\bi U=(2\Lambda)^{-1/2}(\bi Z_1-\bi Z_2)/R$ being independent with $R$ and uniformly distribution on the unit sphere. Then
$$
\begin{array}{lcl}
\displaystyle\bi\Sigma_g&=&\displaystyle\mathbb{E}\left[\frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|}\right]=V\mathbb{E}\left[\frac{(\bi Z_1-\bi Z_2)(\bi Z_1-\bi Z_2)^T}{\|\bi Z_1-\bi Z_2\|}\right]V^T\\\\
&=&\displaystyle V\mathbb{E}\left[\frac{2R^2\Lambda^{1/2}\bi U\bi U^T\Lambda^{1/2}}{\sqrt{2R^2\bi U^T\Lambda^{1/2}\Lambda^{1/2}\bi U}}\right]V^T=\sqrt{2}\mathbb{E} RV\mathbb{E}\left[\frac{\Lambda^{1/2}\bi U\bi U^T\Lambda^{1/2}}{\sqrt{\bi U^T\Lambda\bi U}}\right]V^T.
\end{array}
$$
Denote $\sqrt{2}\mathbb{E} R$ as $c(F)$, the proof is complete.
\hfill$\square$
\vspace{2ex}
\noindent{\bf Proof of Theorem \ref{thm:afeq}}. Multiplying $A$ on the left and $A^T$ on the right to both sides of Equation (\ref{eqn:trgcm2}), we have
$$
A\bi \Sigma_G A^T = \frac{d}{c(F)}\mathbb{E} \frac{A(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^TA^T}{\sqrt{(\bi X_1-\bi X_2)^T\bi \Sigma_G^{-1}(\bi X_1-\bi X_2)}}.
$$
Since $A$ is nonsingular, $A^{-1}$ and $(A^T)^{-1}$ exist. Hence
$$A\bi \Sigma_G A^T= \frac{d}{c(F)}\mathbb{E} \frac{A(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^TA^T}{\sqrt{(\bi X_1-\bi X_2)^TA^T(A\bi \Sigma_GA^T)^{-1}A(\bi X_1-\bi X_2)}}.
$$
It means that $A\bi\Sigma_G A^T$ is the TR version of Gini covariance matrix for $A\bi X+\bi b$, where $\bi X$ is random vector from distribution $F$. \hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Proposition \ref{thm:ifgc}}. The proof is straightforward. Let $\bi Y_1$ and $\bi Y_2$ be independently distributed from $F_{\varepsilon}=(1-\varepsilon)F+\varepsilon \delta_{\bi x}$ and $\bi X_1$ and $\bi X_2$ independently distributed from $F$, then we have
\begin{align*}
&\bi \Sigma_g(F_{\varepsilon}) = \mathbb{E}_{F_\varepsilon}(\bi Y_1-\bi Y_2)\bi s(\bi Y_1-\bi Y_2)^T \\
&= (1-\varepsilon)^2 \mathbb{E}_F\frac{(\bi X_1-\bi X_2)(\bi X_1-\bi X_2)^T}{\|\bi X_1-\bi X_2\|}+
2\varepsilon(1-\varepsilon)\mathbb{E}_F \frac{(\bi X_1-\bi x)(\bi X_1-\bi x)^T}{\|\bi X_1-\bi x\|}.
\end{align*}
Then the result for $IF(\bi x;\bi \Sigma_g, F)$ follows. \hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Corollary \ref{thm:ifafgc}}.
The affine equivariant version of Gini covariance matrix $\bi \Sigma_G$ is a symmetrized M-functional with $w_1(t)=t$ and
$w_2(t)=c(F)/d=c(F_{0})/d$. From Theorem 2 of Sirkia {\em et al.} (2007), we get
\begin{eqnarray*}
\eta_1&=&\frac{(d+1)\mathbb{E}[\|\bi X_1- \bi X_2\|]}{2d(d+2)}=\frac{(d+1)c(F_{0})}{2d(d+2)},\\[1ex]
\eta_2&=&\frac{\mathbb{E}\|\bi X_1- \bi X_2\|}{4d}=\frac{c(F_{0})}{4d}.
\end{eqnarray*}
Thus, the result is obtained.
\hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Corollary \ref{thm:ifkotz}}.
We have the influence function of M-functional $M$ in the form $IF(\bi x; M, F_0)=-2\dot{W}$ where $W=M^{-1/2},$ $\dot{W}=IF(\bi x;W, F_0),$ and
\begin{eqnarray*}
\frac{1}{d}tr(W)&=&-\frac{\frac{1}{d}w_1(\|\bi x\|)-w_2(\|\bi x\|)}{\mathbb{E}[(\frac{1}{d}w^{'}_1(\|\bi Y\|)-w^{'}_2(\|\bi Y\|))\|\bi Y\|]},\\
\dot{W}-\frac{1}{d}tr(W)\bi I_d&=&-\frac{d+2}{2}\frac{w_1(\|\bi x\|)(\frac{\bi x\bi x^T}{\|\bi x\|^2}-\frac{1}{d}\bi I_d)}{\mathbb{E}_{y}[w_1(\|\bi Y\|)+\frac{1}{d}w^{'}_1(\|\bi Y\|)\|\bi Y\|]},
\end{eqnarray*}
where $\bi Y$ is a random vector from the distribution $F_0$ (see pages 220-222 of Huber and Ronchetti (2009)).
With $w_1(t)=t$ and $w_2(t)=1$ along with $w_1^{'}(t)=1$ and $w_2^{'}(t)=0$ for $\bi \Sigma_{K},$ solving for $\dot{W}$ in the above equations we get
\begin{eqnarray*}
\dot{W}&=&\frac{-d(d+2)}{2(d+1)\mathbb{E}\|\bi Y\|}\|\bi x\|\frac{\bi x\bi x^T}{\|\bi x\|^2}+\frac{d+2}{2(d+1)\mathbb{E}\|\bi Y\|}\|\bi x\|\bi I_d-\frac{\|\bi x\|-d}{\mathbb{E}\|\bi Y\|}\bi I_d.
\end{eqnarray*}
Let $c_1(F_0)=\mathbb{E}\|\bi Y\|.$ Therefore, we obtain
\begin{align*}
&IF(\bi x; \bi \Sigma_{K},F_0)=-2\dot{W}\\
&=\frac{d(d+2)}{(d+1)c_1(F_0)}\|\bi x\|\frac{\bi x\bi x^T}{\|\bi x\|^2}+\frac{d}{(d+1)c_1(F_0)}\|\bi x\|\bi I_d-\frac{2d}{c_1(F_0)}\bi I_d.\\
\end{align*}
Hence the result follows.
\hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Proposition \ref{thm:gcmnormal}}. We only prove the asymptotic normality result.
The normality of an U-statistic follows from the central limit theorem on its first order Hoeffding decomposition provided that the U-statistic is non-degenerated. Here we need to show that $\mathbb{E}[\bi \phi_g(\bi X)\bi \phi_g(\bi X)^T] >\bi 0$ and exists. The existence is guaranteed by the assumption of finite second moment. Hence it is sufficient to prove that $\phi_g(\bi X)$ is of full rank almost everywhere. This is true if $P(\bi X \in V)=0$ for any proper linear subspace $V$ ($dim(V)<d$). Particularly, this is true for continuous distribution $F$. \hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Proposition \ref{thm:consistent_trgcm}}. Kotz and TR Gini estimators are examples of the Case 1 considered in D\"{u}mbgen {\em et al.} (2015) with the symmetrization order 1 and 2, respectively. Using the same notations of D\"{u}mbgen {\em et al.} (2015), Kotz and TR Gini estimators are the cases with $\rho(s) = \sqrt{s}, \psi(s) = \frac{1}{2} \sqrt{s}$ and $\psi_2(s)= \frac{1}{4}\sqrt{s}$, which satisfy all conditions on $\rho, \psi$ and $\psi_2$. Under continuous distribution $F_0$ with finite second moments, Theorem 6.11 holds for Kotz and TR Gini estimators, and hence they are $\sqrt{n}$ consistent to $\bi \Sigma_K$ and $\bi \Sigma_G$, respectively. \hfill $\square$
\vspace{2ex}
\noindent{\bf Proof of Proposition \ref{thm:normkotz}}. Proposition \ref{thm:normkotz} follows if the conditions (N1-N4) by Huber (1967) are fulfilled. The notation of this proof will be chose to match Huber's paper. Let $\cal{M}^+$ denote the set of symmetric positive definite $d\times d$ matrices. For $A \in \cal M^+$, we define its norm $\|A\|$ as the spectral norm of $A$, that is $\lambda_1$, where $\lambda_1\geq \lambda_2\geq ...\geq\lambda_d$ are eigenvalues of $A$. Without loss of generality, assume $\bi \mu=\bi 0$. It is clear that the Kotz estimator $\psi(\bi x, M)$ in Huber's paper takes the form of
$$\psi (\bi x, M) =(\bi x^TM^{-1}\bi x)^{-1/2}\bi x\bi x^T-M.$$
Let
$ \lambda(M) = \mathbb{E} \psi(\bi X, M)$ so that the true parameter $I_d$ is defined as $\lambda(I_d) =\bi 0$. Define
$$ U(\bi x, M,\delta) = \sup_{\|M_1-M\|<\delta}\|\psi(\bi x,M_1) -\psi(\bi x, M)\|.$$
According to Huber's Theorem 3 and its corollary, if there exist positive number $b$, $c$ and $\delta_0$ such that $\mathbb{E} U(\bi X, M, \delta)<b\delta$ and $\mathbb{E} U^2(\bi X, M, \delta) < c\delta$ for $\|M-I_d\|+\delta<\delta_0$ and if
$\mathbb{E} (\|\psi(\bi X, I_d)\|^2)$ is nonzero and finite,
then the asymptotic normality of $\hat{\Sigma}_K$ follows.
Note that $U(\bi x, M, \delta)$ is less than
$$\delta+\frac{\|\bi x\bi x^T\|}{\|\bi x\|} \sup_ {\|M_1-M\|<\delta} |(\frac{\bi x^TM_1^{-1}\bi x}{\bi x^T\bi x})^{-1/2}-(\frac{\bi x^TM^{-1}\bi x}{\bi x^T\bi x})^{-1/2}|. $$
Hence, for sufficient small $\delta$, $\mathbb{E} U(\bi X, M, \delta) < \delta+\mathbb{E} \|\bi X\|/d \max(\sqrt{\lambda_1+\delta}-\sqrt{\lambda_d}, \sqrt{\lambda_1}-\sqrt{\lambda_d-\delta} )$, where $\lambda_1$ and $\lambda_d$ are the largest and smallest eigenvalues of $M$, respectively. Since $\|M-I_d\|\leq \delta_0-\delta$, we have $\lambda_1<1+\delta_0-\delta$ and $\lambda_d>1-\delta_0+\delta$. Thus, $\mathbb{E} U(\bi X, M, \delta)< \delta+\mathbb{E} \|\bi X\|/d \max (\sqrt{1+\delta_0}-\sqrt{1-\delta_0+\delta}, \sqrt{1+\delta_0-\delta}-\sqrt{1-\delta_0})$, and it exists a $b>0$ such that $\mathbb{E} U(\bi X, M, \delta)<b\delta$. Similarly, the existence of $c$ can be proved under the assumption of finite second moment, that $\mathbb{E} \|\bi X\|^2<\infty$. Also the result that $\mathbb{E} (\|\psi(\bi X, I_d)\|^2)$ is nonzero and finite follows for continuous $F_0$ with a finite second moment.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,111 |
This entry was posted on Monday, March 29th, 2010 at 3:04 pm and is filed under Paint Schemes. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,662 |
Complicado, complicar y complicación puede referirse a: Alfredo, una persona bastante complicada
Complejo (complejidad, acomplejar) o difícil (dificultad, dificultar); lo contrario de lo simple o sencillo.
Implicado, implicación.
Cómplice, complicidad.
Complicado (película).
Una evolución negativa en una enfermedad o de un proceso de salud. Ejemplo:
Complicaciones del Parto. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,638 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.