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Plusieurs villes ou communes possèdent une rue des Augustins (ou un autre type de voirie). L'odonyme fait référence à la présence - actuelle ou ancienne - d'un couvent de religieux chanoines Augustins, nom populaire des membres de l'Ordre de Saint-Augustin fondé au .
Allemagne
Augustinerstraße à Mayence
Augustinerstraße à Wurtzbourg
Augustinerstraße à Nuremberg
Augustinerstraße à Cologne
Augustinerstraße à Erfurt
Belgique
Drève des Augustins à Auderghem (Bruxelles)
Rue des Augustins (Augustijnenstraat), à Bruxelles
Rue des Augustins à Liège
Rue des Augustins à Tournai
France
Rue des Augustins à Amiens
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Rue des Augustins à Lille
Rue des Augustins à Lyon
Rue des Augustins à Marseille
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Rue des Augustins, ancienne rue de Paris devenue rue d'Argout et rue Hérold
Rue des Augustins à Reims
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Rue des Augustins à Vitré
Suisse
Rue des Augustins à Fribourg
Voir aussi
Quai des Grands-Augustins
Rue des Grands-Augustins
Augustins
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 513
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\section*{ \ \ PROGRAM SUMMARY}
\noindent
{\it Title of program:} QCDF90
\noindent
{\it Computer for which the program is designed:} Any computer
\noindent
{\it Computers under which the program has been tested:}
Silicon Graphics Indigo and PowerChallengeArray, and IBM R6000 58H
{\it Installations:} Boston University, Center for Computational
Science and Department of Physics.
\noindent
{\it Operating systems under which the program has been tested:} IRIX
6.1, and AIX.
\noindent
{\it Programming language used:} Fortran 90
\noindent
{\it No.~of bits in a word:} 64
\noindent
{\it No.~of lines in distributed program, including test data, etc:} 7806
\noindent
{\it Keywords:} QCD, lattice gauge theory
\noindent
{\it Nature of physical problem:} Non-perturbative
computations in QCD
\noindent
{\it Memory required to execute with typical data:}
Varies according to the applications.
Scales proportionally to the lattice volume $NX*NY*NZ*NT$.
On a $16^4$ lattice, the example codes quenched.f90
and propagator.f90 use
approximately 110 Mbytes and 140 Mbytes respectively.
\noindent
{\it Typical running time:} Varies according to the applications.
The example codes quenched.f90 and propagator.f90 take
approximately $45$ microsec to update an SU(3) link,
$8$ microsec to calculate a plaquette,
and $20$ microsec for a CG step per link, using a $16^4$ lattice,
on an SGI Power-Challenge per node.
\vspace*{2.0cm}
\section*{ \ \ LONG WRITE-UP}
\section{Introduction}
\label{intro}
The computer simulation of quantum fluctuations (cf.~for instance
\cite{cjr}, \cite{rebbia}, \cite{creutza}) has been one
of the most powerful tools for obtaining information about the
non-perturbative properties of quantum field theories in general,
and, especially, of Quantum Chromodynamics (QCD) (good accounts
of progress in this field of research can be found in the proceedings of
the yearly international symposia on lattice gauge theories \cite{latconf};
see also \cite{rebbib}). These simulations, which deal with matrix and
vector fields defined over a four-dimensional space-time lattice,
involve huge number of variables and are very demanding in computer
resources. Therefore, good payoffs can be obtained in this
domain of applications from the development of highly-efficient
code. On the other hand, even greater
gains can be achieved through the invention of better algorithms,
which is made much easier by the availability
of high-level, structured programming tools.
High-level programming tools are also invaluable for extracting
physical results from the data collected in the simulations,
which typically requires experimenting with different types of
data analysis and involves substantial amounts of code development.
With the twofold goal of facilitating the development of algorithms
and applications for lattice QCD,
and of maintaining good code performance, we have taken advantage of
the possibilities offered by Fortran90 to write a set of modules for
a high-level, yet efficient implementation of QCD simulations.
Our end product is described in this long write-up, whose main
purpose is to provide researchers with all the information needed
to use our modules. Since this effectively makes the long write-up
a reference document, it is indeed, and necessarily, ``long''.
We have nevertheless striven to be concise, in order to save space
and, especially, because we felt that a concise document would make it
easier for the user to find the relevant information. Most of the
times the functionality provided by our modules will be obvious.
For instance, if {\tt f1} and {\tt f2} are two variables of type
fermi\_field (see later for the precise definition), {\tt f1+f2}
will have as components the sum of the components of the two fields.
Similarly, if {\tt g1} and {\tt g2} are variables of type gauge\_field,
{\tt g1*g2} will have as components the matrix products of the components
of {\tt g1} and {\tt g2}. In other
instances, however, we had to use a bit of creativity in adapting
the symbols of the language to the definitions of some further useful
overloaded operators. Thus, if {\tt f1} and {\tt f2} are again variables
of type fermi\_field, {\tt f1//f2} will be for us a variable of
type gauge\_field having for components the dyadic formed by the
vector components of the two fermi\_fields. For all these less
obvious definitions, there is no substitute to reading the
sections of this article, where all of our overloaded operators
are carefully documented.
We expect that most of the users of our modules will be practitioners of
lattice gauge theory, and as such already quite knowledgeable about
the type of variables that enter QCD simulations.
Having in mind, however, that some of the
users might be application scientists called on to benchmark code
with which they are not too familiar, we decided to include in this
write-up a very concise description of the data structures encountered in
QCD simulations. This is presented in the next section, which summarizes
the kinematics that has been used for lattice QCD since the pioneering
work of Wilson \cite{wilson}. The section that follows discusses
the all-important notion of parallel transport in presence of a gauge
field and our implementation of parallel transport via a generalization
of the C-shift operation, which we call ``U-shift''. Sections \ref{evenand},
\ref{types} and \ref{programmingand} deal with algorithmic issues related
to the ordering of the data, with the description of the data types,
and with some further considerations of programming and efficiency.
The remaining sections are devoted to a detailed description of
our modules and of the functionality which they provide.
\section{Geometry and variables}
\label{geometryand}
We consider a four-dimensional lattice with extent $ \tt NX, NY, NZ$ and
$\tt NT$ in the four directions.
We will assume that $\tt NX,NY,NZ,NT$ are all even.
A lattice site will be labeled by four integer valued variables
$\tt x,y,z,t$ with
\begin{equation}
\tt 0 \le x < NX, \quad 0 \le y < NY, \quad 0 \le z < NZ,
\quad 0 \le t < NT \ .
\label{sites}
\end{equation}
When convenient, we will denote the collection of these four indices
by ${\bf x}$.
We will assume periodic boundary conditions.
The physical system is defined in terms of two types of variables
(also called the dynamical variables):
the gauge fields and the Fermi fields.
The components of a gauge field are $3 \times 3$ unitary, unimodular matrices
(i.e.~elements of the group $SU(3)$, the so called ``color'' group) defined
over the oriented links of the lattice. Later we will see that programming
considerations demand a more involved layout of data, but, conceptually,
a gauge field can be considered as a multidimensional array of complex
variables
\begin{equation}
\tt U(3,3,0:NX-1,0:NY-1,0:NZ-1,0:NT-1,4) \ ,
\label{gaugefield}
\end{equation}
where the first two indices of the generic array element
$\tt U(i,j,x,y,z,t,m)$ are the indices of the $SU(3)$ matrix, whereas
$\tt x,y,z,t$ label a lattice site and $\tt m=1 \dots 4$ labels one of the
four lattice links having origin at the site and oriented in the
positive $\tt m$ direction. When convenient we will use the more
compact notation $U_{ij,{\bf x}}^{\mu}$ to denote the gauge
field elements, or $U_{\bf x}^{\mu}$ to denote the whole matrix defined
over the link (in this compact notation we follow the common practice of
using a Greek letter to denote the direction
of the link). Another useful notation consists in representing by
$\hat\mu$ a four-vector having its $\mu$ component equal to 1,
all other components equal to zero. With this notation, one can say
that the gauge variable $U_{\bf x}^{\mu}$ is defined over the
oriented link from ${\bf x}$ to ${\bf x}+{\hat\mu}$.
The components of a Fermi field are defined over the sites of the
lattice. They are 3-dimensional complex vectors with respect to the
matrices of the color group and carry an additional spin index
$\tt s$ ranging from 1 to 4. Thus the data layout of a Fermi
field can be represented conceptually in terms of an array
of complex variables
\begin{equation}
\tt f(3,0:NX-1,0:NY-1,0:NZ-1,0:NT-1,4) \ ,
\label{fermifield}
\end{equation}
where the first index of the generic array element $\tt f(i,x,y,z,t,s)$
is the color index, $\tt x,y,z,t$ label the site and $\tt s$ is the spin index.
When convenient we will use a more compact notation $\psi_{i,{\bf x},s}$
for the components of a Fermi field, or $\psi_{{\bf x},s}$ for
the whole color vector, or even just $\psi_{\bf x}$.
In the field theoretical definition of the physical system the components
of a Fermi field would be anticommuting elements of a Grassmann algebra
with integration. The rules of integration over elements of a Grassmann
algebra $\psi_a$, $\bar\psi_b$ ($a$ and $b$ stand for any complete
set of indices) have, as their most important consequence, the formula
\begin{equation}
\int \prod_a (d\bar\psi_a \,d\psi_a) \exp[\sum_{a,b} \bar\psi_a
A_{a,b} \psi_b] = {\rm Det}[A] \ .
\label{gaussianint}
\end{equation}
In computational applications ${\rm Det} [A]$ or its derivatives with
respect to the dynamical variables are calculated by means of ordinary
complex variables $\phi_a$, $\bar\phi_b$ making use of the identity
\begin{equation}
{\rm Det}[A] = \int \prod_a ({d\bar\phi_a \,d\phi_a \over \pi})
\exp[\sum_{a,b} \bar\phi_a [A^{-1}]_{a,b} \phi_b] \ .
\label{gaussiancplx}
\end{equation}
Thus effectively one deals with arrays of complex variables as in
(\ref{fermifield}).
\section{The notion of U-shift}
\label{thenotion}
The gauge field serves to define the transport of dynamical variables
between neighboring sites. Gauge theories are characterized by the
property of local gauge invariance. In the present context this means
that it is always possible to redefine the Fermi variables by an $SU(3)$
transformation
\begin{equation}
\psi_{i,{\bf x},s} \to \psi'_{i,{\bf x},s} =
\sum_j G_{ij,{\bf x}} \psi_{j,{\bf x},s} \ ,
\label{gaugetrfm}
\end{equation}
where the elements of the gauge transformation $G_{ij,{\bf x}}$ are
$SU(3)$ matrices defined over the sites. All of the physical quantities
must remain invariant under such transformations.
It is clear that, if the Fermi fields transform according to (\ref{gaugetrfm})
with a $G_{ij,{\bf x}}$ that changes from site to site, a straightforward
finite difference (as one would use in the approximation of a derivative)
\begin{equation}
(\Delta \psi)_{i,{\bf x},s} = \psi_{i,{{\bf x}+\hat\mu},s} -
\psi_{i,{\bf x},s} \
\label{finitediff}
\end{equation}
will produce meaningless results. Rather, one should ``transport''
the variable $\psi_{{\bf x}+\hat\mu}$ from the site
${{\bf x}+\hat\mu}$ to the site $\bf x$ by means of the gauge variable
$U_{\bf x}^{\mu}$ defining a shifted variable
\begin{equation}
\psi^{\rm shifted}_{i,{\bf x},s} = \sum_j U_{ij,{\bf x}}^{\mu}
\psi_{j,{{\bf x}+\hat\mu},s}
\label{shiftfm}
\end{equation}
and then define a gauge covariant finite difference
\begin{equation}
(D \psi)_{i,{\bf x},s} = \psi^{\rm shifted}_{i,{\bf x},s} -
\psi_{i,{\bf x},s} \ .
\label{gfinitediff}
\end{equation}
Under a gauge transformation the gauge field itself changes according
to
\begin{equation}
U_{ij,{\bf x}}^{\mu} \to U_{ij,{\bf x}}^{\prime \mu} =
\sum_{kl} G_{ik,{\bf x}} U_{kl,{\bf x}}^{\mu} [G^{-1}]_{lj,{\bf x}+\hat\mu}
\ .
\label{gaugetru}
\end{equation}
From Eqs.~(\ref{gfinitediff},\ref{gaugetrfm},\ref{gaugetru}) one can verify
that under a gauge transformation the gauge covariant finite difference
changes like $\psi$ itself:
\begin{equation}
(D \psi)_{i,{\bf x},s} \to (D \psi')_{i,{\bf x},s} =
\sum_j G_{ij,{\bf x}} (D \psi)_{j,{\bf x},s} \ .
\label{gaugetrD}
\end{equation}
Thus the gauge covariant finite difference is a meaningful construct
and quantities such as its magnitude or the scalar product
$\sum_{i} \bar\psi_{i,{\bf x},s} (D \psi)_{i,{\bf x},s'}$ are gauge invariant
and thus physically well defined.
It is clear from the above that a circular shift (C-shift) of an array
such as $\tt f(3,0:NX-1,0:NY-1,0:NZ-1,0:NT-1,4)$ will generally be
complemented by multiplication by an element of the gauge field.
We will therefore define a U-shift operation in the following manner.
A U-shift with positive direction parameter $\mu = 1 \dots 4$ of the
Fermi field $\psi_{i,{\bf x},s}$ produces the array
$\psi^{\rm shifted}_{i,{\bf x},s}$ as given by Eq.~(\ref{shiftfm}).
A U-shift with negative direction parameter
$\mu' = -\mu = -1 \dots -4$ of the
Fermi field $\psi_{i,{\bf x},s}$ produces the array
\begin{equation}
\psi^{\rm shifted}_{i,{\bf x},s} = \sum_j U_{ij,{\bf x}-\hat\mu}^{\dagger \mu}
\psi_{i,{{\bf x}-\hat\mu},s} \ ,
\label{negshiftfm}
\end{equation}
this latter equation being motivated by the fact that the transport
factor over a link crossed in the negative direction is the
Hermitian adjoint (or equivalently the
inverse, with a unitary group $U^{\dagger}=U^{-1}$)
of the transport factor for the positively oriented link.
We define a U-shift for the gauge field as well. Since the gauge field
elements have two color indices which should be associated with the
beginning and end of the link (cf.~the gauge transformation properties
of the gauge field variables Eq.~(\ref{gaugetru})) the U-shift of a gauge
field will involve two matrix multiplications. Moreover, it will be
convenient to define its action on a generic gauge field, denoted below by $V$,
not necessarily identical to $U$. The idea is that in general there
will be several variables with the properties of a gauge field (see
the type definitions below) but there will always be one
well defined ``master gauge field'', denoted by $U$, which will serve
to define the transport of all gauge dependent variables. With this
in mind the action of a U-shift on a gauge field is defined as follows.
A U-shift with positive direction parameter $\mu = 1 \dots 4$ of the
gauge field $V_{ij,{\bf x}}^{\nu}$ produces the array
\begin{equation}
V^{{\rm shifted},\nu}_{i,{\bf x}} = \sum_{kl} U_{ik,{\bf x}}^{\mu}
V_{kl,{{\bf x}+\hat\mu}}^{\nu} U_{lj,{\bf x}+\hat\nu}^{\dagger \mu}\ .
\label{shiftv}
\end{equation}
A U-shift with negative direction parameter
$\mu' = -\mu = -1 \dots -4$ of the
gauge field $V_{ij,{\bf x}}^{\nu}$ produces the array
\begin{equation}
V^{{\rm shifted},\nu}_{i,{\bf x}} =
\sum_{kl} U_{ik,{\bf x}-\hat\mu}^{\dagger \mu}
V_{kl,{{\bf x}-\hat\mu}}^{\nu} U_{lj,{\bf x}-\hat\mu+\hat\nu}^{\mu}\ .
\label{negshiftv}
\end{equation}
When acting on field variables which carry no color index (we will define
such field variables below) the U-shift reduces to an ordinary C-shift.
\section{Even and odd components of field variables}
\label{evenand}
All lattice sites can be subdivided into ``even'' and ``odd'' sites
according to whether the sum of the integer valued coordinates
$\tt x+y+z+t$ is even or odd (checkerboard subdivision).
Correspondingly all field variables can
be divided into even and odd variables (for a gauge field variable
we base the subdivision on the origin of the link over which the variable
is defined, i.e. the $\tt x,y,z,t$ indices of the array~(\ref{gaugefield})).
With periodic boundary conditions and with an even lattice size in
all directions, a C-shift or a U-shift of an even field variable produces
an odd field variable and vice versa. There are many algorithms which
demand, especially in the context of a parallel implementation, that
even and odd variables be treated separately. For example, in a Monte
Carlo simulation algorithm all variables at even sites can be upgraded
simultaneously while those at odd sites are kept fixed and vice versa.
We will accommodate these algorithmic demands by defining all of
our field variables as arrays of even or odd field variables.
We will do so by taking advantage of the type definition as follows.
All field variables will be defined through a type. The first component
of the type will be an integer variable $\tt parity$ which will take
values $\tt 0$ and $\tt 1$ for variables defined over even and odd
sites respectively.
It will also be convenient to use the value $\tt -1$ to characterize
a field with parity undefined.
For the gauge variables it will be convenient
to include in the type a single $\mu$ component (of definite parity,
of course). Thus, in addition to the variable $\tt parity$,
the type will contain an integer variable $\tt dir$,
taking values $\tt 1 \dots 4$, to denote the direction of the link
(i.e.~the value of the index $\mu$). It will be convenient to let
$\tt dir$ also take the value $\tt 0$, to characterize $3 \times 3$
complex matrices which are defined over the sites rather than
over the links, such as the $SU(3)$ matrices of the gauge transformation
in Eq.~(\ref{gaugetrfm}).
Finally the type will then contain an array, denoted by $\tt fc$
(for field component) which will contain all the field variables
defined over the sites of a given parity.
Insofar as the indexing of the array is concerned, this is to a large
extent arbitrary, provided that the mapping between the array indices
and the actual Cartesian coordinates of the site is well defined.
For instance, one could collapse two neighboring ``time'' slices
into a single one and use indices $\tt x,y,z,t$ where $\tt t$ ranges
now from $\tt 0$ to $\tt NT2-1=NT/2-1$. On the other hand,
with many architectures efficiency considerations recommend that
the indices $\tt x,y,z,t$ be fused into a single index,
spanning the range $\tt 0$ to $\tt NXYZT2-1=NX*NY*NZ*NT/2-1$.
This will typically be the case when, because of a vectorized or
superscalar architecture, the instructions are pipelined and
longer arrays give rise to better performance. In principle
a good optimizing compiler should recognize when the individual
loops over the $\tt x,y,z$ and $\tt t$ indices can be fused
into a single one and take advantage of this possibility. However,
some compilers may be able to fuse only a limited number of nested loops
or, alternatively, this type of optimization may be hindered by the
the presence of further indices or by a large number of instructions
within the loops. Since the use of types and overloaded
operators makes the actual indexing of the arrays transparent
to the user, we decided to use a single index to label all of
the sites of a definite parity. This index is constructed by
going through the sites of definite parity in a lexicographic
order, increasing the $x$ coordinate first, then $y$, $z$ and $t$,
but, as stated above, the ordering of the sites is largely immaterial.
For all those operations which are performed locally over the sites,
the detailed mapping between the index and the geometry of the lattice
is clearly irrelevant. It is, of course, of consequence for
the implementation of the shift operations and for accessing
the component of a field at a definite Cartesian site.
For such purposes we provide the specifics of the mapping through
some global variables and an initialization subroutine.
We define the following global variables:
{\leftline{\tt INTEGER, DIMENSION(0:NX-1,0:NY-1,0:NZ-1,0:NT-1) ::
xyzt\_index}}
{\leftline{\tt INTEGER, DIMENSION(0:NX-1,0:NY-1,0:NZ-1,0:NT-1) ::
xyzt\_parity}}
{\leftline{\tt INTEGER, DIMENSION(0:NXYZT2-1,0:1,4) ::
xyzt\_cartesian}}
{\leftline{\tt INTEGER, DIMENSION(0:NXYZT2-1,0:1,8) ::
xyzt\_neighbor}}
{\leftline{\tt LOGICAL shift\_initialized}}
{\parindent=0pt where the parameter $\tt NXYZT2=NX*NY*NZ*NT/2$
equals one half of the total number of sites in the lattice.}
The arrays defined above are initialized by executing the subroutine
{\parindent=0pt {\tt shift\_initialization}.
The variable {\tt shift\_initialized}
is initialized to {\tt .FALSE.} All of the function calls
which implement the overloaded shift operators check the value
of {\tt shift\_initialized}. If this is {\tt .FALSE.}, the
subroutine {\tt shift\_initialization} is called and the arrays are
properly initialized. Before returning, {\tt shift\_initialization} sets
{\tt shift\_initialized} to {\tt .TRUE.} From this moment on
the arrays can be used to establish the mapping between the
Cartesian coordinates and the indices within the sublattices
of definite parity. The programmer wishing to use these arrays
before any shift operation is performed can, of course, initialize
them directly via a call to {\tt shift\_initialization}. }
The array component {\tt xyzt\_index(x,y,z,t)} gives the index of the
field component defined over the site with Cartesian coordinates
$\tt x,y,z,t$.
{\parindent=0pt {\tt xyzt\_parity(x,y,z,t)} gives the parity of the
site ($\tt xyzt\_parity(x,y,z,t)$
$\tt = x+y+z+t\; MOD \; 2$).
{\tt xyzt\_cartesian(i,p,m)} gives the Cartesian coordinate ({\tt x,y,z,t}
for {\tt m=1,2,3,4} respectively) of the site with index {\tt i} and
parity {\tt p}. Finally {\tt xyzt\_neighbor(i,p,m)} gives the index
of the nearest neighbor site in direction {\tt m} of a site with index
{\tt i} and parity {\tt p}. The convention is that the
values {\tt m=1,2,3,4}
correspond to the nearest neighbor in the forward $x,y,z,t$
directions, whereas {\tt m=5,6,7,8} correspond to the
nearest neighbor in the backward $x,y,z,t$ directions, respectively.}
\section{Types}
\label{types}
We define the following F90 types.
\subsection{Type gauge\_field}
\label{typeg}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE gauge\_field}
\leftline{\quad INTEGER parity}
\leftline{\quad INTEGER dir}
\leftline{\quad COMPLEX(REAL8),DIMENSION(3,3,0:NXYZT2-1)::fc}
\leftline{END TYPE}
}
As discussed in the previous section, a variable of type gauge\_field
contains the components of a gauge field defined over all the links
of direction $\tt dir$ emerging from the lattice sites
of a given $\tt parity$. The field component $\tt fc(i,j,xyzt)$
is an array of double precision complex variables (the kind $\tt REAL8$ is
defined in the module ``precisions'', see below), where
$\tt i,j$ are the indices of the $SU(3)$ matrix and $\tt xyzt$ labels
the site within the subset of sites of a definite parity.
\subsection{Type full\_gauge\_field}
\label{typeu}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE full\_gauge\_field}
\leftline{\quad TYPE(gauge\_field), DIMENSION(0:1,4) :: uc}
\leftline{END TYPE}
}
A variable of type full\_gauge\_field is meant to store an entire
gauge field configuration, i.e.~8 variables of type gauge\_field
corresponding to the two parity components and the 4 direction
components of a full gauge field. Although the $\tt parity$
and $\tt dir$ components of the individual $\tt uc(i,j)$ components
can be given any value, good programming practice recommends that one sets
$\tt uc(i,j)\%parity = i $, $\tt uc(i,j)\%dir = j $.
\subsection{Type fermi\_field}
\label{typef}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE fermi\_field}
\leftline{\quad INTEGER parity}
\leftline{\quad COMPLEX(REAL8),DIMENSION(3,0:NXYZT2-1,4)::fc}
\leftline{END TYPE}
}
A variable of type fermi\_field
contains the components of a Fermi field defined over the lattice sites
of a given $\tt parity$. The field component $\tt fc(i,xyzt,s)$
is an array of double precision complex variables, where
$\tt i$ is the color index, $\tt xyzt$ labels
the site and $\tt s$ is the spin index of the field.
\subsection{Type complex\_field}
\label{typec}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE complex\_field}
\leftline{\quad INTEGER parity}
\leftline{\quad COMPLEX(REAL8),DIMENSION(0:NXYZT2-1)::fc}
\leftline{END TYPE}
}
The type complex\_field is introduced to store an array of complex
numbers $\tt fc(xyzt)$ defined over the lattice sites
of a given $\tt parity$. Although one could also store such variables
in an array of complex numbers, defining a type has the advantage
that one can record the parity of the field and that it becomes possible
to define overloaded operators (intrinsic operations on intrinsic
types cannot be overloaded). A similar remark applies to the
type real\_field defined below.
\subsection{Type real\_field}
\label{typer}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE real\_field}
\leftline{\quad INTEGER parity}
\leftline{\quad REAL(REAL8),DIMENSION(0:NXYZT2-1)::fc}
\leftline{END TYPE}
}
The type real\_field is introduced to store an array of real
numbers $\tt fc(xyzt)$ defined over the lattice sites
of a given $\tt parity$.
\subsection{Type generator\_field}
\label{typege}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE generator\_field}
\leftline{\quad INTEGER parity}
\leftline{\quad INTEGER dir}
\leftline{\quad REAL(REAL8),DIMENSION(8,0:NXYZT2-1)::fc}
\leftline{END TYPE}
}
Although for computational purposes it is useful to store the components
of an $SU(3)$ gauge field as $3 \times 3$ complex matrices, a general
$SU(3)$ matrix is a function of only 8 real independent parameters.
In particular, given an 8-dimensional real vector with components
$v_k$ one can associate to it the $SU(3)$ matrix
\begin{equation}
U_{ij}= \bigg[\exp \big( \imath \sum_{k=1}^8 v_k \lambda^k \big)
\bigg]_{ij} \ ,
\label{expv}
\end{equation}
where the matrices $\lambda^k$ form a basis in the space
of Hermitian traceless $3 \times 3$ matrices and satisfy the
equations ${\rm Tr} (\lambda^k \lambda^{k'}) = 0$ for
$k \ne k'$, ${\rm Tr} (\lambda^k)^2 = 2$.
The term group generator is commonly used to refer to a traceless
Hermitian matrix, such as
\begin{equation}
H_{ij}= \sum_{k=1}^8 v_k \lambda^k_{ij}
\label{generator}
\end{equation}
in Eq.~(\ref{expv}). For some algorithms it is convenient to deal
directly with the components $v_k$ of a generator, rather with
the exponentiated matrix $U$ or the Hermitian matrix $H$. For this
reason we provide the type generator\_field, aimed at storing generator
components defined over the sites of a given {$\tt parity$}. Since generators
are frequently associated to gauge field variables, we give the
type generator\_field a $\tt dir$ component as well.
\subsection{Type matrix}
\label{typem}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{TYPE matrix}
\leftline{\quad COMPLEX(REAL8),DIMENSION(3,3)::mc}
\leftline{END TYPE}
}
The type matrix is defined for programming convenience,
in order to allow for the overloading of operators and assignments.
For instance, it makes it possible to define an operation $\tt g*m $,
where the variables $\tt g$ and $\tt m$
are of type gauge\_field and matrix respectively,
which implements the matrix product of the components of a gauge
field times a constant matrix.
\section{Programming and efficiency considerations}
\label{programmingand}
\subsection{One layer versus two layer data structure}
\label{onelayer}
Conceptually our variables would be most naturally defined in terms of
a two layer data structure. At the bottom layer we would find objects
such as a single $SU(3)$ matrix or a single color vector, i.e.~three
dimensional complex vector. Overloaded operations such as matrix
multiplication or multiplication of a matrix times a color vector
would also be defined. At the top layer we would then use these
objects to define extended fields, such as the gauge field, consisting
of an array of objects of type matrix. Operators among the objects
of the top layer would be built from the elemental operators
already defined at the bottom layer. However appealing, this organization
of the data would almost certainly imply a huge penalty in efficiency.
It is indeed reasonable to expect that the compiler will implement
overloaded operations in terms of function calls. In a two layer
structure, then, an operation such as the addition of two Fermi
fields would be implemented via repeated calls, site by site, to
the function which adds the color vector components of the two
fields. It is clear that this use of function calls
at very low granularity would imply a heavy computational burden.
The only way to regain efficiency would be to inline the function
calls implementing the elemental operations. While in principle
this is possible, it is not reasonable to expect that compilers
would generally allow inlining of function calls that implement operations
among derived data types over which they have little direct control.
For this reason we decided to forfeit the possibility of defining
a two layer data structure, however conceptually pleasing this
may be, and
organized all of our data into a single layer of user defined types.
Thus the types which we introduce to define
extended fields are, essentially, F90 arrays complemented with one
or two variables ($\tt parity, dir$) specifying their attributes.
As a consequence the computational cost for the use of overloaded
operators between our data structures should not be any bigger than
the cost of a call to a function or subroutine that manipulates large
arrays. On the other hand, the advantages we gain in code structure
and ease of programming are truly remarkable.
\subsection{Overloaded assignments}
\label{overloaded}
The use of overloaded operators may imply the creation of more
temporaries and, consequently, more motion of data than a straightforward
implementation of operations among arrays. Consider for example the
following operation among variables of type fermi\_field:
\begin{equation}
\tt f1=f1+f2+f3 \ .
\label{addfff}
\end{equation}
(We will formally define the addition of Fermi fields later, but it
will perform the obvious operation of adding the $\tt fc$ components
of the fields.)
With ordinary arrays the compiler might put the result of $\tt f1+f2$
in a temporary $\tt t1$ and then add $\tt t1$ and $\tt f3$ placing
the result in $\tt f1$. Thus there would be two write-to-memory
operations per
component of the arrays. (A good optimizing compiler could even use
registers, dispensing with the creation of the temporary and
of one of the copies to memory.) However, if the overloaded addition
of Fermi fields is implemented via function calls, what we expect to
happen is that the function implementing $\tt f1+f2$ places the
result into a temporary $\tt t1$ returning the address of the
corresponding data structure to the calling program. The compiler
at this point will probably copy $\tt t1$ into a temporary $\tt t2$, since
it would not be safe to pass the addresses of $\tt t1$ and $\tt f3$
to the add function which will likely put
the result into $\tt t1$ again. Finally, the result will be copied
into $\tt f1$. If implemented in this manner, the entire operation
involves four write-to-memory operations: to $\tt t1$, to $\tt t2$,
to $\tt t1$ again and to $\tt f1$. (Of course, all of the above is
implementation dependent. As far as we know, F90 does not specify how the
variables should be passed in function calls. An operating system could
let the calling program pass to a function the address where it expects
the result, making the
call $\tt a=function(b,c)$ effectively identical to
$\tt CALL \; subroutine(a,b,c)$. In this case
the composite operation~(\ref{addfff})
could be implemented with two copies to memory only.)
The procedure could be drastically simplified through the use of
an overloaded assignment $\tt +=$. Instruction~(\ref{addfff}) could be
written
\begin{equation}
\tt f1\,+=f2+f3 \ ,
\label{pefff}
\end{equation}
which the compiler would implement by issuing first a call to a function
that adds $\tt f2$ and $\tt f3$ returning the result in $\tt t1$. The
addresses of $\tt f1$ and $\tt t1$ would then be passed to a subroutine,
e.g.~$\tt plus\_eq(a,b)$ that implements the operation $\tt f1=f1+t1$
among the components of the data types. The required number of copies
to memory would be only two.
In order to allow for these possible gains in efficiency, we have defined
a large set of overloaded assignments, which will be detailed in the
description of the module ``assign'' given below. Since F90 permits
only the use of the $\tt =$ symbol for the assignment, we have implemented
its overloading by defining two global variables: a character variable
$\tt assign\_type$ and an integer variable $\tt assign\_spec$ (for assign
specification, introduced to accommodate assignments of a more
elaborate nature). The default values of these variables are $\tt '='$
and $\tt 0$. They are initialized with these values and reset to
their default values at the end of all overloaded assignments. We
follow this procedure to avoid the occurrence of accidental erroneous
assignments. When $\tt assign\_type$ equals $\tt '='$ the result of
the assignment between variables of identical type is the expected
copy of the data structure at the r.h.s. into the variable at the l.h.s..
(We also define overloaded $\tt '='$ assignments between variables
of different type; the results of such assignments are explained
in the description of the module ``assign''.) Overloaded
assignments such as $\tt a\,+=b$ are obtained by setting
$\tt assign\_type$ (and possibly $\tt assign\_spec$) to the appropriate
value immediately before the assignment. We recommend the following
pattern for the instructions:
\begin{equation}
\tt assign\_type='+'; \quad a=b
\label{asgnone}
\end{equation}
or (this implements a U-shift from direction $\tt n$)
\begin{equation}
\tt assign\_type='u'; \quad assign\_spec=n; \quad a=b
\label{asgntwo}
\end{equation}
The overloaded assignments are implemented via case constructs, which
make reference to the values of the global variables $\tt assign\_type$,
$\tt assign\_spec$. A simplified version of the code for an assignment
would be as follows:
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{SUBROUTINE typea\_eq\_typeb(a,b)}
\leftline{\quad TYPE(typea), INTENT(INOUT) :: a}
\leftline{\quad TYPE(typeb), INTENT(IN) :: b}
\leftline{\quad SELECT CASE(assign\_type)}
\leftline{\quad CASE('=')}
\leftline{\quad \quad implements a=b}
\leftline{\quad CASE('+')}
\leftline{\quad \quad implements a=a+b}
\leftline{\quad CASE DEFAULT}
\leftline{\quad \quad returns an error message and stops execution
if the value}
\leftline{\quad \quad of assign\_type does not correspond
to any defined assignment}
\leftline{\quad END SELECT}
\leftline{\quad assign\_type='='; \quad assign\_spec=0}
\leftline{END SUBROUTINE typea\_eq\_typeb}
}
We wish to emphasize that the structure of
data and operations which we have introduced may still cause loss
of efficiency with some compilers, even with an optimizing one.
It might happen that code performing the same calculations as a
code written in terms of our data structures, but formulated without
use of any derived data types, is converted, upon compilation, into a more
efficient executable. However, we designed our data structure
and defined our operators and assignments in a way which should present
no barrier to a highly efficient, parallelizing compilation. It will
be an interesting experiment to verify how different compilers respond
to it.
\section{Modules}
\label{modules}
\subsection{Module precisions}
\label{precisions}
This module defines two kind parameters, $\tt REAL8 $ and $\tt LONG $.
These parameters store the kind of an 8-byte floating point variable
and of an 8-byte integer variable. They are used to render the
kind definitions machine independent. $\tt INTEGER(LONG) $ variables
are used only for the parallel generation of pseudorandom numbers
in a system independent way (cf.~the module ``random\_numbers'').
If 8-byte integers are not supported by the architecture, the module
random\_numbers should be modified to run with shorter integers
or to use system supplied parallel pseudorandom numbers, and the
definition of $\tt LONG$ should be changed accordingly.
\subsection{Module global\_module}
\label{global}
This module defines the integer constants $\tt NX$, $\tt NY$, $\tt NZ$
and $\tt NT$ which specify the size of the lattice. $\tt NX$, $\tt NY$,
$\tt NZ$, $\tt NT$ must all be even. It defines the reduced
temporal extent $\tt NT2= NT/2$, and the products
$\tt NXYZT = NX*NY*NZ*NT$, $\tt NXYZT2 = NX*NY*NZ*NT2$.
It also defines for convenience the constants
$\tt NCGV = 9*NXYZT2$, $\tt NCFV = 12*NXYZT2$,
$\tt NRGV= 2*NCGV$, $\tt NRFV= 2*NCFV$, $\tt NRGEV= 8*NXYZT2$,
which are equal to the number of complex or, respectively, real variables
in the $\tt fc$ components of the types gauge\_field, fermi\_field
and generator\_field.
All of the types introduced in Sect.~\ref{types} are declared in this module.
Finally the module declares a few global variables, namely, the master
gauge field:
\leftline{$\tt TYPE(full\_gauge\_field) \quad u $ }
{\parindent=0pt
the assignment variables (cf.~Sect.~\ref{overloaded}):}
\leftline{$\tt CHARACTER \quad assign\_type $}
\leftline{$\tt INTEGER \quad assign\_spec $}
{\parindent=0pt
the arrays {\tt xyzt\_index}, {\tt xyzt\_parity}, {\tt xyzt\_cartesian},
{\tt xyzt\_neighbor} and the logical variable {\tt shift\_initialized},
already mentioned in Sect.~\ref{evenand},}
{\parindent=0pt
the context logical array, used in conditional operations (cf.~the module
``conditionals''):}
\leftline{$\tt LOGICAL, DIMENSION(0:NXYZT2-1) ::
\; context $}
{\parindent=0pt
and the variables used for the generation of pseudorandom numbers
(see the module ``random\_numbers''):}
\leftline{$\tt INTEGER \quad seed\_a, seed\_b $}
\leftline{$\tt INTEGER, DIMENSION(0:NXYZT2-1) ::
\; seeds $}
{\parindent=0pt The module contains the subroutine {\tt shift\_initialization}
(see Sect.~\ref{evenand}).}
\subsection{Module constants}
\label{constants}
This module defines some useful parameters, making them available
to all program units which use it. Namely, the following real constants
are defined: $\tt PI$ ($\pi$),
$\tt PI2$ ($\pi/2$), $\tt TWOPI$ ($2 \pi$),
$\tt SQRT2$ ($\sqrt{2}$), $\tt SQRT22$ ($\sqrt{2}/2$),
$\tt SQRT3$ ($\sqrt{3}$), $\tt SQRT33$ ($\sqrt{3}/3$),
$\tt TWOSQRT33$ ($2 \sqrt{3}/3$), the complex constant $\tt IU$
($\imath$), and the arrays:
\leftline{$\tt COMPLEX(REAL8), DIMENSION(3,3) :: ZERO\_m, UNIT, IU\_m$}
\leftline{$\tt COMPLEX(REAL8), DIMENSION(3) :: ZERO\_v$}
\leftline{$\tt REAL(REAL8), DIMENSION(8) :: ZERO\_ge$}
\leftline{$\tt COMPLEX(REAL8), DIMENSION(3,3,8) :: LAMBDA$}
\leftline{$\tt COMPLEX(REAL8), DIMENSION(4,4,5) :: GAMMA$}
$\tt UNIT$ and $\tt IU\_m$ are set equal to the unit matrix,
and to $\imath$ times the unit matrix, respectively.
$\tt ZERO\_m$, $\tt ZERO\_v$, $\tt ZERO\_ge$ have all
components equal to zero. The array $\tt LAMBDA$ stores
the components of the $\lambda$ matrices:
\leftline{ ${\tt LAMBDA(i,j,k)}=\lambda_{i,j}^k$,}
{\parindent=0pt
and the array $\tt GAMMA$ stores the components of
Dirac's $\gamma$ matrices,}
\leftline{${\tt GAMMA(s1,s2,m)}=
\gamma_{s{\scriptscriptstyle 1},s{\scriptscriptstyle 2}}^m,\;
m=1 \dots 5 \;,$
in our chosen representation.}
{\parindent=0pt (We follow the convention $\gamma^5 = \gamma^1 \gamma^2
\gamma^3 \gamma^4$.)}
Notice that we do not make any distinction between upper and lower indices
for the $\lambda$ and $\gamma$ matrices: $\lambda^k=\lambda_k$,
$\gamma^m=\gamma_m$ and the use of upper or lower indices is only dictated
by notational convenience.
\subsection{Module field\_algebra}
\label{fieldalgebra}
This module defines several overloaded operators that perform
arithmetic operations between fields and other variables.
We describe here all the operations which are defined. For conciseness
we introduce notational conventions. We use the symbols $\tt g,$
$\tt u,$ $\tt f,$ $\tt c,$ $\tt r,$ $\tt ge$ and $\tt m$
to denote variables of type gauge\_field, full\_gauge\_field,
fermi\_field, complex\_field, real\_field,
generator\_field and matrix, respectively, and the
symbols $\tt complex$ and $\tt real$ to denote a complex or real variable
of kind $\tt REAL8$ (cf.~Sect.~\ref{precisions}). When necessary,
we will use subscripts, e.g.~$\tt f_1, f_2$, to distinguish between two
variables of the same type.
All operators obey the following general rules:
i) When the result of the operation is a field, if the two operands
have a $\tt parity$ component, the $\tt parity$ of the result is
the $\tt parity$ of the operands if they have the same $\tt parity$,
otherwise it is undefined (i.e.~$= -1$). If a single operand
has a $\tt parity$ component, then the $\tt parity$ of the result
takes the same value. A similar rule applies to the direction
component of the variables of type gauge\_field and generator\_field: if
both operands have the same $\tt dir$ or a single operand carries a $\tt dir$
component, then the $\tt dir$ component of the result is set to this
value. Otherwise it is set to $0$.
ii) When the operator acts between fields, the operation is performed
site by site and the result is again a variable of field type. When the
operator acts between a variable of type field and a global variable
(i.e.~$\tt m$, $\tt complex$ and $\tt real$) the site variable
is combined with the global variable. For example, the operations
$\tt c=c_1+c_2$ and $\tt c=c_1+complex$ would be implemented as
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{\quad c\%fc(xyzt)=c1\%fc(xyzt)+c2\%fc(xyzt)}
\leftline{END DO}
}
and
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{\quad c\%fc(xyzt)=c1\%fc(xyzt)+complex}
\leftline{END DO}
}
\leftline{respectively.}
The following operations are defined and have the obvious meaning,
implicit in the symbol:
\vskip 1mm
\leftline{$\tt g_1+g_2$,\quad $\tt g_1-g_2$,\quad $\tt g_1*g_2$,
\quad $\tt g*f$,\quad $\tt f*g$,\quad $\tt g*c$,\quad $\tt c*g$,
\quad $\tt g/c$,}
\leftline{$\tt g*r$,\quad $\tt r*g$,\quad $\tt g/r$,\quad $\tt g+m$,
\quad $\tt m+g$,\quad $\tt g-m$,\quad $\tt m-g$,\quad $\tt g*m$,
\quad $\tt m*g$,}
\leftline{$\tt g*complex$,\quad $\tt complex*g$,\quad $\tt g/complex$,
\quad $\tt g*real$,\quad $\tt real*g$,\quad $\tt g/real$;}
\vskip 3mm
\leftline{$\tt f_1+f_2$,\quad $\tt f_1-f_2$,\quad $\tt f*c$,\quad $\tt c*f$,
\quad $\tt f/c$,\quad $\tt f*r$,\quad $\tt r*f$,\quad $\tt f/r$,
$\tt \; f*m$, $\tt \; m*f$,}
\leftline{$\tt f*complex$,\quad $\tt complex*f$,\quad $\tt f/complex$,
\quad $\tt f*real$,\quad $\tt real*f$, \quad $\tt f/real$;}
\vskip 3mm
\leftline{$\tt c_1+c_2$,\quad $\tt c_1-c_2$,\quad $\tt c_1*c_2$,
\quad $\tt c_1/c_2$, \quad $\tt c+r$,\quad $\tt r+c$,\quad $\tt c-r$,
\quad $\tt r-c$,}
\leftline{$\tt c*r$,\quad $\tt r*c$,\quad $\tt c/r$,\quad $\tt r/c$,
\quad $\tt c+complex$,\quad $\tt complex+c$,\quad $\tt c-complex$,}
\leftline{$\tt complex-c$,\quad $\tt c*complex$,\quad $\tt complex*c$,
\quad $\tt c/complex$,\quad $\tt complex/c$,}
\leftline{$\tt c+real$,\quad $\tt real+c$,\quad $\tt c-real$,
\quad $\tt real-c$,\quad $\tt c*real$,\quad $\tt real*c$,}
\leftline{$\tt c/real$,\quad $\tt real/c$;}
\vskip 3mm
\leftline{$\tt r_1+r_2$,\quad $\tt r_1-r_2$,\quad $\tt r_1*r_2$,
\quad $\tt r_1/r_2$, \quad $\tt r+real$,\quad $\tt real+r$,}
\leftline{$\tt r-real$,\quad $\tt real-r$,\quad $\tt r*real$,
\quad $\tt real*r$,\quad $\tt r/real$,\quad $\tt real/r$;}
\vskip 3mm
\leftline{$\tt ge_1+ge_2$,\quad $\tt ge_1-ge_2$,\quad $\tt ge*r$,
\quad $\tt r*ge$,\quad $\tt ge/r$,\quad $\tt ge*real$,}
\leftline{$\tt real*ge$,\quad $\tt ge/real$;}
\vskip 3mm
\leftline{$\tt m_1+m_2$,\quad $\tt m_1-m_2$,\quad $\tt m_1*m_2$,
\quad $\tt m*complex$,\quad $\tt complex*m$,\quad $\tt m/complex$,}
\leftline{$\tt m*real$,\quad $\tt real*m$,\quad $\tt m/real$;}
\vskip 2mm
We do not provide any clarification about the operations
listed above (it would be truly superfluous)
but for the observation that the symbol $\tt *$ implies matrix multiplication
when acting between operands of type gauge\_field or matrix,
and matrix by vector or vector by matrix when one of the operand is
a fermi\_field and the other a gauge\_field or a matrix. Notice that there
is no implicit complex conjugation of the vector at the r.h.s. of a
vector by matrix multiplication, i.e. $\tt f=f_1 * m$ translates into
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO s=1,4}
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{DO i=1,3}
\leftline{\quad f\%fc(i,xyzt,s)=f1\%fc(1,xyzt,s)*m\%mc(1,i)\quad \&}
\leftline{\quad \quad \quad +f1\%fc(2,xyzt,s)*m\%mc(2,i)
+f1\%fc(3,xyzt,s)*m\%mc(3,i)}
\leftline{END DO}
\leftline{END DO}
\leftline{END DO}
}
The following additional operations have a special meaning:
\leftline{$\tt g_1/g_2$ :}
{\parindent=0pt
the gauge field $\tt g_1$ is multiplied, site by site, by the Hermitian
adjoint of the gauge field $\tt g_2$ (the notation is motivated by the fact
that, with unitary matrices, the Hermitian adjoint of a matrix is also
its inverse; however, there is no restriction that the variables stored
in a gauge field must represent unitary matrices).}
\leftline{$\tt m/g_2$ and $\tt g_1/m$ : same as above,
but with $\tt m$ a matrix rather than a gauge}
\leftline{field.}
\leftline{$\tt g_1//g_2$ : the Hermitian adjoint of the gauge
field $\tt g_1$ is multiplied, site by}
\leftline{site, by the gauge field $\tt g_2$.}
\leftline{$\tt g_1//m$ and $\tt m//g_2$ : same as above,
but with $\tt m$ a matrix rather than a gauge}
\leftline{field.}
\leftline{$\tt f/g \;$ and $\tt g//f $ : the Fermi field $\tt f$ is
right or left multiplied, site by site, by}
\leftline{the Hermitian adjoint of the gauge field $\tt g$.}
\leftline{$\tt f/m \;$ and $\tt m//f $ : same as above,
but with $\tt m$ a matrix rather than a gauge}
\leftline{field.}
\leftline{$\tt f_1*f_2$ :}
{\parindent=0pt
this operation returns a complex field having as site components
the scalar product, taken over the color and the spin indices, of
the complex conjugate of $\tt f_1$ and $\tt f_2$. Explicitly,
$\tt c=f_1*f_2$ would be implemented as}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{\quad c\%fc(xyzt)=0}
\leftline{DO s=1,4}
\leftline{DO i=1,3}
\leftline{\quad c\%fc(xyzt)=c\%fc(xyzt) \quad \&}
\leftline{\quad \quad \quad +CONJG(f1\%fc(i,xyzt,s))*f2\%fc(i,xyzt,s)}
\leftline{END DO}
\leftline{END DO}
\leftline{END DO}
}
\leftline{$\tt f_1//f_2$ :}
{\parindent=0pt
this operation returns a variable of type gauge\_field having as
site components the dyadic (over the color indices) of
$\tt f_1$ and the complex conjugate of $\tt f_2$. The spin indices
are summed over. Explicitly, $\tt g=f_1//f_2$ would be implemented as}
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{DO i=1,3}
\leftline{DO j=1,3}
\leftline{\quad g\%fc(i,j,xyzt)=f1\%fc(i,xyzt,1)
*CONJG(f2\%fc(j,xyzt,1))}
\leftline{\quad DO s=2,4}
\leftline{\quad \quad g\%fc(i,j,xyzt)=g\%fc(i,j,xyzt) \quad \&}
\leftline{\quad \quad \quad \quad
+f1\%fc(i,xyzt,s)*CONJG(f2\%fc(j,xyzt,s))}
\leftline{\quad END DO}
\leftline{END DO}
\leftline{END DO}
\leftline{END DO}
}
\leftline{$\tt ge_1*ge_2$ : this operation returns a real field having
as site components the}
\leftline{scalar product of the site components of the generators.}
\leftline{$\tt g_1.Dot.g_2$ : this operation returns a real field having
as site components the}
\leftline{the real part of the trace of the product of the Hermitian
adjoint of the site}
\leftline{components of the gauge field $\tt g_1$ with the site components
of the gauge}
\leftline{field $\tt g_2$.}
The following named operators are also defined:
{\parindent=0pt
$\tt i.Gamma.f$ , where $\tt i$ is a scalar integer.
This operation returns a Fermi field
having as site components the product of a single $\gamma$ matrix
or of a pair of $\gamma$ matrices times the site components $\psi_{\bf x}$
of the Fermi field $\tt f$. Our convention is as follows. The integer
variable $\tt i$ can take value $1$ through $5$ or value
$10*i_1+i_2$, where $i_1$ and $i_2$ can again range from
$1$ to $5$. In the former case the operator implements the product
$\gamma_i \psi_{\bf x}$. In the latter case the pair $i_1,i_2$
stands for two indices labeling a matrix
$\gamma_{i{\scriptscriptstyle1}\,i{\scriptscriptstyle2}}$, where
$\gamma_{i{\scriptscriptstyle1}\,i{\scriptscriptstyle2}}=
{\imath \over 2} [\gamma_{i{\scriptscriptstyle1}}
\gamma_{i{\scriptscriptstyle2}}-
\gamma_{i{\scriptscriptstyle2}}
\gamma_{i{\scriptscriptstyle1}}]$,
$\gamma_{i\,5}=-\gamma_{5\,i}=\gamma_i \gamma_5$ with
$i,i_1,i_2 = 1 \dots 4$, and the operator implements the product
$\gamma_{i{\scriptscriptstyle1}\,i{\scriptscriptstyle2}}\psi_{\bf x}$.
Thus, for instance, $\tt i=25; f1=i.Gamma.f2$ would implement
$\psi_{1{\bf x}} = \gamma_2 \gamma_5 \psi_{2{\bf x}}$. Products
of $\gamma$ matrices have been explicitly incorporated in the
definition of the $\tt .Gamma.$ operator because they are
frequently encountered in the evaluation of matrix elements
of fermionic variables.
$\tt f.Gamma.i$ , where $\tt i$ is a scalar integer.
This operation returns a Fermi field
having as site components the product of site components
of the Fermi field $\tt f$ times a single $\gamma$ matrix
or of a pair of $\gamma$ matrices, following the same convention
about the values of $\tt i$ as above.
$\tt i.Lambda.g$ , where $\tt i$ is a scalar integer.
This operation returns a gauge field
having as site components the product of the matrix $\lambda_i$
times the site components of the gauge field $\tt g$.
$\tt g.Lambda.i$ , where $\tt i$ is a scalar integer.
This operation returns a gauge field
having as site components the product of the site components
of the gauge field $\tt g$ times the matrix $\lambda_i$.}
In addition we define the following unary operators:
\centerline{$\tt .I.$,\quad $\tt .Minus.$,\quad $\tt .Conjg.$,
\quad $\tt .Adj.$, \quad $\tt .Ctr.$ \quad $\tt .Tr.$ \quad
$\tt .Sqrt.$ \quad and \quad $\tt .Exp.$}
When acting on a variable of type gauge\_field, fermi\_field or
complex\_field $\tt .I.$ returns $\imath$ times the variable. When
acting on a variable of type real\_field it returns a complex
field given by $\imath$ times the real field. This is introduced
for efficiency, since the operator is implemented by switching
real and imaginary parts with the appropriate change of sign, rather than
through a complex multiplication.
When acting on a variable of type gauge\_field, fermi\_field,
complex\_field, real\_field or generator\_field,
$\tt .Minus.$ returns the negative of the variable.
When acting on a variable of type gauge\_field, fermi\_field,
complex\_field or matrix $\tt .Conjg.$ returns the complex conjugate
of the variable, i.e. a variable whose complex components are the
complex conjugate of the original one.
When acting on a variable of type gauge\_field or matrix $\tt .Adj.$
returns the Hermitian adjoint of the variable.
When acting on a variable of type gauge\_field or matrix $\tt .Ctr.$
returns a complex\_field or complex number, respectively,
equal to the trace (at each site in the case of a field) of
the operand.
When acting on a variable of type gauge\_field or matrix $\tt .Tr.$
returns a real\_field or real number, respectively,
equal to the real part of the trace (at each site in the case of a field)
of the operand.
When acting on a variable of type real\_field $\tt .Sqrt.$
returns a real\_field having as site components the square root
of the absolute value of the site components of the operand.
At the same time the global variable context is set to $\tt .TRUE.$
at all sites where the operand is non-negative and to $\tt .FALSE.$
at all other sites.
When acting on a variable of type real\_field $\tt .Exp.$
returns a real\_field having as site components the exponential
of the site components of the operand.
\subsection{Modules assign\_isotype1, assign\_isotype2, \par\indent
assign\_isotype3 and assign\_mixed}
\label{assign}
These modules define the normal assignment and a variety of
overloaded assignments which are defined for efficiency
(cf.~Sect.~\ref{overloaded} above) and programming convenience.
They are presented as four separate modules (assign\_isotype1,
assign\_isotype2 and assign\_isotype3 define assignments
between variables of the same type,
assign\_mixed between variables of different type) to
reduce the overall length of the individual modules.
We reproduce here all the
available assignments. We use the notational conventions
we introduced at the beginning of Sect.~\ref{fieldalgebra}.
Namely, we use the symbols $\tt g,$ $\tt u,$
$\tt f,$ $\tt c,$ $\tt r,$ $\tt ge$ and $\tt m$
to denote variables of type gauge\_field, full\_gauge\_field,
fermi\_field, complex\_field, real\_field,
generator\_field and matrix, respectively, and the
symbols $\tt complex$ and $\tt real$ to denote a complex or real variable
of kind $\tt REAL8$ (cf.~Sect.~\ref{precisions}).
Also, we use subscripts, e.g.~$\tt f_1, f_2$, to distinguish between two
variables of the same type.
When the assignment relies on the the fact that the global
variables $\tt assign\_type$
and $\tt assign\_spec$ have a value different from their default
values $'='$ and $0$, we will denote this fact by the
using the combined symbols $\tt assign\_type (assign\_spec) = $
to denote the assignment. For example, we would use $\tt f_1 \; += f_2 $ or
$\tt f_1 \; U(2)= f_2 $ to denote assignments which in the actual coding
would be implemented as
\leftline{{\tt assign\_type='+'; f1=f2} , or}
\leftline{{\tt assign\_type='U'; assign\_spec=2; f1=f2} , respectively.}
A general rule is that all assignments set the global
variables $\tt assign\_type$ and $\tt assign\_spec$
equal to their default values $'='$ and $0$,
no matter what the assignment does. As discussed
in Sect.~\ref{overloaded}, this is done in order
to avoid the accidental use of erroneous assignments.
For the $\tt parity$ component,
the rule is that, if the destination is not an operand in the assignment
(i.e.~it is a variable with strict $\tt INTENT(OUT)$),
the $\tt parity$ component (if present)
of the variable at the l.h.s.~of the assignment (destination)
is set equal to the $\tt parity$ of the variable at the
r.h.s.~of the assignment (source), or set to $\tt -1$ if the
source has no $\tt parity$.
Similarly, when the destination is not an operand in the
assignment and has a $\tt dir$ component, this is set equal
to the $\tt dir$ of the source or to $\tt 0$
if the source has no $\tt dir$.
An exception to the rule above about the $\tt parity$ component occurs with
the $\tt assign\_type = 'u'$, $\tt assign\_type = 'w'$
and $\tt assign\_type = 'x'$ assignments,
which copy into the destination a shifted source. In this case,
if the $\tt parity$ of the source is defined, the $\tt parity$ of
the destination is set to the opposite value.
If the destination is an operand in
the assignment (i.e.~it is a variable with $\tt INTENT(INOUT)$)
$\tt parity$ and $\tt dir$ are treated in a manner similar to
what happens in the definition of the operators implemented by the
overloaded assignment. Typically, if the destination and the
other operand have the same $\tt parity$, this is preserved, otherwise
the $\tt parity$ of the destination is set to $\tt -1$ (undefined).
An exception is found in the assignments $\tt U=$ and $\tt W=$
which implement the sum of the destination with a shifted operand,
in which case the $\tt parity$ of the destination is preserved
if the other operand has the opposite $\tt parity$ (as is the
case in a geometrically meaningful operation) and is returned
undefined otherwise.
In what follows we list all of the available assignments
and define their action,
appending a few words of explanation when appropriate.
When the assignment is not followed by further clarifications,
it means that it is a straightforward assignment
(with $\tt assign\_type \; '='$) copying
the content of the source into the destination. Also, whenever the
assignment implements operations which can be performed by using
overloaded operators, we illustrate its action
simply by reformulating it in terms of these operators. We refer to the
sections detailing the modules where the overloaded operators
are defined for clarification of their action.
The assignments are listed in order of destination type, first,
and then of source type. The ordering of the types is the same as
their order of introduction in Sect.~\ref{types}.
Available assignments:
\leftline{$\tt g_1=g_2$}
\leftline{$\tt g_1\;+=g_2$ \quad \quad ($\tt g_1=g_1+g_2$)}
\leftline{$\tt g_1\;-=g_2$ \quad \quad ($\tt g_1=g_1-g_2$)}
\leftline{$\tt g_1\;*(0)=g_2$ \quad \quad ($\tt g_1=g_1*g_2$)}
\leftline{$\tt g_1\;*(-1)=g_2$ \quad \quad ($\tt g_1=g_2*g_1$)}
Notice how the $\tt assign\_spec$ variable is used, above and
immediately below, to specify the
order of the operands in the non-commutative matrix multiplication.
\leftline{$\tt g_1\;/(0)=g_2$ \quad \quad ($\tt g_1=g_1/g_2$)}
\leftline{$\tt g_1\;/(-1)=g_2$ \quad \quad ($\tt g_1=g_2//g_1$)}
\leftline{$\tt g_1\;u(dir)=g_2$ \quad \quad ($\tt g_1=dir.Ushift.g_2$)}
\leftline{$\tt g_1\;U(dir)=g_2$ \quad \quad ($\tt g_1=g_1+(dir.Ushift.g_2)$)}
\leftline{$\tt g_1\;t=g_2$ \quad \quad ($\tt g_1=g_2$ where $\tt context$
is $\tt .TRUE.$)}
\leftline{$\tt g_1\;f=g_2$ \quad \quad ($\tt g_1=g_2$ where $\tt context$
is $\tt .FALSE.$)}
\leftline{$\tt g_1\;A=g_2$ \quad \quad ($\tt g_1=.Adj.g_2$)}
\leftline{$\tt g_1\;C=g_2$ \quad \quad ($\tt g_1=.Conjg.g_2$)}
\leftline{$\tt g_1\;I=g_2$ \quad \quad ($\tt g_1=.I.g_2$)}
\leftline{$\tt g_1\;M=g_2$ \quad \quad ($\tt g_1=.Minus.g_2$)}
\leftline{$\tt g=u$ \quad \quad ($\tt u\%uc(g\%parity,g\%dir)$ is copied to
$\tt g\%fc$)}
\leftline{$\tt g\;t=u$ \quad \quad (same as above, but only
where $\tt context$ is $\tt .TRUE.$)}
\leftline{$\tt g\;f=u$ \quad \quad (same as two lines above, but only
where $\tt context$ is $\tt .FALSE.$)}
\leftline{$\tt g=ge$ \quad \quad ($\tt g=.Matrix.ge$)}
\leftline{$\tt g\;E=ge$ \quad \quad ($\tt g=.Exp.ge$)}
\leftline{$\tt g=m$ \quad \quad (all elements of g are set equal to $\tt m$)}
\leftline{$\tt g\;*(0)=m$ \quad \quad ($\tt g=g*m$)}
\leftline{$\tt g\;*(-1)=m$ \quad \quad ($\tt g=m*g$)}
\leftline{$\tt g\;*=complex$ \quad \quad ($\tt g=g*complex$)}
\leftline{$\tt g\;*=real$ \quad \quad ($\tt g=g*real$)}
\leftline{$\tt g\;/=complex$ \quad \quad ($\tt g=g/complex$)}
\leftline{$\tt g\;/=real$ \quad \quad ($\tt g=g/real$)}
\leftline{$\tt u=g$ \quad \quad ($\tt g\%fc$ is copied to
$\tt u\%uc(g\%parity,g\%dir)$ )}
\leftline{$\tt u\;t=g$ \quad \quad (same as above, but only
where $\tt context$ is $\tt .TRUE.$)}
\leftline{$\tt u\;f=g$ \quad \quad (same as two lines above, but only
where $\tt context$ is $\tt .FALSE.$)}
\leftline{$\tt u_1=u_2$}
\leftline{$\tt f\;*(0)=g$\quad \quad ($\tt f=f*g$)}
\leftline{$\tt f\;*(-1)=g$\quad \quad ($\tt f=g*f$)}
\leftline{$\tt f\;/(0)=g$\quad \quad ($\tt f=f/g$)}
\leftline{$\tt f\;/(-1)=g$\quad \quad ($\tt f=g//f$)}
(Note the function played by $\tt assign\_spec$ in the four preceding
assignments.)
\leftline{$\tt f_1=f_2$}
\leftline{$\tt f_1\;+=f_2$ \quad \quad ($\tt f_1=f_1+f_2$)}
\leftline{$\tt f_1\;-=f_2$ \quad \quad ($\tt f_1=f_1-f_2$)}
\leftline{$\tt f_1\;u(dir)=f_2$ \quad \quad ($\tt f_1=dir.Ushift.f_2$)}
\leftline{$\tt f_1\;U(dir)=f_2$ \quad \quad ($\tt f_1=f_1+(dir.Ushift.f_2)$)}
\leftline{$\tt f_1\;w(dir)=f_2$ \quad \quad ($\tt f_1=dir.Wshift.f_2$)}
\leftline{$\tt f_1\;W(dir)=f_2$ \quad \quad ($\tt f_1=f_1+(dir.Wshift.f_2)$)}
\leftline{$\tt f_1\;x(dir)=f_2$ \quad \quad ($\tt f_1=dir.Xshift.f_2$)}
\leftline{$\tt f_1\;X(dir)=f_2$ \quad \quad ($\tt f_1=f_1+(dir.Xshift.f_2)$)}
\leftline{$\tt f_1\;C=f_2$ \quad \quad ($\tt f_1=.Conjg.f_2$)}
\leftline{$\tt f_1\;I=f_2$ \quad \quad ($\tt f_1=.I.f_2$)}
\leftline{$\tt f_1\;M=f_2$ \quad \quad ($\tt f_1=.Minus.f_2$)}
\leftline{$\tt f\;*=c$ \quad \quad ($\tt f=f*c$)}
\leftline{$\tt f\;/=c$ \quad \quad ($\tt f=f/c$)}
\leftline{$\tt f\;*=r$ \quad \quad ($\tt f=f*r$)}
\leftline{$\tt f\;/=r$ \quad \quad ($\tt f=f/r$)}
\leftline{$\tt f\;*=complex$ \quad \quad ($\tt f=f*complex$)}
\leftline{$\tt f\;*=real$ \quad \quad ($\tt f=f*real$)}
\leftline{$\tt f\;/=complex$ \quad \quad ($\tt f=f/complex$)}
\leftline{$\tt f\;/=real$ \quad \quad ($\tt f=f/real$)}
\leftline{$\tt c=g$ \quad \quad ($\tt c=.Ctr.g$)}
\leftline{$\tt c_1=c_2$}
\leftline{$\tt c_1\;+=c_2$ \quad \quad ($\tt c_1=c_1+c_2$)}
\leftline{$\tt c_1\;-=c_2$ \quad \quad ($\tt c_1=c_1-c_2$)}
\leftline{$\tt c_1\;*=c_2$ \quad \quad ($\tt c_1=c_1*c_2$)}
\leftline{$\tt c_1\;/=c_2$ \quad \quad ($\tt c_1=c_1/c_2$)}
\leftline{$\tt c_1\;C=c_2$ \quad \quad ($\tt c_1=.Conjg.c_2$)}
\leftline{$\tt c_1\;M=c_2$ \quad \quad ($\tt c_1=.Minus.c_2$)}
\leftline{$\tt c_1\;I=c_2$ \quad \quad ($\tt c_1=.I.c_2$)}
\leftline{$\tt c=r$}
\leftline{$\tt c\;+=r$ \quad \quad ($\tt c=c+r$)}
\leftline{$\tt c\;-=r$ \quad \quad ($\tt c=c-r$)}
\leftline{$\tt c\;*=r$ \quad \quad ($\tt c=c*r$)}
\leftline{$\tt c\;/=r$ \quad \quad ($\tt c=c/r$)}
\leftline{$\tt c\;M=r$ \quad \quad ($\tt c=.Minus.r$)}
\leftline{$\tt c=complex$}
\leftline{$\tt c\;+=complex$ \quad \quad ($\tt c=c+complex$)}
\leftline{$\tt c\;-=complex$ \quad \quad ($\tt c=c-complex$)}
\leftline{$\tt c\;*=complex$ \quad \quad ($\tt c=c*complex$)}
\leftline{$\tt c\;/=complex$ \quad \quad ($\tt c=c/complex$)}
\leftline{$\tt c\;M=complex$ \quad \quad ($\tt c=.Minus.complex$)}
\leftline{$\tt c=real$}
\leftline{$\tt c\;+=real$ \quad \quad ($\tt c=c+real$)}
\leftline{$\tt c\;-=real$ \quad \quad ($\tt c=c-real$)}
\leftline{$\tt c\;*=real$ \quad \quad ($\tt c=c*real$)}
\leftline{$\tt c\;/=real$ \quad \quad ($\tt c=c/real$)}
\leftline{$\tt c\;M=real$ \quad \quad ($\tt c=.Minus.real$)}
\leftline{$\tt r=g$ \quad \quad ($\tt r=.Tr.g$)}
\leftline{$\tt r=f$ \quad \quad ($\tt r=f*f$)}
\leftline{$\tt r=c$ \quad \quad the elements of $\tt r$ are set equal to the
real part of the elements of $\tt c$}
\leftline{$\tt r_1=r_2$}
\leftline{$\tt r_1\;+=r_2$ \quad \quad ($\tt r_1=r_1+r_2$)}
\leftline{$\tt r_1\;-=r_2$ \quad \quad ($\tt r_1=r_1-r_2$)}
\leftline{$\tt r_1\;*=r_2$ \quad \quad ($\tt r_1=r_1*r_2$)}
\leftline{$\tt r_1\;/=r_2$ \quad \quad ($\tt r_1=r_1/r_2$)}
\leftline{$\tt r_1\;M=r_2$ \quad \quad ($\tt r_1=.Minus.r_2$)}
\leftline{$\tt r_1\;R=r_2$ \quad \quad ($\tt r_1=.Sqrt.r_2$)}
\leftline{$\tt r_1\;E=r_2$ \quad \quad ($\tt r_1=.Exp.r_2$)}
\leftline{$\tt r=real$}
\leftline{$\tt r\;+=real$ \quad \quad ($\tt r=r+real$)}
\leftline{$\tt r\;-=real$ \quad \quad ($\tt r=r-real$)}
\leftline{$\tt r\;*=real$ \quad \quad ($\tt r=r*real$)}
\leftline{$\tt r\;/=real$ \quad \quad ($\tt r=r/real$)}
\leftline{$\tt r\;M=real$ \quad \quad ($\tt r=.Minus.real$)}
\leftline{$\tt ge=g$ \quad \quad ($\tt ge=.Generator.g $)}
\leftline{$\tt ge_1=ge_2$}
\leftline{$\tt ge_1\;+=ge_2$ \quad \quad ($\tt ge_1=ge_1+ge_2$)}
\leftline{$\tt ge_1\;-=ge_2$ \quad \quad ($\tt ge_1=ge_1-ge_2$)}
\leftline{$\tt ge_1\;M=ge_2$ \quad \quad ($\tt ge_1=.Minus.ge_2$)}
\leftline{$\tt ge_1\;S=ge_2$ \quad \quad ($\tt ge_1=.Sq.ge_2$)}
\leftline{$\tt ge\;*=r$ \quad \quad ($\tt ge=ge*r$)}
\leftline{$\tt ge\;/=r$ \quad \quad ($\tt ge=ge/r$)}
\leftline{$\tt ge\;*=real$ \quad \quad ($\tt ge=ge*real$)}
\leftline{$\tt ge\;/=real$ \quad \quad ($\tt ge=ge/real$)}
The following assignments perform global reductions, either absolute
or restricted to the lattice sites where $\tt context$ is $\tt .TRUE.$
or $\tt .FALSE.$:
\leftline{$\tt complex=c$ \quad \quad
($\tt complex=\sum_{xyzt} c(xyzt)$)}
\leftline{$\tt complex\;t=c$ \quad \quad
($\tt complex=\sum_{WHERE(context(xyzt))} c(xyzt)$)}
\leftline{$\tt complex\;f=c$ \quad \quad
($\tt complex=\sum_{WHERE(.NOT.context(xyzt))} c(xyzt)$)}
\leftline{$\tt real=c$ \quad \quad
($\tt real=\sum_{xyzt} Real[c(xyzt)]$)}
\leftline{$\tt real\;t=c$ \quad \quad
($\tt real=\sum_{WHERE(context(xyzt))} Real[c(xyzt)]$)}
\leftline{$\tt real\;f=c$ \quad \quad
($\tt real=\sum_{WHERE(.NOT.context(xyzt))} Real[c(xyzt)]$)}
\leftline{$\tt real=r$ \quad \quad
($\tt real=\sum_{xyzt} r(xyzt)$)}
\leftline{$\tt real\;t=r$ \quad \quad
($\tt real=\sum_{WHERE(context(xyzt))} r(xyzt)$)}
\leftline{$\tt real\;f=r$ \quad \quad
($\tt real=\sum_{WHERE(.NOT.context(xyzt))} r(xyzt)$)}
\subsection{Module shifts}
\label{shifts}
This module defines the operators $\tt .Cshift.$, $\tt .Ushift.$,
$\tt .Wshift.$ and $\tt .Xshift.$ The left operand for all these
operators is an integer $\tt m$ which must take one of the values
$\tt 1,2,3,4,-1,-2,-3,-4$ and specifies the direction and
orientation of the shift. The right operand can be any variable of
field type for $\tt .Cshift.$. It can be any variable of field type with the
exception of the type generator\_field for $\tt .Ushift.$, while it must be
a variable of type fermi\_field for $\tt .Wshift.$ and $\tt .Xshift.$
The $\tt parity$
component of the right operand must be defined, i.e.~take value $\tt 0$
or $\tt 1$. If the $\tt parity$ is not defined or if $\tt m$ does
not take one of the values specified above the function call
implementing the operator returns an error message and stops the program.
All of these operators return a field variable of the same type as
the left operand and opposite $\tt parity$. If the right operand
is of the type gauge\_field or generator\_field and thus has also
a $\tt dir$ component, this is passed to the result unchanged.
$\tt .Cshift.$ implements an ordinary C-shift of the field, but with
respect to the Cartesian geometry of the lattice. This is why the
parity is interchanged. Given a site with Cartesian coordinates
$\bf x$ in the sublattice of the parity of the result, the operator
copies into the corresponding element of the result the element of the
right operand which is defined over the lattice site ${\bf x} +s \hat\mu$,
where $s$ is the sign of $\tt m$ and $\mu = {\rm abs}({\tt m})$.
$\tt .Ushift.$ moves the data in a manner similar to $\tt .Cshift.$,
but with the inclusion of the appropriate transport factors, defined
in terms of the global field $\tt U$ (cf.~``global module'').
For the variable of type gauge\_field and fermi\_field the U-shift
operation has been defined in Sect.~\ref{thenotion} (cf.~Eqs.~(\ref{shiftv}),
(\ref{negshiftv}), and ~Eqs.~(\ref{shiftfm}), (\ref{negshiftfm}),
for gauge fields and Fermi fields, respectively).
The action of a U-shift on variables
of type complex\_field and real\_field reduces to a C-shift.
The U-shift of a generator field is not normally encountered in
QCD simulations and for this reason it is not explicitly implemented here.
It can be implemented by using the functionality provided by the
module generator\_algebra to re-express the generator field as a field
of hermitian matrices (i.e.~of type gauge\_field), shifting the latter,
and converting it again to a generator\_field.
The operator $\tt .Wshift.$ acts only on Fermi fields and it is a
combination of a $\tt .Ushift.$ and the product with a $\gamma$
matrix. Precisely, if we again define $s$ to be the sign of $\tt m$
and define $\mu$ or $\tt mu$ to be the absolute value of $\tt m$,
then the operation $\tt f2=m.Wshift.f1$ is equivalent to
\leftline{\tt f2=m.Ushift.f1-s*(mu.Gamma.(m.Ushift.f1))}
Equivalently
\begin{equation}
f_{2,{\bf x}}= (1-\gamma_{\mu})U^{\mu}_{{\bf x}}
f_{1,{\bf x}+\hat\mu}
\label{pwilson}
\end{equation}
for positive $\tt m$, and
\begin{equation}
f_{2,{\bf x}}= (1+\gamma_{\mu})
U^{\dagger \mu}_{{\bf x}-\hat\mu} f_{1,{\bf x}-\hat\mu}
\label{mwilson}
\end{equation}
for negative $\tt m$.
The operator $\tt .Xshift.$ also acts only on Fermi fields and it is
equivalent to a W-shift bracketed by two matrices $\gamma_5$
(where $\gamma_5=\gamma_1 \gamma_2 \gamma_3 \gamma_4$):
\begin{equation}
f_{2,{\bf x}}= \gamma_5 (1-\gamma_{\mu}) \gamma_5 U^{\mu}_{{\bf x}}
f_{1,{\bf x}+\hat\mu}
\label{pxwilson}
\end{equation}
for positive $\tt m$, and
\begin{equation}
f_{2,{\bf x}}= \gamma_5 (1+\gamma_{\mu}) \gamma_5
U^{\dagger \mu}_{{\bf x}-\hat\mu} f_{1,{\bf x}-\hat\mu}
\label{mxwilson}
\end{equation}
for negative $\tt m$.
The reason for the explicit introduction of the W-shift and X-shift
operators is that the combinations (\ref{pwilson}-\ref{mxwilson})
appear in the Wilson
discretization of the Dirac operator, which is a very
widely used lattice Dirac operator, and related algebra,
and that the calculation of the l.h.s.~of Eqs.~(\ref{pwilson}-\ref{mxwilson})
is one of the most
time consuming tasks of any QCD simulation. Moreover, the combinations
$1 \pm \gamma_{\mu}$ appearing in (\ref{pwilson}-\ref{mxwilson})
are projection operators, which effectively limits the U-shift to
a subspace of the spin space of dimensionality two. Thus a direct
implementation of the W-shift, rather than via a combination of
the $\tt .Ushift.$ and $\tt .Gamma.$ operators, entails substantial
advantages of efficiency.
\subsection{Module Dirac\_operator}
\label{diracop}
The (Wilson) lattice Dirac operator, acting on a Fermi field
$f_1$, produces a Fermi field
$f_2$, given by
\begin{equation}
f_{2,{\bf x}}= \sum_{\mu} [(1-\gamma_{\mu})U^{\mu}_{{\bf x}}
f_{1,{\bf x}+\hat\mu} + (1+\gamma_{\mu})
U^{\dagger \mu}_{{\bf x}-\hat\mu} f_{1,{\bf x}-\hat\mu}]
\label{dirac}
\end{equation}
It is obvious from this equation that the lattice Dirac operator
only connects components of Fermi fields of opposite parity.
The unary operator $\tt .Dirac.$ accepts as operand a variable of
type fermi\_field, which must have a definite $\tt parity$,
and returns a variable of the same type and opposite $\tt parity$
given by the action of the lattice Dirac operator~(\ref{dirac})
on the operand. The unary operator $\tt .XDirac.$ implements the
action of the Dirac operators bracketed by two matrices $\gamma_5$,
i.e.~, if $\tt f$ is a variable of type fermi\_field,
$\tt .XDirac.f$ returns the same results as
$\tt i5.Gamma.(.XDirac.(i5.Gamma.f))$, where the integer variable
$\tt i5$ equals 5.
In this module the operators $\tt .Dirac.$ and $\tt .XDirac.$ are
implemented using the operators $\tt .Wshift.$ and $\tt .Xshift.$,
introduced in the module field\_algebra. We have defined them
as separate operators for convenience of coding and
also because, the application of these operators being the most
CPU intensive part for the majority of applications, this module
isolates the code whose optimization would produce the largest
returns. A programmer striving for exceptional efficiency might
want to code this module as a highly optimized, self-standing
implementation of the lattice Dirac operator. Even if this route
is chosen, we are certain that the advantages of having a module
written at a higher level against which to compare the results
of the optimized module are not lost on the practicing
computational scientist.
\subsection{Module generator\_algebra}
\label{generatoralgebra}
This module defines the unary operators $\tt .Matrix.$, $\tt .Generator.$,
$\tt .Sq. $ and $\tt .Exp.$ which perform some special operations
involving generator fields. The operators accept arguments of the
type generator\_field or gauge\_field and return as result a variable
of one of these types. The $\tt parity$ and $\tt dir$ components of the
argument are passed on to the result.
$\tt .Matrix.$ accepts an argument $\tt ge$ of type generator\_field
and returns a result $\tt v$ of type gauge\_field containing, site by site,
the Hermitian matrix
\begin{equation}
v_{ij,{\bf x}} =\sum_k \lambda^k_{ij}\, ge_{k,{\bf x}} \ .
\label{matrix}
\end{equation}
$\tt .Generator.$ accepts an argument $\tt v$ of the type gauge\_field
and returns a result $\tt ge$ of type generator\_field containing,
site by site, the traceless, antihermitian part of the argument:
\begin{equation}
ge_{k,{\bf x}} = -{\imath \over 4} \sum_{ij}\lambda^k_{ij}\, (v_{ji,\bf x}
-v^*_{ij,\bf x}) \ .
\label{gen}
\end{equation}
Notice that $\tt .Generator.$ and $\tt .Matrix.$ are not inverse
operators. However $\tt .Generator.(.Matrix.(IU*ge))$ does return $\tt ge$,
while $\tt .Matrix.(.Generator.v)$ returns the traceless,
antihermitian part of $\tt v$: $v_{\scriptscriptstyle{AH}}=
(v-v^{\dagger}) / (2 \imath)$.
$\tt .Sq.$ accepts an argument of type generator\_field $\tt ge_1$
and returns a result $\tt ge_2$ of the same type containing, site by site,
the generator corresponding to the traceless part of the square
of $\tt .Matrix.ge1$:
\begin{equation}
(ge_2)_{l,{\bf x}} =
{ 1 \over 2} \sum_{ijk} \big[ \lambda^l_{ij} \,
\big( \sum_m \lambda^m_{jk}\, (ge_1)_{ m,{\bf x}} \big) \,
\big( \sum_n \lambda^n_{ki}\, (ge_1)_{ n,{\bf x}} \big)
\big]\ .
\label{sq}
\end{equation}
$\tt .Exp.$ accepts an argument $\tt ge$ of type generator\_field
and returns a result $\tt v$ of type gauge\_field containing, site by site,
the exponentiated generator component:
\begin{equation}
v_{ij,{\bf x}} =\big[ \exp \big(\imath \sum_k \lambda^k\, ge_{k,{\bf x}}\big)
\big]_{ij} \ .
\label{exp}
\end{equation}
The algorithm used for this exponentiation deserves a few words of explanation.
Let us define $H=\sum_k \lambda^k_{ij}\, ge_{k,{\bf x}}$, $q={\rm Tr} H^2$
and $p={\rm Det} H = ({\rm Tr} H^3)/3$. From the characteristic
equation (recall that ${\rm Tr} H =0$)
\begin{equation}
H^3 - {q \over 2} H - p I =0 \ ,
\label{charact}
\end{equation}
$I$ being the identity matrix, satisfied by $H$ and
therefore by its eigenvalues $h_n$, we can
easily calculate the eigenvalues as
\begin{equation}
h_n =a \cos[\alpha + 2 \pi (n-1)/3], \quad n=1,2,3 \ ,
\label{eigenval}
\end{equation}
with $a=\sqrt{2 q / 3}$,$\; \alpha = [\cos^{-1}(4 p / a^3)]/3$.
We order the eigenvalues so that $|h_1| \ge |h_2| \ge |h_3|$.
In a basis where $H$ is diagonal, it is easy to express it as
a linear combination of two matrices of type ``$\lambda^3\,$''
(one 0 eigenvalue) and ``$\lambda^8\,$'' (two degenerate eigenvalues),
respectively. Of the six different ways in which this can be done
we use the decomposition
\begin{equation}
H = S + K
\label{spk}
\end{equation}
with $S = {\rm diag} (-h_1-2 h_2, h_1+2 h_2, 0)$,
$K = {\rm diag} (2(h_1+ h_2), -h_1-h_2, -h_1-h_2)$.
By using the eigenvalues determined above, it is
straightforward to express $S$ in the form
\begin{equation}
S = c_1 H + c_2 (H^2 - q I /3) \ .
\label{scc}
\end{equation}
Through their dependence on the
eigenvalues and Eq.~(\ref{eigenval}), however, $c_1$ and $c_2$ are
functions of the invariants $q$ and $p$ only. It follows that
Eqs.~(\ref{scc}) and (\ref{spk}) provide a decomposition into
two matrices of type ``$\lambda^3\,$'' and ``$\lambda^8\,$'' irrespective
of the basis. On the other hand, with a matrix
of type ``$\lambda^3\,$'' it is straightforward to calculate
$\exp(\imath S)$ expressing it as a linear combination
of $I$, $S$ and $S^2$. Similarly $\exp(\imath K)$ can be expressed
as a linear combination of $I$ and $K$. $\exp (\imath H)$ can be
finally calculated as product of the two commuting
matrices $\exp (\imath S)$ and $\exp (\imath K)$.
It is very important to have an efficient algorithm for the
exponentiation of a matrix, since this operation can be a time
consuming component of several QCD calculations. The algorithm
outlined above has been implemented in the module
``generator\_algebra'' by performing a substantial amount of the
algebra directly in terms of generator components and inlining all of
the operations. The exponentiation can thus be done with a
reasonably contained number of arithmetic operations, in particular
approximately 300 explicit real multiplications. By way of
comparison, just one product of $3 \times 3$ complex matrices requires
108 real multiplications (i.e.~27 complex multiplications -- these could
also be performed with 81 real multiplications, but then with a much larger
number of adds).
\subsection{Module random\_numbers}
\label{randomnumbers}
This module implements a parallelizable version of the unix pseudorandom
number generator erand48 and provides added functionality.
erand48 is a congruential pseudorandom number generator based on the
iterative formula
\begin{equation}
s_{i+1}=a_1*s_i+b_1 \quad {\rm mod} \; m \ ,
\label{erand}
\end{equation}
where $a_1=\tt 0x5DEECE66D$, $b_1=\tt 0xB$, $m=2^{48}$, $s_i$
and $s_{i+1}$ are integers of at least 48 bits of precision.
The ``seeds'' $s_i$ are converted
to real pseudorandom numbers $r_i$ with uniform distribution
between $0$ and $1$ by $r_i=2^{-48}\, s_i$.
As presented above, the algorithm is intrinsically serial. However it
follows from Eq.~(\ref{erand}) that the $\rm N^{th}$ iterate $s_{i+N}$
is still of the form $s_{i+N}=a_N*s_i+b_N \; {\rm mod} \; 2^{48}$
with integers
$a_N$ and $b_N$ which are uniquely determined by $a_1$, $b_1$.
The module takes advantage of this fact and of the existence of
the global variable $\tt seeds$ (cf.~global\_module) to generate
pseudorandom numbers in a parallelizable fashion.
The module defines the following unary operators: $\tt .Seed.$,
$\tt .Rand.$, $\tt .Gauss.$ and $\tt .Ggauss.$.
$\tt .Seed.$ must be used to initialize the generation of pseudorandom
numbers. When invoked with an argument $\tt saved\_seed$ of kind
$\tt LONG$ (8-byte integer, defined in the module precisions)
$\tt .Seed.$ sets the global variable $\tt seeds$ to the
sequence~(\ref{erand})
beginning with $\tt saved\_seed$ and also sets the global variables
$\tt seed\_a$, $\tt seed\_b$ to the appropriate multiplier and constant
term, $a_N$ and $b_N$, needed to produce increments by $N=NX*NY*NZ*NT/2$
in the sequence of pseudorandom numbers. It returns $\tt .TRUE.$.
When acting on a logical variable equal to $\tt .TRUE.$, $\tt .Seed.$
returns the current seed (=$\tt seeds(0,0,0,0)$), which must
be used to restart the sequence of pseudorandom numbers.
If the argument is $\tt .FALSE.$, $\tt .Seed.$ returns $0$.
The unary operator $\tt .Rand.$, if invoked with a real argument $\tt real$
of kind $\tt REAL8$, returns a real\_field of pseudorandom numbers
with uniform distribution between 0 and $\tt real$. At the same time
it upgrades the global variables $\tt seeds$ using the multiplier $a_N$
(i.e.~$\tt seed\_a$) and constant term $b_N$ (i.e.~$\tt seed\_b$).
It follows that subsequent calls to $\tt .Rand.$ produce real fields
with the same distribution of pseudorandom numbers which one would
have obtained invoking erand48 sequentially within nested DO loops:
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO xyzt=0,NXYZT2-1}
\leftline{...}
}
\leftline{The parity of the results is undefined.}
If $\tt .Rand.$ has an argument of type real\_field, it returns
a real\_field of pseudorandom numbers uniformely distributed between
0 and the corresponding component of the argument. The parity of
the result is the same as the parity of the argument.
The unary operator $\tt .Gauss.$ returns a real field of
pseudorandom numbers with gaussian distribution
of mean square deviation equal to the argument of
$\tt .Gauss.$ and upgrades the global variable $\tt seeds$.
The argument can again be a variable of kind $\tt REAL8$
or of type real\_field and the parity of the result is undefined
or equal to the parity of the argument, respectively.
The unary operator $\tt .Ggauss.$ works like $\tt .Gauss.$ but fills with
gaussian random numbers the components of a generator\_field,
setting its direction equal to 0. Precisely, the instruction
$\tt ge=.Ggauss.r $, although in the module it is implemented
differently, would be equivalent to
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO i=1,8}
\leftline{\quad auxr=.Gauss.r}
\leftline{\quad ge\%fc(i,:,:,:,:)=auxr\%fc}
\leftline{END DO}
\leftline{ge\%parity=auxr\%parity}
\leftline{ge\%dir=0}
}
\leftline{where $\tt auxr$ is a variable of type real\_field}
This module assumes the availability of long (8-byte) integers and the fact
that a multiplication of long integers will return the lowest 8 bytes of the
product (i.e. $a*b\; {\rm mod}\; 2^{64}$) without producing an arithmetic error
when the product exceeds the maximum long integer. If these assumptions
are not satisfied, the module should be replaced with some other suitable
module. Also, we would like to point out that the algorithm of
Eq.~(\ref{erand}) produces pseudorandom numbers of reasonably good quality
and period ($ \approx 10^{14}$). However, a computer capable of
100 Gflops sustained running a program that makes use of one pseudorandom
number every thousand floating point operations would exhaust the
whole set of pseudorandom numbers in one million seconds, which is not
a very long time. Thus for calculations that are very computer intensive
or which demand pseudorandom numbers of exceptionally good
quality, the module should be modified to meet the more
stringent demands. Two improvements which can be made with
minimal changes would consist in the use of a larger $m$
in Eq.~(\ref{erand}) (with appropriate $a_1$ and $b_1$) and/or of
a reshuffle of the pseudorandom numbers produced by the algorithm.
Of course, one could also make use of the F90 RANDOM\_NUMBER
subroutine, but the results would no longer be machine independent.
\subsection{Module conditionals}
\label{conditionals}
This module defines six overloaded relational operators, $\tt >$, $\tt >=$,
$\tt < $, $\tt <= $, $\tt == $, $\tt /=$, and the operator $\tt .Xor.$
The relational operators take as operands two real\_fields or one
real\_field and one real variable of kind $\tt REAL8$.
They return a logical variable which is set to $\tt .TRUE.$ if
the two fields have the same (defined) parity or if the single
field operand has defined parity, and is set to $\tt .FALSE.$ otherwise.
At the same time the global variable $\tt context$
is set to $\tt .TRUE.$ at all sites where the relation is satisfied,
to $\tt .FALSE.$ at all other sites. For example, the function
implementing the relational operator $\tt a>b$, with
$\tt a$ and $\tt b$ of type real\_field, could contain a line:
$\tt\; context=a\%fc>b\%fc \;$, which produces the action mentioned above.
The operator $\tt .Xor.$ accepts as operands a pair of fields
of the same type and returns a field, also of the same type,
having as elements the corresponding elements of the first operand
at the sites where the global variable $\tt context$ is $\tt .TRUE.$,
the elements of the second operand at the sites where $\tt context$
is $\tt .FALSE.$. For clarification, the function $\tt g\_xor\_g$
implementing the operation $\tt g1 .Xor. g2$, where $\tt g1$
and $\tt g2$ are fields of type gauge\_field, would contain
the code
\vskip 4mm
{\baselineskip 5mm \tt
\leftline{DO j=1,3}
\leftline{DO i=1,3}
\leftline{\quad WHERE(context)}
\leftline{\quad \quad g\_xor\_g\%fc(i,j,:,:,:,:)=g1\%fc(i,j,:,:,:,:)}
\leftline{\quad ELSEWHERE}
\leftline{\quad \quad g\_xor\_g\%fc(i,j,:,:,:,:)=g2\%fc(i,j,:,:,:,:)}
\leftline{\quad END WHERE}
\leftline{END DO}
\leftline{END DO}
}
The $\tt parity$ of the field returned by $\tt .Xor.$ is the common
parity of the two operands if they have the same $\tt parity$,
otherwise it is undefined. In addition, for operands of type
gauge\_field, the $\tt dir$ component of the returned field is the
common $\tt dir $ of the operands if they have the same $\tt dir$,
otherwise it is set to 0.
The operators provided by the module ``conditionals'' can be very
conveniently used to select elements out of two fields according
to some local condition, an operation which lies at the foundation
of stochastic simulation techniques.
\subsection{Subroutine write\_conf and read\_conf}
\label{wrconf}
The file ``write\_read\_conf.f90'' contains two subroutines which
serve to store and retrieve an entire SU(3) gauge field configuration,
written in a portable, compressed ASCII format. Only the first two
columns of the gauge field matrices are stored, because
the third one can be recovered from the unitarity and unimodularity
constraints. The write\_conf subroutine takes advantage of the fact
that all of the elements of the gauge field matrices have magnitude
smaller or equal to 1 to re-express their real and imaginary parts
as 48bit integers. These integers are then written in base 64,
with the digits being given by the ASCII collating sequence
starting from 0. Thus, 8 characters are needed to represent either
the real or the imaginary part of the gauge field matrix elements
and an entire gauge field matrix is represented by 96 ASCII characters,
without loss of numerical information. A detailed description of
the contents of the file generated by write\_conf and of the
standard used for coding the information is written, as a header,
at the beginning of the file itself. This makes the file with
the compressed gauge configuration portable and usable, irrespective
of the availability of the write\_conf and read\_conf subroutines or
of a separate description of their implementation.
\section{Example code}
\label{excode}
In order to illustrate the power of the modules we developed, we
present here two programs which implement a multihit Metropolis
simulation of quenched QCD and the calculation of a quark propagator.
Anybody familiar with the complexity of the programs for
lattice QCD simulations will appreciate the conciseness of our
examples. It is also to be noticed that a large amount of the
code in the programs performs peripheral functions, such as
initialization and printout. If we consider the Metropolis
simulation program, for instance, the section of the code which
performs the actual upgrading steps consists of only 28 lines!
It is our hope that researchers interested in using our modules
will find it easy to become familiar with their functionality
and that, not being hindered by inessential programming burdens,
they will thus be able to make much faster progress in the development
of far-reaching QCD applications.
\subsection{quenched.f90}
{\chardef \other = 12
\def\deactivate{%
\catcode `\\ = \other \catcode`\{ = \other
\catcode `\} = \other \catcode`\$ = \other
\catcode `\& = \other \catcode`\# = \other
\catcode `\% = \other \catcode`\~ = \other
\catcode `\^ = \other \catcode`\_ = \other
}
\def\makeactive#1{\catcode`#1 = \active \ignorespaces}
{%
\makeactive\^^M %
\gdef\obeywhitespace{%
\makeactive\^^M %
\let^^M = \par\indent %
\aftergroup\setbox0=\lastbox %
\obeyspaces %
}%
}
\def\par\indent{\par\indent}
\def\setbox0=\lastbox{\setbox0=\lastbox}
\def\verbatim{\par\begingroup\deactivate\obeywhitespace\tt\parindent = 0in
\baselineskip=5mm \catcode `\| = 0 %
}
\def\endgroup{\endgroup}
\def\|{|}
\verbatim
! Program Qcdf90_quenched
! Copyright by Indranil Dasgupta, Andrea R. Levi, Vittorio Lubicz
! and Claudio Rebbi - Boston University - May 1996
! This program may be freely copied and used as long as this notice
! is retained.
PROGRAM Qcdf90_quenched
USE precisions
USE constants
USE global_module
USE field_algebra
USE generator_algebra
USE conditionals
USE shift
USE random_numbers
USE assign_mixed
USE assign_isotype1
IMPLICIT NONE
TYPE(gauge_field):: staple,g_old,g_new
TYPE(real_field):: plaq_old,plaq_new,bf_ratio,rand
TYPE(generator_field):: ge
TYPE(matrix) :: zero_matrix, unit_matrix
LOGICAL l_test,l_seed
REAL(REAL8) clock_dcl,clock_upd,clock_plaq
REAL(REAL8) beta,saved_beta,hp,av_plaq,aux,range_small,range_unit
CHARACTER(LEN=64) in_filename,out_filename
CHARACTER(LEN=16) id
INTEGER(LONG) saved_seed,inp_seed
INTEGER clock_rate,clock_1,clock_2
INTEGER hotcoldread,save,num_upd,p,m,sign,nu,i,hit,num_hit
! input variables:
WRITE (*,'("Lattice size: ",4I5)') NX,NY,NZ,NT
WRITE (*,'("Enter beta: ")',ADVANCE='NO')
READ *,beta
WRITE (*,'("Enter number of updates: ")',ADVANCE='NO')
READ *,num_upd
WRITE (*,'("Select the starting configuration. Enter 0 for&
&a hot start ")')
WRITE (*,'("1 for a cold start, 2 to read from Disk: ")',ADVANCE='NO')
READ *,hotcoldread
! other useful variables:
num_hit=6 ! number of Metropolis multiple hits
range_unit=1._REAL8 ! unitary range for the random numbers
range_small=0.1_REAL8 ! range for the random numbers
inp_seed=1 ! input seed for random numbers generator
zero_matri
in_filename= 'configuration.in'
out_filename='configuration.out'
! initializing system clock
CALL SYSTEM_CLOCK(clock_1,clock_rate)
clock_dcl=1._REAL8/clock_rate
clock_upd=0._REAL8
clock_plaq=0._REAL8
! initializing random generator and gauge configuration
SELECT CASE(hotcoldread)
CASE(0)
l_seed=.Seed.inp_seed
DO p=0,1
DO m=1,4
ge=.Ggauss.range_unit
END DO
END DO
CASE(1)
l_seed=.Seed.inp_seed
unit_matri
DO p=0,1
DO m=1,4
END DO
END DO
CASE(2)
CALL read_conf(saved_beta,id,hp,saved_seed,in_filename)
IF(inp_seed==0) THEN
WRITE (*,'("saved_seed=",I15)') saved_seed
l_seed=.Seed.saved_seed
ELSE
l_seed=.Seed.inp_seed
WRITE (*,'("seed re-initialized")')
ENDIF
CASE DEFAULT
WRITE (*,'("hotcoldread must only be 0,1 or 2")')
STOP
END SELECT
DO i=1,num_upd ! Main Loop
! Metropolis update
CALL SYSTEM_CLOCK(clock_1)
DO p=0,1
DO m=1,4
! Staple
staple=zero_matrix
stapl
stapl
DO nu=1,4
IF(nu.EQ.m) CYCLE
DO sign=-1,1,2
staple=staple+((nu*sign).Ushift.
END DO
END DO
g_old=
DO hit=1,num_hit
plaq_old=g_old.Dot.staple
ge=.Ggauss.range_small
g
g
g_new=(.Exp.ge)*g_old
plaq_new=g_new.Dot.staple
bf_ratio=.Exp.(beta/3._REAL8*(plaq_new-plaq_old))
rand=.Rand.range_unit
l_test=rand<bf_ratio
assign_type='t'; g_old=g_new
END DO
END DO
END DO
CALL SYSTEM_CLOCK(clock_2)
clock_upd=clock_upd+(clock_2-clock_1)*clock_dcl
! Plaquette
CALL SYSTEM_CLOCK(clock_1)
av_plaq=0._REAL8
DO p=0,1
DO m=1,3
DO nu=m+1,4
aux=
av_plaq=av_plaq+aux
END DO
END DO
END DO
av_plaq=av_plaq/REAL(18*NXYZT,REAL8)
CALL SYSTEM_CLOCK(clock_2)
clock_plaq=clock_plaq+(clock_2-clock_1)*clock_dcl
WRITE (*,'("iteration ",I5," av. plaq.= ",F10.6)') i,av_plaq
END DO ! End Main Loop
! Save configuration on disk
WRITE (*,'("Save configuration on disk ? (Yes=1, &
&No=0): ")',ADVANCE='NO')
READ *,save
IF(save==1) THEN
WRITE (*,'("saving the configuration")')
id='conf 0.0.0'
hp=0.0
l_seed=.TRUE.
saved_seed=.Seed.l_seed
WRITE (*,'(" saved_seed = ",I15)') saved_seed
CALL write_conf(beta,id,hp,saved_seed,out_filename)
ENDIF
! Print timing
WRITE (*,'("Av. upgrade time in microsecs per link",F9.3)') &
(1000000*clock_upd)/(4*NXYZT*num_upd)
WRITE (*,'("Av. measure time in microsecs per plaquette",F9.3)')&
(1000000*clock_plaq)/(6*NXYZT*num_upd)
END
|endverbatim
\subsection{propagator.f90}
\verbatim
! Program Qcdf90_propagator
! Copyright by Indranil Dasgupta, Andrea R. Levi, Vittorio Lubicz
! and Claudio Rebbi - Boston University - January 1996
! This program may be freely copied and used as long as this notice
! is retained.
PROGRAM Qcdf90_propagator
USE precisions
USE constants
USE global_module
USE field_algebra
USE generator_algebra
USE conditionals
USE shift
USE dirac
USE random_numbers
USE assign_mixed
USE assign_isotype1
USE assign_isotype2
IMPLICIT NONE
TYPE(fermi_field):: psi,chi,grad,h,m_h,mp_m_h
REAL(REAL8) clock_dcl,clock_cg
REAL(REAL8) kappa,tolerance,residue,saved_beta,hp
REAL(REAL8) alpha,old_alpha,g_2,g_old_2,beta_cg,old_beta_cg
REAL(REAL8) h_a_h,norm_psi
CHARACTER(LEN=64) in_filename
CHARACTER(LEN=16) id
INTEGER(LONG) saved_seed
INTEGER clock_rate,clock_1,clock_2
INTEGER iter,nsteps,niter,init_niter,stop_flag,init_stop_flag
INTEGER i,xyzt,s
! input variables:
WRITE (*,'("Enter kappa: ")',ADVANCE='NO')
READ *,kappa
WRITE (*,'("Enter max numbers of cg steps: ")',ADVANCE='NO')
READ *,nsteps
WRITE (*,'("Enter tolerance: ")',ADVANCE='NO')
READ *,tolerance
! the conjugated gradient will
! run until the residue<tolerance
! or for a maximum of nsteps
! other useful variables:
in_filename= 'configuration.in'
init_stop_flag=2
init_niter=4
! gauge configuration is read from the disk
CALL read_conf(saved_beta,id,hp,saved_seed,in_filename)
! the source chi (in the even sites) is set arbitrarly in this example
DO i=1,3
DO s=1,4
DO xyzt=0,NXYZT2-1
ch
END DO
END DO
END DO
ch
! psi must be initialized as the starting trial configuration.
! the simplest choice is psi=chi
psi=chi
! initializing system clock
CALL SYSTEM_CLOCK(clock_1,clock_rate)
clock_dcl=1._REAL8/clock_rate
! Calculate psi as the solution of: M*psi=chi,
! where M is the fermion matrix, and chi is a given source.
! The residue is printed to monitor the convergence.
stop_flag=init_stop_flag
niter=init_niter
iter=0
m_h=psi-(kappa**2)*(.Dirac.(.Dirac.psi))
mp_m_h=m_h-(kappa**2)*(.Xdirac.(.Xdirac.m_h))
grad=chi-mp_m_h
g_2=grad*grad
h=grad
norm_psi=psi*psi
residue=g_2/norm_psi
WRITE (*,'("residue= ",F20.16," at step:",I5)') residue,iter
old_alpha=0._REAL8
old_beta_cg=1._REAL8
DO iter=1,nsteps
m_h=h-(kappa**2)*(.Dirac.(.Dirac.h))
h_a_h=m_h*m_h
beta_cg=g_2/h_a_h
psi=psi+beta_cg*h
norm_psi=psi*psi
IF(mod(iter,niter)==0 .AND. g_2/norm_psi<tolerance) THEN
stop_flag=stop_flag-1
m_h=psi-(kappa**2)*(.Dirac.(.Dirac.psi))
mp_m_h=m_h-(kappa**2)*(.Xdirac.(.Xdirac.m_h))
grad=chi-mp_m_h
g_2=grad*grad
h=grad
g_old_2=g_2
residue=g_2/norm_psi
WRITE (*,'("residue= ",F20.16," at step:",I5)') residue,iter
IF(stop_flag == 0) EXIT
ELSE
mp_m_h=m_h-(kappa**2)*(.Xdirac.(.Xdirac.m_h))
grad=grad-beta_cg*mp_m_h
g_old_2=g_2
g_2=grad*grad
alpha=g_2/g_old_2
h=grad+alpha*h
norm_psi=psi*psi
residue=g_2/norm_psi
IF(mod(iter,niter) == 0) THEN
WRITE (*,'("residue= ",F20.16," at step:",I5)') residue,iter
ENDIF
old_beta_cg=beta_cg
old_alpha=alpha
END IF
END DO
CALL SYSTEM_CLOCK(clock_2)
clock_cg=(clock_2-clock_1)*clock_dcl
!test solution:
m_h=psi-(kappa**2)*(.Dirac.(.Dirac.psi))
mp_m_h=m_h-(kappa**2)*(.Xdirac.(.Xdirac.m_h))
grad=chi-mp_m_h
norm_psi=psi*psi
g_2=grad*grad
residue=g_2/norm_psi
WRITE (*,'("final residue= ",F20.16)') residue
! Print timing
WRITE (*,'("Cg time per iteration per link in microsecs",F9.3)') &
(1000000*clock_cg)/(iter*4*NXYZT)
END
|endverbatim
\subsection{Compilation and sample run output}
The code has been tested and compiled on a SGI PowerChallengeArray
with 90 MHz processor nodes, using IRIX 6.1 Fortran 90,
with a single processor -O3 optimization flags or
with the flags -O3 -pfa -mp to implement multiprocessing;
on a SGI Indigo using the IRIX 6.1 Fortran 90;
and on the IBM R6000 58H model 7013 at 55 MHz,
with the xlf90 IBM compiler using the -O3 optimization flags.
The run of the example programs produce the following
outputs when running on a single processor of
the SGI PowerChallengeArray.
{\it Output of quenched.f90}
\verbatim
Lattice size: 8 8 8 8
Enter beta: 6.0
Enter number of updates: 15
Select the starting configuration. Enter 0 for a hot start
1 for a cold start, 2 to read from Disk: 1
iteration 1 av. plaq.= 0.849923
iteration 2 av. plaq.= 0.773278
iteration 3 av. plaq.= 0.727001
iteration 4 av. plaq.= 0.699791
iteration 5 av. plaq.= 0.677709
iteration 6 av. plaq.= 0.664358
iteration 7 av. plaq.= 0.654980
iteration 8 av. plaq.= 0.645880
iteration 9 av. plaq.= 0.638568
iteration 10 av. plaq.= 0.635049
iteration 11 av. plaq.= 0.631868
iteration 12 av. plaq.= 0.628131
iteration 13 av. plaq.= 0.624450
iteration 14 av. plaq.= 0.621757
iteration 15 av. plaq.= 0.619540
Save configuration on disk ? (Yes=1, No=0): 1
saving the configuration
saved_seed = 182618478903297
Av. upgrade time in microsecs per link 275.533
Av. measure time in microsecs per plaquette 8.624
|endverbatim
{\it Output of propagator.f90}
\verbatim
Enter kappa: 0.155
Enter max numbers of cg steps: 2000
Enter tolerance: 1.e-14
residue= 0.2417309784323778 at step: 0
residue= 0.0047398663491857 at step: 4
residue= 0.0007379739612745 at step: 8
residue= 0.0001969957469930 at step: 12
residue= 0.0000640891236947 at step: 16
residue= 0.0000235032623593 at step: 20
residue= 0.0000079006298388 at step: 24
residue= 0.0000030559375109 at step: 28
residue= 0.0000014286464825 at step: 32
residue= 0.0000006891018149 at step: 36
residue= 0.0000003502268064 at step: 40
residue= 0.0000002296348520 at step: 44
residue= 0.0000001023143421 at step: 48
residue= 0.0000000357544848 at step: 52
residue= 0.0000000114945632 at step: 56
residue= 0.0000000034230144 at step: 60
residue= 0.0000000010030646 at step: 64
residue= 0.0000000005153800 at step: 68
residue= 0.0000000003507320 at step: 72
residue= 0.0000000002150572 at step: 76
residue= 0.0000000000880532 at step: 80
residue= 0.0000000000289906 at step: 84
residue= 0.0000000000095210 at step: 88
residue= 0.0000000000042035 at step: 92
residue= 0.0000000000016253 at step: 96
residue= 0.0000000000004389 at step: 100
residue= 0.0000000000001207 at step: 104
residue= 0.0000000000000298 at step: 108
residue= 0.0000000000000066 at step: 112
residue= 0.0000000000000022 at step: 116
final residue= 0.0000000000000022
Cg time per iteration per link in microsecs 21.548
|endverbatim
}
\section*{Acknowledgments}
This research was supported in part under DOE grant DE-FG02-91ER40676.
We are grateful to the Center of Computational Science and the Office
of Information Technology for support and access to the Boston
University supercomputer facility.
V.L. acknowledges the support of an INFN post-doctoral fellowship.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,789
|
The Portal uses the open source application VuFind to index and search the metadata records describing members' rare, unique, and uncommon materials. The process begins with the acquisition of MARC and/or EAD files from participating institutions. VuFind's underlying indexer, Solr/Lucene, is then used to index all records. Once indexed, the system supports free text and field-specific searches. Results are displayed with cover art, when available, and provide a number of Web 2.0 features such as facet browse, "Add to my favorites," and "Make a comment."
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,181
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Caleb Bak grew up in Columbia Heights, Minnesota.
At Concordia Academy, Caleb Bak earned All-Tri Metro Conference honors as a junior and senior. He was named All-Metro by the Minneapolis Star Tribune in 2009 and honorable mention Associated Press all-state selection. Caleb Bak notched 71 total tackles, including 31 solo tackles as a senior.
In 2010, it will be Caleb Bak's freshman season with Minnesota. Caleb redshirted during the entire 2010 season.
Order your Minnesota Golden Gopher football tickets today and help cheer on Caleb Bak at TCF Bank Stadium.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,078
|
<?xml version="1.0" encoding="UTF-8"?>
<beans xmlns="http://www.springframework.org/schema/beans"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://www.springframework.org/schema/beans
http://www.springframework.org/schema/beans/spring-beans.xsd">
<!--SolrServer -->
<bean id = "httpSolrServer" class="org.apache.solr.client.solrj.impl.HttpSolrServer">
<constructor-arg index="0" value="http://192.168.25.133:8888/solr/collection1"/>
</bean>
</beans>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,575
|
Q: SL ItemsControl, command on ViewModel not firing from ItemsControl (CheckBox) I'm using PRISM v2, CAL, SL4 and MVVM and have a delegate command on my ViewModel called CheckCommand. The ItemsControl contains a checkbox and I'm trying to get the items in ItemsControl/Checkbox to fire this command when it's checked - but it's not communication back to the viewmodel!
I think it's because each items 'datacontext' is the individual object the item is bound to, rather than the ViewModel?
- My suspicion is actually correct, cause if I move my DelegateCommand out of the viewmodel and into the class defining the items in itemscontrol I can see the commands/methods beeing fired!
View:
<ListBox x:Name="BasketListBox" ItemsSource="{Binding BasketCollection}" MinWidth="200">
<ListBox.ItemTemplate>
<DataTemplate>
<CheckBox commands:Checked.Command="{Binding CheckCommand}" IsChecked="False" </CheckBox>
</DataTemplate>
</ListBox.ItemTemplate>
Can anyone point me in the right direction please?
Cheers, Mcad.
EDIT 1:
The commanding now works, see solution below. BUT, I now run into another problem:
"An exception occurred while creating a region with name 'basketRegion'. The exception was: System.InvalidOperationException: ItemsControl's ItemsSource property is not empty. This control is being associated with a region, but the control is already bound to something else. If you did not explicitly set the control's ItemSource property, this exception may be caused by a change in the value of the inherited RegionManager attached property"
Created seperate question for this problem to make it more clean:
PRISM-MVVM, ItemsControl problem with View injection
A: You want every CheckBox to fire the same command? You could:
<CheckBox commands:Checked.Command="{Binding DataContext.CheckCommand, ElementName=BasketListBox}"
Or you could have every child view model expose the command via their own property.
A: Thanx Kent. You put me on the right path to solve this, ended up doing this:
<ListBox x:Name="basketListBox" ItemsSource="{Binding basketcollection}" MinWidth="200">
<ListBox.ItemTemplate>
<DataTemplate>
<CheckBox commands:Checked1.Command="{Binding DataContext.CheckCommand, ElementName=basketListBox}" Content="{Binding basketName}"> </CheckBox>
</DataTemplate>
</ListBox.ItemTemplate>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,860
|
Q: Using reference variable to point to a Comparator? I ran into a new way of creating Comparator while doing exercises. Can anyone explain this a little bit?
class Checker{
public Comparator<Player> desc = new Comparator<Player>() {
public int compare(Player A, Player B){
if(A.score < B.score){
return 1;
}
else if(A.score == B.score){
return B.name.compareTo(A.name);
}
else{
return -1;
}
}
};
}
In the past, I have only seen people doing this:
class Checker implements Comparator{
@Override
public int compare(Player A, Player B){
..........
..........
}
}
So the first example really seems novel to me(because I am a beginner?). It does make sense: desc could be an property/instance variable of class Checker which points to a new instance of Comparator class/interface. However are there more stories behind these two different ways of doing things? They all require creating a different class so I don't see how either one could be more organized.
A: Both the syntax are absolutely correct. In first case you are simply using concept of anonymous class. In second you created a class Checker which implements compare method.
As a beginner it is much easier to understand second syntax other than that there is no difference between the two.
You can study more about anonymous class here -
https://docs.oracle.com/javase/tutorial/java/javaOO/anonymousclasses.html
If you want to use comparator at more places its better to use solution in separate class than anonymous one. Anonymous vs class solution is something like inline css style vs style classes.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,357
|
\section{Introduction}
Various testse control measures in the vicinity of a game reserve are
experimented within a simulation context. The {\em G. brevipalpis} and {\em G.
austeni} populations in and around the Hluhluwe--iMfolozi Game Reserve are the
object of interest. The problem posed is essentially that of the influence of a
reserve of a particular size and geometry on tsetse population levels outside
its confines and what can be done about it. It is based on the premise that the
reserve is a problem and that it is the cause of unusually high tsetse numbers
in the adjacent agricultural areas. The animals in the reserve are, moreover,
considered to be a reservoir of trypanosomes and, particularly, more lethal
strains.
Little is known of the tsetse species in question and so--called worst--case
values must be assumed. The implementation accordingly does not correspond
exactly to reality, instead, is rather simplistic and more humble than the
competancy of the model itself allows.
\subsubsection*{Vector Competence}
The predominant infection is that of {\em Trypanasoma Congolense}, {\em T.
Vivax} being prevalent to a far lesser extent. It is noteworthy that out of 900
{\em G. brevipalpis} tenerals split into 3 equal groups and respectively allowed
to feed on a different parasitaemic animal, the midgut of 4 \% and the preboscis
of 0 \% were found to be infected ({\sc Motloang}, {\sc Masumu}, {\sc Van Den
Bossche}, {\sc Majiwa} and {\sc Latif}
\cite{MotloangMasumuVanDenBosscheMajiwaLatif}). The prevalence of preboscal
infection for the same experiment involving {\em G. austeni} tenerals was, in
contrast, 12 \% and a further 19 \% were found to have an infected midgut ({\sc
Motloang}, {\sc Masumu}, {\sc Van Den Bossche}, {\sc Majiwa} and {\sc Latif}
\cite{MotloangMasumuVanDenBosscheMajiwaLatif}).
{\sc Motloang}, {\sc Masumu}, {\sc Van Den Bossche}, {\sc Majiwa} and {\sc
Latif} \cite{MotloangMasumuVanDenBosscheMajiwaLatif} also conducted a second
experiment in which they challenged each of 7 susceptable bovines with a
different {\em G. brevipalpis} catch taken from the wild in an insect--proof
facility (the combined catches totalling 468 specimens). No infection resulted.
The same trial was then conducted by challenging each of 2 bovines and 1 goat
with a different {\em G. austeni} catch taken from the wild (the combined
catches totalling a mere 43 specimens only). All three challenges resulted in
infection.
Both prevalence and transmission rates are therefore exceptionally high for {\em
G. austeni} while they are virtually non--existant in the case of {\em G.
brevipalpis}. The issue of mechanical infection by {\em G. brevipalpis} is
currently under investigation by the same authors.
\subsubsection*{The Hluhluwe--iMfolozi Game Reserve}
The Hluhluwe--iMfolozi Game Reserve is located in the southern vicinity of
28$^o$S and 32$^o$E, in KwaZulu--Natal, South Africa. The reserve measures some
960 km$^2$.
\begin{figure}
\begin{center}
\includegraphics[height=15.5cm, angle=90, clip = true]{hluhluweImfoloziSmaller.jpg}
\caption{Satellite Image of the Hluhluwe--iMfolozi Game Reserve and its surroundings.} \label{HluhluweImfoloziSatelliteImage}
\end{center}
\end{figure}
Inland of the coastal plain and set in the foothills of the escarpment, the
temperature of the region is somewhat elevated for its latitutde. The once
mighty Black and White iMfolozi rivers meander through the reserve to their
confluence in the iMfolozi sector (see Figure
\ref{HluhluweImfoloziSatelliteImage}), while the Hluhluwe river passes through
the reserve at the Hluhluwe end (see Figure
\ref{HluhluweImfoloziSatelliteImage}). It is noteworthy, with regard to both
{\em G. brevipalpis} and {\em G. austeni}, that the Hluhluwe river has a flood
plain within the reserve and that the backwater of the Hluhluwe Dam also extends
well into it. At around this position, the reserve is approximately only 25 km
from the St. Lucia estuary, a world heritage site which lies to the East.
The geomorphology of the iMfolozi lowlands is one of deep, sweeping, valleys
carved out by ancient rivers, which recedes into more rugged hill--terrain. Open
grasslands, savannah and woodlands characterise the iMfolozi sector. In
contrast, the Hluhluwe sector gives rise to ridges as high as 540 metres above
mean sea level. Precipitation at these levels is sufficient to sustain coastal,
scarp forest with lush valley bushveld below. The vegetation type of the whole
reserve could broadly be described as typical Zululand Lowveld, interspersed
with minor Northern Zululand Sourveld. Riverine forest and thicket make the
reserve the habitat of both {\em G. brevipalpis} and {\em G. austeni} and these
two species, in association with large populations of buffalo and other wild
animals, lead it to be something of a thorn in the side of neighbouring
agriculture. Habitat outside the reserve is degraded to the extent that the
boundary of the reserve is discernable in satellite images.
The Hluhluwe--iMfolozi Game Reserve has the distinction of being the oldest
proclaimed game reserve in Africa. It is a lucrative tourist attraction as well
as a protected area in terms of the National Environmental Management Protected
Areas Act (Act no. 57 of 2003). Any tsetse control within the reserve is
considered to be highly undesireable, or even out of the question from the point
of view of substantial legislation and agreements cited by Ezemvelo K.Z.N.
Wildlife. The reserve falls within what was once the very extensive habitat of
{\em G. pallidipes}, a species which was completely eradicated from
KwaZulu--Natal in the first half of the 20th century.
\subsubsection*{Kolmogoroff--Petrovsky--Piscounoff Equations}
{\sc Williams}, {\sc Dransfield} and {\sc Brightwell} \cite{Williams2}
originally entertained the idea of using Fisher's equation to model the
distribution of tsetse populations and {\sc Hargrove} \cite{Hargrove6} devised
the best implementation his circumstances permitted. The model entertained in
this work is based on a very similar equation and differs mostly in the exact
specification of population density in the logistic part. It belongs to a more
general category of partial differential equations known as
Kolmogoroff--Petrovsky--Piscounoff (K.P.P.) equations. Such partial differential
equations also happen to be parabolic. In this regard, it is important to note
that one cannot simply solve a parabolic, partial differential equation with a
forward difference in time, nor should one use finite differences for
non--rectangular geometries. The former is widely accepted as a faux pas and
even in the event of circumstances which favour a correct solution, it has no
credibility whatsoever. As such, the problem is ideally suited to the
application of the finite element method. Fisher's equation is both parabolic
and nonlinear.
Dispersal is assumed to occur by a process similar to diffusion,
self--regulating growth is assumed to be logistic and a straight--forward linear
rate is used to model any artificially imposed mortality. The flies are assumed
to move with some kind of Brownian motion down a diffusion gradient based on the
random nature of their movement (observed by {\sc Bursell} \cite{Bursell4} and
demonstrated by {\sc Rogers} \cite{Rogers1}). Fisher's equation is not perfectly
suited to tsetse application owing to the large puparial duration which
characterises the {\em Glossina} genus. Any growth in the present fly population
has its origins in an historical fly population; one which existed one puparial
duration ago. This puparial duration is temperature dependent. While Fisher's
equation was largely solved for academic reasons, it offered some interesting
insights.
The model itself exceeded 5000 lines of fairly extensively commented Fortran,
while the mesh generator exceeded 1700 lines.
\subsubsection*{{\em G. brevipalpis} and {\em G. austeni}}
{\em G. brevipalpis} and {\em G. austeni} are, in all likelihood, not the most
ideal species for this type of application. Both forest species are thought to
be fairly specialised and therefore habitat--specific. This observation is
independently born out by the {\sc Rogers} and {\sc Robinson}
\cite{RogersAndRobinson} study (based on {\sc Ford} and {\sc Katondo}
\cite{FordAndKatondo}'s maps) as well as the pupal water loss model in {\sc
Childs} \cite{Childs2}. {\em G. brevipalpis} would, more generally, appear to be
closely associated with the riverine forest, or thicket, adjacent to drainage
lines. While its pupal habitat appears to be more stringently confined than that
of {\em G. austeni} (according to demonstrations of the pupal water loss model
in {\sc Childs} \cite{Childs2}), the present work will suggest {\em G.
brevipalpis} to be more far--ranging than {\em G. austeni}. The latter is
thought to be relatively sedentary in addition to being fickle in habitat. Both
species might therefore not subscribe well to the diffusion model in venturing
outside preferred habitat. {\sc Rogers}' \cite{Rogers1} experiments with {\em G.
fuscipes fuscipes} were in fairly uniform habitat and even then there was
light--sensitive preference. Nothing appears to be known about the diffusion
rates of {\em G. brevipalpis} and {\em G. austeni}. The pupal durations also
subscribe worst ({\sc Parker} \cite{Parker1}) to the quantitative work done by
{\sc Phelps} and {\sc Burrows} \cite{phelpsAndBurrows1} (subsequently modified
by {\sc Hargrove} \cite{Hargrove3}). One would certainly prefer to be modelling
savannah species.
\section{Derivation of a Model}
The aim of the model is to predict how a tsetse population becomes distributed
in space and how this distribution changes over time, through migration,
self--regulating growth and artificially--imposed mortality. The intention is to
predict a population density, $\rho(\mathbf{x},t)$ (in which $\mathbf{x}$ and
$t$ are space and time respectively), based on the these phenomena.
The subject of the intended model is the vector of trypanosomiasis, namely adult
tsetse flies. Pupae neither migrate, nor do they (or any flies belonging to the
nulliparous cohort, for that matter) form any part of the actively reproductive
population. For these reasons the population density, $\rho(\mathbf{x},t)$, is
defined not to include pupae. While it is tempting to also exclude any flies
belonging to the nulliparous cohorts from a reproductive point of view, such
flies are mobile and subject to the external, artificial mortality to be
imposed; indeed, the subject of this investigation. While the correspondence of
the reproductive population to the mobile and vulnerable population is not
perfect, it is suitably close.
\subsection{Describing the Three Dynamics of Interest}
How might one model the population change brought about through migration,
self--regulating growth and artificially--imposed mortality? If all three effects can be regarded as being mutually independent of one another, they can be considered in isolation.
\subsubsection{Migration}
{\sc Bursell} \cite{Bursell4} put forward the theory that the movement of tsetse
was of a random nature, not unlike Brownian motion, and {\sc Rogers}
\cite{Rogers1} proved these assertions quantitatively. If one can conceive of a
gas as a continuum, it is only slightly more abstract to conceive of a fly
population as a continuum.
Consider the hypothetical scenario of a mobile population in the absence of
either reproduction or mortality. Biomass should therefore be conserved and the
standard continuum--mechanical result for mass conservation pertains. It can be
manipulated to give a result not unlike the Reynolds transport theorem and
Fick's first law applied. (A full exposition is provided in the addendum.) The resulting rate for the effect of migration in isolation is
\begin{eqnarray*}
\frac{\partial \rho}{\partial t} &=& \lambda \ \mathop{\rm div} \nabla \rho
\end{eqnarray*}
in which $\lambda$ is the diffusion coefficient.
\subsubsection{Self--Regulating Growth}
The logistic model needs little introduction to a biological audience. The
population is assumed to grow at a rate which is some proportion, $r$, of some
parent population $\rho^*$. Reason dictates that this growth rate must
necessarily also be constrained by the carrying capacity, $K$, of the
environment. The simplest and most obvious construct which will accomplish this
is a factor $1 - \frac{\rho^*}{K}$. Such a factor allows the growth rate to drop
off linearly as the population level rises. The resulting rate for
self--regulating growth in isolation could therefore be modelled as
\begin{eqnarray*}
\frac{\partial \rho}{\partial t} &=& r \rho^* \left( 1 - \frac{\rho^*}{K} \right).
\end{eqnarray*}
Larval production is clearly dependent on the population which existed one
puparial duration ago in the tsetse context. What about population density in
the second factor of the logistic term; the one limiting the growth rate? The
pertinent population is not as obvious in this instance. Combined pupal and
teneral mortality is an order of magnitude higher than adult mortality ({\sc
Hargrove} \cite{Hargrove3}) and {\sc Vale} \cite{Vale1} seems to think that
parasitism alone accounts for between 40\% to 60\% of the overall pupal
mortality, under usual circumstances in a favourable environment. Quantitative
work linking predation and parasitism to the density at pupal sites has been
carried out by {\sc Rogers} and {\sc Randolph} \cite{RogersAndRandolph1}. That
work could therefore be taken to recommend a logistic term based entirely on an
historical population density, that which existed at the time of larval
deposition and parturition.
Can such a model be reconciled with the other, remaining causes of pupal
mortality? Although fat loss\footnotemark[1] and water loss\footnotemark[1]
\footnotetext[1]{Teneral mortality from both fat loss and water loss is thought
to be high and is often the cumulative effect of temperature and humidity
conditions which prevailed during the pupal phase.} are determined by external
variables of temperature and humidity, indirect dependence on population density
is possible in the event of a shortage of available breeding sites. Spatial
variation of temperature and humidity is incorporated, up to a point, in the
carrying capacity and growth rate of each environment. It is important to note,
however, that the effects of any temporal variation in the growth rate, $r$, are
beyond the scope of the standard logistic model, although it does allow for a
time--dependent carrying capacity, $K$. In this particular model pupal
metabolism and development are temperature--dependent, therefore
time--dependent. Including this dependence allows the origins of presently
ecloding pupae to be traced to their parent population.
The fact that an historical population level was responsible for both larval
production and subsequent, density--dependent mortality is taken into account in
this particular model. Both larval production and subsequent natural mortality
are dependent on the historic population level, that which existed one puparial
duration ago. The population density at the time of larval deposition was
\begin{eqnarray*} \label{3}
\rho^*(\mathbf{x},t) \equiv \rho(\mathbf{x},t - {\bar \tau}),
\end{eqnarray*}
in which ${\bar \tau}$ is the relevant puparial duration.
The correspondence of the reproductive population to the mobile and vulnerable
population is not perfect, however, this can readilly be taken into account by
assuming a fixed age profile for the growth rate.
\begin{assumption} \label{assumption1} {\bf \em The age profile of the
population is fixed.}
\end{assumption}
How reasonable is this assumption? One consequence is that the imposition of any
artificial, adult--selective mortality gives rise to a damped logistic response
from the model above $K/2$ and an over--reactive one below $K/2$. In reality, a
smaller proportion of reproductive adults than the model supposes, exists. The
greater part of the logistic curve may be approximated as linear, as can the
entire curve, locally. No problems should be expected for reasoable use.
``Reasonable use'' does not, however, include scenarios such as aerial spraying.
At some point what is presently being couched in terms of a slightly extended
first interlarval period becomes the phase entrainment of the reproductive
population. Tsetse reproduction is not continuous. Instead, it occurs at
discrete intervals dictated by the first and subsequent interlarval periods.
Problems could therefore be aniticpated, should the age profile of the general
population become significantly altered by any artificially imposed,
adult--selective carnage (e.g aerial spraying) and the subsequent population not
be allowed sufficient time to re--equilibrate. Pupae, about to emerge, would not
be susceptable to the aforementioned mortality. Shortly afterwards, a dramatic
change in the age profile of the population is induced by the addition of
abnormally large proportions of young flies of an unreproductive age. The model
fails to recognize the abnormally youthful population as such and assumes it to
be reproductive. What in reality should be a reproductive `shadow', a dearth of
reproductive flies (which the model does not recognize), leads to an
over--estimated eclosion exactly one puparial duration later. These fictitious,
newly--emerged flies, nonetheless, assist in compensating for what, in reality,
is yet another skewed age profile; this time a surplus of now--mature and
reproductive flies. Their maturity would otherwise lead to an underestimated
eclosion, yet another puparial duration later, and so on. Fortunately the tsetse
fly is a $K$--strategist and, given enough time, the model is expected to
re--equilibrate. It would obviously not recover from a sequence of such events
in close succession. This limitation is negligeable in comparison to one
presently to be discused in connection with a model based on Fisher's equation.
\subsubsection{Artificially Imposed Mortality}
Suppose that the effect of targets, pour--ons etc. is to cause the population density to decline according to $\rho \delta t$, where $\delta$ is independent of the population density. Then the resulting rate for artificially imposed mortality in isolation is
\begin{eqnarray*}
\frac{\partial \rho}{\partial t} &=& - \delta \rho.
\end{eqnarray*}
\subsection{A Governing Equation}
The combined effect of all three phenomena is additive and a model can therefore
be based on the following equation. Two alternatives arise based on the exact
specification of the parent population density, $\rho^*(\mathbf{x},t)$. In the
equation
\begin{eqnarray} \label{1}
\frac{\partial \rho(\mathbf{x},t)}{\partial t} &=& \lambda \mathop{\rm div}
\nabla \rho(\mathbf{x},t) + r \rho^*(\mathbf{x},t) \left( 1 -
\frac{\rho^*(\mathbf{x},t)}{K} \right) - \delta \rho(\mathbf{x},t),
\end{eqnarray}
$\rho(\mathbf{x},t)$ is otherwise the current population density (in which
$\mathbf{x}$ and $t$ are space and time respectively), $\lambda$ is an diffusion
rate, $r$ is the population growth rate, $K$ is the carrying capacity of the
environment and $\delta$ is an artificially imposed mortality. The quantity
$\rho^*(\mathbf{x},t)$ is either an historical population density or the current
population density, depending on the model preferred.
{\sc Remark:} Notice that in the special case of $\rho^*(\mathbf{x},t) = \rho(\mathbf{x},t)$ and $\delta = 0$, equation \ref{1} becomes immediately recognizable as Fisher's equation in its classical form. It is otherwise part of a more general and widely inclusive family, known as Kolmogoroff--Petrovsky--Piscounoff equations.
\subsubsection{Fisher's Equation}
As {\sc Williams}, {\sc Dransfield} and {\sc Brightwell} \cite{Williams2} rightly observe, equation \ref{1} can, in the special case of $\rho^*(\mathbf{x},t) = \rho(\mathbf{x},t)$, be manipulated to a Fisher's equation in which there are new, modified carrying capacities and growth rates,
\begin{eqnarray*}
K' = K \left( 1 - \frac{\delta}{r} \right) \hspace{5mm} \mbox{and} \hspace{5mm} r' = r \left( 1 - \frac{\delta}{r} \right),
\end{eqnarray*}
respectively. A logistic term dependent on the present population density is,
however, something known to be incorrect. The larval deposition responsible for
growth in the present population took place a significant time previously. In
the Fisher's--equation model the existance of the pupal phase is denied,
alternatively, pupae are assumed to migrate and reproduce.
Why use an equation which is less appropriate? Firstly, it would be interesting
to know if there is any difference between the K.P.P.--model results and those
arising from the unquestioning application of Fisher's equation; given the long
periods to equilibrate that this particular application will allow. Secondly,
convergence with little or no iteration will indicate the steady state. Thirdly,
there is a certain amount of academic interest as the mathematics is more
challenging. If the numerical techniques employed are powerful enough to solve
a nonlinear Fisher's equation, they will, logically, solve an equation with any
other variants of the logistic term contemplated. (One might later wish to model
another vector, like {\em cullicoides}.) Fourthly, a point already mentioned is
that if the age profile in the model of interest is altered in such a way that
it contains a significantly higher than usual proportion of flies of a
nulliparous age, then the growth rate (which is based on a fixed age profile) in
the model may no longer be appropriate. Just how reasonable is the assumption of
a fixed age profile? Fisher's equation makes a far worse assumption and a
comparison between results might indicate the extent of the problem.
\section{Scaling}
Suppose that $T$ is (only for the present) a unit of time, $X$ is a unit of length and $\eta$ is a characteristic population density. The scaled variables are then
\begin{eqnarray*}
{\mathbf{x}} = { \bar {\mathbf{x}} }X, \hspace{5mm} t = {\bar t } T \hspace{5mm} \mbox{and} \hspace{5mm} \rho = {\bar \rho} \eta \mbox{ \ \ (including }K = {\bar K} \eta \mbox{).} &&
\end{eqnarray*}
The operators become
\begin{eqnarray*}
\frac{\partial}{\partial {\mathbf{x}}} = \frac{1}X
\frac{\partial}{\partial {\bar {\mathbf{x}}}} &\Rightarrow& \mathop{{\rm div}} {\nabla} {\rho } =
\displaystyle \frac{\eta}{X^2} \mathop{\bar {\rm div}} {\bar \nabla} {\bar
\rho}
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{\partial}{\partial t} = \frac{1}{T} \frac{\partial}{\partial
{\bar t}} &\Rightarrow& \frac{\partial \rho }{\partial t} =
\frac{\eta}{T} \frac{\partial {\bar \rho}}{\partial {\bar t}}
\end{eqnarray*}
in terms of the scaled variables. Thus
\begin{eqnarray*}
\frac{\eta}{T} \frac{\partial {\bar \rho}}{\partial {\bar t}} &=&
\frac{\eta}{X^2} \ \lambda \ \mathop{\bar {\rm div}} {\bar \nabla} {\bar \rho} +
r \eta {\bar \rho}^* \left( 1 - \frac{ {\bar \rho}^* }{\bar K} \right) - \delta
\eta {\bar \rho}.
\end{eqnarray*}
All of this suggests using $T = \displaystyle \frac{X^2}{\lambda}$ and $X = \displaystyle \sqrt{\frac{\lambda}{r}}$ so
that the above equation becomes
\begin{eqnarray*} \label{20}
\frac{\partial {\bar \rho}}{\partial {\bar t}} &=& \mathop{\bar {\rm div}} {\bar
\nabla} {\bar \rho} + {\bar \rho}^* \left( 1 - \frac{\bar \rho^*}{\bar K}
\right) - \frac{ \delta }{r} {\bar \rho}.
\end{eqnarray*}
If the mesh is in units of kilometres, for example, then it must be converted by
dividing through by $\sqrt{\frac{\lambda}{r}}$ kilometres. One would imagine the
desire to multiply positions by $\sqrt{\frac{\lambda}{r}}$ when outputting the
solution, thereby returning to kilometres.
{\bf Complication:} If one intends including any environmental variation in the rates of diffusion and growth, the scaled equation will entail different and therefore irreconcileable time steps.
\begin{eqnarray} \label{10}
\frac{\partial {\bar \rho}}{\partial {\bar t}} &=&
\frac{\lambda}{\lambda_{\scriptsize scale}} \mathop{\bar {\rm div}} {\bar
\nabla} {\bar \rho} + \frac{r}{r_{\scriptsize scale}}{\bar \rho}^* \left( 1 -
\frac{\bar \rho^*}{\bar K} \right) - \frac{ \delta }{r_{\scriptsize scale}} {\bar \rho}.
\end{eqnarray}
The above equation allows a time discretisation which conforms.
\section{Variational Formulation} \label{156}
A variational formulation of equation \ref{10} is obtained in the usual fashion:
Premultiplying the primitive variable equation by an arbitrary function, $w$,
and integrating over the domain, $\Omega$, to obtain
\begin{eqnarray} \label{31}
\int_{ \Omega } w \frac{\partial \rho}{\partial t} {d{\Omega}} &=&
\frac{\lambda}{\lambda_{\scriptsize scale}} \int_{ \Omega } w \ \mathop{\rm div}{{\nabla} {\rho }} {d {\Omega}} +
\frac{r}{r_{\scriptsize scale}} \int_{ \Omega } w \rho^* \left( 1 - \frac{ \rho^* }{K} \right) {d
{\Omega}} - \frac{ \delta }{r_{\scriptsize scale}} \int_{ \Omega } w \rho {d {\Omega}} \nonumber \\
\end{eqnarray}
The approximation--wise cumbersome second derivatives appearing in this
equation can also be avoided in the usual fashion. The term which contains the
divergence of ${\nabla} {\rho }$ can be regarded as one part of a differentiated
product and the divergence theorem applied so that
\begin{eqnarray*}
\int_{ \Omega } w \frac{\partial {
\rho}_{,i}}{\partial x_i} \ {d {\Omega}} &=& \int_{ \Gamma}
w {\rho}_{,i} {n}_i \ d{\Gamma} - \int_{
\Omega } {w}_{,i} {\rho}_{,i} \ {d {\Omega}},
\end{eqnarray*}
in which $\mathbf{n}$ is the outward unit normal and $\Gamma$ is the domain
boundary. Of the two terms resulting, one is a boundary integral. The integrand
obviously vanishes for a von Neumann, `$\mathbf{n} \cdot \nabla \rho = 0$'--type
boundary condition. The arbitrary vector of the formulation can otherwise be
assigned a value of zero on boundaries where boundary conditions are Dirichlet
(and variational formulation is consequently not required). The boundary
integral is zero along such boundaries.
\section{Discretisation} \label{136}
A backward difference is used for the temporal discretisation, while the finite
element method is used for the spatial discretisation. The solution at time $t$
is accordingly assumed to be a linear combination of shape functions,
${\mathbf{\psi}}({\mathbf{x}})$. That is,
\[
\rho({\mathbf{x}}) {\mid}_t = \sum_{i=1}^{N}
{c}_i \psi_i({\mathbf{x}}),
\]
where the $c_i$ are the constants of the finite element approximation (the nodal
solutions) and $N$ is the total number of nodes. The problem on each element is
calculated in terms of a standard, master element coordinate system, $\{
{\mathbf{\xi}} \}$.
\subsection{Finite Element Construction}
The approximate equation, to be solved for the nodal population densities,
$P^e_j$ (pertaining to element $e$), are consequently
\begin{eqnarray}
&& \hspace{-25mm} {\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ \left\{ \ \frac{1}{\Delta t}
\int_{\hat{\Omega}} \phi_{i} \phi_{j} J^e {d{\hat{\Omega}}} \ + \
\frac{\lambda}{\lambda_{\scriptsize \mbox{scale}}} \ \int_{\hat{\Omega}}
\frac{\partial \phi_{i}}{\partial x_k} \frac{\partial \phi_{j}}{\partial x_k}
J^e {d{\hat{\Omega}}} \ + \ \frac{ \delta }{ r_{\scriptsize \mbox{scale}} } \int_{\hat{\Omega}} \phi_{i} \phi_{j} J^e {d{\hat{\Omega}}} \right\} {\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ P^e_j
\hspace{0mm} \nonumber \\
\hspace{10mm} &=& \ {\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ \left\{
\frac{1}{{\Delta}t} \int_{\hat{\Omega}} \phi_{i} \phi_{m} J^e {d{\hat{\Omega}}}
\ P^e_m{\mid}_{t - {\Delta}t} \ + \ \frac{r}{r_{\scriptsize \mbox{scale}}} \
\int_{\hat{\Omega}} \phi_{i} \phi_{n} J^e {d{\hat{\Omega}}} \ P_n^{e}{\mid}_{t -
{\bar \tau}} \right. \nonumber \\
\hspace{10mm} && - \ \frac{r}{r_{\scriptsize \mbox{scale}}} \ \left.
\int_{\hat{\Omega}} \phi_{i} \phi_{l} \frac{ \phi_{j} }{ K } J^e
{d{\hat{\Omega}}} \ P_l^{e}{\mid}_{t - {\bar \tau}} \ \
P_j^{e}{\mid}_{t - {\bar \tau}} \ \right\},
\end{eqnarray}
in which ${\mathop{\mbox{\LARGE\bf\sf A}}}$ is the element assembly
operator, $E$ is the total number of elements, $e$, into which the
domain has been subdivided, $\hat{\Omega}$ is the master element
domain, $\Delta t$ is the length of the time step, the
$\phi_i(\mathbf{\xi})$ are the basis functions,
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{eqnarray*}
\frac{\partial \phi_{i}}{\partial x_j}( {\mathbf{\xi}} ) = \frac{\partial \phi_{i}}{\partial{\xi}_k}
\frac{{\xi}_k}{\partial x_j} \ , \hspace{10mm} {J^e} = \left| \frac{\partial {\mathbf{x}}}{\partial \mbox{\boldmath{$\xi$}} } \right| \ \mbox{for element e},
\end{eqnarray*}
$\lambda$ is the rate of diffusion, $\lambda_{\scriptsize \mbox{scale}}$ is a
diffusion--rate scale, $r$ is the population growth rate, $r_{\scriptsize
\mbox{scale}}$ is a population growth--rate scale, $K$ is the carrying capacity
of the environment and $\delta$ is an artificially imposed mortality.
$P_n^{e}{\mid}_{t - {\bar \tau}}$ denotes the solution at the time of larval
deposition (that which led to the present eclosion), ${\bar \tau}$ being the
relevant average of puparial durations.
\subsubsection*{Fisher's Equation}
The analogous finite element implementation for Fisher's equation is
\begin{eqnarray}
&& {\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ \left\{ \ \frac{1}{\Delta t}
\int_{\hat{\Omega}} \phi_{i} \phi_{j} J^e {d{\hat{\Omega}}} \right. \ + \
\frac{\lambda}{\lambda_{\scriptsize \mbox{scale}}} \ \int_{\hat{\Omega}} \frac{\partial \phi_{i}}{\partial x_k} \frac{\partial
\phi_{j}}{\partial x_k} J^e {d{\hat{\Omega}}} \ - \ \frac{r}{r_{\scriptsize \mbox{scale}}} \ \int_{\hat{\Omega}} \phi_{i}
\phi_{j} J^e {d{\hat{\Omega}}} \nonumber \\
&& \hspace{0mm} + \ \left. \frac{r}{r_{\scriptsize \mbox{scale}}} \ \int_{\hat{\Omega}} \phi_{i} \phi_{j} \frac{
\phi_{l} }{ K } J^e {d{\hat{\Omega}}} \ P_l^{e \ linearisation} \ + \ \frac{ \delta }{ r_{\scriptsize \mbox{scale}} } \int_{\hat{\Omega}} \phi_{i} \phi_{j} J^e {d{\hat{\Omega}}} \right\}
{\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ P^e_j \nonumber \\
&& \hspace{30mm} = {\mathop{\mbox{\LARGE\bf\sf A}}}_{e=1}^E \ \left\{
\frac{1}{{\Delta}t} \int_{\hat{\Omega}} \phi_{i} \phi_{m} J^e {d{\hat{\Omega}}}
\ P^e_m{\mid}_{t - {\Delta}t} \right\},
\end{eqnarray}
in which
\[
\mathbf{P}^{e \ linearisation} = 2 \mathbf{P}^e \mid_t - \mathbf{P}^e \mid_{t - \Delta t},
\]
the second order accurate linearisation originally used in {\sc Childs} \cite{Childs1}.
\renewcommand{\thefootnote}{\arabic{footnote}}
\section{The Relevant Parental Population}
Determining the puparial duration leading to the present eclosion is a minor problem in its own right. It is known that at a given temperature, $T$, the puparial duration, in days, can be calculated according to the formula
\begin{eqnarray} \label{2}
\tau(T) &=& \frac{ 1 + e^{a + bT} }{\kappa},
\end{eqnarray}
({\sc Phelps and Burrows} \cite{phelpsAndBurrows1}). For females, $\kappa =
0.057 \pm 0.001$, $a = 5.5 \pm 0.2$ and $b = -0.25 \pm 0.01$. For males, $\kappa
= 0.053 \pm 0.001$, $a = 5.3 \pm 0.2$ and $b = -0.24 \pm 0.01$ ({\sc Hargrove}
\cite{Hargrove3}). The puparial durations of all species, with the exception of
{\em G. brevipalpis}, are thought to lie within 10\% of the value predicted by
this formula ({\sc Parker} \cite{Parker1}). {\em G. brevipalpis} takes a little
longer. The shortest puparial duration is that of {\em G. austeni}.
If ${\bar \tau}$ is the relevant average of puparial durations (which is, of course, dependent on itself) then
\begin{eqnarray*}
{\bar \tau} &\equiv& \frac{1}{\bar \tau} \left[ \left[ \frac{}{} t - \mbox{floor}\left\{ t \right\} \right] \tau(T_{\scriptsize \mbox{ceil}\left\{ t \right\}}) \ + \sum_{i=\mbox{\scriptsize floor}\left\{ t \right\}}^{ \mbox{\scriptsize ceil}\left\{ t - \tau + 1 \right\} } \tau(T_{\mbox{\scriptsize }{\scriptsize i}}) \right. \nonumber \\
&& \hspace{52mm} + \ \left. \left[ \frac{}{} \mbox{ceil}\left\{ t - \tau \right\} - (t - \tau) \right] \ \tau(T_{{\mbox{\scriptsize ceil}} \left\{ t - \tau \right\}}) \displaystyle \right],
\end{eqnarray*}
in which $\tau(T)$ is given by the formula (\ref{2}) and $T$ is the mean daily
temperature on the day indicated by the subscript. Newton's method is used in
solving the above equation. The relevant parental population at the time $t -
{\bar \tau}$ is a weighted average of the nearest two solutions since a backward
difference was used for the temporal discretisation.
\section{Implementation in the Context of the Reserve and its Surroundings}
A suitable finite element mesh had to be generated and pertinent carrying
capacities, growth rates and diffusion coefficients associated with it. Little
is known of the tsetse species in question and so--called worst--case values
must be assumed. The implementation accordingly does not correspond exactly to
reality, instead, is rather simplistic and more humble than the competancy of
the model itself allows.
The carrying capacities were also the initial, start--up values assumed, with
the exception of the northern and western boundaries, where the population was
assumed to be 0\%. A constant supply of flies maintaining the eastern and
southern boundaries of the region at a 5\% level was assumed (regardless of the
controls experimented with). The model also requires daily temperature data.
The lack of statistically significant data and the simplicity of an initial case
study were deemed to mitigate the following assumption.
\begin{assumption} \label{assumption2}
{\bf \em There are no seasonal, or temporal, changes in the environment.}
\end{assumption}
In other words the model assumes that any non--density--dependent causes of
pupal mortality are constant in time, although the potential to vary carrying
capacity does exist in the model; both $K$ and $r$ are constant in time. This is
not to say that the environment and climate do not vary spatially. Spatial
variation of the environment is incorporated into the parameters of growth rate
and carrying capacity. Of course in reality there are seasonal changes, however,
one would hope that the forest environment would be more insulated than most.
Pupal survival for {\em G. brevipalpis} and {\em G.austeni} is more especially
sensitive to environmental conditions ({\sc Childs} \cite{Childs2}); the
environmental conditions which are incorporated into the parameters of growth
rate and carrying capacity and which are, in this application, assumed not to vary in time.
\subsection{The Finite Element Mesh}
Nine--noded quadrilateral elements and the associated $Q_2$ element basis were used. A program to generate the mesh was written based on {\sc Childs} and {\sc Reddy} \cite{Childs3}.
\begin{figure}
\vspace{-10mm}
\begin{center}
\includegraphics[height=20cm, angle=0, clip = true]{mesh.jpg}
\vspace{-25mm}
\caption{The finite element mesh.} \label{mesh}
\end{center}
\vspace{-10mm}
\end{figure}
\subsection{The Carrying Capacity, $K$}
The {\em Glossina} genus is a $K$-strategist and carrying capacities are
therefore a deciding factor in the implementation. The problem posed is,
fundamentally, that of the influence of a reserve of a particular size and
geometry on tsetse population levels outside its confines, a reserve which is a
reservoir of more lethal strains and higher levels of trypanosomiasis, and what
can be done about it. It was with this in mind that the decision was taken to
model the carrying capacity of the region, rather simplistically, as in Table
\ref{carryingCapacity} and Figure \ref{startUp}. {\em G. brevipalpis} data
(Figure \ref{brevipalpisProbability}) are indeed suggestive of such a reality,
however, the {\em G. austeni} data (Figure \ref{austeniProbability}) do not correspond as well to such a simplistic distribution.
\begin{table}[H]
\begin{center}
\begin{tabular}{c | c c c c}
& & & & \\
$\approx d$ / km \hspace{5mm} & \hspace{5mm} $d \le 0$ & \hspace{5mm} $0 < d \le 2.5$ & \hspace{5mm} $2.5 < d \le 5$& \hspace{5mm}$d > 5$ \\
& & & & \\ \hline
& & & & \\
$K$ / \% \hspace{5mm} & \hspace{5mm} 100 & \hspace{5mm} 20 & \hspace{5mm} 10 & \hspace{5mm} 5 \\
& & & & \\
\end{tabular}
\caption{The carrying capacity, $K$, designated according to the approximate distance, $d$, from the reserve boundary.} \label{carryingCapacity}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04startUp.jpg}
\caption{Start--up values and carrying capacity (with the exception of the northern and western boundaries).} \label{startUp}
\end{center}
\end{figure}
In reality, the South--Western, iMfolozi section of the reserve is hotter and
drier. That environment is consequently not as hospitable to {\em G.
brevipalpis} and {\em G. austeni} as the lush, North--Eastern, Hluhluwe side. In
reality, some suitable habitat exists outside the confines of the reserve. In
reality, tsetse numbers in the East are supplemented by St. Lucia and the
coastal area below it, whereas populations in the North are sometimes limited by
higher ground.
\subsection{The Maximum Growth Rate, $r$}
Growth rates were assigned in such a way as to correpond to these self--same
zones. At Hluhluwe--iMfolozi mean annual temperatures, each female would produce
4 pupae based on {\sc Glasgow} \cite{Glasgow1}'s 49-day, average, adult
life--span. The population would therefore grow by 2.4 \% day$^{-1}$ in the
absence of any early mortality, assuming an equal ratio of the sexes. Of course,
in reality there is massive pupal and teneral mortality. In reality, the growth
rate is probably closer to 0.85\% day$^{-1}$ ({\sc Hargrove} \cite{Hargrove3}).
A worst--case value of 1.1\% day$^{-1}$ in good habitat was used for the
purposes of this investigation.
\begin{table}[H]
\begin{center}
\begin{tabular}{c | c c c c}
& & & & \\
$\approx d$ / km \hspace{5mm} & \hspace{5mm} $d \le 0$ & \hspace{5mm} $0 < d \le 2.5$ & \hspace{5mm} $2.5 < d \le 5$& \hspace{5mm}$d > 5$ \\
& & & & \\ \hline
& & & & \\
$r$ / \% day$^{-1}$ \hspace{5mm} & \hspace{5mm} 1.1 & \hspace{5mm} 0.5 & \hspace{5mm} 0.2 & \hspace{5mm} 0.1 \\
& & & & \\
\end{tabular}
\caption{The growth rate, $r$, designated according to the approximate distance, $d$, from the reserve boundary.} \label{growthRate}
\end{center}
\end{table}
\subsection{The Diffusion Coefficient, $\lambda$}
No information would appear to be available on the diffusion rates of either
{\em G. brevipalpis} or {\em G. austeni}, whatsoever. The most comprehensive set
of measurements are probably those for {\em G. morsitans} recorded by {\sc
Jackson} \cite{Jackson1} (reported in {\sc Rogers} \cite{Rogers1}). The rate at
which {\em G. morsitans} dispersed in {\em G. swynnertoni} habitat was found to
be a consistant 0.153 km$^2$day$^{-1}$ if the initial stages of the experiment
are omitted. Other work, mostly by the same author (also summarised in {\sc
Rogers} \cite{Rogers1}), suggests {\em G. morsitans} was possibly uncomfortable
in {\em G. swynnertoni} habitat. The very low end of the {\em G. morsitans}
range would appear to be about 0.04 km$^2$day$^{-1}$. The point is that
different habitats can have different coefficients, as do different species, and
one would imagine temperature plays a role.
\begin{figure}
\begin{center}
\includegraphics[height=13cm, angle=90, clip = true]{brevipalpisProbability.jpg}
\caption{{\em G. brevipalpis} risk. The large concentration of pink triangles coincides with the position of the Hluhluwe Dam and its backwater. The minor habitat to the NW of the reserve and the diagonal of red crosses leading down from it, through the reserve, are associated with the Black iMfolozi River. (Source: {\sc Hendrickx} \cite{Hendrickx})} \label{brevipalpisProbability}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04steadyState.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08steadyState.jpg}
\caption{Computed steady states for $\lambda = 0.04 \mbox{ km}^2 \mbox{day}^{-1}$, at left, and $\lambda = 0.08 \mbox{ km}^2 \mbox{day}^{-1}$, at right, after 2 years.} \label{1st}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt16steadyState.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32steadyState.jpg}
\caption{Computed steady states for $\lambda = 0.16 \mbox{ km}^2 \mbox{day}^{-1}$, at left, and $\lambda = 0.32 \mbox{ km}^2 \mbox{day}^{-1}$, at right, after 2 years. Notice that the boundary conditions are very close and starting to effect the computed zone of influence at these high diffusion rates.} \label{2nd}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=13cm, angle=90, clip = true]{austeniProbability.jpg}
\caption{{\em G. austeni} risk. The large concentration of pink triangles indicates the position of the Hluhluwe Dam and its backwater. The minor habitat evident in the South of the reserve is in the vicinity of the confluence of the Black and White iMfolozi Rivers. (Source: {\sc Hendrickx} \cite{Hendrickx})} \label{austeniProbability}
\end{center}
\end{figure}
{\em G. brevipalpis} and {\em G. austeni} are profoundly different species to
{\em G. morsitans}, in both size and habitat. {\em G. morsitans} is of an
intermediate size, while {\em G. brevipalpis} is one of the largest flies known.
{\em G. austeni} is the smallest of the tsetse flies. {\em G. brevipalpis} and
{\em G. austeni} are both forest--dwelling, whereas {\em G. morsitans} is a
savannah species. Under such circumstances one is required to be more
resourceful. The premise of this work is that the reserve is a problem, that it
is the cause of unusually high tsetse numbers in the adjacent agricultural
areas. Indeed, the difference in habitat which is visibly discernable in
satellite images and the tsetse risk map, Figure \ref{brevipalpisProbability},
certainly are suggestive of a reserve with a zone of influence in the case of a
very habitat--specific {\em G. brevipalpis}. The distribution of {\em G.
austeni} (Figure \ref{austeniProbability}) is less well understood. In the case
of {\em G. austeni} it is arguable whether a likely zone of influence can be
detected. (There appear to be areas of good {\em G. austeni} habitat outside the
reserve.) An haven with a zone of influence is nevertheless a premise on which
to proceed and one might speculate a smaller range i.e. a lower `diffusion'
rate, as a point of departure, due to the small size of this fly.
One strategy would be to proceed by trial and error with different values of the
coefficient until matching zones of influence to those evidenced by Figure
\ref{brevipalpisProbability} and Figure \ref{austeniProbability} are produced.
This is, in fact, exactly what was resorted to. The value of the diffusion
coefficient was either halved, or doubled, until a suitable zone of influence
was generated. The match was obviously not perfect due to higher ground to the
North, suitable habitat outside the reserve (which was not modelled) and
supplementation from the St. Lucia populations. With hindsight, a mesh which
included a bigger area would have been preferred. The crude technique suggested
some very acceptable values.
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
The diffusion rate of {\em G. austeni} is probably 0.04 km$^2$day$^{-1}$, that
of a very comfortable {\em G. morsitans} population. Intuitively, this makes
sense because of the small size of the species. At worst, one might speculate
that it could reach 0.08 km$^2$day$^{-1}$. The diffusion coefficient of {\em G.
brevipalpis}, however, came as something of a surprise for a forest--dwelling
species. At somewhere between 0.16 km$^2$day$^{-1}$\footnotemark[2]
\footnotetext[2]{that of an uncomfortable {\em G. morsitans} population.} and,
very possibly as high as 0.32 km$^2$day$^{-1}$ (an order of magnitude larger
than {\em G. austeni}) it approaches {\sc Rogers} \cite{Rogers1}'s observations
of {\em G. fuscipes fuscipes}, a fly of similar habitat, though smaller size.
\renewcommand{\thefootnote}{\arabic{footnote}}
\subsection{The Temperature, $T$}
The South African Meteorological Services quote a mean annual temperature of
22.1$^o$C for Mpila, inside the reserve (based on data collected during the
1980's and 1990's). This is consistant with the data of {\sc Schulze} and {\sc
Maharaj} \cite{SchulzeAndMaharaj}, who define the overall area as being of a
temperature greater than 22$^o$C. The Mpila value is further corroborated by
ARC-ISCW automatic weather stations situated between Mtunzini and Pongola
(operational since 2004). They suggest an average daily temperature of 22$^o$C,
according to the Department of Agriculture.
\section{Results}
The results of the simulations are intended to address the following questions:
\begin{enumerate}
\item What are the likely, worst--case diffusion rates of {\em G. brevipalpis} and {\em G. austeni}?
\item What will the long--term effect of the temporary, 2--year use of pour--ons in the surrounding areas be?
\item Can the influence of the Hluhluwe--iMfolozi Game Reserve on surrounding
tsetse population levels be negated? If so, what measures will this require?
\item Can the tsetse populations of the Hluhluwe--iMfolozi Game Reserve and
their associated trypanosomiasis be completely isolated from the surrounding
areas? If so, what measures will this require?
\item What is a practical barrier width?
\item Can the populations within the reserve be `pumped out' to extinction from outide the reserve; failing that, down to the 20\% level?
\item In what way are diffusion rates relevant to containment, eradication and any subsequent rebound?
\end{enumerate}
For each simulation the model was first run for two years to allow it to settle
down to a steady--state. Two years was deemed more than adequate time for the
model to equilibrate. This was decided on the basis of a visual inspection of
the values (which were no longer changing) as well as the marked drop in the
number of iterations required to solve the alternative model, based on Fisher's
equation. The model was then run for another two years\footnotemark[1]
\footnotetext[1]{Deltamethrin pour--ons can not safely be used on cattle for any
longer than two years without compromising their resistance to tick--bourne
diseases, consequently an enzootic condition} with various controls in place,
all of which were modelled at a 2 \% day$^{-1}$ mortality to start with. The
tsetse population was then allowed to rebound for a further two years, with or
without controls still in place.
Barriers with a width greater than 4 km were not experimented with, even though they might be more optimal in terms of the required number of targets. This is since they were deemed to be a self--defeating waste of land.
\subsection{A 2.5 km--wide Barrier Surrounding the Reserve (with a 2 \% day$^{-1}$ Mortality Throughout)}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04twoPoint5KmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08twoPoint5KmBarrier.jpg}
\caption{The results of an approximately 2.5 km--wide barrier after 2 years.
At left, $\lambda = 0.04 \mbox{ km}^2 \mbox{day}^{-1}$. At right, $\lambda =
0.08 \mbox{ km}^2 \mbox{day}^{-1}$.} \label{2.5km}
\end{center}
\end{figure}
\newpage
\subsection{A 5 km--wide Barrier Surrounding the Reserve (with a 2 \% day$^{-1}$ Mortality Throughout)}
\vspace{5mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04fiveKmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08fiveKmBarrier.jpg}
\caption{The results of an approximately 5 km--wide barrier after 2 years.
At left, $\lambda = 0.04 \mbox{ km}^2 \mbox{day}^{-1}$. At right, $\lambda =
0.08 \mbox{ km}^2 \mbox{day}^{-1}$.} \label{5kmA}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt16fiveKmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32fiveKmBarrier.jpg}
\caption{The results of an approximately 5 km--wide barrier after 2 years.
At left, $\lambda = 0.16 \mbox{ km}^2 \mbox{day}^{-1}$. At right, $\lambda =
0.32 \mbox{ km}^2 \mbox{day}^{-1}$.} \label{5kmB}
\end{center}
\end{figure}
\subsection{The Quest for an Impenetrable, 5 km--wide, Surrounding Barrier}
The influence of the reserve on surrounding tsetse populations is negated in both scenarios on the left, however, flies with higher levels and more lethal strains of trypanosomiasis are still able to commute. In both right--hand side scenarios the barrier zone is completely vacant, which means no flies leave the barrier, which, in turn, means no flies ever cross it.
\vspace{5mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08fiveKmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08fiveKm4percent.jpg}
\caption{An approximately 5 km--wide barrier, after 2 years, for $\lambda = 0.08
\mbox{ km}^2 \mbox{day}^{-1}$. At left, a 2 \% day$^{-1}$ mortality throughout
the barrier. At right, a 4 \% day$^{-1}$ mortality throughout.} \label{impenetrableA}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32fiveKm4percentBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32fiveKm12percent.jpg}
\caption{An approximately 5 km--wide barrier, after 2 years, for $\lambda = 0.32
\mbox{ km}^2 \mbox{day}^{-1}$. At left, a 4 \% day$^{-1}$ mortality throughout
the barrier. At right, a 12 \% day$^{-1}$ mortality. (At 8 \% day$^{-1}$ the
barrier zone along the concave boundary was deemed to be not entirely vacant.)}
\label{impenetrableB}
\end{center}
\end{figure}
\subsection{Pour-Ons and the Subsequent Rebound, With or Without Suppression}
\subsubsection{$\lambda = 0.08 \mbox{ km}^2 \mbox{day}^{-1}$}
\vspace{5mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08everywhereOutside.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08rebound.jpg}
\caption{The result of a 2 \% day$^{-1}$ mortality imposed everywhere outside the reserve for a period of 2 years (left); the rebound after a further 2 years (right).} \label{pourOnsA1}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08rebound2Point5kmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt08rebound5kmBarrier.jpg}
\caption{The rebound suppressed by a barrier with a mortality of 2 \% day$^{-1}$ everywhere inside it. At left, an approximately 2.5 km--wide barrier; At right, an approximately 5 km--wide barrier.} \label{pourOnsA2}
\end{center}
\end{figure}
\subsubsection{$\lambda = 0.32 \mbox{ km}^2 \mbox{day}^{-1}$}
\vspace{5mm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32everywhereOutside.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32rebound.jpg}
\caption{The result of a 2 \% day$^{-1}$ mortality imposed everywhere outside the reserve for a period of 2 years (left); the rebound after a further 2 years (right).} \label{pourOnsB1}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32rebound5kmBarrier.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32rebound5km8percentBarrier.jpg}
\caption{The rebound suppressed by an approximately 5 km--wide barrier. At left,
a 2 \% day$^{-1}$ mortality throughout the barrier. At right, an 8 \% day$^{-1}$
mortality throughout the barrier.} \label{pourOnsB2}
\end{center}
\end{figure}
\subsection{The Implications of the Diffusion Coefficient for Eradication and Any Subsequent Rebound}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04everywhere.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt04everywhereRebound.jpg}
\caption{$\lambda = 0.04 \mbox{ km}^2 \mbox{day}^{-1}$. The result of a 2 \% day$^{-1}$ mortality imposed everywhere for a period of 2 years (left); the population rebound after a further two years (right).} \label{eradicationA}
\vspace{5mm}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32everywhere.jpg}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32everywhereRebound.jpg}
\caption{$\lambda = 0.32 \mbox{ km}^2 \mbox{day}^{-1}$. The result of a 2 \% day$^{-1}$ mortality imposed everywhere for a period of 2 years (left); the population rebound after a further two years (right).} \label{eradicationB}
\end{center}
\end{figure}
\subsection{`Pumping' Out the Reserve Population from the Boundary}
\begin{figure}[H]
\begin{center}
\includegraphics[width=7.7cm, angle=0, clip = true]{pt32fiveKm50percent.jpg}
\caption{The result of a 5 km--wide barrier in which a 50 \% day$^{-1}$ mortality is imposed for a period of 2 years ($\lambda = 0.32 \mbox{ km}^2 \mbox{day}^{-1}$).} \label{pumping}
\end{center}
\end{figure}
\section{Improvising Barrier Mortality Rates}
Odour--baited targets and cattle treated with so--called `pour--ons' are the means by which tsetse barriers can be constructed.
\subsection{Translating Barrier Mortality into Odour--Baited Targets and the Possible Revelation of Competition}
The definitive experimental work involving target barriers for {\em G. austeni} and {\em G. brevipalpis}
is that of {\sc Esterhuizen}, {\sc Kappmeier Green}, {\sc Nevill} and {\sc Van
Den Bossche} \cite{EsterhuizenKappmeierGreenNevillVanDenBossche}. Essentially
they barricaded a small peninsula with targets. They also placed targets on the
peninsula itself and measured the decline of {\em G. austeni} and {\em G.
brevipalpis} in relation to the target density on the peninsula. At a target
density of 8 km$^{-2}$ {\em G. austeni} was found to decline at a rate of around
- 0.014 day$^{-1}$.
Despite good estimates of the rate of {\em G. austeni} decline, choosing other
parameters in {\sc Williams}, {\sc Dransfield} and {\sc Brightwell}
\cite{Williams1}'s seminal equation is something of a heuristic exercise. If one
takes cognizance of {\sc Rogers} and {\sc Randolph} \cite{RogersAndRandolph1}'s
findings on predation, the pupal water loss model of {\sc Childs} \cite{Childs2}
etc., a 2 \% day$^{-1}$ pupal mortality rate would certainly not be unreasonable
for the species in question. Natural mortality for the nulliparous cohort is
known to be high and, although it does not fall victim to targets in the same
proportions as adults, some still do ({\sc Hargrove} \cite{Hargrove9}). With
this fact in mind the nulliparous stage flies were assigned a natural mortality
of 2.2 \% day$^{-1}$ and assumed to have a target--related mortality one half
that of adults. At 22.1$^o$C the relevant formulae\footnotemark[1]
\footnotetext[1]{{\sc Hargrove} \cite{Hargrove3}'s improved {\sc East African
High Commision} \cite{Anonymous} formulae.} for the first and subsequent
interlarval periods predict 17.5 days and 10.5 days respectively. The formula
\ref{2} gives a puparial duration of 34.6 days for females. A miscarriage rate
of 5 \%, and therefore a fecundity of 0.475 was used (in keeping with {\sc
Williams}, {\sc Dransfield} and {\sc Brightwell} \cite{Williams1}).
Solving {\sc Williams}, {\sc Dransfield} and {\sc Brightwell} \cite{Williams1}'s
equation using the aforementioned parameters and Newton's method suggested the 8
km$^{-2}$ target density of {\sc Esterhuizen}, {\sc Kappmeier Green}, {\sc
Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche} was equivalent to an
artificially imposed mortality of 2.39 \% day$^{-1}$ (0.30 \% day$^{-1}$ per
target). Target--related mortality is obviously much lower for these forest
species. (By comparison, a single odour--baited target\footnotemark[1]
\footnotetext[1]{as specified in {\sc Vale, Hargrove, Cockbill} and {\sc Phelps}
\cite{ValeHargroveCockbillAndPhelps}} kills 2\% day$^{-1}$ of the female {\em G.
pallidipes} population {\sc Hargrove} \cite{Hargrove6}.)
So far as {\em G. brevipalpis} is concerned, {\sc Esterhuizen}, {\sc Kappmeier
Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche}'s results are not as clear.
Indeed, the results of this work suggest their barrier--zone might have been
completely ineffectual against a very mobile {\em G. brevipalpis}. Another
possibility is that {\em G. brevipalpis} is completely impartial to
odour--baited targets. Yet something certainly did happen in both
target--containing sectors when the concentration of targets reached a density
of 8 km$^{-2}$. {\em G. brevipalpis} initially declined at a rate indicative of
an imposed mortality of 0.63 \% day$^{-1}$ (again, using {\sc Williams}, {\sc
Dransfield} and {\sc Brightwell} \cite{Williams1}'s esteemed equation). A
subsequent reversal of this decline then coincided with the demise of {\em G.
austeni} and the {\em G. brevipalpis} population grew to levels never previously
attained; in spite of the targets. A number of explanations spring to mind. One
argument is that the data is too poor, that what is being observed is simply
random noise, should be ignored. Another possibility is that there was a delay
in recolonization by this highly mobile species. Certainly one has good reason
to suspect an element of diffusion to be operative, even if not the overriding
analysis. Why then, the delay? There could be reasons.
An alternative explanation is that the reversal in fortune of the one species
coincided with the demise of the other due to the two being in competition: So
deleterious was the pressence of {\em G. austeni} to {\em G. brevipalpis} that
its removal is able to counteract the imposition of a 0.63 \% day$^{-1}$
mortality on {\em G. brevipalpis} (a decline rate of - 0.0039 day$^{-1}$). In
retrospect, such a situation might have been anticipated. Indeed, one of the
posits of this model is that the limitations on growth at pupal sites are
density dependent. Pupal habitat for {\em G. brevipalpis} is more stringently
confined than for {\em G. austeni} (according to demonstrations of the pupal
water loss model in {\sc Childs} \cite{Childs2}) and the {\em G. austeni}
puparial duration is a full 20 \% shorter than that of {\em G. brevipalpis}
({\sc Parker} \cite{Parker1}). One would imagine {\em G. brevipalpis} also has
an adverse effect on {\em G. austeni}. Just how severe and whether or not it can
be exploited, is not evident. Further experimentation is required. That {\sc
Esterhuizen}, {\sc Kappmeier Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche} were simply not able to
measure a true value for the target--related mortality of {\em G. brevipalpis},
owing to high diffusion rates, is their own conclusion.
If the accepted wisdom that the effect of uniformly distributed targets is
additive, then a given mortality may be designed in terms of Table
\ref{targetMortality} as follows.
\begin{table}[H]
\begin{center}
\begin{tabular}{c | c c c c}
& & & & \\
$\delta$ / day$^{-1}$ \hspace{5mm} & \hspace{5mm} 0.02 & \hspace{5mm} 0.04 & \hspace{5mm} 0.08 & \hspace{5mm} 0.12 \\
& & & & \\ \hline
& & & & \\
{\em G. austeni} \hspace{5mm} & \hspace{5mm} 7 & \hspace{5mm} 13 & \hspace{5mm} 27 & \hspace{5mm} 40 \\
& & & & \\
{\em G. brevipalpis} \hspace{5mm} & \hspace{5mm} 25 & \hspace{5mm} 51 & \hspace{5mm} 102 & \hspace{5mm} 152 \\
& & & & \\
{\em G. pallidipes} \hspace{5mm} & \hspace{5mm} 1 & \hspace{5mm} 2 & \hspace{5mm} 4 & \hspace{5mm} 6 \\
& & & & \\
\end{tabular}
\caption{The number of targets per km$^2$ which will produce a required daily mortality, $\delta$, for each species.} \label{targetMortality}
\end{center}
\end{table}
\subsection{Tethered, Treated Cattle}
Unpublished experiments by S. J. Torr (reported in {\sc Hargrove}, {\sc Torr}
and {\sc Kindness} \cite{HargroveTorrAndKindness}) suggest that a single
odour--baited target kills the equivalent number of {\em G. pallidipes} females
in 1 km$^2$ as an insecticide--treated ox of weight 400kg does in a day. Since
{\sc Esterhuizen}, {\sc Kappmeier Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche} used the same 1.5 $\times$ 1
m, black--blue--black targets (manufactured by Bonar Industries,
Harare)\footnotemark[1], \footnotetext[1]{{\sc Esterhuizen} \cite{Esterhuizen1}
and {\sc Hargrove} \cite{Hargrove9}} the corresponding target--related mortality
should apply to the ox for {\em G. austeni} and {\em G. brevipalpis}; assuming
these species do not discriminate any differently between the chemical
signatures of the beast and the target.
\subsection{Treated Herds}
In {\sc Hargrove}, {\sc Holloway}, {\sc Vale}, {\sc Gough} and {\sc Hall}
\cite{HargroveHollowayValeGoughAndHall} it was determined that tsetse catches
changed with the tonnage of cattle, $m$, in a ventilated shed and could be
described by
\begin{eqnarray*}
\delta \propto 4 m^{0.475}.
\end{eqnarray*}
Torr's experiment (reported in {\sc Hargrove}, {\sc Torr}
and {\sc Kindness} \cite{HargroveTorrAndKindness}) allows the constants in the simplistic model,
\begin{eqnarray*}
\left[ \begin{array}{c} 0.0030 \\ 0.00079 \\ 0.02 \end{array} \right] &=& 4 \ \left[ \begin{array}{c} c_{\scriptsize \mbox{austeni}} \\ c_{\scriptsize \mbox{brevipalpis}} \\ c_{\scriptsize \mbox{pallidipes}} \end{array} \right] \ 0.4^{0.475},
\end{eqnarray*}
to be determined. The minimum tonnage of cattle required to induce a given daily mortality in a square kilometre is therefore given by Table \ref{tonnage}.
\begin{table}[H]
\begin{center}
\begin{tabular}{c | c c c }
& & & \\
species & {\em G. austeni} & {\em G. brevipalpis} & {\em G. pallidipes} \\
& & & \\ \hline
& & & \\
herd mass / tons km$^{-2}$ & $\displaystyle 0.4 \left( \frac{\delta}{0.0030} \right)^{\frac{1}{0.475}}$ & $\displaystyle 0.4 \left( \frac{\delta}{0.00079} \right)^{\frac{1}{0.475}}$ & $\displaystyle 0.4 \left( \frac{\delta}{0.02} \right)^{\frac{1}{0.475}}$ \\
& & & \\
\end{tabular}
\caption{The treated herd mass required to bring about a given mortality, $\delta$, in each species.} \label{tonnage}
\end{center}
\end{table}
\section{Conclusions}
The premise that the entire reserve, and it alone, is a problem is not as valid
for {\em G. austeni} as it is in the case of {\em G. brevipalpis} (Figures
\ref{austeniProbability} and \ref{brevipalpisProbability}). In the case of {\em
G. austeni} it may well be worth singling out individual locii for barricading
(e.g. the Hluhluwe Dam, its backwater and the area in the vicinity of the
confluence of the Black iMfolozi and White iMfolozi rivers), based on {\sc
Childs} \cite{Childs2} and {\sc Hendrickx} \cite{Hendrickx}.
The diffusion rate of {\em G. austeni} is probably around 0.04 km$^2$
day$^{-1}$, if the reserve exerts an influence on surrounding population levels
7 to 10 km from its boundaries. If, however, the influence is as great as 10 to
15 km, the diffusion rate is as high as 0.08 km$^2$ day$^{-1}$ (Figures
\ref{1st} and \ref{austeniProbability}). The diffusion rate of {\em G.
brevipalpis} can be assumed to be around 0.32 km$^2$ day$^{-1}$, if the reserve
exerts an influence on surrounding population levels as distant as 20 to 25 km
from its boundaries. If, however, this influence is as little as 15 to 20 km,
the diffusion rate might be as low as 0.16 km$^2$ day$^{-1}$ (Figures
\ref{brevipalpisProbability} and \ref{2nd}).
Based on the worst--case values in terms of which the problem was phrased, the
simulations suggest that the temporary imposition of a 2 \% day$^{-1}$ mortality
everywhere outside the reserve for a period of 2 years will have no lasting
effect on the influence of the reserve when it comes to either population;
although it certainly will eradicate tsetse from areas of poor habitat, outside
the reserve (Figures \ref{pourOnsA1} to \ref{pourOnsB2}). It is doubtful whether
the populations within the reserve can be `siphoned' or `pumped out' to
extinction, or even the 20\% level, from outide the reserve boundary (Figure
\ref{pumping}).
The influence of the reserve on surrounding {\em G. austeni} population levels
can be completely negated by a 5 km--wide barrier in which there is a mortality
of 2 \% day$^{-1}$ throughout (Figure \ref{5kmA}). A 2.5 km--wide barrier of
targets will, however, not suffice (Figure \ref{2.5km}). For {\em G.
brevipalpis} a 5 km--wide barrier to the same end will require a mortality of 4
\% day$^{-1}$ throughout (Figure \ref{impenetrableB}). Notice, however, that
these measures are not in any way able to address the likelihood of more lethal
strains and a higher level of trypanosome infection in flies close to the
reserve boundary, regardless of any reduction in their numbers.
A 5 km--wide barrier of odour--baited targets with a mortality of 4 \%
day$^{-1}$, throughout, should succeed in completely isolating a worst--case,
Hluhluwe--iMfolozi {\em G. austeni} population and its associated more lethal
strains and higher levels of trypanosomiasis from the surrounding areas (Figure
\ref{impenetrableA}). A more optimistic estimate of its mobility suggests a
mortality of 2 \% day$^{-1}$ might suffice (Figure \ref{5kmA}). In the event
that {\em G. brevipalpis} is shown to be a vector of trypanosomiasis, a
worst--case mortality of 12 \% day$^{-1}$, throughout, will be required to
achieve the same end of complete isolation (Figure \ref{impenetrableB}). A
mortality of 8 \% day$^{-1}$ can be used at non--concave boundaries. It is
further recommended that any barrier include the surroundings of the Hluhluwe
dam and its backwater, as if it were part of the reserve, based on {\sc Childs}
\cite{Childs2}.
For a given mortality, more mobile species are found to be more vulnerable to
eradication than more sedentary species while the opposite is true for
containment. The scenarios depicted in Figures \ref{eradicationA} and
\ref{eradicationB} demonstrate, firstly, that high diffusion rates are more
amenable to eradication, since the same local consistancy is not required.
Species with high diffusion rates are vulnerable to controls which are
geographically more remote. Secondly, high diffusion rates lead to a much weaker
recovery from levels close to extinction\footnotemark[1]
\footnotetext[1]{Although this could be an artefact of assuming more mobile
species have a growth rate the same as more sedentary species.}. The reason is
that there is a tendency to disperse which is not efficacious at the lower
levels of logistic growth. This might seem obvious to the reader, yet one
popular theory for the phenomenal Lambwe Valley rebound (reported in {\sc Turner
and Brightwell} \cite{TurnerAndBrightwell} and {\sc Hargrove} \cite{Hargrove7})
is re--invasion, usually quoting the high mobility of {\em G. pallidipes}. A
very large neighbouring population would be required for such a rebound to be
attributed to re--invasion. More likely explanations would be the
over--estimation of the temperature at pupal sites, or a problem with
insecticide application.
Susceptability to odour--baited targets and existing population distributions
aside, one is now presented with a scenario in which {\em G. brevipalpis} may be
more vulnerable to eradication than containment and vice versa for {\em G.
austeni}. Yet whether or not {\em G. brevipalpis} is even an agent of infection
is still a moot point ({\sc Motloang}, {\sc Masumu}, {\sc Van Den Bossche}, {\sc
Majiwa} and {\sc Latif} \cite{MotloangMasumuVanDenBosscheMajiwaLatif}). {\em G.
austeni}, in contrast, is without the slightest doubt a highly competent vector
of trypanosomiasis.
The possibility that eliminating {\em G. brevipalpis} will create further
opportunity for {\em G. austeni} and, consequently, trypanosomiasis needs to be
considered. The experimental results of {\sc Esterhuizen}, {\sc Kappmeier
Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche} can be interpretted to lend
credence to exactly such a theory. They could suggest intense competition
between the two species, to the extent that {\em G. brevipalpis} may actually
benefit from odour--baited targets should their density be sufficient to
eliminate {\em G. austeni} only. The existance of a reciprocal effect on {\em G.
austeni} may be well worth investigating. Then again, what is observed could
simply be a delayed invasion response or even random noise. That {\sc
Esterhuizen}, {\sc Kappmeier Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche} were simply not able to
measure the target--related mortality of {\em G. brevipalpis}, owing to high
diffusion rates, is an alternative conclusion.
The required density of treated herds is not practical for the purposes of
barriers and containment. While tethered, deltamethrin--treated cattle can be
used as an alternative to odour--baited targets for savannah species, the
substitution is impractical in the case of forest species. (Individually
tethered, treated cattle distributed uniformly throughout a barrier zone may be
less likely than targets to fall victim to the tragedy--of--commons type
mentality known to prevail among the local population.)
The K. P. P. equation can be solved by way of the application of the finite element method for the spatial discretisation and a backward difference for the temporal discretisation. This strategy in combination with the linearisation and iteration of the nonlinear term also worked well for Fisher's equation, with good convergence for the range of conditions investigated.
A combination of warm temperatures, low imposed mortalities and long two year
cycles gave the population ample time to re--equilibrate with the result that
there was no discernable difference in the results obtained from either model.
The reader should, nonetheless, be cautioned against the use of a model based on
an unmodified Fisher's equation when faced with rather more severe circumstances
e.g. the catastrophic mortalities imposed by aerial spraying and lower
temperatures. Attributing subsequent growth to a current, as opposed to
historical, population would be profoundly incorrect under such circumstances.
Under the same circumstances, the model based on historical parentage would not
take the subsequent reproductive phase entrainment and altered age profile into
account. Reproductive rates would initially be over--estimated, later,
under--estimated and so on. Unlike Fisher's equation, however, the model is
expected to recover.
What of the danger of an altered age profile in the light of a longer than usual first interlarval period? Comparison of the results of the two models suggests that if circumstances allow the population to re--equilibrate there are unlikely to be any problems.
\subsection{Recommendations to Management}
In the case of {\em G. austeni} a permanent, 5 km--wide isolating barrier should
be maintained on the Eastern side of the reserve, around the Hluhluwe Dam and
its backwater, the vicinity of the confluence of the Black iMfolozi and White
iMfolozi rivers as well as other locii outside the reserve. The required
mortality of the barrier should be no less than 4 \% day$^{-1}$ for the
worst--case scenario. This value corresponds to a deployment of odour--baited
targets\footnotemark[1] \footnotetext[1]{As used by {\sc Esterhuizen}, {\sc
Kappmeier Green}, {\sc Nevill} and {\sc Van Den Bossche}
\cite{EsterhuizenKappmeierGreenNevillVanDenBossche}} with a minimum density of
13 km$^{-2}$. If, however, {\em G. austeni} is only diffusing at a rate as low
as 0.04 km$^2$ day$^{-1}$, then the required number of odour--baited
targets\footnotemark[1] is 7 km$^{-2}$. Tetherd, treated cattle can also be used
as substitutes. Periodically rotating them in and out of the barrier zone would
prevent a loss of resistance to tick--bourne diseases and an enzootic condition.
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
That {\sc Esterhuizen}, {\sc Kappmeier Green}, {\sc Nevill} and {\sc Van Den
Bossche} \cite{EsterhuizenKappmeierGreenNevillVanDenBossche} were simply not
able to measure the target--related mortality of {\em G. brevipalpis}, owing to
high diffusion rates, is one conclusion. Another is that the two species may be
in competition, a possibility which may or may not be exploitable. The
impartiallity of {\em G. brevipalpis} to odour--baited targets is, in any event,
obviously a concern. Should this species be conclusively shown to be a vector, a
permanent, 5 km--wide barrier around the reserve as well as that part of the
Hluhluwe dam and its backwater lying outside the reserve would need to be
maintained. The required mortality of the barrier could be as high as 12 \%
day$^{-1}$\footnotemark[2] \footnotetext[2]{12 \% day$^{-1}$ where the boundary
is concave and 8 \% day$^{-1}$ where convex} to achieve complete isolation. This
is obviously not practical in terms of what one can only surmise is the
mortality of current odour--baited target technology. The less ambitious goal of
negating the reserves influence on the surrounding {\em G. brevipalpis}
population would require a mortality of 4 \% day$^{-1}$, throughout. This value
corresponds to a deployment of odour--baited targets with a minimum density of
51 km$^{-2}$; again a clearly impractical proposition.
\renewcommand{\thefootnote}{\arabic{footnote}}
\section{Acknowledgements}
Abdalla Latif and the Onderstepoort Veterinary Institute are once again thanked
for their generousity in co-funding this research. Guy Hendrickx is gratefully
acknowledged for donating the two maps of tsetse risk and Andrew Parker once
again thanked for the information on puparial durations. Other, general
information on the iMfolozi--Hluhluwe game reserve was supplied by Ezemvelo
K.Z.N. Wildlife and the satellite image was kindly supplied by Marina Faber of
Peace Parks Foundation. Brian Williams is thanked for taking a general interest
in this work and John Hargrove, for sharing his vast knowledge of tsetse.
This work is obviously a synthesis of the monumental research efforts and
pioneering work carried out by the likes of Jackson, Bursell, Phelps, Vale,
Rogers, Williams and Hargrove, to name but a few. This work is, to a large
extent, theirs with the appropriate numerical methods applied.
\nocite{Murray1}
\nocite{Anonymous2}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,791
|
package smodelkit;
import java.io.File;
import java.io.FileNotFoundException;
import java.io.Serializable;
import java.util.ArrayList;
import java.util.Collections;
import java.util.Iterator;
import java.util.List;
import java.util.Map.Entry;
import java.util.Random;
import java.util.Scanner;
import java.util.Set;
import java.util.TreeMap;
import java.util.TreeSet;
import java.util.stream.Stream;
import java.util.stream.StreamSupport;
import smodelkit.util.Counter;
import smodelkit.util.Helper;
import smodelkit.util.Pair;
import smodelkit.util.Range;
import smodelkit.util.Tuple2Comp;
/**
* Represents a dataset.
*
*/
public class Matrix implements Serializable, Iterable<Vector>
{
private static final long serialVersionUID = 1L;
String relationName;
String comments;
// Stores instance rows and their weights.
ArrayList<Vector> data;
// Meta-data
ArrayList<String> attrNames;
ArrayList<TreeMap<String, Integer>> strToEnum;
ArrayList<TreeMap<Integer, String>> enumToStr;
// This stores the number of columns that are used in each categorical distribution
// when NominalToCategorical is used. This assumes all columns are either converted
// from nominal values or are all real valued, but not mixed.
List<Integer> numCatagoricalCols;
/**
* This is the number of columns (from the right side) that are output columns.
* This is set using -c in the relation name. The default is 1.
*/
private int numLabelColumns;
/**
* Creates a 0x0 matrix. You should call loadARFF or setSize next.
*/
public Matrix()
{
relationName = null;
data = new ArrayList<>();
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
numCatagoricalCols = new ArrayList<Integer>();
numLabelColumns = 1;
}
/**
* Copy constructor. Everything is copied except the internal arrays in Vectors,
* which are immutable.
* @param other
*/
public Matrix(Matrix other)
{
relationName = other.relationName;
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
for (int i = 0; i < other.cols(); i++)
{
attrNames.add(other.getAttrName(i));
strToEnum.add(other.strToEnum.get(i));
enumToStr.add(other.enumToStr.get(i));
}
numCatagoricalCols = new ArrayList<Integer>();
for (int i : other.numCatagoricalCols)
{
numCatagoricalCols.add(i);
}
data = new ArrayList<>();
for (int j = 0; j < other.rows(); j++)
{
addRow(other.row(j));
}
numLabelColumns = other.numLabelColumns;
}
/**
* Copies the specified portion of that matrix into this matrix
*/
public Matrix(Matrix other, int rowStart, int colStart, int rowCount,
int colCount)
{
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
for (int i = 0; i < colCount; i++)
{
attrNames.add(other.getAttrName(colStart + i));
strToEnum.add(other.strToEnum.get(colStart + i));
enumToStr.add(other.enumToStr.get(colStart + i));
}
if (other.numCatagoricalCols.size() != 0)
throw new UnsupportedOperationException("Categorical distributions are not supported when copying " +
" a part of a matrix.");
numCatagoricalCols = new ArrayList<Integer>();
data = new ArrayList<>();
for (int j = 0; j < rowCount; j++)
{
Vector rowSrc = other.row(rowStart + j);
double[] rowDest = new double[colCount];
for (int i = 0; i < colCount; i++)
rowDest[i] = rowSrc.get(colStart + i);
addRow(new Vector(rowDest, rowSrc.getWeight()));
}
numLabelColumns = 0;
}
public void copyMetadata(Matrix other)
{
relationName = other.relationName;
attrNames = new ArrayList<String>(other.attrNames);
strToEnum = new ArrayList<TreeMap<String, Integer>>(other.strToEnum);
enumToStr = new ArrayList<TreeMap<Integer, String>>(other.enumToStr);
numCatagoricalCols = new ArrayList<Integer>(other.numCatagoricalCols);
numLabelColumns = other.numLabelColumns;
}
/**
* Adds an empty attribute column as the last column in this matrix.
* By default this column is continuous. To make it nominal, add attribute
* values by calling addAttributeValue.
*/
public void addEmptyColumn(String attributeName)
{
if (attrNames.contains(attributeName))
throw new IllegalArgumentException("This matrix alread contains attribute name: " + attributeName);
attrNames.add(attributeName);
strToEnum.add(new TreeMap<>());
enumToStr.add(new TreeMap<>());
}
/**
* Add an attribute value to the specified column.
* @param column
* @param attributeValueName
*/
public void addAttributeValue(int column, String attributeValueName)
{
if (strToEnum.get(column).keySet().contains(attributeValueName))
throw new IllegalArgumentException("Column " + column + " alread contains attribute value: " + attributeValueName);
int largestAttrIndex = enumToStr.get(column).isEmpty() ? 0 :
Collections.max(enumToStr.get(column).keySet()) + 1;
strToEnum.get(column).put(attributeValueName, largestAttrIndex);
enumToStr.get(column).put(largestAttrIndex, attributeValueName);
}
public void addAttributeValueIfItDoesNotExist(int column, String attributeValueName)
{
if (strToEnum.get(column).keySet().contains(attributeValueName))
return;
int largestAttrIndex = enumToStr.get(column).isEmpty() ? 0 :
Collections.max(enumToStr.get(column).keySet()) + 1;
strToEnum.get(column).put(attributeValueName, largestAttrIndex);
enumToStr.get(column).put(largestAttrIndex, attributeValueName);
}
public int getNumLabelColumns()
{
return numLabelColumns;
}
public void setNumLabelColumns(int numLabelColumns)
{
this.numLabelColumns = numLabelColumns;
}
/** Adds a copy of the specified portion of that matrix to this matrix
*
* The number of columns that will be copied to this matrix is this.cols().
*
* @param other
* @param rowStart
* @param colStart
* @param rowCount
* @throws Exception
*/
public void add(Matrix other, int rowStart, int colStart, int rowCount)
{
if (colStart + cols() > other.cols())
throw new IllegalArgumentException("out of range");
for (int i = 0; i < cols(); i++)
{
if (other.getValueCount(colStart + i) != getValueCount(i))
throw new IllegalArgumentException("incompatible relations");
}
for (int j = 0; j < rowCount; j++)
{
Vector rowSrc = other.row(rowStart + j);
double[] rowDest = new double[cols()];
for (int i = 0; i < cols(); i++)
rowDest[i] = rowSrc.get(colStart + i);
addRow(new Vector(rowDest, rowSrc.getWeight()));
}
}
public void addRows(Matrix other, int num)
{
addRows(other, 0, num);
}
public void addRows(Matrix other, int start, int num)
{
for (int i = start; i < start + num; i++)
addRow(other.row(i));
}
public void addRows(Matrix other, List<Integer> rowsToCopy)
{
for (int i : rowsToCopy)
{
addRow(other.row(i));
}
}
public void removeRow(int row)
{
data.remove(row);
}
public void removeColumn(int colNumber)
{
if (numCatagoricalCols.size() != 0)
throw new UnsupportedOperationException("Cannot remove columns from a matrix that has been filtered "
+ "by NominalToCategorical.java");
attrNames.remove(colNumber);
strToEnum.remove(colNumber);
enumToStr.remove(colNumber);
for(int i = 0; i < rows(); i++)
{
data.get(i).remove(colNumber);
}
assert attrNames.size() == row(0).size();
}
/**
* Copies num columns from that to this beginning at start. Meta-data is copied too.
*
* If this is empty, then instance weights are set to those from other. If this is not empty,
* then instance weights are not changed.
*/
public void copyColumns(Matrix other, int start, int num)
{
if (rows() > 0 && rows() != other.rows())
throw new IllegalArgumentException("other must have the same number of rows as this.");
if (start + num > other.cols())
{
throw new IllegalArgumentException("Index out of range.");
}
if (num == 0)
{
throw new IllegalArgumentException("Did you mean to copy 0 columns?");
}
addListRange(other.attrNames, attrNames, start, num);
addListRange(other.strToEnum, strToEnum, start, num);
addListRange(other.enumToStr, enumToStr, start, num);
if (other.numCatagoricalCols.size() > 0)
{
// Make sure we are copying over a whole categorical distribution. Currently I do
// not support copying more than one categorical distribution.
int catStart = 0;
int i = 0;
while(catStart < start)
{
catStart += other.numCatagoricalCols.get(i);
i++;
}
if (catStart != start)
throw new IllegalArgumentException("Argument give for start is not at the beginning of a "
+ "categorical distribution.");
if (other.numCatagoricalCols.get(i) != num)
throw new IllegalArgumentException("Given num=" + num + ", which is not the size of the categorical"
+ " distribution (" + other.numCatagoricalCols.get(i) + ")."
+ "\"start\"=" + start + ".");
numCatagoricalCols.add(other.numCatagoricalCols.get(i));
}
if (data.size() > 0)
{
for(int r = 0; r < other.rows(); r++)
{
double[] toAdd = new double[num];
for (int c = 0; c < toAdd.length; c++)
toAdd[c] = other.row(r).get(start + c);
row(r).addAll(new Vector(toAdd));
}
}
else
{
// I don't have any rows.
for(int r = 0; r < other.rows(); r++)
{
addRow(other.row(r).subVector(start, start + num));
}
}
}
/**
* Creates a new matrix like this one but with only the specified columns.
* @return
*/
public Matrix selectColumns(List<Integer> columns)
{
Matrix result = new Matrix();
for (int c : columns)
{
result.copyColumns(this, c, 1);
}
return result;
}
/**
* Returns a copy of the specified columns of this matrix. Meta-data is copied too.
*/
public Matrix getColumns(int start, int num)
{
Matrix result = new Matrix();
result.copyColumns(this, start, num);
return result;
}
/**
* Returns the index of the column of the given attribute name, or -1 if
* it is not found.
*/
public int getAttributeColumnIndex(String attrName)
{
return attrNames.indexOf(attrName);
}
/**
* Gets the value from the specified row corresponding to the specified attribute.
*/
public double get(int rowNumber, String attrName)
{
int index = attrNames.indexOf(attrName);
if (index == -1)
throw new IllegalArgumentException("Attribute name \"" + attrName + "\" not found.");
return row(rowNumber).get(index);
}
public Matrix getColumnsIgnoringFilteredCatagoricalColumns(int start, int num)
{
Matrix result = new Matrix();
this.numCatagoricalCols = new ArrayList<>();
result.copyColumns(this, start, num);
return result;
}
/**
* Adds num elements to the end of dest, starting with source.get(start).
*/
private <T> void addListRange(List<T> source, List<T> dest, int start, int num)
{
for (int i = start; i < num + start; i++)
{
dest.add(source.get(i));
}
}
/*
* Adds a copy of the given vector to this datset.
*/
public void addRow(Vector v)
{
v = new Vector(v);
// Verify the given row.
if (v.size() != cols())
throw new IllegalArgumentException("The given row is not the expected size for this dataset.");
for (int i : new Range(v.size()))
{
if (!isContinuous(i))
{
if (v.get(i) < 0)
throw new IllegalArgumentException("Nominal values cannot be negative.");
if (v.get(i) >= getValueCount(i) && !Vector.isUnknown(v.get(i)))
{
throw new IllegalArgumentException("Nominal value is out of range.");
}
}
}
data.add(v);
}
public void checkCompatibility(Matrix other)
{
int c = cols();
if(other.cols() != c)
throw new IllegalArgumentException("Matrices have different number of columns");
for(int i = 0; i < c; i++)
{
if(getValueCount(i) != other.getValueCount(i))
throw new IllegalArgumentException("Column " + i + " has mis-matching number of values");
}
}
public void clear()
{
attrNames.clear();
strToEnum.clear();
enumToStr.clear();
data.clear();
}
/**
* Resizes this matrix (and sets all attributes to be continuous, with instance weights of 1)
*/
public void setSize(int rows, int cols)
{
data = new ArrayList<>();
for (int j = 0; j < rows; j++)
{
double[] row = new double[cols];
addRow(new Vector(row, 1.0));
}
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
for (int i = 0; i < cols; i++)
{
attrNames.add("");
strToEnum.add(new TreeMap<String, Integer>());
enumToStr.add(new TreeMap<Integer, String>());
}
}
/**
* Loads from an ARFF file
* @param filename
* @param loadComments If true, any comments before the relation name will be loaded.
*/
public void loadFromArffFile(String filename, boolean loadComments)
{
try (Scanner s = new Scanner(new File(filename)))
{
loadArff(s, loadComments);
}
catch (FileNotFoundException e)
{
throw new RuntimeException(e);
}
}
/**
* Loads from an ARFF file
* @param filename
*/
public void loadFromArffFile(String filename)
{
loadFromArffFile(filename, false);
}
/**
* Loads from a string containing data in the arff format.
* @param filename
* @throws FileNotFoundException
*/
public void loadFromArffString(String content)
{
try (Scanner s = new Scanner(content))
{
loadArff(s, false);
}
}
/**
* Loads from an ARFF file
* @param filename
* @throws FileNotFoundException
*/
private void loadArff(Scanner s, boolean loadComments)
{
data = new ArrayList<>();
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
boolean READDATA = false;
StringBuilder commentsBuilder = new StringBuilder();
while (s.hasNext())
{
String line = s.nextLine().trim();
if (line.length() > 0)
{
if (line.charAt(0) == '%')
{
if (loadComments)
{
commentsBuilder.append(line);
commentsBuilder.append("\n");
}
}
else
{
if (!READDATA)
{
try (Scanner t = new Scanner(line))
{
String firstToken = t.next().toUpperCase();
if (firstToken.equals("@RELATION"))
{
QuoteParser u = new QuoteParser(line);
u.next();
relationName = u.next().replace("'", "").replace("\"", "");
if (relationName.toUpperCase().contains("-C"))
{
QuoteParser p = new QuoteParser(relationName);
while (!p.next().toUpperCase().equals("-C"))
{
}
String labelColsStr = p.next();
// Get the number of label columns.
numLabelColumns = Integer.parseInt(labelColsStr);
if (numLabelColumns > 0)
throw new IllegalArgumentException("Label columns at the beginning of "
+ "a dataset are not supported, so arguments given to -c in"
+ " arff files must be negative. This is necessary to make dataset compatible"
+ " with Meka.");
// I need to be able to add a negative sign, and ignore it, to make my datasets compatible with Meka.
numLabelColumns = Math.abs(numLabelColumns);
}
if (relationName.contains(":"))
{
relationName = relationName.split(":")[0];
}
t.nextLine();
}
if (firstToken.equals("@ATTRIBUTE"))
{
TreeMap<String, Integer> ste = new TreeMap<String, Integer>();
strToEnum.add(ste);
TreeMap<Integer, String> ets = new TreeMap<Integer, String>();
enumToStr.add(ets);
QuoteParser parser = new QuoteParser(line);
parser.next();
String attributeName = parser.next();
if (attributeName.equals("?"))
throw new IllegalArgumentException("\"?\" is a reserved token. Found in line: " + line);
attrNames.add(attributeName);
int vals = 0;
String type = parser.next().toUpperCase();
if (type.equals("REAL")
|| type.equals("CONTINUOUS")
|| type.equals("INTEGER")
|| type.equals("NUMERIC"))
{
}
else
{
if (!line.contains("{"))
throw new RuntimeException("Missing \"{\" from line: " + line );
if (!line.contains("}"))
throw new RuntimeException("Missing \"}\"");
String values = line.substring(
line.indexOf("{") + 1,
line.indexOf("}"));
try (Scanner v = new Scanner(values))
{
v.useDelimiter(",");
while (v.hasNext())
{
String value = v.next().trim();
if (value.length() > 0)
{
if (value.equals("?"))
throw new IllegalArgumentException("\"?\" is a reserved token. Found in line: " + line);
ste.put(value, new Integer(
vals));
ets.put(new Integer(vals),
value);
vals++;
}
}
}
}
}
if (firstToken.equals("@DATA"))
{
READDATA = true;
}
}
}
else
{
loadDataRow(line, Collections.emptyList());
}
}
}
}
comments = commentsBuilder.toString();
validate();
}
/**
* Throws an exception if this matrix is not valid.
*/
private void validate()
{
// Check for duplicate attribute names.
Set<String> prev = new TreeSet<>();
int i = 0;
for (String name : attrNames)
{
if (prev.contains(name))
throw new IllegalArgumentException("Dupilcate attribute names are not allowed in arff"
+ " format. Duplicate name: " + name + ", index: " + i);
prev.add(name);
i++;
}
}
private void loadDataRow(String line, List<Integer> ignoredColumns)
{
double[] newRow = new double[cols()];
double instanceWeight = 1.0;
// There are 2 ways to store a data row: sparse or not sparse. The non-sparse way stores
// all values separated by commas.
// The sparse way stores key value pairs separate by commas, where they key is the index
// of the attribute, and the value is the value it has. Indexes start at 0. If an attribute
// does not have a value specified, it will be zero, or the first nominal value.
line = line.trim();
if (line.startsWith("{") && line.endsWith("}"))
{
// Data is stored in key value pairs.
try (Scanner t = new Scanner(line))
{
t.useDelimiter(",");
boolean atLeastOnePieceOfDataRead = false;
while (t.hasNext())
{
String next = t.next().trim();
if (!t.hasNext() && next.startsWith("{") && next.endsWith("}") && atLeastOnePieceOfDataRead)
{
instanceWeight = parseInstanceWeight(next, line);
continue;
}
// Remove the curly brackets if present.
if (next.charAt(0) == '{')
next = next.substring(1, next.length());
if (next.charAt(next.length() - 1) == '}')
next = next.substring(0, next.length() - 1);
// Check for the case where a row is just {}, verses has empty key-value pairs such as {,}.
if (next.trim().isEmpty())
{
if (line.contains(","))
throw new IllegalArgumentException("Value \"" + next + "\" does not specify a key-value pair. Line: " + line);
else
break;
}
String[] parts = next.split(" ");
if (parts.length != 2)
throw new IllegalArgumentException("Value \"" + next + "\" does not specify a key-value pair. Line: " + line);
int attrIndex;
try
{
attrIndex = Integer.parseInt(parts[0]);
}
catch(NumberFormatException e)
{
throw new IllegalArgumentException("Cannot parse attribute index \"" + parts[0] + "\" from line: " + line);
}
if (attrIndex >= strToEnum.size())
{
throw new IllegalArgumentException("Index \"" + attrIndex + "\" is out of range in line: " + line);
}
if (isContinuous(attrIndex))
{
try
{
newRow[attrIndex] = Double.parseDouble(parts[1]);
}
catch(NumberFormatException e)
{
throw new IllegalArgumentException("Expected a continuous value, but got \"" + parts[1] + "\" in line: " + line);
}
}
else
{
Integer attrValueAsInteger = strToEnum.get(attrIndex).get(parts[1]);
if (attrValueAsInteger == null)
{
throw new IllegalArgumentException("Unrecognized attribute value \"" + parts[1]
+ "\" for attribute \"" + getAttrName(attrIndex) + "\" in line: " + line);
}
double doubleValue = (int)attrValueAsInteger;
if (doubleValue == -1)
{
throw new IllegalArgumentException("Error parsing the value '" + parts[1] + "' on line: " + line);
}
if (newRow[attrIndex] != 0)
{
throw new IllegalArgumentException("Attribute " + attrIndex + " is specified multiple times in line: " + line);
}
newRow[attrIndex] = doubleValue;
}
atLeastOnePieceOfDataRead = true;
}
}
}
else
{
// Data is stored with every value specified separated by commas.
int curPos = 0;
int curPosInAttributeNames = 0;
try (Scanner t = new Scanner(line))
{
t.useDelimiter(",");
while (t.hasNext())
{
String textValue = t.next().trim();
if (textValue.isEmpty())
throw new RuntimeException("Line contains empty string in column " + curPos + ": " + line);
if (!t.hasNext() && textValue.startsWith("{") && textValue.endsWith("}"))
{
instanceWeight = parseInstanceWeight(textValue, line);
continue;
}
if (ignoredColumns.contains(curPos))
{
curPos++;
continue;
}
double doubleValue;
int vals;
try
{
vals = enumToStr.get(curPosInAttributeNames).size();
}
catch(IndexOutOfBoundsException e)
{
throw new IllegalArgumentException("The given line has more entries than their are attributes. Line: " + line);
}
// Missing instances appear in the dataset
// as a double defined in Vector.getUnknownValue().
if (textValue.equals("?"))
{
doubleValue = Vector.getUnknownValue();
}
// Continuous values appear in the instance
// vector as they are
else if (vals == 0)
{
doubleValue = Double
.parseDouble(textValue);
}
// Discrete values appear as an index to the "name"
// of that value in the "attributeValue" structure
else
{
if (!strToEnum.get(curPosInAttributeNames).containsKey(textValue))
{
throw new RuntimeException(String.format(
"Unknown attribute value \"%s\" for attribute \"%s\" in line:\n%s",
textValue, getAttrName(curPosInAttributeNames), line));
}
doubleValue = strToEnum.get(curPosInAttributeNames)
.get(textValue);
if (doubleValue == -1)
{
throw new RuntimeException(
"Error parsing the value '"
+ textValue
+ "' on line: "
+ line);
}
}
newRow[curPosInAttributeNames] = doubleValue;
curPos++;
curPosInAttributeNames++;
}
}
}
addRow(new Vector(newRow, instanceWeight));
}
/**
* Parses an instance weight from between { }.
* @param line For debugging only.
*/
private double parseInstanceWeight(String str, String line)
{
try
{
return Double.parseDouble(str.substring(1, str.length() - 1));
}
catch(NumberFormatException e)
{
throw new NumberFormatException("Unable to parse instance weight in line: " + line);
}
}
/**
* Loads from a .names and .data or .test file.
* @throws FileNotFoundException
*/
public void loadFromNamesFormat(String namesFilename, String dataFilename) throws FileNotFoundException
{
List<Integer> ignoredColumns = loadNamesFile(namesFilename);
loadDataFile(dataFilename, ignoredColumns);
validate();
}
/**
* @return Columns that should be ignored when reading the data file.
* @throws FileNotFoundException
*/
private List<Integer> loadNamesFile(String namesFilename) throws FileNotFoundException
{
data = new ArrayList<>();
attrNames = new ArrayList<String>();
strToEnum = new ArrayList<TreeMap<String, Integer>>();
enumToStr = new ArrayList<TreeMap<Integer, String>>();
ArrayList<String> output_attr_name = new ArrayList<String>();
ArrayList<TreeMap<String, Integer>> output_str_to_enum = new ArrayList<TreeMap<String, Integer>>();
ArrayList<TreeMap<Integer, String>> output_enum_to_str = new ArrayList<TreeMap<Integer, String>>();
int outputCols = 1;
int lineNumber = 0;
List<Integer> ignoredColumns = new ArrayList<Integer>();
try (Scanner s = new Scanner(new File(namesFilename)))
{
while (s.hasNext())
{
lineNumber++;
String line = s.nextLine();
line = line.replaceAll("\\s|\\.", "");
if (line.isEmpty())
continue;
// Remove comments.
int commentStartIndex = line.indexOf('|');
if (commentStartIndex != -1)
line = line.substring(0, commentStartIndex);
if (lineNumber == 1)
{
if (Helper.isInteger(line))
{
// This dataset has multiple target classes.
outputCols = Integer.parseInt(line);
}
}
else if (output_attr_name.size() < outputCols)
{
// Load the output column meta-data.
String colName = "class" + (output_attr_name.size() + 1);
output_attr_name.add(colName);
TreeMap<String, Integer> ste = new TreeMap<String, Integer>();
TreeMap<Integer, String> ets = new TreeMap<Integer, String>();
String[] attrValues = line.split(",");
for (int i = 0; i < attrValues.length; i++)
{
ste.put(attrValues[i], i);
ets.put(i, attrValues[i]);
}
output_str_to_enum.add(ste);
output_enum_to_str.add(ets);
}
else
{
// Load feature meta-data.
String[] parts = line.split(":");
if (parts.length != 2)
throw new IllegalArgumentException("Expected exactly 1 ':' in the line: " + line);
if (parts[1].equals("ignore"))
{
ignoredColumns.add(attrNames.size() + ignoredColumns.size());
}
else if (parts[1].equals("continuous"))
{
attrNames.add(parts[0]);
strToEnum.add(new TreeMap<String, Integer>());
enumToStr.add(new TreeMap<Integer, String>());
}
else
{
attrNames.add(parts[0]);
// Get the attribute names.
TreeMap<String, Integer> ste = new TreeMap<String, Integer>();
TreeMap<Integer, String> ets = new TreeMap<Integer, String>();
String[] attrValues = parts[1].split(",");
for (int i = 0; i < attrValues.length; i++)
{
ste.put(attrValues[i], i);
ets.put(i, attrValues[i]);
}
strToEnum.add(ste);
enumToStr.add(ets);
}
}
}
}
attrNames.addAll(output_attr_name);
strToEnum.addAll(output_str_to_enum);
enumToStr.addAll(output_enum_to_str);
return ignoredColumns;
}
private void loadDataFile(String dataFilename, List<Integer> ignoredColumns) throws FileNotFoundException
{
try (Scanner s = new Scanner(new File(dataFilename)))
{
while (s.hasNext())
{
String line = s.next();
// Remove this temp code when done matching Xinchuan's results.
// // This is for extracting Xinchuan's QT data.
// double qt;
// if (dataFilename.endsWith(".test"))
// qt = 1.0;
// else
// qt = Double.NEGATIVE_INFINITY;
// String[] parts = line.split(",");
// if (Double.parseDouble(parts[parts.length - 1]) >= qt)
// {
// parts = Arrays.copyOf(parts, parts.length -1);
// line = StringUtils.join(parts, ",");
// loadDataRow(line.replace(", " + parts[parts.length - 1], ""), ignoredColumns);
// }
loadDataRow(line, ignoredColumns);
}
}
}
// Returns the number of rows in the matrix
public int rows()
{
return data.size();
}
// Returns the number of columns (or attributes) in the matrix
public int cols()
{
return attrNames.size();
}
/**
* Returns the specified row. The values in the result must not be changed because
* an array of values for an instance might be shared by other parts of the code.
* @param r The index of the row to return.
*/
public Vector row(int r)
{
return data.get(r);
}
// Returns the name of the specified attribute
public String getAttrName(int col)
{
return attrNames.get(col);
}
// Returns the name of the specified value
public String getAttrValueName(int attr, int val)
{
String result = enumToStr.get(attr).get(val);
if (result == null)
throw new IndexOutOfBoundsException(
String.format("Attribute \"%s\" does not have the value %s.", getAttrName(attr), val));
return result;
}
/**
* Returns the index of the specified attribute name in the specified attribute column.
*/
public int getAttrValueIndex(int attr, String attrValueName)
{
return strToEnum.get(attr).get(attrValueName);
}
/**
* Returns the number of values associated with the specified attribute (or column)
* 0=continuous, 2=binary, 3=trinary, etc.
*/
public int getValueCount(int col)
{
return enumToStr.get(col).size();
}
public boolean isContinuous(int column)
{
return getValueCount(column) == 0;
}
public int countValues(int column, double value)
{
int count = 0;
for (int r = 0; r < rows(); r++)
{
if (row(r).get(column) == value)
count++;
}
return count;
}
/**
* Shuffles the row order
*/
public void shuffle(Random rand)
{
for (int n = rows(); n > 0; n--)
{
int i = rand.nextInt(n);
Vector tmp = data.get(n - 1);
data.set(n - 1, data.get(i));
data.set(i, tmp);
}
}
public void shuffle(Random rand, Matrix buddy)
{
for (int n = rows(); n > 0; n--)
{
int i = rand.nextInt(n);
Vector tmp = data.get(n - 1);
data.set(n - 1, data.get(i));
data.set(i, tmp);
tmp = buddy.data.get(n - 1);
buddy.data.set(n - 1, buddy.data.get(i));
buddy.data.set(i, tmp);
}
}
// Returns the mean of the specified column
public double findMean(int col)
{
double sum = 0;
int count = 0;
for (int i = 0; i < rows(); i++)
{
double v = row(i).get(col);
if (!Vector.isUnknown(v))
{
sum += v;
count++;
}
}
return sum / count;
}
// Returns the min value in the specified column
public double findMin(int col)
{
double m = Vector.getUnknownValue();
for (int i = 0; i < rows(); i++)
{
double v = row(i).get(col);
if (!Vector.isUnknown(v))
{
if (Vector.isUnknown(m) || v < m)
m = v;
}
}
return m;
}
// Returns the max value in the specified column
public double findMax(int col)
{
double m = Vector.getUnknownValue();
for (int i = 0; i < rows(); i++)
{
double v = row(i).get(col);
if (!Vector.isUnknown(v))
{
if (Vector.isUnknown(m) || v > m)
m = v;
}
}
return m;
}
/**
* Returns the most common value in the specified column
*/
public double findMode(int col)
{
TreeMap<Double, Integer> tm = new TreeMap<Double, Integer>();
for (int i = 0; i < rows(); i++)
{
double v = row(i).get(col);
if (!Vector.isUnknown(v))
{
Integer count = tm.get(v);
if (count == null)
tm.put(v, new Integer(1));
else
tm.put(v, new Integer(count.intValue() + 1));
}
}
int maxCount = 0;
double val = Vector.getUnknownValue();
Iterator<Entry<Double, Integer>> it = tm.entrySet().iterator();
while (it.hasNext())
{
Entry<Double, Integer> e = it.next();
if (e.getValue() > maxCount)
{
maxCount = e.getValue();
val = e.getKey();
}
}
return val;
}
/**
* Returns a copy of all of the columns in this matrix which correspond to the nominal class
* specified by labelClass. This assumes that the NominalToCatagorical filter was used to
* create this matrix.
*/
public Matrix getCategoricalizedLabelCols(int labelClass)
{
if (labelClass > numCatagoricalCols.size())
throw new IllegalArgumentException("labelClass out of range");
int start = 0;
for (int i = 0; i < labelClass; i++)
{
start += numCatagoricalCols.get(i);
}
Matrix result = new Matrix();
result.copyColumns(this, start, numCatagoricalCols.get(labelClass));
return result;
}
/**
* Finds all column number in this matrix corresponding to the nominal value which
* col is a column from.
* @return A pair containing the first (inclusive) and last (exclusive) columns.
*/
public Tuple2Comp<Integer, Integer> getColumnsCorespondingToFilteredNominalLabelInColumn(int col)
{
// Find the column number where the filtered nominal value starts.
int first = 0;
int i = 0;
for (; first + numCatagoricalCols.get(i) < col; i++)
{
first += numCatagoricalCols.get(i);
}
return new Tuple2Comp<>(first, first + numCatagoricalCols.get(i));
}
public List<Integer> getNumCatagoricalCols() { return Collections.unmodifiableList(numCatagoricalCols); }
public void setNumCatagoricalCols(List<Integer> numCatagoricalCols)
{
this.numCatagoricalCols = new ArrayList<Integer>(numCatagoricalCols);
}
/**
* If this matrix has been filtered using NominalToCatagorical, this will return the number of
* nominal column in the original matrix (before filtering). Otherwise this will be 0.
*/
public int getFilteredNominalColsTotal() { return numCatagoricalCols.size(); }
/**
* Counts the number of nominal columns in this matrix. If the filter NominalToCatagorical has been
* applied, this should be 0.
* @return
*/
public int countNominalCols()
{
int count = 0;
for (int i = 0; i < cols(); i++)
{
if (!isContinuous(i))
{
count++;
}
}
return count;
}
public boolean hasNominalCols()
{
for (int i = 0; i < cols(); i++)
{
if (!isContinuous(i))
return true;
}
return false;
}
public boolean hasContinuousCols()
{
for (int i = 0; i < cols(); i++)
{
if (isContinuous(i))
return true;
}
return false;
}
public boolean containsUnknowns()
{
for (Vector vec : this)
{
for (int d = 0; d < vec.size(); d++)
{
if (Vector.isUnknown(vec.get(d)))
return true;
}
}
return false;
}
public String getRelationName()
{
return relationName;
}
public void setRelationName(String name)
{
relationName = name;
}
public String getComments()
{
return comments;
}
public void setComments(String comments)
{
this.comments = comments;
}
/**
* Returns a new matrix in which classes are balanced by oversampling
* the rows corresponding to under-represented classes.
*
* This will give strange results if the class (last column) is real
* valued and takes on many values.
*/
public Matrix oversample(Random rand)
{
if (numLabelColumns > 1)
throw new UnsupportedOperationException("Oversampling of mutli-dimensional datasets is not"
+ " supported.");
Counter<Double> counter = new Counter<Double>();
for (int r : new Range(this.rows()))
{
counter.increment(row(r).get(this.cols() - 1));
}
Matrix result = new Matrix(this);
int maxCount = counter.maxCount();
// For each label value in the counter:
for (Double output : counter.keySet())
{
// Find all rows with that label value.
List<Vector> rowsWithOutput = new ArrayList<>();
for (int r : new Range(this.rows()))
{
if (row(r).get(this.cols() - 1) == output)
{
rowsWithOutput.add(data.get(r));
}
}
// Sample from those rows until the count equals maxCount.
while(counter.getCount(output) < maxCount)
{
Vector instanceAndWeight = rowsWithOutput.get(rand.nextInt(rowsWithOutput.size()));
result.addRow(instanceAndWeight);
counter.increment(output);
}
}
return result;
}
/**
* Splits the inputs and labels into new Matrixes.
* @return The first element is the inputs. The second is the labels.
*/
public Pair<Matrix> splitInputsAndLabels()
{
Matrix inputs = new Matrix(this, 0, 0, this.rows(), this.cols() - this.getNumLabelColumns());
Matrix labels = new Matrix(this, 0, this.cols() - this.getNumLabelColumns(), this.rows(),
this.getNumLabelColumns());
return new Pair<>(inputs, labels);
}
@Override
public String toString()
{
StringBuilder result = metaDataToString();
result.append("@DATA\n");
for (int i = 0; i < rows(); i++)
{
result.append(rowToString(i));
result.append("\n");
}
return result.toString();
}
/**
* Converts this matrix to a string using a sparse format for attribute values.
* @return
*/
public String toStringSparse()
{
StringBuilder result = metaDataToString();
result.append("@DATA\n");
for (int i = 0; i < rows(); i++)
{
result.append(rowToStringSparse(row(i)));
result.append("\n");
}
return result.toString();
}
private StringBuilder metaDataToString()
{
StringBuilder result = new StringBuilder();
if (comments != null && !comments.isEmpty())
{
result.append(comments);
result.append("\n");
}
if (numLabelColumns > 1)
{
result.append(String.format("@RELATION '%s: -c %d '\n", relationName, -numLabelColumns));
}
else
{
result.append(String.format("@RELATION %s\n", relationName));
}
result.append("\n");
for (int i = 0; i < attrNames.size(); i++)
{
if (Helper.iteratorToList(new QuoteParser(attrNames.get(i))).size() > 1)
{
// This attribute name will be parsed into multiple tokens, meaning it has unqoated white space.
result.append("@ATTRIBUTE \"" + attrNames.get(i) + "\"\t");
}
else
{
result.append("@ATTRIBUTE " + attrNames.get(i) + "\t");
}
int vals = getValueCount(i);
if (vals == 0)
result.append("NUMERIC\n");
else
{
result.append("{");
for (int j = 0; j < vals; j++)
{
if (j > 0)
result.append(", ");
result.append(enumToStr.get(i).get(j));
}
result.append("}\n");
}
}
result.append("\n");
return result;
}
public String rowToString(int rowNumber)
{
return rowToString(row(rowNumber));
}
public String rowToString(Vector rowToPrint)
{
StringBuilder result = new StringBuilder();
for (int j = 0; j < rowToPrint.size(); j++)
{
if (j > 0)
result.append(",");
if (getValueCount(j) == 0)
{
if (Vector.isUnknown(rowToPrint.get(j)))
{
result.append("?");
}
else
{
if ((int)rowToPrint.get(j) == rowToPrint.get(j))
result.append((int)rowToPrint.get(j));
else
result.append(rowToPrint.get(j));
}
}
else
{
String valueName = enumToStr.get(j).get((int) rowToPrint.get(j));
if (valueName == null)
{
if (Vector.isUnknown(rowToPrint.get(j)))
valueName = "?";
else
throw new IllegalArgumentException("Unknown value " + rowToPrint.get(j)
+ " for attribute with index " + j);
}
result.append(valueName);
}
}
if (rowToPrint.getWeight() != 1.0)
{
result.append(", {");
result.append(rowToPrint.getWeight());
result.append("}");
}
return result.toString();
}
public String rowToStringSparse(Vector rowToPrint)
{
StringBuilder result = new StringBuilder();
result.append("{");
boolean aValueHasBeenWritten = false;
for (int j = 0; j < rowToPrint.size(); j++)
{
// Only store values which are not 0 since 0 is the default when loading a sparse matrix.
if (rowToPrint.get(j) != 0)
{
if (aValueHasBeenWritten)
result.append(",");
else
aValueHasBeenWritten = true;
result.append(j + " ");
if (getValueCount(j) == 0)
{
if (Vector.isUnknown(rowToPrint.get(j)))
{
result.append("?");
}
else
{
if ((int)rowToPrint.get(j) == rowToPrint.get(j))
result.append((int)rowToPrint.get(j));
else
result.append(rowToPrint.get(j));
}
}
else
{
String valueName = enumToStr.get(j).get((int) rowToPrint.get(j));
valueName = valueName == null ? "?" : valueName;
result.append(valueName);
}
}
}
result.append("}");
if (rowToPrint.getWeight() != 1.0)
{
result.append(", {");
result.append(rowToPrint.getWeight());
result.append("}");
}
return result.toString();
}
@Override
public Iterator<Vector> iterator()
{
return new Iterator<Vector>()
{
private int nextIndex = 0;
@Override
public boolean hasNext()
{
return nextIndex < data.size();
}
@Override
public Vector next()
{
Vector result = data.get(nextIndex);
nextIndex++;
return result;
}
};
}
public Stream<Vector> stream()
{
return StreamSupport.stream(spliterator(), false);
}
/**
* Determines if this matrix has any instance whose weight is not 1 (the default).
*/
public boolean hasInstanceWeightsNot1()
{
for (Vector v: data)
{
if (v.getWeight() != 1.0)
{
return true;
}
}
return false;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,130
|
Q: Explaining Python and converting into Java In Darius Bacon's code, on the line 11 and 12, there is the following code:
prefixes = set(word[:i] for word in words for i in range(2, len(word)+1))
I'm trying to translate his program into Java and I'm having difficulties with this one.
What does this do?
A: Expanding the list comprehension:
prefixes = set()
for word in words:
for i in range(2, len(word)+1)
prefixes.add(word[:i])
word[:i] is word up to but not including index i
A: Try this in Java
Set<String> prefixes = new HashSet<String>();
for(String word:words){
for(int i=1;i<word.length;i++){
prefixes.add(word.substring(0,i));
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,698
|
{"url":"https:\/\/www.bristolmathsresearch.org\/seminar\/mariusz-lemanczyk\/","text":"# Mariusz Lemanczyk\n\nNicolaus Copernicus University in Torun\n\n### Moebius orthogonality in dynamics\n\nErgodic Theory and Dynamical Systems Seminar\n30th November 2017, 3:00 pm \u2013 4:00 pm\nHoward House, 4th Floor Seminar Room\n\nIn 2010 P. Sarnak formulated the following conjecture: All deterministic sequences are orthogonal to the classical Moebius function \\mu. That is:\n$$\\lim_{N\\to\\infty}\\frac1N\\sum_{n\\leq N}f(T^nx)\\mu(n)=0$$\nfor each topological dynamical system $(X,T)$ of zero entropy and $f\\in C(X)$, $x\\in X$. The original (main) motivation for this conjecture was that the famous Chowla's conjecture on autocorrelations of $\\mu$ implies Sarnak's conjecture. During my talk, I'd rather not be overviewing concrete instances of the validity of Sarnak's conjecture, but will concentrate on presenting it as one more evidence of deep connections between ergodic theory and number theory. In particular, I'll show to which extent, we know now about equivalence of Sarnak and Chowla's conjectures.","date":"2019-03-24 01:20:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7607770562171936, \"perplexity\": 1692.101466472034}, \"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-2019-13\/segments\/1552912203123.91\/warc\/CC-MAIN-20190324002035-20190324024035-00402.warc.gz\"}"}
| null | null |
Hyles stricta är en fjärilsart som beskrevs av Tutt 1904. Hyles stricta ingår i släktet Hyles och familjen svärmare. Inga underarter finns listade i Catalogue of Life.
Källor
Svärmare
stricta
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,746
|
\section{Introduction}
\label{sec:intro}
Strongly-interacting theories featuring new non-Abelian gauge groups, where confinement in a ``dark sector'' (DS) at some confinement scale $\Lambda$ leads to stable composite states, offer an interesting alternative explanation of Dark Matter (DM).
In these theories, the stability of the DM candidate can be ensured either by imposing additional symmetries or, more naturally, result from accidental symmetries of the theory. The dark matter candidates can thus be dark pions, baryons or glueballs depending on the exact setup.
Such theories can be realised in a variety of non-Abelian gauge groups and can even help explain observed discrepancies between observation and simulations at cosmic scales via so-called DM self-interactions~\cite{Hochberg:2014dra,Tsai:2020vpi,Lee:2015gsa,Choi:2016hid,Cline:2013zca,Boddy:2014yra,Boddy:2014qxa,Soni:2016gzf}. Moreover, in addition to DM candidate(s), much like Standard Model (SM) QCD, such dark non-Abelian gauge theories feature a spectra of bound states. While such composite DM candidates may exist as a result of strong dynamics, whether and how these DS theories communicate with the SM remains an interesting open question. To this end, one could introduce new SM--DS mediators~\cite{Beauchesne:2018myj,Hochberg:2018rjs,Bernreuther:2019pfb,Mies:2020mzw,Renner:2018fhh}, or charge DS fermions under some of the SM gauge group~\cite{Buckley:2012ky,Appelquist:2015yfa,Bai:2010qg,Kribs:2009fy,Appelquist:2013ms}. The latter scenarios were also realised in the context of technicolor theories e.g.~\cite{Nussinov:1985xr, 1990NuPhB.329..445C,Barr:1990ca,Ryttov:2008xe,Lewis:2011zb}, at least some of which are now under siege after the discovery of the SM Higgs boson; when the DS fermions carry a SM charge, careful consideration of existing electroweak precision tests is required. For a review on fundamental composite dynamics and discussions of SM electroweak constraints see~\cite{Cacciapaglia:2020kgq}.
Once a non-Abelian gauge group with fixed number of flavours and colours is chosen, together with a fermionic representation and a SM--DS mediation mechanism, theoretical predictions for the mass spectra of bound states are obtained by means of lattice simulations. These simulations can predict several useful quantities for a phenomenological analysis in three different regimes;
(i) the chiral regime where dark quarks can be assumed massless $m_{q_D} \ll \Lambda$, (ii) the comparable scales regime $m_{q_D} \sim \Lambda$, and
(iii) the heavy quark/quarkonia regime $m_{q_D} \gg \Lambda$.
Along with the dark hadron mass spectra, lattice simulations can also predict a variety of useful inputs, such as decay constants or matrix elements,
useful for computing cross-sections. These predictions are then taken as an input for a low-energy effective theory in order to devise experimental searches and evaluate sensitivities. It should be noted that lattice simulations do not predict exact mass-scales. However they can predict bound state masses in terms of some common mass scale, which may be freely chosen. For an excellent review pertaining to this discussion see~\cite{Kribs:2016cew}.
Experimental searches for such strongly-interacting dark sectors depend on the mediator mechanisms as well as the comparative mass scales. For example, at the LHC, cases where dark quark masses ($m_{q_D}$) and corresponding confinement scale are much smaller than the collider centre-of-mass energy ($m_{q_D} \lesssim \Lambda \ll \sqrt{s}$) lead to spectacular signatures in terms of semi-visible jets or emerging jets~\cite{Cohen:2015toa,Daci:2015hca,Schwaller:2015gea}. If on the other hand if the three scales are comparable ($m_{q_D} \sim \Lambda \sim \sqrt{s}$), then resonance-like searches may prove useful, depending on the relevant production mechanisms~\cite{Kribs:2018ilo,Hochberg:2015vrg,Kribs:2016cew}. Finally, cases where $m_{q_D} \gg \Lambda, m_{q_D} \lesssim \sqrt{s}$ lead to unusual signals known as quirks~\cite{Knapen:2016hky, Harnik:2008ax}. If the strongly-interacting sector is non-QCD like, other signatures such as Soft Unclustered Energy Patterns are also possible~\cite{Knapen:2016hky, Harnik:2008ax}.
In the vast program of exploring strongly-interacting theories, direct searches for such scenarios have been a focus of the experimental program~\cite{Sirunyan:2018njd, CMS:2020kwj}.
In this work, we instead demonstrate the power of precision measurements of SM-like final states, by taking the so-called ``stealth dark matter" scenarios~\cite{Kribs:2018ilo} as an example theory. Stealth dark matter scenarios are realised in \sund theories with even \nd. In such theories, the baryonic DM candidate is a scalar particle and is stable on account of dark baryon number conservation.
Along with the dark baryon, the theory also features dark pions and mesons as bound states, which are lighter than the dark baryon.
Dark sector interactions with the SM are realised by charging part (or all) of the dark sector under the SM electroweak gauge group.
This leads to signals at direct-detection experiments via Higgs exchange, and the dark rho (\rhod) mixing with the SM gauge bosons leads to signals at the LHC.
Kribs et al~\cite{Kribs:2018ilo} considered such a theory and performed generic lattice simulations for $\nd = 4$ in the comparable scales regime
$m_q \sim \Lambda$ and the quenched limit. They furthermore constructed concrete realisations of such a model where dark quarks respect exact custodial SU(2) symmetry~\cite{Appelquist:2015yfa}. In \cite{Kribs:2018oad} they constructed effective theories for dark mesons in such theories while in \cite{Kribs:2018ilo}, they confronted the meson sector with LHC searches. In doing so, they have provided a complete setup from microscopic theory of dark quarks to macroscopic theory of scalar dark matter and meson bound states. With a mass scale ranging from $\mathcal{O}(100) \rm{GeV}$ to TeV, this theory is an ideal candidate with which to explore the implications of LHC cross-section measurements for strongly-interacting DM scenarios. In this paper we use \CONTUR~\cite{Butterworth:2016sqg,Buckley:2021neu} to study the impact such bound states would have had on existing LHC measurements, and using the lattice calculations of Appelquist et al~\cite{Appelquist:2014jch,Appelquist:2015yfa}, connect this to the relevant cosmological and direct-detection limits.
The \CONTUR method makes use of the bank of LHC measurements (and a few searches) whose results and selection logic are preserved in runnable \RIVET~\cite{Bierlich:2019rhm} routines.
Hundreds of measurements are preserved in this way, which allows generated signal events to quickly and efficiently be confronted with the observed data in a wide variety of final states.
In simple terms, if a new-physics contribution would have modified a SM spectrum beyond its measured uncertainties, ``we would have seen it''; \CONTUR uses the CL$_{s}$~\cite{Read:2002hq} method to quantify this exclusion, making use of bin-to-bin uncertainty correlation information from LHC measurements where available.
This approach has been shown to be highly complementary to the direct search programme~\cite{Buckley:2020wzk,Butterworth:2020vnb}. For models such as those discussed in this paper, where the expected signature at a $pp$ collider changes drastically depending on the model parameter choices, direct searches may be inefficient and hard to motivate. \CONTUR offers a comprehensive and robust way to probe the parameter space.
The paper is structured as follows. In the next section the models are summarised, while Section~\ref{sec:collider} is dedicated to the resulting collider phenomenology. In Section~\ref{sec:contur} we
present and discuss the implications of LHC measurements for the putative dark mesons. In Section~\ref{sec:dm} we then translate these constraints
into constraints on the DM candidate and discuss the impact on DM phenomenology more generally, before concluding.
\section{Model details}
\label{sec:model}
The principal model considered here is a \sund gauge theory with $\nd = 4$ and $N_f = 4$ (Weyl flavours), and the dark quarks (fermions) are in the fundamental representation of the dark colour gauge group \sund\footnote{We will also briefly discuss the case $\nd=2$}.
The dark quarks are further charged under the SM gauge group and transform in a vector-like representation. As vector-like fermions, they have a mass term
which is independent of any electroweak symmetry breaking mechanism. Charging them under the SM gauge group nevertheless generates interactions with
the SM Higgs boson, and lead to masses originating from electroweak symmetry breaking just like any other SM fermion masses.
It is possible to write down these renormalisable vector and chiral mass terms in $N_f = 4$ theory, though not in the $N_f = 2$ theory~\cite{Kribs:2018oad}. For simplicity in the theory, $m$ characterises a common vector-like mass term with $\Delta$ introducing the splitting, while $yv$ characterises the chiral mass with a small factor $\epsilon$ enabling splitting of the chiral masses. Here we will assume $\epsilon$ is negligible.
Electroweak precision tests and Higgs coupling measurements constrain the electroweak symmetry breaking mass term to
be small compared to the vector masses ($yv \ll m$).
In the absence of charges under the SM -- in other words when the theory is taken in isolation -- it exhibits
SU($N_D^{\textrm{fund}}$) $\times$ SU($N_D^{\textrm{anti}}$) chiral symmetry, where $N_D^{\textrm{fund}}$ denotes fundamental representation and $N_D^{\textrm{anti}}$ the anti-fundamental.
When couplings to the SM Higgs boson are turned on, some of the flavour symmetries are explicitly broken.
The model under consideration here will however preserve custodial SU(2) which is the residual accidental global symmetry of the Higgs multiplet after it acquires a vacuum expectation value. Interactions with the Higgs connect flavor symmetries of the fermionic sector with the $O(4) \simeq \sutl \times \sutr$ global symmetry Higgs potential. Out of these the \sutr group contains the U(1)$_\textrm{Y}$ subgroup of the SM via the $t_3$ generator of the SU(2) gauge group.
Below the dark confinement scale, the theory becomes a low-energy effective field theory and is described in terms of mesons and baryons of the sector.
Confinement spontaneously breaks the chiral symmetry of the dark fermions down to the diagonal subgroup $SU(N_D^{\textrm{fund}}) \times SU(N_D^{\textrm{anti}}) \to \sund_V$,
where dark pions live. In total there are 15 pions corresponding to $N_f^2 - 1$ broken generators.
However, for the purposes of the phenomenology, only the lightest pions of the theory, \pid, are considered.
Similarly there are dark rho mesons, \rhod. The \pid and \rhod form triplets under \sutl or \sutr . Should the \sutl be gauged, the \rhod
mix with all three SM weak gauge bosons, and can thus be produced at the LHC via the Drell-Yan (DY) process.
For the \sutr case only the \rhodz
can be produced this way, since when one chooses to gauge the U(1)$_{\textrm{Y}}$ subgroup only the \rhodz mixes,
in this case with the SM $B$ field.
The decays of \pid are also interesting and are intimately connected to the mass spectrum and symmetries of the theory.
The dark pions (to be precise, dark kaons) of the theory mix with the Goldstones of the SM and thus generate couplings with the SM gauge bosons as well with the Higgs boson.
It can be shown with the help of chiral perturbation theory that decays of the \pid to gauge bosons are suppressed by
$\sim m_h^2/m^2_{K_D}$, in both \sutl and \sutr scenarios.
(Here $m_{K_D}$ is the mass of dark kaon,
which is assumed to be not much heavier than \mpid, the mass of the \pid.)
The models of greatest interest to our discussion are therefore referred to as `gaugephobic' \sutl and \sutr scenarios.
Conceptually this small coupling to the gauge bosons can be understood as the Higgs mixing with the kaons, which are doublets,
leading to a suppression by a factor of $m_h^2/(m^2_{K_D}-m_h^2)$, which approximates to $\sim m_h^2/m^2_{K_D}$ when the dark kaon
masses are heavier than the Higgs mass.
\begin{figure}[tbp]
\begin{centering}
\includegraphics[width=0.7\textwidth]{figures/model_details.pdf}
\caption{Flowchart of the model parameters details. Throughout this work, we will use low-energy parametrization of the theory. Fields depicted as $\pi^D_L, \rho^D_L, \pi^D_R, \rho^D_R$ are referred to as \pid and \rhod in the text without $L, R$ indices, since they serve no purpose other
than to illustrate multiplicity. }
\label{fig:model_details}
\end{centering}
\end{figure}
Finally, the origin of DM phenomenology in this model deserves some discussion.
The model features a scalar dark baryon which is stable by virtue of dark baryon symmetry.
As the dark quark masses have contributions from electroweak symmetry breaking, it also gives rise to scalar baryon (DM) interactions
with the Higgs boson, leading to a Higgs-mediated signal at direct-detection experiments.
The two mass terms also imply that dark quarks, and consequently the dark baryon, have a tuneable coupling to the Higgs boson.
Due to the strongly-interacting nature of the theory, lattice computations prove to be useful for determining the inputs for the low-energy effective
theory. For this model, such lattice simulations were performed for quenched masses in comparable regime scenario $\Lambda \sim m_{q_D}$.
These simulations provide us with the spectra of \pid, \rhod and dark baryon masses for certain values of the pion to rho mass ratio
($\mpid/\mrhod \equiv \eta$) in units of a dimensionful scaling parameter.
Along with this, the calculations provide the matrix element for dark baryon scattering via Higgs at direct-detection experiments, parametrised
as $f^{DM}_f$ for the same values of $\eta$. We use these quantities as inputs for our study.
We will ultimately derive constraints the tuneable Higgs to dark quark coupling.
The above discussion is summarised as a flowchart in Figure~\ref{fig:model_details}. For the sake of clarity we use explicit left and right indices for $\pi_D, \rho_D$ in the figure. In the discussions however, we drop these indices assuming that the dark rhos and pions belong to the representation of choice.
In practical terms, we use the effective Lagrangians as implemented in \cite{Kribs:2018ilo}\footnote{Except for changing $N_D$, $\eta$, \mpid we keep all other inputs in the model files to be default.}. This Lagrangian does not contain dark kaon production, despite them being at a similar mass scale to the dark pion. The dark kaons are expected to be produced via their mixing with the SM Higgs bosons and will in general lead to small a production cross section, due to the mixing suppression and high masses. Thus, only phenomenology of dark pions and rhos is considered.
For the low-energy parametrization, one can write an effective Lagrangians where dark pion couplings to gauge bosons are suppressed by $\xi \sim m_h^2/m^2_{K_D}$, as discussed above, or take $\xi$ to be a free parameter and set it to 1. When $\xi = 1$, the pion couplings to gauge bosons are unsuppressed and this leads to a so-called `gaugephilic' scenario. For such a scenario to exist, the dark pions should be in the $SU(2)_L$ representation and hence (only) the low-energy parametrization of the $SU(2)_L$ model involves a gaugephilic scenario as well.
An example of such a gaugephilic scenario can be realised in, for example, two-flavour chiral theory, and is discussed at length in \cite{Kribs:2018oad}. For theories containing two flavours, there is no scalar baryon hence no dark matter candidate.
We will discuss gaugephilic scenarios from the point of view of their collider phenomenology; however no connection to the DM phenomenology
will be made.
\section{Dark mesons and collider phenomenology}
\label{sec:collider}
As discussed above, the kinematic mixing of the \rhod with SM gauge bosons opens the possibility of its production via quark-antiquark annihilation (DY).
If kinematically allowed, the decay of the \rhod to \pid pairs is possible. The \rhod may also decay directly to SM fermion-antifermion pairs.
In the gaugephobic case, the \pid decays primarily to pairs of fermions, while in gaugephilic cases it may also decay via $W+h$ or $Z+h$
if this is kinematically allowed.
\subsection{\rhod and \pid production and decay at the LHC}
In our study we calculate the cross-section for \rhod and \pid production using \HERWIG~\cite{Bellm:2019zci} to simulate events based upon the
Feynrules model files~\cite{Degrande:2011ua} provided by Kribs et al~\cite{Kribs:2018ilo}. We scan over various parameter planes of the model,
generating all leading-order \TwoToTwo processes in which at least one dark meson is either an outgoing leg,
or an $s$-channel resonance.
The most important dark meson production mechanism at the LHC is single \rhod production, often in association with another hard particle,
with the cross-sections for \sutl models typically an order of magnitude larger than for \sutr models.
Assuming 13~TeV $pp$ collisions, the largest cross-sections come from \rhod particles produced with a quark or gluon
(for a $\mrhod \sim 1$~TeV, of the order of 100~fb each for \rhodz,
\rhodp and \rhodm for \sutl,
and of the order of 10~fb, for \rhodz only, in \sutr).
The \rhod can also be produced in association with a weak vector boson, with a cross-section about an order of magnitude less than for quarks and
gluons in the \sutl case. In the \sutr case, production with weak bosons is negligible.
Finally, the \rhod can be produced with a photon, with cross-sections around $1$ and $0.1$~fb for \sutl and \sutr respectively.
\HERWIG will also generate \pid pair production mediated via an $s$-channel \rhod, taking into account the calculated \rhod
width\footnote{There is an overlap between these processes and the $\rhod$+jet processes, which are divergent at low jet transverse momentum.
\HERWIG regulates this by imposing a cut-off on the transverse momentum of the outgoing partons which defaults to 20~GeV.
The total visible cross-section, and the sensitivities derived in the next section, have a very weak dependence on this cut-off for
values of a few tens of GeV.}.
\begin{figure}[tbp]
\vspace{-0.4cm}
\subfloat[$pp \rightarrow \rhodz q$ (\sutl)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GiL_v4/p_p_to_rho0_q.pdf}}
\subfloat[$pp \rightarrow \rhodz q$ (\sutr)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GoR_v4/p_p_to_rho0_q.pdf}}
\includegraphics[height=4.5cm]{figures/cross_sections/GoL_v4/cbar.pdf} \\
\subfloat[$pp \rightarrow \pidp \pidm$ (\sutl)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GiL_v4/p_p_to_DP_ch_DP_ch.pdf}}
\subfloat[$pp \rightarrow \pidp \pidm$ (\sutr)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GoR_v4/p_p_to_DP_ch_DP_ch.pdf}}
\includegraphics[height=4.5cm]{figures/cross_sections/GoL_v4/cbar.pdf} \\
\subfloat[$s$-channel $pp\rightarrow \rhod \rightarrow l^+ l^- $ (\sutl)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GiL_v4/p_p_to_rho0_to_l_l.pdf}}
\subfloat[$s$-channel $pp\rightarrow \rhod \rightarrow l^+ l^- $ (\sutr)]{\includegraphics[width=0.40\textwidth]{figures/cross_sections/GoR_v4/p_p_to_rho0_to_l_l.pdf}}
\includegraphics[height=4.5cm]{figures/cross_sections/GoL_v4/cbar.pdf} \\
\caption{Selected BSM cross-sections as a function of \mpid and $\eta$, for 13~TeV $pp$ collisions.
The left and right columns shows cross-sections for \sutl and \sutr respectively. The shapes are qualitatively similar, but the cross-sections for the latter are about an order of magnitude smaller.
The gaugephilic/gaugephobic distinction does not make a qualitative difference.
Below the $\eta=0.5$ boundary, \rhod decays primarily to \pid pairs, and this is enhanced at the threshold.
Above, \rhod decays to pairs of fermions dominate. White areas indicate regions where the estimated cross-section is below $10^{-3}$ fb.}
\label{fig:bsm-production}
\end{figure}
Once produced, the decay of a \rhod depends on the mass hierarchy of the dark mesons. If $\rhod \rightarrow \pid\pid$ is kinematically allowed
($\eta\leq 0.5$), then this is by far the dominant decay mechanism, with over 99\% branching fraction (and sub-percent level fractions of decays to
fermions or quarks). On the other hand, if the \rhod cannot decay to \pid, it decays 25\% of the time to each generation of quark,
with the remaining branching fraction is shared equally between decays to each generation of leptons.
Pair-production of \pid via a virtual \rhod is maximised when \mrhod is twice \mpid ($\eta = 0.5$), as expected.
For the \sutl model, this process can reach about $10^4$~fb for $\mrhod\sim 1$~TeV, or $10^3$~fb in the \sutr model.
Dark pions can also be singly-produced with a quark or gluon.
This process is independent of the mass of the \rhod, and is typically of the order of 1-10~fb for a pion mass of 1~TeV.
Finally, $s$-channel production of pairs of leptons or quarks (and also $t$-channel for quarks) via a \rhod can be large,
around 10~fb in the \sutr model and 10-100~fb for the \sutl model, where in addition lepton-neutrino pairs can be produced if the intermediate
particle was a charged \rhod.
Examples of some of the subprocess cross-sections as a function of $\eta$ and \mpid for the processes mentioned above are shown in
Fig.~\ref{fig:bsm-production}, for 13~TeV $pp$ collisions.
\subsection{Expected experimental signatures at the LHC}
Given the cross-section results, the LHC experimental signatures which are expected to be the most sensitive to these models will also
depend on the mass hierarchy of the \rhod and \pid, as well as the handedness and the gaugephobic or gaugephilic nature of the model.
For $\eta > 0.5$, the singly-produced \rhodz decaying to leptons can provide a clean signature that should be visible in high-mass
DY measurements and searches. This is the case for all types of model considered.
One may also expect a large cross-section of dijet, or $t\bar{t}$-like signatures (from \rhod decays to quarks),
but such hadronic-only signatures would be far more difficult to distinguish from the QCD background at the LHC.
For $\eta < 0.5$, the expected signatures depend more on the \pid decay modes: in gaugephobic models, the dominant decay of charged and neutral \pid
is to a mixture of third-generation quarks (once above the kinematic threshold), leading to a multijet signature, suggesting that measurements
of $t\bar{t}$, and/or final states involving $b$-tagged jets, will be most sensitive.
Otherwise, one would need to rely on the accompanying decay of a vector boson to select events:
this means that $Z+$jets or $W$+jets signatures might be expected to give good sensitivity.
On the other hand, for gaugephilic models there are regons where the dominant branching fraction for \pid particles involves a vector boson
and a Higgs boson, with decays to third generation quarks still important depending upon the \pid mass region.
In this case, $Z+$jets or $W$+jets signatures can also be expected to play a role.
\section{Collider constraints on Dark Meson Production}
\label{sec:contur}
We scan over the parameter planes of the model, and use \CONTUR v2.0.2 to identify parameter points for which an observably
significant number of events would have entered the fiducial phase space of the measurements available in \rivet 3.1.4,.
\CONTUR evaluates the discrepancy this would have caused, under the assumption that the measured values, which have all been shown to
be consistent with the SM, are identical to it. This is used to derive an exclusion for each parameter
point, taking into account correlations between experimental uncertainties where available. The ATLAS run 2 dilepton resonance search~\cite{Aad:2019fac}
is also available in \rivet and is used, with some caveats since in this case the SM background is modelled by a fit
to data rather than a precision calculation; the impact of this will be discussed below.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[width=0.32\textwidth]{figures/exclusions/GiL/HighMass/dominantPools0.pdf}\label{fig:gil}}
\subfloat[]{\includegraphics[width=0.32\textwidth]{figures/exclusions/GoL/HighMass/dominantPools0.pdf}\label{fig:gol}}
\subfloat[]{\includegraphics[width=0.32\textwidth]{figures/exclusions/GoR/HighMass/dominantPools0.pdf}\label{fig:gor}}
\caption{Scans in $\eta \-- \mpid$ for three sub-models.
(a) Gaugephilic \sutl, (b) Gaugephobic \sutl (c) Gaugephobic \sutr. The colours
indicate the dominant signature pool giving the sensitivity. The white solid line is the 95\%
exclusion and the white dashed line is the 68\% exclusion.
\label{fig:scans}}
\begin{tabular}{llll}
\swatch{seagreen}~CMS high-mass Drell-Yan $\ell\ell$ &
\swatch{darkolivegreen}~ATLAS high-mass Drell-Yan $\ell\ell$ &
\swatch{turquoise}~ATLAS $\ell_1\ell_2$+\ensuremath{E_T^{\rm miss}}{}+jet \\
\swatch{orangered}~ATLAS $ee$+jet &
\swatch{green}~ATLAS \ensuremath{E_T^{\rm miss}}{}+jet
\swatch{silver}~ATLAS jets
\end{tabular}
\end{figure}
Scans are performed in the $\eta \-- \mpid$ plane for the Gaugephilic \sutl,
Gaugephobic \sutl and Gaugephobic \sutr sub-models, and shown in Fig.~\ref{fig:scans}.
The expected change in behaviour around $\eta = 0.5$ is clearly seen.
The dominant exclusion for much of the plane comes from the 139 fb$^{-1}$ ATLAS dilepton search,
with the CMS measurement using 3.2 fb$^{-1}$~\cite{Sirunyan:2018owv} and the ATLAS 7 and 8~TeV measurements~\cite{Aad:2013iua,Aad:2016zzw} all
having an impact at lower \mpid.
No dilepton measurements using the full run 2 integrated luminosity from the LHC are
yet available.
For the $\eta > 0.5$ region, the \rhodz decay does indeed produce a resonant signature, as may be seen for an example point in Fig.\ref{fig:gol_hmdy},
and so this
limit can be taken as a good estimate. In this case, \rhod masses as high as 3.8~TeV are excluded for $\eta$ close to unity
in the left-handed cases, with this limit being reduced to around 3.4~TeV for the \sutr model due to the generally lower cross-section.
The other measurement showing good sensitvity across a broad range of parameters is
the recent ATLAS measurement of multijet event shapes~\cite{Aad:2020fch}.
This has a significant impact for $\eta < 0.5$, consistent with the expectations from \pid decay.
This comes from expected distortions in the event shapes, for example the transverse sphericity shown in
Fig.~\ref{fig:gol_jets}, at high jet multiplicity and high transverse mass. Even at the high scales at which the measurement
is made, multijet production is a challenging cross-section for which to obtain precise SM predictions, and this may be
seen in the spread of predictions shown in \cite{Aad:2020fch}. The default \CONTUR assumption that the SM is equal to the data may
therefore overestimate the exclusion. However, the \pid decay signature does overwhelmingly impact the six-or-more-jet
distributions, with little impact on the three, four or five jet event shapes, something that seems diffcult to absorb
into a QCD uncertainty. This is similar to saying that if one is prepared to treat the lower multiplicity distributions
as a ``control region'' from which the six-jet distribution is then extrapolated, the \CONTUR exclusion should be a good estimate.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[width=0.35\textwidth]{figures/rivet/ATLAS_2019_I1725190/d01-x01-y01.pdf}\label{fig:gol_hmdy}}
\subfloat[]{\includegraphics[width=0.35\textwidth]{figures/rivet/ATLAS_2020_I1808726/d36-x01-y01.pdf}\label{fig:gol_jets}}\\
\subfloat[]{\includegraphics[width=0.35\textwidth]{figures/rivet/ATLAS_2018_I1705857/d01-x01-y01.pdf}\label{fig:gol_ttbb}}
\subfloat[]{\includegraphics[width=0.35\textwidth]{figures/rivet/ATLAS_2014_I1279489/d03-x01-y01.pdf}\label{fig:gol_lljet}}
\caption{
Example kinematic distributions for the Gaugephobic \sutl model.
(a) Di-electron mass distribution from \cite{Aad:2019fac}, $\mpid=2$~TeV, $\eta=0.7$.
The uncertainties on the data and on the BSM
are statisistical only, those on the background fit include the systematics.
(b) Transverse sphericity in multijet event shapes~\cite{Aad:2020fch}, $\mpid=750$~GeV, $\eta=0.45$.
Full uncertainties are shown on the data.
(c) $t\bar{t}b\bar{b}$ cross-sections~\cite{Aaboud:2018eki},
$\mpid=390$~GeV, $\eta=0.45$. Full uncertainties are shown on the data, and by the
yellow band in the lower ratio plot.
(d) ATLAS dilepton+jets measurement~\cite{Aad:2014dta}, $\mpid=260$~GeV, $\eta=0.4$.
Uncertainties shown as in (c)
\label{fig:rivet}}
\end{figure}
This challenging $\eta < 0.5$ region requires further discussion, however, and hidden beneath
the multijet sensitivites in Fig.~\ref{fig:scans} are exclusions from a wide variety of other final states
reflecting the many possible \pid decay chains.
For the \sutl cases, there is also some apparent sensitivity in this region from the Drell-Yan search.
However, while lepton pairs may be produced in \pid decays,
for example via top quarks, they are no longer resonant at the \rhod mass and so the ``bump hunt'' approach of \cite{Aad:2019fac}
is no longer valid.
In this case, we revert to the default \CONTUR mode, which uses only particle-level measurements. We also turn off
the multijet analysis, and re-scan the region $\eta<0.5$ at low \mpid. This allows a more detailed investigation of
the different \pid decay signatures.
The results are shown in Fig.~\ref{fig:loweta_scans}.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[]{figures/exclusions/GiL/LowEta/dominantPools0.pdf}\label{fig:gil_loweta}}
\subfloat[]{\includegraphics[]{figures/exclusions/GoL/LowEta/dominantPools0.pdf}\label{fig:gol_loweta}}
\caption{Scans in $\eta \-- \mpid$ for two sub-models, focussing on the $\eta<0.5$ region
and excluding the ATLAS dilepton search.
(a) Gaugephilic \sutl, (b) Gaugephobic \sutl
The colours indicate the dominant signature pool giving the sensitivity. The white solid line is the 95\%
exclusion and the white dashed line is the 68\% exclusion.
\label{fig:loweta_scans}}
\begin{tabular}{llll}
\swatch{lightsalmon}~CMS $ee$+jet &
\swatch{darksalmon}~CMS $\mu\mu$+jet &
\swatch{orangered}~ATLAS $ee$+jet \\
\swatch{orange}~ATLAS $\ell\ell$+jet &
\swatch{darkorange}~ATLAS $\mu\mu$+jet &
\swatch{blue}~ATLAS $\ell$+\ensuremath{E_T^{\rm miss}}{}+jet \\
\swatch{navy}~ATLAS $\mu$+\ensuremath{E_T^{\rm miss}}{}+jet &
\swatch{cadetblue}~ATLAS $e$+\ensuremath{E_T^{\rm miss}}{}+jet &
\swatch{powderblue}~CMS $\ell$+\ensuremath{E_T^{\rm miss}}{}+jet \\
\swatch{snow}~ATLAS Hadronic $t\bar{t}$ &
\swatch{turquoise}~ATLAS $\ell_1\ell_2$+\ensuremath{E_T^{\rm miss}}{}+jet &
\swatch{darkolivegreen}~ATLAS high-mass Drell-Yan $\ell\ell$ \\
\swatch{seagreen}~CMS high-mass Drell-Yan $\ell\ell$ &
\swatch{magenta}~ATLAS 4$\ell$ &
\swatch{yellow}~ATLAS $\gamma$ \\
\swatch{mediumseagreen}~ATLAS $\ell\ell\gamma$ &
\swatch{darkgoldenrod}~ATLAS $\gamma$+\ensuremath{E_T^{\rm miss}}{} &
\swatch{silver}~ATLAS jets
\end{tabular}
\end{figure}
With the ATLAS multijet event shape measurement removed, the sensitivity in the jet channel is based mostly on inclusive and
dijet analyses and is signficantly reduced, which then reduces the combined sensitivity at higher \mpid values.
However, a mix of other analyses can now be seen to be contributing to the exclusion.
\begin{itemize}
\item when \mpid falls in the Higgs mass window of the ATLAS $H \rightarrow \gamma\gamma$ fiducial cross-section measurement~\cite{Aad:2014lwa},
the $\pid \rightarrow \gamma \gamma$ decays populate the cross-section and, despite their suppression due to dark flavour
symmetry~\cite{Kribs:2018oad}, would have led to an observable excess at high $p_T^{\gamma\gamma}$, had they been present.
\item for $\eta \approx 0.2$ and $\mpid \approx 220$~GeV, the boosted hadronic top measurement~\cite{Aaboud:2018eqg} is most sensitive;
tops are produced in the dominant decay modes of both neutral and charged \pid in this region, and for low $\eta$,
where the \rhod is much heavier than the \pid, they will indeed be boosted.
\item at low \mpid and $\eta$ just below 0.5, the most sensitive measurements are of $t\bar{t}$ production in the $e\mu$ channel~\cite{Aad:2019hzw},
where an excess at low transverse momentum of the $e\mu$ system would have been particularly prominent.
\item at higher $\eta$ values, and especially in the gaugephilic case, the full run 2 four-lepton measurement~\cite{Aad:2021ebo} is sensitive
for $\mpid \approx 200$~GeV and above. In this region $\pid\pid \rightarrow Z h t \bar{b}$ is significant, and can give rise to such
signatures, as can $\pid\pid \rightarrow hhZW$ and $\pid\pid \rightarrow hhWW$ (in the gaugephilic case) if kinematically possible.
\item at low \mpid and $\eta$, and in the gaugephobic case also at higher $\eta$ and $\mpid \approx 400$~GeV,
lepton missing transverse energy and jet final states are important. In both cases this is due to
$\pid\pid \rightarrow t\bar{t}t\bar{b}$, with the ATLAS $t\bar{t} + b$-jets measurement~\cite{Aaboud:2018eki} being especially powerful,
where a significant excess in the $\ell + \geq 4b$ cross-section should have been visible, as shown in Fig.~\ref{fig:gol_ttbb}.
\item in the $0.3 < \eta < 0.5$ region for $\mpid > 220$~GeV, measurements of dilepton plus jets\cite{Aad:2014dta,Khachatryan:2016iob}
are sensitive, especially in the gaugephilic case. This is due the $\pid\pid \rightarrow Z h t \bar{b}$
(in both models) and the $\pid\pid \rightarrow hhZW$ and $\pid\pid \rightarrow hhWW$ decay modes (in the gaugephilic case).
An example distribution is shown in Fig.~\ref{fig:gol_lljet}.
\item The high mass Drell-Yan measurements~\cite{Aad:2016zzw,Sirunyan:2018owv} also still play a role in the total exclusion, although they
are rarely dominant. This indicates that future measurements should be powerful.
\end{itemize}
\section{Implications for Dark Matter Phenomenology}
\label{sec:dm}
Taking the searches considered in \cite{Kribs:2018ilo}, along with the constraints derived in the previous two
sections, we now consider the implications for stealth DM models. In Ref.~\cite{Appelquist:2015yfa}, the authors demonstrate potential ways to connect collider limits with DM analysis.
We follow this strategy here in order to connect the different regimes together.
It should be noted that unlike WIMP theories consisting of elementary particles, the LHC signals in our scenarios
are not directly related to the production of dark matter.
It is by using the underlying fundamental theory of dark quarks in the SU(4) fundamental representation,
which fixes the mass spectra, that we can connect seemingly unrelated LHC signatures to DM phenomenology.
From our results of the previous section, we obtain excluded values of $\eta$ and $\mpid$.
These can be used to obtain the corresponding DM mass \mdm via
\begin{equation}
m_{S0}(\eta) = m_B(\eta) = \mdm(\eta) = \frac{amS0(\eta)}{amps(\eta)}\times \mpid(\eta)
\end{equation}
where $amS0, amps$ are the masses predicted by lattice simulations, and \mpid is the excluded pseudo-scalar mass obtained
in the previous section. The exact values of $amS0, amps$ are summarised in Table~\ref{tab:lattice_inputs}.
We interpolate between different $\eta$ values in order to get Fig. \ref{fig:DM}.
Finally, when $\eta \to 1$, we approach the heavy quark limit. In this limit, the masses of bound states are simply sum of the
masses of the constituent quarks.
In this case the scalar baryon mass, being made up of four quarks, should be twice \mpid, which has two constituent quarks.
Having observed that such a limit is already reached for $\eta = 0.77$, we keep the mass ratios constant beyond for higher $\eta$ values.
\begin{table}[h!]
\centering
\begin{tabular}{ |c|c|c|c|c|}
\hline
$\eta$ & $amps$ & $amv$ & $amS0$ & $f^{DM}_f$ \\
\hline
0.77 & 0.3477 & 0.4549 & 0.9828 & 0.153 \\
0.70 & 0.2886 & 0.4170 & 0.8831 & 0.262 \\
0.50 & 0.2066 & 0.3783 & 0.7687 & 0.338 \\
\hline
\end{tabular}
\caption{Lattice inputs for $\beta = 11.028$ on $32^3 \times 64$ lattices taken from~\cite{Appelquist:2014jch} for this work. $amps, amv$ and $amS0$ represent dimensionless pseudo-scalar, vector and dark baryon masses while $f^{(DM)}_f$ is lattice input for computing DM direct-detection cross-section via Higgs exchange.}
\label{tab:lattice_inputs}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[scale = 0.5]{figures/DM/dm_excl.pdf}
\caption{Limits on the mass of dark matter given constraints on \mpid and \mrhod for the \sutl model (green) and the \sutr model (red).
As discussed in the text, only gaugephobic limits are used. The solid lines represent the region where lattice results
are available, whilst the dashed lines represent our extrapolation assuming the heavy quark limit has been achieved.
\label{fig:DM}
}
\end{figure}
The results are illustrated in Fig. \ref{fig:DM}, where we demonstrate exclusion limits on \mdm as a function of $\eta$.
The two curves correspond to \sutl (green) and \sutr (red) gaugephobic models respectively.
The dashed lines represent our extension of the provided lattice results to the heavy quark limits.
We can see that the two limits differ significantly for smaller \mdm, where limits on $\eta$ in the \sutl theory are
very different to those from \sutr theory, as discussed in the previous section.
For higher \mdm the two scenarios lead to more similar limits on $\eta$, pushing the allowed \mdm to the multi-TeV regime.
The limits from \sutr scenarios are always somewhat weaker than corresponding limits in \sutl theories.
\section{Combined Constraints}
As discussed previously, the DM candidate couples to the SM Higgs and scatters off nuclei at direct-detection experiments.
The dark matter-nucleus scattering element constitutes of two parts, one corresponding to matrix element of the fermions in the scalar baryons ($f_f^{DM}$) which is computed via lattice simulation \cite{Appelquist:2014jch} and second, the Yukawa coupling between Higgs and dark quarks. Unfortunately, the Yukawa coupling as defined in Section~\ref{sec:model} and also introduced in Fig.~\ref{fig:model_details} can not be directly constrained and hence needs to be recast into so-called \yeff parameter which is independent of dark fermion mass \cite{Appelquist:2015yfa}.
We identify that the so-called linear case as discussed in \cite{Appelquist:2015yfa} corresponds to the gaugephobic \sutl case, while the quadratic case
corresponds to the \sutr gaugephobic case. This can be understood as follows. As explained in Section~\ref{sec:model}, the dark quarks obtain their masses from the vector as well as EWSB mass terms. The physical masses are thus a mixture of the two mass terms after mass diagonalization. The coupling of Higgs to these physical quark masses will be proportional to the Higgs--dark quark Yukawa coupling times the mass of the dark quark, just like in the SM. If the dark quark masses are dominated by EWBS (chiral) mass term, one gets a quadratic dependence of Higgs--dark quark coupling, otherwise a linear dependence. Therefore, in the linear (\sutl) case, EWSB is dominantly (but not entirely) responsible for quark mass splitting, while the situation reverses in the other limit. It should be noted that \yeff is a free parameter of the theory and can be constrained using experimental data which is exactly what we will do in this section by combining latest Xenon1T limits with the LHC limits and obtain maximum allowed coupling \yeff between the dark quarks and Higgs.
We briefly outline the exact procedure we follow in order to compute the DM--nucleus scattering cross-sections which closely follows\cite{Appelquist:2014jch, Appelquist:2015yfa}.
The dark matter nucleus spin-independent scattering cross-section is given by
\begin{equation}
\sigma(\textrm{DM},N) = \displaystyle\frac{\mu(m_{\textrm{DM}}, m_N)^2}{\pi} \left(Z \mathcal{M}_p + (A-Z) \mathcal{M}_n \right)^2,
\end{equation}
where $N$ is the nucleus, $\mu(m_{\textrm{DM}}, m_N)$ represents the nucleus - dark matter reduced mass\footnote{$\mu(m_1, m_2) = m_1 m_2/(m_1+m_2)$}, Z, A are the atomic and mass numbers, here taken for Xenon and finally $\mathcal{M}_{p,n}$ are the amplitudes for scattering off proton or neutron.
This quantity is dependent on the experiment target, in order to ease the comparison among different experiments the cross-section is conveniently expressed as
\begin{equation}
\sigma_0(\textrm{DM}, a) = \sigma(\textrm{DM}, N)\displaystyle\frac{\mu(m_{\textrm{DM}}, m_a)^2}{\mu(m_{\textrm{DM}}, m_N)^2\,A^2},
\end{equation}
where $\sigma_0(\textrm{DM}, a)$ is the scattering cross-section per nucleon at zero momentum transfer.
The DM--nucleus scattering cross-section mediated by the Higgs exchange contains two parts, the first corresponding to the Higgs--SM quark current while the other contains
Higgs--DM exchange.
In particular the amplitude is given by\footnote{Note: we need to compute the squared amplitude, thu the expression for $\mathcal{M}_{p,n}$ must be squared.}
\begin{equation}
\mathcal{M}_{p,n} = \frac{g_{p,n}\,g_{DM}}{m_h^2}
\end{equation}
with
\begin{equation}
g_{p,n} = \frac{m_{p,n}}{v}\left[\sum_{q = u,d,s}f_q^{(p,n)} + \frac{6}{27}\left( 1 - \sum_{q = u,d,s}f_q^{(p,n)} \right) \right] \\
\end{equation}
and
\begin{equation}
g_{DM} \simeq
\begin{cases}
\yeff\, f_f^{DM}, & \text{\sutl case}\\
\\
\yeff^2\,\displaystyle\frac{v}{\mdm} f_f^{DM}, & \text{\sutr case}
\end{cases}
\label{eq:yeff}
\end{equation}
The quantity $f_f^{DM}$ is extracted from lattice calculations and precise values are tabulated in Table~\ref{tab:lattice_inputs}, while \yeff is the effective Yukawa coupling independent of the dark quark masses. Using the definitions above and the lattice parameters for $f_q^{(p,n)}$ as defined in \cite{Hill:2011be, Belanger:2008sj}, we update the constraints on \yeff, as shown
in Fig.\ref{fig:DM_combined} and Fig.\ref{fig:DM_combined_yeff}\footnote{We thank O.Witzel, E. Neil and G. Kribs for confirming that the original constraints for quadratic case correspond to contours of $\yeff^2$ instead of $\yeff$.}.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2L_77.pdf}}
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2R_77.pdf}}\\
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2L_70.pdf}}
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2R_70.pdf}}\\
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2L_55.pdf}}
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/xenon_xsec_SU2R_55.pdf}}
\caption{(left panel) Constraints for \sutr model, (right panel) constraints $SU(2)_L$ model for $\eta = 0.77$. Colored contours show DM--SM scattering cross-sections for fixed value of \yeff, vertical lines represent \mdm limits derived from LEP limits on pion mass (grey dot-dashed line) and updated LHC constraints derived in this work (black dot-dashed line). Also overlaid are the recent Xenon1T constraints on DM--nucleon coherent scattering. \cite{Aprile:2018dbl}.
\label{fig:DM_combined}
}
\end{figure}
To begin with, in Fig.\ref{fig:DM_combined}, we show contours of \yeff in the \mdm and DM--nucleon scattering cross-section plane for \sutr model (left panel) and \sutl case (right panel) for fixed value of $\eta$ of 0.77 (top panel), 0.70 (middle panel) and 0.55 (bottom panel). The values of $\eta$ correspond to those for which lattice calculations are performed. We also overlay the latest Xenon1T DM--nucleon elastic scattering cross-section limits~\cite{Aprile:2018dbl}. The way to interpret this plot would be to find values of \yeff which lie below the Xenon1T curve for a given \mdm. These correspond to direct detection allowed parameter spaces. Finally, we also show illustrate exclusions on \mdm obtained from LEP II limits~\footnote{Although we note some caveats to the oft-quoted
generic 100~GeV limit on charged fermions from LEP-II~\cite{Egana-Ugrinovic:2018roi}, they do not apply here.} in~\cite{Kribs:2018oad} (dot-dashed grey line) and from our LHC analysis as discussed in section above (dot-dashed black line). Combining this with the DM--nucleon scattering cross-section limits, one can then derive the maximum value of \yeff which obeys all limits. Stringent limits on \yeff imply weaker coupling between DM and Higgs and thus in general weaker DM--SM interactions.
Before discussing the ultimate lessons we learn from this, we first discuss Fig.\ref{fig:DM_combined}. As expected from Eq.\ref{eq:yeff}, for the \sutr case the DM--nucleon scattering cross-section is independent of \mdm while for \sutl case it is inversely proportional to \mdm for a given value of \yeff. For \sutr case, the Xenon1T allowed value of \yeff decreases as $\eta$ increases, this is because $f^{DM}_f$ increases as $\eta$ increases. The dependence is more complicated for \sutl case where $g_{DM}$ depends on both $f^{DM}_f$ and \mdm which increase as $\eta$ increases. However $f^{DM}_f$ increases faster than \mdm and hence \yeff decreases albeit at a smaller rate.
Finally, the limits from DM--nucleon elastic scattering are complemented by the limits on \mdm derived by LEP and LHC analyses. In particular, for each $\eta$, the corresponding figure illustrates the impact of updated LHC limits compared to the LEP exclusion. The LHC limits as obtained by us in this analysis together with the Xenon1T limits thus either demand smaller values of \yeff or large \mdm to be compatible with current experimental situation. This is precisely reflected in Fig.~\ref{fig:DM_combined_yeff}, where we plot the maximum allowed values of \yeff for various value of $\eta$ (solid lines) and overlay the limits obtained from LHC (dot-dashed line). It should be noted that lower values of $\eta$ push the theory into the chiral (massless quark) limit where no lattice results are available, and we refrain from deriving any limits in this region.
\begin{figure}[h]
\centering
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/ysol_su2L.pdf}}
\subfloat[]{\includegraphics[scale = 0.5]{figures/DM/ysol_su2R.pdf}}
\caption{(left panel) Maximum allowed value of \yeff for given value of $\eta$ for \sutl model, (right panel) constraints \sutr model. The black dot dashed line shows LHC limits derived in this work. Unlike in \cite{Appelquist:2015yfa}, we do not extend our LHC limits in the chiral limit (ie low $\eta$) leading to an abrupt cut off of the LHC limits around \mdm of about 7 TeV (left panel) and 6 TeV (right panel) respectively.
\label{fig:DM_combined_yeff}
}
\end{figure}
\section{Conclusions}
The analysis presented in this paper is significant in two respects. First, we take into account the updated Xenon1T limits which are almost an order of magnitude stronger than the previous
iteration. Second, the updated LHC limits (now including those from measurements as well as searches) push \mdm to the multi-TeV regime, and much stronger than any
previously obtained.
Global investigations, first carried out in Ref.~\cite{Kribs:2018oad, Appelquist:2015yfa, Kribs:2018ilo} and this analysis will have implications
for possible relic density generation mechanisms in the underlying composite SU(4) theories.
While the exact details need lattice calculation, it was argued~\cite{Appelquist:2015yfa} that if DM relic density is to be generated by asymmetric abundance,
a few TeV \mdm scale would satisfy the observed amount.
Given current LHC limits which are much beyond this value, one can expect difficulties in realising asymmetric abundance scenarios for relic density generation.
Finally, we mention that the LHC limits as derived in~\cite{Kribs:2018ilo} and in this work are relatively weak for small $\eta$,
where $\rhod \to \pid\pid$ decays are open.
Small values of $\eta$ also correspond to the chiral limit of the theory. Therefore we conclude that on the theory side,
more efforts are needed to constrain chiral limit of such strongly-interacting scenarios.
From the experimental point of view, limits on this region can be expected to improve as more precise measurements of a variety
of final states are made, including tops, $b$-quarks and dileptons.
As pointed out in~\cite{Kribs:2018ilo}, for $\mpid < 150$~GeV or so, final states involving $\tau$ leptons are important.
No such measurements are currently available; should they be made in future, they could have significant impact.
Overall, while some the general properties of these models can be surmised, the detailed phenomenology has many uncertainties, in particular
driven by the unknown strong dynamics.
This makes dedicated search analyses for specific benchmark scenarios of limited interest.
While scenarios (as mentioned in the introduction) which lead to non-SM like final states will still require dedicated search strategies,
the uncertainty in the phenomenology strongly motivates model-independent measurements of a wide variety
of SM-like final states. These measurements can be then be analysed for discrepancies and, in their absence, rapidly interpretated as limits
over a very broad range of parameters.
\section*{Acknowledgements}
SK thanks Axel Maas for several useful discussions and E. Neil, O. Witzel and G. Kribs for clarifications about the direct-detection constraints. We thank Chi Leung for precursor investigations
of the model as part of her final year undergraduate project.
JMB has received funding from the European Union's Horizon 2020 research and innovation
programme as part of the Marie Skłodowska-Curie Innovative Training
Network MCnetITN3 (grant agreement no. 722104) and from a UKRI Science and Technology Facilities Council
(STFC) consolidated grant for experimental particle physics. SK is supported by Elise-Richter grant project number V592-N27.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,877
|
Q: Child container failed in JAX_RS and jersey web service It is simple web service using JAX_RS and Jersey. But there is an error A child container failed during start.
This the error log.
A child container failed during start
java.util.concurrent.ExecutionException: org.apache.catalina.LifecycleException: Failed to start component [StandardEngine[Catalina].StandardHost[localhost].StandardContext[/RestfulWebServiceServer]]
at java.util.concurrent.FutureTask.report(FutureTask.java:122)
at java.util.concurrent.FutureTask.get(FutureTask.java:192)
at org.apache.catalina.core.ContainerBase.startInternal(ContainerBase.java:943)
at org.apache.catalina.core.StandardHost.startInternal(StandardHost.java:871)
at org.apache.catalina.util.LifecycleBase.start(LifecycleBase.java:145)
at org.apache.catalina.core.ContainerBase$StartChild.call(ContainerBase.java:1408)
at org.apache.catalina.core.ContainerBase$StartChild.call(ContainerBase.java:1398)
at java.util.concurrent.FutureTask.run(FutureTask.java:266)
at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1142)
at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:617)
at java.lang.Thread.run(Thread.java:745)
Caused by: org.apache.catalina.LifecycleException: Failed to start component [StandardEngine[Catalina].StandardHost[localhost].StandardContext[/RestfulWebServiceServer]]
at org.apache.catalina.util.LifecycleBase.start(LifecycleBase.java:162)
pom.xml
<!-- https://mvnrepository.com/artifact/com.sun.jersey/jersey-core -->
<dependencies>
<dependency>
<groupId>com.sun.jersey</groupId>
<artifactId>jersey-core</artifactId>
<version>1.19</version>
</dependency>
<!-- https://mvnrepository.com/artifact/asm/asm -->
<dependency>
<groupId>asm</groupId>
<artifactId>asm</artifactId>
<version>3.1</version>
</dependency>
<!-- https://mvnrepository.com/artifact/com.sun.jersey/jersey-server -->
<dependency>
<groupId>com.sun.jersey</groupId>
<artifactId>jersey-server</artifactId>
<version>1.19</version>
</dependency>
<!-- https://mvnrepository.com/artifact/junit/junit -->
<dependency>
<groupId>junit</groupId>
<artifactId>junit</artifactId>
<version>4.12</version>
</dependency>
<!-- https://mvnrepository.com/artifact/com.sun.jersey/jersey-servlet -->
<dependency>
<groupId>com.sun.jersey</groupId>
<artifactId>jersey-servlet</artifactId>
<version>1.19</version>
</dependency>
<dependency>
<groupId>com.sun.jersey</groupId>
<artifactId>jersey-json</artifactId>
<version>1.19</version>
</dependency>
</dependencies>
<build>
<sourceDirectory>src</sourceDirectory>
<plugins>
<plugin>
<artifactId>maven-war-plugin</artifactId>
<version>2.6</version>
<configuration>
<warSourceDirectory>WebContent</warSourceDirectory>
<failOnMissingWebXml>false</failOnMissingWebXml>
</configuration>
</plugin>
<plugin>
<artifactId>maven-compiler-plugin</artifactId>
<version>3.5.1</version>
<configuration>
<source>1.8</source>
<target>1.8</target>
</configuration>
</plugin>
</plugins>
</build>
</project>
User.java
package entities;
public class User{
private String name;
private String id;
private String address;
private String email;
public String getName() {
return name;
}
public void setName(String name) {
this.name = name;
}
public String getId() {
return id;
}
public void setId(String id) {
this.id = id;
}
public String getAddress() {
return address;
}
public void setAddress(String address) {
this.address = address;
}
public String getEmail() {
return email;
}
public void setEmail(String email) {
this.email = email;
}
public User(String name, String id, String address, String email) {
super();
this.name = name;
this.id = id;
this.address = address;
this.email = email;
}
public User() {
super();
// TODO Auto-generated constructor stub
}
}
UserJSONService.java
package ws;
import javax.ws.rs.Consumes;
import javax.ws.rs.GET;
import javax.ws.rs.POST;
import javax.ws.rs.Path;
import javax.ws.rs.Produces;
import javax.ws.rs.core.MediaType;
import javax.ws.rs.core.Response;
import entities.User;
@Path("/json/user")
public class UserJSONService {
@GET
@Path("/get")
@Produces(MediaType.APPLICATION_JSON)
public User getUser(){
User u1 = new User();
u1.setName("John Doe");
u1.setId("1");
u1.setAddress("Kolkata");
u1.setEmail("someone@gmail.com");
return u1;
}
@POST
@Path("/post")
@Consumes(MediaType.APPLICATION_JSON)
public Response sendUserInJSON(User user) {
String result = "User saved : " + user;
return Response.status(201).entity(result).build();
}
public static double JAVA_VERSION = getVersion ();
static double getVersion () {
String version = System.getProperty("java.version");
int pos = version.indexOf('.');
pos = version.indexOf('.', pos+1);
return Double.parseDouble (version.substring (0, pos));
}
}
web.xml
<?xml version="1.0" encoding="UTF-8"?>
<web-app xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://xmlns.jcp.org/xml/ns/javaee" xsi:schemaLocation="http://xmlns.jcp.org/xml/ns/javaee http://xmlns.jcp.org/xml/ns/javaee/web-app_3_1.xsd" id="WebApp_ID" version="3.1">
<display-name>RestfulWebServiceServer</display-name>
<welcome-file-list>
<welcome-file>index.html</welcome-file>
<welcome-file>index.htm</welcome-file>
<welcome-file>index.jsp</welcome-file>
<welcome-file>default.html</welcome-file>
<welcome-file>default.htm</welcome-file>
<welcome-file>default.jsp</welcome-file>
</welcome-file-list>
<servlet>
<servlet-name>jersey-serlvet</servlet-name>
<servlet- class>com.sun.jersey.spi.container.servlet.ServletContainer</servlet-class>
<init-param>
<param-name>com.sun.jersey.config.property.packages</param-name>
<param-value>ws</param-value>
</init-param>
<init-param>
<param-name>com.sun.jersey.api.json.POJOMappingFeature</param-name>
<param-value>true</param-value>
</init-param>
<load-on-startup>1</load-on-startup>
</servlet>
<servlet-mapping>
<servlet-name>jersey-serlvet</servlet-name>
<url-pattern>/rest/*</url-pattern>
</servlet-mapping>
</web-app>
A: using the correct dependency
<!-- https://mvnrepository.com/artifact/asm/asm -->
<dependency>
<groupId>asm</groupId>
<artifactId>asm</artifactId>
<version>3.3.1</version>
</dependency>
instead of asm 3.1 solved my issue.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,147
|
\section*{Acknowledgments}
\section{Introduction}
\definecolor{magenta}{HTML}{FF00FF}
\definecolor{bluebar}{HTML}{0000ff}
\definecolor{grey}{HTML}{666666}
\definecolor{orange}{HTML}{ff9900}
\begin{figure}[]
\centering
\includegraphics[width=.48\textwidth]{img/stochastic_vs_deterministic_ranker.pdf}
\caption{While deterministic neural rankers output a point estimate probability ({\color{magenta}magenta} values) of relevance for a combination of query ({\color{bluebar}blue} bars) and document ({\color{grey}grey} bars), stochastic neural rankers output a predictive distribution{} ({\color{orange}orange} curves). The dispersion of the predictive distribution{} provides an estimation of the model uncertainty.}
\label{fig:stochastic_vs_deterministic}
\end{figure}
According to the Probability Ranking Principle (PRP)~\cite{robertson1977probability}, ranking documents in decreasing order of their probability of relevance leads to an optimal document ranking for ad-hoc retrieval\footnote{Standard retrieval task where the user specifies his information need through a query which initiates a search by the system for documents that are likely relevant~\cite{baeza1999modern}.}. \citet{gordon1991utility} discussed that for the PRP to hold, ranking models must at least meet the following conditions: \textbf{[C1]}{} assign well calibrated probabilities of relevance, i.e. if we gather all documents for which the model predicts relevance with a probability of e.g. 30\%, the amount of relevant documents should be 30\%, and \textbf{[C2]}{} report certain predictions, i.e. only point estimates such as e.g. 80\% probability of relevance.
DNNs have been shown to outperform classic Information Retrieval (IR) ranking models over the past few years in setups where considerable training data is available. It has been shown that DNNs are not well calibrated in the context of computer vision~\cite{guo2017calibration}. If the same is true for neural L2R{} models for IR, e.g. transformer-based models for ranking~\cite{nogueira2019passage}, \textbf{[C1]}{} is not met. Additionally, there are a number of sources of uncertainty in the training process of neural networks~\cite{gal2016uncertainty} that make it unreasonable to assume that neural ranking models fulfill \textbf{[C2]}{}: \emph{parameter uncertainty} (different combinations of weights that explain the data equally well), \emph{structural uncertainty} (which neural architecture to use for neural ranking), and \emph{aleatoric uncertainty} (noisy data). Given these sources of uncertainty, using point estimate predictions and ranking according to the PRP might not achieve the optimal ranking for retrieval. While the effectiveness benefits of risk-aware models~\cite{wang2009mean,wang2009portfolio} which take into account the risk\footnote{In this paper we use risk and uncertainty interchangeably.} have been shown for non-neural IR approaches, this has not yet been explored for neural L2R{} models.
In this paper we first analyze the calibration of neural rankers, specifically BERT-based rankers{} for IR tasks to evaluate how calibrated they are. Then, to model the uncertainty of BERT-based rankers{}, we propose \emph{stochastic} neural ranking models (see Figure~\ref{fig:stochastic_vs_deterministic}), by applying different techniques to model uncertainty of DNNs, namely MC Dropout~\cite{gal2016dropout} and Deep Ensembles~\cite{lakshminarayanan2017simple} which are agnostic to the particular DNN.
In our experiments, we test models under \emph{distributional shift}, i.e. the test data distribution is different from the training data also referred to as out-of-distribution (OOD) examples~\cite{lee2018simple}. In real-world settings, there are often inputs that are shifted due to factors such as non-stationarity and sample bias. Additionally, this experimental setup provides a way of measuring whether the DNN \textit{"know what it knows"}~\cite{ovadia2019can}, e.g. output high uncertainty for OOD examples.
We find that BERT-based rankers{} are not robustly calibrated. Stochastic BERT-based rankers{} have 14\% less calibration error on average than BERT-based rankers{}. Uncertainty estimation from stochastic BERT-based rankers{} is advantageous for downstream applications as shown by our experiments for risk-aware neural ranking (2\% more effective on average relative to a model without risk-awareness) and for predicting unanswerable conversational contexts (improves classification by 33\% on average of all conditions).
\section{Related Work}
\subsubsection*{Calibration and Uncertainty in IR}
Even though to optimally rank documents according to the PRP~\cite{robertson1977probability} requires the model to be calibrated~\cite{gordon1991utility} (\textbf{[C1]}{}), the calibration of ranking models has received little attention in IR. In contrast, in the machine learning community there have been a number of studies about calibration~\cite{ovadia2019can,maddox2019simple}, due to the larger decision making pipelines DNNs are often part of and their importance for model interpretability~\cite{thiagarajan2020calibrating}. For instance, in the automated medical domain it is important to provide a calibrated confidence measure besides the prediction of a disease diagnosis to provide clinicians with sufficient information~\cite{jiang2012calibrating}.~\citet{guo2017calibration} has shown that DNNs are not well calibrated in the context of computer vision, motivating our study of the calibration of neural L2R{} models.
The second condition (\textbf{[C2]}{}) for optimal retrieval when ranking according to the PRP~\cite{gordon1991utility} is that models report predictions with certainty. While the (un)certainty has not been studied in neural L2R{} models, there are classic approaches in IR that model the uncertainty. Such approaches have been mostly inspired by economics theory, treating variance as a measure of uncertainty~\cite{varian1999economics}. Following such ideas, non-neural ranking models that take uncertainty into account (i.e. risk-aware models), and thus do not follow the PRP~\cite{robertson1977probability}, have been proposed~\cite{zhu2009risky,wang2009portfolio}, showing significant effectiveness improvements compared to the models that do not model uncertainty. Uncertainty estimation is a difficult task that has other applications in IR besides improving the ranking effectiveness: it can be employed to decide between asking clarifying questions and providing a potential answer in conversational search~\cite{aliannejadi2019asking}; to perform dynamic query reformulation~\cite{lin2020query} for queries where the intent is uncertain; and to predict questions with no correct answers~\cite{feng-etal-2020-none}.
\subsubsection*{Bayesian Neural Networks}
Unlike standard algorithms to train neural networks, e.g. SGD, that fit point estimate weights given the observed data, Bayesian Neural Networks (BNNs) infer a distribution over the weights given the observed data. \citet{Denker1987LargeAL} contains one of the earliest mentions of choosing probability over weights of a model. An advantage of the Bayesian treatment of neural networks~\cite{mackay1992practical,neal2012bayesian,blundell2015weight} is that they are better at representing existing uncertainties in the training procedure. One limitation of BNNs is that they are computationally expensive compared to DNNs. This has lead to the development of techniques that scale well, and do not require modifications of the neural net architecture and training procedure. \citet{gal2016dropout} proposed a way to approximate Bayesian inference by relying on dropout~\cite{srivastava2014dropout}. While dropout is a regularization technique that ignores units with probability $p$ during every training iteration and is disabled at test time, \texttt{Dropout}{}~\cite{gal2016dropout} employs dropout at both train and test time and generates a predictive distribution{} after a number of forward passes. \citet{lakshminarayanan2017simple} proposed an alternative: they employ ensembles of models (\texttt{Ensemble}{}) to obtain a predictive distribution{}. \citet{ovadia2019can} showed that \texttt{Ensemble}{} are able to produce well calibrated uncertainty estimates that are robust to dataset shift.
\subsubsection*{Conversational Search}
\emph{Conversational search} is concerned with creating agents that fulfill an information need by means of a \emph{mixed-initiative} conversation through natural language interaction. A popular approach to conversational search is its modeling as an ad-hoc retrieval task: given an ongoing conversation and a large corpus of historic conversations, retrieve the response that is best suited from the corpus (this is also known as conversation response ranking~\cite{wu2017sequential, yang2018response,penha2020curriculum,gu2020speaker,lu2020improving}). This retrieval-based approach does not require task-specific semantics by domain experts~\cite{henderson2019convert}, and it avoids the difficult task of dialogue generation, which often suffers from uninformative, generic responses~\cite{li2016diversity} or responses that are incoherent given the dialogue context~\cite{li2016persona}. One of the challenges of conversational search is identifying unanswerable questions~\cite{feng-etal-2020-none}, which can trigger for instance clarifying questions~\cite{aliannejadi2019asking}. Identifying unanswerable conversational
contexts is one of the applications we employ uncertainty estimation for. Intuitively, if the system has high uncertainty in the available responses, there may be no correct response available. In this paper we focus on pointwise BERT for ranking, which is a competitive approach for the conversation response ranking task\footnote{BERT-based rankers{} are currently the top performing models across three conversation response ranking benchmarks: \url{https://bit.ly/34RTJ2r}.}.
\section{Method}
In this section we introduce the methods used for answering the following research questions: \textbf{RQ1} \textit{How calibrated are deterministic and stochastic BERT-based rankers{}?} \textbf{RQ2} \textit{Are the uncertainty estimates from stochastic BERT-based rankers{} useful for risk-aware ranking?} \textbf{RQ3} \textit{Are the uncertainty estimates obtained from stochastic BERT-based rankers{} useful for identifying unanswerable queries?} We first describe how to measure the calibration of neural rankers (\textbf{[C1]}{}), followed by our approach for modeling and ranking under uncertainty (\textbf{[C2]}{}), and then we describe how we evaluate their robustness to distributional shift.
\subsection{Measuring Calibration} \label{section:calibration}
To evaluate the \emph{calibration} of neural rankers (\textbf{RQ1}) we resort to the Empirical Calibration Error (ECE)~\cite{naeini2015obtaining}. ECE is an intuitive way of measuring to what extent the confidence scores from neural networks align with the true correctness likelihood. It measures the difference between the observed reliability curve~\cite{degroot1983comparison} and the ideal one. More formally, we sort the predictions of the model, divide them into $c$ buckets $\{B_{0}, ..., B_{c}\}$, and take the weighted average between the average predicted probability of relevance $avg(B_{i})$ and the fraction of relevant documents $\frac{rel(B_{i})}{|B_{i}|}$ in the bucket:
\begin{equation*}
ECE = \sum_{i=0}^{c} \frac{|B_{i}|}{n} \bigg| avg(B_{i}) - \frac{rel(B_{i})}{|B_{i}|} \bigg| ,
\end{equation*}
where $n$ is the total number of test examples.
\subsection{Modeling Uncertainty}
First we define the ranking problem we focus on, followed by the deterministic BERT-based ranker{} baseline model (\texttt{BERT}{}). Having set the foundations, we move to the methods we propose to answer \textbf{RQ2} and \textbf{RQ3}: a stochastic BERT-based ranker{} to \emph{model uncertainty} (\texttt{S-BERT}) and a risk-aware BERT-based ranker{} to \emph{take into account uncertainty provided by \texttt{S-BERT} when ranking} (\texttt{RA-BERT}).
\subsubsection{Conversation Response Ranking}
The task of conversation response ranking~\cite{zhang2018modeling,gu2019utterance,tao2019multi,henderson2019convert,penha2020curriculum,yang2020iart} (also known as \emph{next utterance selection}), concerns retrieving the best response given the dialogue context. We choose this specific task due to the large-scale training data available, suitable for the training of neural L2R{} models. Formally, let $\set{D}=\{(\set{U}_i, \set{R}_i, \set{Y}_i)\}_{i=1}^{N}$ be a data set consisting of $N$ triplets: dialogue context, response candidates and response relevance labels. The dialogue context $\set{U}_i$ is composed of the previous utterances $\{u^1, u^2, ... , u^{\tau}\}$ at the turn $\tau$ of the dialogue. The candidate responses $\set{R}_i = \{r^1, r^2, ..., r^k\}$ are either ground-truth responses or negative sampled candidates, indicated by the relevance labels $\set{Y}_i = \{y^1, y^2, ..., y^k\}$\footnote{Typically, the number of candidates $k \ll K$, where $K$ is the number of available responses and by design the number of ground-truth responses is usually one, the observed response in the conversational data. In our experiments k=10.}. The task is then to learn a ranking function $f(.)$ that is able to generate a ranked list for the set of candidate responses $\set{R}_i$ based on their predicted relevance scores $f(\set{U}_i,r)$.
\subsubsection{Deterministic \texttt{BERT} Ranker}
We use BERT for learning the function $f(\set{U}_i,r)$, based on the representation learned by the \texttt{[CLS]}{} token in a pointwise manner. The input for BERT is the concatenation of the context $\set{U}_i$ and the response $r$, separated by SEP tokens. This is the equivalent of early adaptations of BERT for ad-hoc retrieval~\cite{yang2019simple} transported to conversation response ranking. Formally the input sentence to BERT is $concat(\set{U}_i,r) = u^1 \; | \; [U] \; | \; u^2 \; | \; [T] \; | \; ... \; | \; u^{\tau} \; | \; [SEP] \; | \; r$, where $|$ indicates the concatenation operation. The utterances from the context $\set{U}_i$ are concatenated with special separator tokens $[U]$ and $[T]$ indicating end of utterances and turns. The response $r$ is concatenated with the context using BERT's standard sentence separator $[SEP]$. We fine-tune BERT on the target conversational corpus and make predictions as follows: $f(\set{U}_i,r) = \sigma(FFN(BERT_{CLS}(concat(\set{U}_i,r)))),$ where $BERT_{CLS}$ is the pooling operation that extracts the representation of the \texttt{[CLS]}{} token from the last layer and $FFN$ is a feed-forward network that outputs logits for two classes (relevant and non-relevant). We pass the logits through a softmax transformation $\sigma$ that gives us a probability of relevance. Since $f(\set{U}_i,r)$ outputs a point estimate value of relevance probability, we refer to it as \texttt{BERT}{}.
\subsubsection{Stochastic \texttt{S-BERT} Ranker}
In order to obtain a predictive distribution{}, $R_{r} = \{f(\set{U}_i,r)^{0}, f(\set{U}_i,r)^{1}, ... , f(\set{U}_i,r)^{n}\}$, which allows us to extract uncertainty estimates, we rely on two techniques, namely \texttt{Ensemble}{}~\cite{lakshminarayanan2017simple} and \texttt{Dropout}{}~\cite{gal2016dropout}. Both techniques scale well and do not require modifications on the architecture or training of BERT.
\paragraph{Using Deep Ensembles (\texttt{S-BERT$^{E}$}{})}
We train $M$ models using different random seeds without changing the training data, each with its own set of parameters $\{\theta_{m}\}_{m=1}^M$ and make predictions with each one of them to generate $M$ predicted values: $R_{r}^{E} = \{f(\set{U}_i,r)^{0}, f(\set{U}_i,r)^{1}, ... , f(\set{U}_i,r)^{M}\}$. The mean of the predicted values is used as the predicted probability of relevance: $\text{\texttt{S-BERT$^{E}$}{}}(\set{U}_i,r) = E[R_{r}^{E}],$ and the variance $var[R_{r}^{E}]$ gives us a measure of the uncertainty in the prediction.
\paragraph{Using MC Dropout (\texttt{S-BERT$^{D}$}{})}
We train a single model with parameters $\theta$ and employ dropout at test time and generate stochastic predictions of relevance by conducting $T$ forward passes: $R_{r}^{D} = \{f(\set{U}_i,r)^{0}, f(\set{U}_i,r)^{1}, ... , f(\set{U}_i,r)^{T}\}$. The mean of the predicted values is used as the predicted probability of relevance: $\text{\texttt{S-BERT$^{D}$}{}}(\set{U}_i,r) = E[R_{r}^{D}],$ and the variance $var[R_{r}^{D}]$ gives us a measure of the uncertainty.
\subsubsection{Risk-Aware \texttt{RA-BERT} Ranker}
\label{section:riskaware}
Given the predictive distribution{} $R_{r}$, obtained either by \texttt{Ensemble}{} or \texttt{Dropout}{}, we use the following function to rank responses with risk-awareness:
\begin{equation*}
\begin{split}
\text{\texttt{RA-BERT}}(\set{U}_i,r) = E[R_{r}] - b * var[R_{r}] \\
- 2b\sum_{i}^{n-1}cov[R_{r}, R_{r_{i}}],
\end{split}
\end{equation*}
where $E[R_{r}]$ is the mean of the predictive distribution{}, and $b$ is a hyperparameter that controls the aversion or predilection towards risk. Unlike \cite{zuccon2011back}, we are not combining different runs that encompass different model architectures. We instead take a Bayesian interpretation of the process of generating a predictive distribution{} from a single model architecture. We refer to the rankers as \texttt{RA-BERT$^{D}$}{} and \texttt{RA-BERT$^{E}$}{}, when using \texttt{S-BERT$^{D}$}{}'s predictive distribution{} and \texttt{S-BERT$^{E}$}{}'s predictive distribution{} respectively.
\subsection{Robustness to Distributional Shift}
In order to evaluate whether we can trust the model's calibration and uncertainty estimates, similar to \cite{ovadia2019can} we evaluate how robust the models are to different types of shift in the test data. We do so by training the model using one setting and applying it in a different setting. Specifically for all three research questions we test the models under the following two settings: cross-domain and cross-NS.
\subsubsection{Cross Domain} We train the model using the training set from one domain, i.e. dataset, known as the source domain $\set{D_{S}}$ and evaluate it on the test set of a different domain, known as the target domain $\set{D_{T}}$. This is also known as the problem of domain generalization~\cite{gulrajani2020search}.
\subsubsection{Cross Negative Sampling} Pointwise L2R{} models are trained pairs of query and relevant document and pairs of query and non relevant documents~\cite{lucchese2017impact}. Selecting the non-relevant documents requires a \emph{negative sampling} (NS) strategy. For the cross-NS{} condition, we test models on negative documents that were sampled using a different NS strategy than during training, evaluating the generalization of the models on a shifted distribution of candidate documents. We use three NS strategies. In \textbf{\texttt{NS\textsubscript{random}}{}} we randomly select a response $r$ from the list of all responses. For \textbf{\texttt{NS\textsubscript{classic}}} we retrieve negative samples using the conversational context $\set{U}_{i}$ as query to a conventional retrieval model and all the responses $r$ as documents. In \textbf{NS\textsubscript{sentenceEmb}} we represent both $\set{U}_{i}$ and all the responses $r$ with a sentence embedding technique and retrieve candidate responses using a similarity measure.
\section{Experimental Setup}
We consider three large-scale information-seeking conversation datasets\footnote{\texttt{MSDialog}{} is available at~\url{https://ciir.cs.umass.edu/downloads/msdialog/}; \texttt{MANTiS}{} is available at~\url{https://guzpenha.github.io/MANtIS/}; \texttt{UDC\textsubscript{DSTC8}}{} is available at ~\url{https://github.com/dstc8-track2/NOESIS-II}.} that allow the training of neural ranking models for conversation response ranking: \textbf{\texttt{MSDialog}{}}~\cite{qu2018analyzing} contains 246K context-response pairs, built from 35.5K information seeking conversations from the Microsoft Answer community, a QA forum for several Microsoft products; \textbf{\texttt{MANTiS}{}}~\cite{penha2019introducing} contains 1.3 million context-response pairs built from conversations of 14 Stack Exchange sites, such as \textit{askubuntu} and \textit{travel}; \textbf{\texttt{UDC\textsubscript{DSTC8}}{}}~\cite{Kummerfeld_2019} contains 184k context-response pairs of disentangled Ubuntu IRC dialogues.
\subsection{Implementation Details}
We fine-tune BERT~\cite{devlin2019bert} (\textit{bert-base-cased}) for conversation response ranking using the \textit{huggingface-transformers}~\cite{wolf2019huggingface}. We follow recent research in IR that employed fine-tuned BERT for retrieval tasks~\cite{nogueira2019passage,yang2019simple}, including conversation response ranking~\cite{penha2020curriculum,vig2019comparison,whang2019domain}. When training BERT we employ a balanced number of relevant and non-relevant---sampled using BM25~\cite{robertson1994some}---context and response pairs. The sentence embeddings we use for cross-NS{} is \texttt{sentenceBERT}~\cite{reimers2019sentence} and we employ dot product calculation from FAISS~\cite{JDH17}. We consider each dataset as a different domain for cross-NS{}. We use cross entropy loss and the Adam optimizer~\cite{kingma2014adam} with $lr=5^{-6}$ and $\epsilon = 1^{-8}$, we train with a batch size of $6$ and fine-tune the model for 1 epoch. This baseline BERT-based ranker{} setup yields comparable effectiveness with SOTA methods\footnote{We obtain 0.834 $R_{10}@1$ on \texttt{UDC\textsubscript{DSTC8}}{} with our baseline BERT model, c.f. Table~\ref{table:effectiveness}, while \texttt{SA-BERT}~\cite{gu2020speaker} achieves 0.830. The best performing model of the DSTC8~\cite{kim2019eighth} also employed a fine-tuned BERT}.
\subsection{Evaluation}
To evaluate the \emph{effectiveness} of the neural rankers we resort to a standard evaluation metric in conversation response ranking~\cite{yuan2019multi,gu2020speaker,tao2019multi}: recall at position $K$ with $n$ candidates\footnote{For example $R_{10}@1$ indicates the number of relevant responses found at the first position when the model has to rank 10 candidate responses.}: $R_n@K$. To evaluate the \emph{calibration} of the models, we resort to the Empirical Calibration Error (cf. \S\ref{section:calibration}, using $C=10$). Throughout, we report the test set results for each dataset. To evaluate the \emph{quality of the uncertainty estimation} we rely on two downstream tasks. The first is to improve conversation response ranking itself via Risk-Aware ranking (cf. \S\ref{section:riskaware}). The second, which fits well with conversation response ranking, is to predict unanswerable conversational contexts. Formally the task is to predict whether there is a correct answer in the candidates list $\set{R}$ or not. In our experiments, for half of the instances we remove the relevant response from the list, setting the label as None Of The Above (NOTA). The other half of the data has label 0 indicating that there is a suitable answer in the candidates list, for which we remove one of the negative samples instead. Similar to~\citet{feng-etal-2020-none}, who proposed to use the outputs (logits) of a LSTM-based model in order to predict NOTA, we use the uncertainties as additional features to the classifier for NOTA prediction. The input space with the additional features is fed to a learning algorithm (Random Forest), and we evaluate it with a 5 fold cross-validation procedure using F1-Macro.
\section{Results}
\begin{figure}[]
\centering
\includegraphics[width=.4\textwidth]{img/calibration_by_n_docs.pdf}
\caption{Calibration of \texttt{BERT}{} trained on a balanced number of relevant and non-relevant documents, and tested on unbalanced data with more non-relevant (\texttt{\#-non-rel}) than relevant (1 per query) documents. A fully calibrated model is represented by the dotted diagonal---for every bucket of confidence in relevance, the percentage of relevant documents found in that bucket is the confidence.}
\label{fig:calibration_by_n_docs}
\vspace{-0.5cm}
\end{figure}
\input{tables/risk_aware_table}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{img/risk_aware_ranking.pdf}
\caption{Gains of the Risk-Aware BERT-ranker for different values of risk aversion $b$.}
\label{fig:risk_aware_by_b}
\end{figure*}
\subsection{Calibration of Neural Rankers (RQ1)}
In order to answer our first research question about the calibration of neural rankers, let us first analyze \texttt{BERT}{} under standard settings (no distributional shift). Our results show that \texttt{BERT}{} is both effective and calibrated under no distributional shift conditions. In Table~\ref{table:effectiveness} we see that when the target data (\textit{Test on $\rightarrow$}) is the same as the source data (\textit{Train on $\downarrow$})---indicated by underlined values---we obtain the highest effectiveness (on average 0.70 $R_{10}@1$) and the lowest calibration error (on average 0.036 ECE). When plotting the calibration curves of the model in Figure~\ref{fig:calibration_by_n_docs}, we observe the curves to be almost diagonal (i.e. having near perfect calibration) when there are an equal number of relevant and non-relevant candidates ($\texttt{\#-non-rel} = 1$).
However, when we make the conditions more realistic\footnote{In a production system, the retrieval stage would be executed over all candidate responses. As a consequence, the data is highly unbalanced, i.e. only a few relevant responses among potentially millions of non-relevant responses.} by having multiple non-relevant candidates for each conversational context, we observe in Figure~\ref{fig:calibration_by_n_docs} that the calibration errors start to increase, moving away from the diagonal. Additionally, when we challenge the model in cross-domain{} and cross-NS{} settings, the calibration error increases significantly as evident in Table~\ref{table:effectiveness}. On average, the ECE is \textbf{4.6} times higher for cross-domain{} and \textbf{7.9} times higher for cross-NS{}. Thus \textbf{answering the first part of our first research question about the calibration of deterministic BERT-based rankers{}, indicating that they do not have robust calibrated predictions}, failing on the scenarios where there is a distributional shift.
In order to answer the remaining part of RQ1, on how calibrated are \emph{stochastic} BERT-based rankers{}, we consider Table~\ref{table:calibration}. It displays the improvements (relative drop in ECE) over \texttt{BERT}{} in terms of calibration. We see that \texttt{S-BERT$^{E}$}{} is on average \textbf{14}\% better (has less calibration error) than \texttt{BERT}{}, while \texttt{S-BERT$^{D}$}{} is on average \textbf{10}\% better than \texttt{BERT}{}, \textbf{answering our first research question: stochastic BERT-based rankers{} have better calibration than deterministic BERT-based ranker{}}. We hypothesize that \texttt{S-BERT$^{E}$}{} lead to less ECE than \texttt{S-BERT$^{D}$}{} because it better captures the model uncertainty in the training procedure, since it combines different weights that explain equally well the prediction of relevance given the inputs. On the next section we focus on evaluating the effectiveness of such models that are better calibrated and also taking into account uncertainty when ranking.
\input{tables/nota_cross_domain}
\input{tables/nota_cross_ns}
\subsection{Uncertainty Estimates for Risk-Aware Neural Ranking (RQ2)}
In order to evaluate the quality of the uncertainty estimations, we first resort to using them as a measure of the risk through risk-aware neural ranking (\texttt{RA-BERT$^{D}$}{} and \texttt{RA-BERT$^{E}$}{}). Figure \ref{fig:risk_aware_by_b} displays the effectiveness in terms of $R_{10}@1$ gains over \texttt{BERT}{} for the different settings (cross-domain{} and cross-NS{}) when varying the risk aversion $b$.
We note that when $b=0$, we are using the mean of the predictive distribution{} and disregard the risk, which is equivalent to \texttt{S-BERT$^{D}$}{} and \texttt{S-BERT$^{E}$}{}. The ensemble based average \texttt{S-BERT$^{E}$}{} is more effective than the baseline \texttt{BERT}{} for almost all combinations and \texttt{S-BERT$^{D}$}{} is equivalent to the baseline. When using $b<0$, we are ranking with risk predilection (the opposite of risk aversion), and in all conditions we found that the effectiveness was significantly worse than when $b=0$ and thus $b<0$ is not displayed in Figure \ref{fig:risk_aware_by_b}.
When increasing the risk aversion ($b>0$), we see that it has different effects depending on the combination of domain and NS. For instance, when training in \texttt{MSDialog}{} and applying on \texttt{UDC\textsubscript{DSTC8}}{}, increasing the risk aversion improves effectiveness of \texttt{RA-BERT$^{E}$}{} until $b$ reaches 0.25 and after that the effectiveness drops, meaning that too much risk aversion is not effective. In order to investigate whether ranking with risk aversion is more effective than using the predictive distribution{} mean, we select $b$ based on the best value observed on the validation set.
Table ~\ref{table:risk-aware-ranking} displays the results of this experiment, showing the improvements of \texttt{RA-BERT$^{D}$}{} and \texttt{RA-BERT$^{E}$}{} over \texttt{S-BERT$^{D}$}{} and \texttt{S-BERT$^{E}$}{} respectively. The results show that in a few cases (8 out of 30) the best value of $b$ is 0, for which risk-aversion is not the best option in the development set. We obtain effectiveness improvements primarily on the cross-NS{} condition (up to 17.2\% improvement of $R_{10}@1$), which is the hardest condition (when the models are most ineffective, c.f. Table~\ref{table:effectiveness}). \textbf{This answers our third research question, indicating that the uncertainties obtained from stochastic neural rankers are useful for risk-aware ranking, specially in the cross-NS{} setting where the baseline model is quite ineffective.} \texttt{RA-BERT$^{E}$}{} is on average 2\% more effective than \texttt{S-BERT$^{E}$}{}, while \texttt{RA-BERT$^{D}$}{} is on average 1.7\% more effective than \texttt{S-BERT$^{D}$}{}.
\subsection{Uncertainty Estimates for NOTA prediction (RQ3)}
Besides using the uncertainty estimation for risk-aware ranking, we also employ it for the NOTA (None of the Above) prediction task. We compare here different input spaces for the NOTA classifier. $E[R^{D}]${} stands for the input space that only uses the mean of the predictive distribution for the $k$ candidate responses in $\set{R}$ using \texttt{S-BERT$^{D}$}{}, +$var[R^{E}]${} uses both $E[R^{D}]${} and the uncertainties of \texttt{S-BERT$^{E}$}{} for the $k$ candidates and +$var[R^{D}]${} uses both the scores $E[R^{D}]${} and the uncertainties of \texttt{S-BERT$^{D}$}{}. Our results show that the uncertainties from \texttt{S-BERT$^{D}$}{} and of \texttt{S-BERT$^{E}$}{} significantly improve the F1 for NOTA prediction for both cross-domain{} (Table~\ref{table:nota_cross_domain}, improvement of 24\% on average when using \texttt{S-BERT$^{D}$}{}) and cross-NS{} settings (Table~\ref{table:nota_cross_ns}, improvement of 46\% on average when using \texttt{S-BERT$^{D}$}{}) \textbf{which answers our last research questions that the uncertainty estimates from stochastic neural rankers do improve the effectiveness of the NOTA prediction task (by an average of 33\% for all conditions considered).}
\section{Conclusions}
In this work we study the calibration and uncertainty estimation of neural rankers, specifically BERT-based rankers{}. We first show that deterministic BERT-based ranker{} is not robustly calibrated for the task of conversation response ranking and we improve its calibration with two techniques to estimate uncertainty through \emph{stochastic neural ranking}. We also show the benefits of estimating uncertainty using risk-aware neural ranking and for predicting unanswerable conversational contexts. As future work, investigating other applications of stochastic neural rankers are important, e.g. for other neural L2R{} architectures, for other retrieval tasks~\cite{guo2019deep}, for fair retrieval~\cite{fern2020evaluating}, for ensembling neural rankers~\cite{zuccon2011back} and for query reformulation~\cite{lin2020query}.
\section*{Acknowledgements}
This research has been supported by NWO projects SearchX (639.022.722) and NWO Aspasia (015.013.027).
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{"url":"http:\/\/tex.stackexchange.com\/questions\/163992\/displaying-fractions-nicely","text":"# Displaying fractions nicely\n\nI'd like the following to look nicer:\n\n$\\frac{1 - \\left(\\frac{r-3}{r-1} \\right)^{k-(p+q)+1}}{1 - \\left(\\frac{r-3}{r-1} \\right)}$\n\n\nAs of now, the horizontal line is way too long.\n\n-\nWelcome to TeX.SX. A tip: If you indent lines by 4 spaces, then they're marked as a code sample. You can also highlight the code and click the \"code\" button ({}) or hit Ctrl+K. \u2013\u00a0 Claudio Fiandrino Mar 6 '14 at 12:16\nThere is the \\dfrac command, but I can't remember in which package it can be found. \u2013\u00a0 Christian Hupfer Mar 6 '14 at 12:26\n@ChristianH. amsmath. \u2013\u00a0 Gonzalo Medina Mar 6 '14 at 12:30\nIf the horizontal line does not include the exponent, then I'd be unsure whether the exponent applies to the whole fraction. \u2013\u00a0 MvG Mar 6 '14 at 17:53\n\nYou can use the command \\mathrlap from the mathtools package.\n\nMWE:\n\n\\documentclass{article}\n\n\\usepackage{mathtools}\n\n\\begin{document}\n\nDisplaystyle version\n\n$\\dfrac{1 - \\left(\\dfrac{r-3}{r-1} \\right)^{\\mathrlap{k-(p+q)+1}}}{1 - \\left(\\dfrac{r-3}{r-1} \\right)}$\n\n\\bigskip\n\nHere is the inline version $\\frac{1 - \\left(\\frac{r-3}{r-1} \\right)^{\\mathrlap{k-(p+q)+1}}}{1 - \\left(\\frac{r-3}{r-1} \\right)}$\n\n\\end{document}\n\n\n-\n\nYou could certainly use a trick like this:\n\n${\\frac{1 - \\left(\\frac{r-3}{r-1} \\right)}{1 - \\left(\\frac{r-3}{r-1} \\right)}}^{\\mbox{\\tiny$k-(p+q)+1$}}$\n\n-\n+1, but you can tweak it even better by adding \\!\\! at the beginning of the tiny mbox, as ^{\\mbox{\\tiny $\\!\\!k-(p+q)+1$}} \u2013\u00a0 Steven B. Segletes Mar 6 '14 at 13:15","date":"2015-09-04 04:15:03","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\": 2, \"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\": 0.90687096118927, \"perplexity\": 3688.3677341215507}, \"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-2015-35\/segments\/1440645335509.77\/warc\/CC-MAIN-20150827031535-00053-ip-10-171-96-226.ec2.internal.warc.gz\"}"}
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\section{Introduction}
Recently a simple minisuperspace describing the Oppenheimer -
Snyder (OS) collapsing star was found [1]. The semiclassical wave
function of that model (e.g. the semiclassical
solution of the Wheeler-DeWitt equation)
is a bound state. This leads to quantization conditions. The
corresponding (Bohr-Sommerfeld) quantization condition can be
written in the form
\begin{equation}
F(M,R_{0}) = \hbar \left( n + \frac{1}{2} \right) ~~,~~
n = 0,1,2,...
\end{equation}
where $M$ is the mass of the collapsing star, $R_{0}$ is its
initial radius, and $F(M,R_{0})$ is a function of $M$ and
$R_{0}$ to be given later [1]. For fixed $R_{0}$, (1.1) implies
mass quantization.
The idea of mass quantization
is an old one. Using general arguments
from quantum mechanics (of adiabatic variables) and general
relativity, Bekenstein [2] got the black hole
area quantization condition
\begin{equation}
M_{ir}^{2} = \frac{1}{2} \hbar n ~~,~~ n = 1,2,3,...
\end{equation}
where $M_{ir}$ is the irreducible mass of the black hole, which
is related to its surface area , $ {\cal A}
= 16 \pi M_{ir}^{2} $ . This discrete spectrum can be related
to the thermodynamic properties of black holes [3].
We will show that in the case of black holes, one can get (1.2)
from (1.1), which is correct for any OS star.
Beside the quantization condition (1.2), one can find in this
explicit model also the general
quantum states (the solutions of the Wheeler-DeWitt equation)
forming this star. By that one can hope to understand
better not just the quantization conditions,
but also the thermodynamical properties of black holes.
This is the purpose of this paper.
While it has been known for two decades, there are still some open
questions concerning the entropy of black holes.
The classical considerations [4], which use the
analogy between some geometrical properties of black holes
and thermodynamics, gives the ``laws of black hole
thermodynamics", from which one can get the entropy.
Semiclassical considerations [5], on the other hand,
use the (formal) path integrals (approach)
to find the partition function, and then the entropy. But both
approaches do not use basic statistical mechanics reasoning,
namely, finding the entropy by calculating the number of different
``microscopic states" that correspond to the same ``macroscopic state"
that we call a black hole. This ``missing link" is very important,
because it requires the
understanding of {\em microscopic} states forming
a (macroscopic) black hole. Those
microscopic states are the quantum states, so their importance
to the understanding of black holes thermodynamics is obvious.
The OS model gives a one dimensional minisuperspace,
in which of course one cannot hope to get a degeneracy that
will give a nonzero entropy. So one must extend the model.
One such an extension is the inclusion of inhomogeneous (spherically
symmetric dust) distributions. This was done a long time ago by
Lund [6]. Lund used the dust matter as a ``clock" and then fixed
the gauge completely, reducing the constrained matter-gravity
theory to an unconstrained one.
We will use Lund's infinite dimensional ``midisuperspace", and
apply it to the collapsing star case.
Though infinite dimensional, Lund's midisuperspace shares some
resemblance with the OS model, so one can analyse it
in the same manner and find the quantum
states that correspond to a classical black hole.
Knowing the microscopic (quantum) states, one can calculate
their degeneracy, and find the entropy of the black hole. Then
by using the standard thermodynamical definitions one can
get the other thermodynamical quantities (e.g. the temperature).
We will consider both the static ``eternal black hole picture" [7],
and the dynamic Hawking evaporation one [8]. They both can be
studied in our framework, and the results that we get are
in agreement with the known ones.
In this work we use the {\em semiclassical} approximation
only. This is for two reasons: first, the OS model as well as
Lund's one, are correct only semiclassicaly. And second, we
use Einstein gravity (coupled to matter) which should be
(at least) a good approximation semiclassicaly.
We use geometrical units $G = c = 1$.
The paper is organized as follows: in chapter 2 we describe
the OS model, solve (semiclassicaly) the corresponding
Wheller-DeWitt equation, and get the mass and area quantization
conditions. In chapter 3 we describe Lund's midisuperspace and
find the (semiclassical) solutions to the Wheeler-DeWitt equation.
In chapter 4 we describe the midisuperspace of a collapsing star,
and find the quantum states forming this star (black hole).
In chapter 5 we study the thermodynamical properties of black
holes in this framework, and chapter 6 presents some concluding
remarks.
\vspace{1cm}
\section{The OS Model}
\subsection{The OS Minisuperspace}
\setcounter{equation}{0}
In 1939 Oppenheimer and Snyder [9] found a very simple solution
(of Einstein gravity couple to dust matter)
describing a collapse of a spherically symmetric homogeneous dust star.
In their solution the Schwarzschild exterior is smoothly connected
to the interior region, which is a slice of a Friedmann universe.
The interior region is described by the Friedmann line element
\begin{equation}
d s^{2} = -N^{2}(t) dt^{2} + a^{2}(t) [ d{\chi}^{2} +
sin^{2}{\chi} d {\Omega^{2}_{2}} ]
\end{equation}
The range of $\chi$ is $0 \leq \chi \leq \chi_{0}$ , where
$\chi_{0} \leq \pi/2$ . At $\chi = \chi_{0}$ the interior is
matched to the exterior Schwarzschild solution. If $M$ and $R_{0}$
are the mass and initial radius of the star, then the matching
conditions are
\begin{eqnarray}
M &=& \frac{1}{2} a_{0} sin^{3}\chi_{0} \nonumber \\
R_{0} &=& a_{0} sin \chi_{0}
\end{eqnarray}
where $a_{0}$ is the initial Friedmann radius.
The gravitational Lagrangian may be split into its interior and
exterior parts,
\begin{equation}
L_{G} = 4\pi \int_{0}^{\chi_{0}} sin^{2}\chi d \chi
\left[ \frac{3a}{N}{\dot{a}}^{2} - 3Na \right] +
\int_{r \geq r_{s}} \sqrt{-g}R d^{3}x
\end{equation}
where $r_{s}$ is the surface radius of the collapsing star.
The matter Lagrangian is
\begin{equation}
L_{M} = - 8\pi \int \sqrt{-g} \rho U^{\mu}U_{\mu} d^{3}x
\end{equation}
where $\rho$ is the density of the star and $U_{\mu}$ is the
four-velocity of the matter particles. Energy momentum
conservation, $\nabla_{\mu} T^{\mu \nu} = 0$, implies
$\rho = \rho_{0}/a^{3}$, where $\rho_{0}$ is a constant to
be determined by the initial conditions. The OS model
requires $\rho_{0} = 3a_{0}/8\pi$. So using (2.3),(2.4) and
$ U^{\mu}U_{\mu} = -1 $ we get
the total Lagrangian
\begin{equation}
L = L_{G} + L_{M} = 12 \pi \int_{0}^{\chi_{0}} sin^{2}\chi \left[
\frac{a \dot{a}^{2}}{N} - N (a - a_{0}) \right] + \int_{r>r_{s}}
\sqrt{- g} R d^{3}x
\end{equation}
The Hamiltonian corresponding to (2.5) is
\begin{equation}
H = N \left[ \frac{1}{4\alpha_{0} a} P_{a}^{2}
+ \alpha_{0} (a - a_{0}) \right]
+ \int_{r>r_{s}} {\cal H} d^{3} x
\end{equation}
where $ P_{a} = \partial L / \partial \dot{a} $, and $
\alpha_{0} = 12 \pi\int_{0}^{\chi_{0}} sin^{2}\chi d \chi $ .
Because the classical solution for $ r > r_{s} $ is the Schwarzschild
space-time, for which $ R_{Sch.}=0 $, only the first term in (2.5)
(or (2.6)) will contribute to the semiclassical dynamics\footnote{Using
the path integral approach, the semiclassical wave function is
\begin{eqnarray*}
\psi_{_{WKB}} = A exp[i S_{Class.}/\hbar ]
\end{eqnarray*}
and we see that only the first term in (2.5) will contribute.}.
The Wheeler-DeWitt equation is the quantum version of the
classical Hamiltonian constraint, $\partial H / \partial N
= 0 $ , in the coordinate representation: $ |\Psi> = \psi(a)
{}~,~ P_{a} = -i\hbar \partial / \partial a $ . Using (2.6) we
get the ``Schr\"{o}dinger equation"
\begin{equation}
\left( - \hbar^{2} \frac{d^{2}}{d a^{2}} + V(a) \right) \psi(a) = 0
\end{equation}
where $ V(a) = 4\alpha_{0}^{2} a(a - a_{0}) $ . If we define
$ x \equiv a - a_{0}/2 $, we get
\begin{equation}
\left( - \hbar^{2} \frac{d^{2}}{d x^{2}} + \frac{1}{4}
\omega^{2} x^{2} \right) \psi(x) = E \psi(x)
\end{equation}
where $ \omega = 4\alpha_{0} $ and $ E = \alpha_{0}^{2} a_{0}^{2} $.
As we can see from (2.8), $\psi $ describes an harmonic oscillator
(with mass $m = 1/2$). So the semiclassical wave function
describes a one-dimensional harmonic oscillator, which is of
course a bound state.
As in the Hartle-Hawking case [10], the solution of (2.8)
describes a superposition of two ``universes"\footnote{
The solution can be written as
\begin{eqnarray*}
\psi(x) = A (e^{ip(x)/\hbar} + e^{-ip(x)/\hbar})
\end{eqnarray*}
where $ p(x) = \int_{x_{0}}^{x} \sqrt{|V(x')|}dx' $.}, one that
collapsses to form a black hole, and one that expands, a white hole.
\vspace{0.5cm}
\subsection{Mass and Area Quantization}
Because the wave function of the OS model describes
a bound state, the spectrum is quantized.
Semiclassically we should use the
Bohr-Sommerfeld quantization condition, but in the case of an
harmonic oscillator it is exact,
\begin{equation}
E(n) = \hbar \omega \left( n + \frac{1}{2} \right)
{}~~,~~ n = 0,1,2,...
\end{equation}
Using the definition of $E$ and $\omega$ (see below (2.8)) we have
\begin{equation}
\frac{1}{4} \alpha_{0} a_{0}^{2} = \hbar \left( n +
\frac{1}{2} \right)
\end{equation}
In [1] we consider only the case $ R_{0} >> 2M $ which corresponds
to the usual cosmological situation. In that case we have
$ \alpha_{0} \simeq 4 \pi \chi_{0}^{3} $, and using (2.2) and (2.10)
we get [1]
\begin{equation}
M R_{0}^{3}(n) = \frac{1}{2\pi^{2}} \hbar^{2}
{\left( n + \frac{1}{2} \right)}^{2}
\end{equation}
For fixed initial radius, (2.11) gives mass quantization.
Of course all the above describes a dynamical process: the collapse
of the star. In this work we try to understand the quantum properties
of a ``static black hole," as seen by an outside observer, and
especially to find its entropy and
temperature (as measured by that observer).
For an outside observer, the above picture can be static only if
$ R_{0} \leq 2M $ . In that case the geometry outside the horizon
is always Schwarzschild, which is of course static. The
OS model requires $ R_{0} \geq 2M $ [9], so
only if $ R_{0} = 2M $ this model can describe a static geometry
everywhere outside the horizon.
In the case $ R_{0} = 2M $ we have $ \chi_{0} =
\pi / 2 $, and we get from (2.2) and (2.10)
\begin{equation}
M^{2}(n) = \frac{1}{3 \pi^{2}} \hbar \left( n +
\frac{1}{2} \right) ~,
\end{equation}
a surface area quantization:
the surface area of the black hole\footnote{ For a
Schwarzschild black hole $M_{ir} = M$, so ${\cal A} = 16 \pi
M^{2}$.} goes {\em linearly} with the quantum number $n$.
This is Bekenstein's result [2], but we got it by using an
explicit model, and by solving the corresponding Wheeler-DeWitt
equation. The fact that
we use a very simple (and even a non-realistic) model (the
OS star), and still get his results,
suggests that these are quite general.
In Bekenstein's original paper [2] the prefactor for $M^{2}$
was $ \hbar / 2 $ , but further considerations by Mukhanov [3]
suggest that the prefactor should
be $ \hbar ln 2 / 4 \pi $. In our case the prefactor is $ \hbar
/ 3 \pi^{2} $. This prefactor is a model dependent,
and the best that we
can hope (using our simple model\footnote{ For example,
we take only $R_{0} = 2M$. A more reasonable model should
take some average between $R_{0} = 0$ and $R_{0} = 2M$.}) is to
get the same order of magnitude. This is in fact what we got.
In the OS model, the (microscopic) state of a black hole with
a (macroscopic) mass $M$, is $|\Psi_{n}>$, where $n$ satisfy (2.12).
So in this model, for each macroscopic state (labeled by the mass $M$,
or the area ${\cal A}$) there is only one microscopic state\footnote{
We have a one dimensional harmonic oscillator, so there is no
degeneracy.} (labeled by the quantum number $n$). So the entropy of
the black hole in the OS model is zero\footnote{The entropy
goes like
\begin{eqnarray*}
S \sim ln(\hbox{number of microscopic states}) = ln(1) = 0
\end{eqnarray*}}. This is because the OS minisuperspace
is ``too small" (one dimensional). If we want to understand
the thermodynamics of black holes, we must extend the model.
In the next chapter we describe a
much bigger midisuperspace (an infinite dimensional one), which
will turn out to be
much more appropriate for studying black holes thermodynamics.
\vspace{1cm}
\section{Lund's Midisuperspace}
\subsection{The ADM Reduction Process}
\setcounter{equation}{0}
For a global hyperbolic space-time, $M = R \times \Sigma^{(3)}$,
one can use the ADM splitting [11], and write the Hamiltonian of
a dust matter coupled to Einstein gravity in the form [6]
\begin{equation}
H = \int d^{3}x dt \left[ N ( {\cal H}^{0} + {\cal E} )
+ N_{i} ( {\cal H}^{i} + {\cal P}^{i} ) \right]
\end{equation}
where $N$ and $N_{i}$ are the lapse function and shift vector
respectively. ${\cal H}^{0}$ and ${\cal H}^{i}$ are the
gravitational super-hamiltonian and super-momentum
\begin{eqnarray}
{\cal H}^{0} &=& h^{1/2} \left[ \pi^{ij}\pi_{ij} - \frac{1}{2}
\pi^{2} - h R^{(\Sigma)} \right] \\
{\cal H}^{i} &=& -2 h^{1/2} D_{j} \pi^{ij}
\end{eqnarray}
where $h_{ij}$ is the induced 3-metric on $\Sigma^{(3)}$
($ h = det(h_{ij})$ ), and $\pi^{ij}$ its conjugate momenta
($ \pi = {\pi^{i}}_{i}$ ). ${\cal E}$ and ${\cal P}^{i}$ are
the dust matter Hamiltonian and momentum respectively
\begin{eqnarray}
{\cal E} &=& h^{1/2} n^{\mu} n^{\nu} T_{\mu \nu} \\
{\cal P}^{i} &=& h^{1/2} n^{\mu} h^{ij} T_{\mu j}
\end{eqnarray}
where $n^{\mu}$ is a unit normal vector to the hypersurface
$\Sigma^{(3)}$, and $T_{\mu \nu}$ is (using (2.4))
\begin{equation}
T_{\mu \nu} = 16 \pi \rho U_{\mu} U_{\nu}
\end{equation}
If we define the scalar field $\phi$ ,
\begin{equation}
U_{\mu} \equiv \nabla_{\mu} \phi
\end{equation}
we can treat it as a dynamical variable (and $\rho$, which is
not a dynamical variable, will be a function of $h_{ij},~\pi^{ij},
{}~\phi,~\nabla_{\mu} \phi $, to be determined later.).
In the spherically symmetric case one can use the
$(R,\theta,\phi)$-coordinates on $\Sigma^{(3)}$ in which
\begin{equation}
ds_{(3)}^{2} = e^{2\mu} dR^{2} + e^{2\lambda} d{\Omega}^{2}_{2}
\end{equation}
where $\mu$ and $\lambda$ are functions of $t$ and $R$, and
$d\Omega^{2}_{2}$ is the volume element in $S^{2}$.
If we define $\pi_{\mu}$ and $\pi_{\lambda}$ such that
\begin{equation}
\pi^{ij} = \mbox{diag}\left( \frac{1}{2} e^{-2\mu}\pi_{\mu},~
\frac{1}{4} e^{-2\lambda}\pi_{\lambda},~\frac{1}{4} e^{-2\lambda}
sin^{-2}\theta \pi_{\lambda} \right)
\end{equation}
we get
\begin{eqnarray}
{\cal H}^{0} &=& e^{-(\mu + 2\lambda)} \left( \pi_{\mu}^{2} / 8
- \pi_{\mu} \pi_{\lambda} / 4 - 2 e^{2(\mu + 2\lambda)}
\left[ e^{-2\lambda} - \right. \right. \nonumber \\
& & \left. \left. e^{-2\mu} \left( 2 \lambda" -2\lambda' \mu'
+ 3 (\lambda')^{2} \right) \right] \right) \\
{\cal H}^{R} &=& -e^{-2\mu} ( \pi_{\mu}' - \mu' \pi_{\mu} -
\lambda' \pi_{\lambda} ) \\
{\cal E} &=& {\left( 16 \pi \rho h^{1/2} \right)}^{-1} p_{\phi}^{2}\\
{\cal P}^{R} &=& p_{\phi} h^{11} \phi'
\end{eqnarray}
where prime denote differentiation with respect to $R$, and
$p_{\phi} = \partial L / \partial \dot{\phi} = -16 \pi N h^{1/2}
U^{0} \rho $. We see that if we choose the coordinates for which
$ N^{R} = 0$, and using $U^{\mu} U_{\mu} = -1$, we get $ \rho
= {(16 \pi h^{1/2})}^{-1} (1 + h^{11} (\phi')^{2})^{-1/2} p_{\phi}$ ,
so from (3.12) we get
\begin{equation}
{\cal E} = {\left( 1 + h^{11} (\phi')^{2} \right)}^{\frac{1}{2}}
p_{\phi}
\end{equation}
and we see that ${\cal E}$ goes {\em linearly} with $p_{\phi}$.
This suggests that we can use $\phi$ as a time variable. And
indeed taking $\phi = -t ~,~ N^{R}=0 $ gives (using (3.10)-(3.13))
the known general solutions [12].
To complete the gauge fixing (the reduction process)
one must choose also the $R$-coordinate.
{}From the equations of motion (derive from (3.10)-(3.13))
one can get that $\lambda' e^{\lambda -\mu}$ is
a function of $R$ only, so one can choose
\begin{equation}
R = \lambda' e^{\lambda - \mu}
\end{equation}
Now the conjugate momenta are
\begin{eqnarray}
\pi_{R} &=& -{\left( \lambda' e^{\lambda-\mu} \right)}^{-1}
\pi_{\mu} \\
\bar{\pi}_{\lambda} &=& \pi_{\lambda} - e^{\lambda}
{\left( (\lambda')^{-1} e^{-\lambda} \pi_{\mu} \right)}'
\end{eqnarray}
Using (3.11) and (3.13), the supermomentum constraint,
$ {\cal H}^{R} + {\cal P}^{R} = 0 $, is now
\begin{equation}
\pi_{R} + \bar{\pi}_{\lambda} \lambda' + p_{\phi} \phi' = 0
\end{equation}
After solving the constraints, one ends up with the reduced
Lagrangian (or Hamiltonian)
\begin{equation}
S_{red} = 4\pi \int dt dR (\pi_{y}\dot{y} - {\cal H}_{ADM} )
\end{equation}
where
\begin{eqnarray}
y &=& 8 e^{\lambda} \\
\pi_{y} &=& \frac{1}{8} e^{-\lambda} \bar{\pi}_{\lambda}
\end{eqnarray}
and
\begin{equation}
{\cal H}_{ADM} = R^{2} \left( \frac{1}{y} \pi_{y}^{2} +
{(2R)}^{-2} ( R^{-2} -1 ) y \right)
\end{equation}
So we end up with ${\infty}^{1}$ unconstraint degrees of freedom,
the $y(r)$ field, with the Hamiltonian (3.22). The space of all
$y(r)$-field solutions is what we call ``Lund's midisuperspace".
\vspace{0.5cm}
\subsection{Quantum States}
The reduced Hamiltonian is
\begin{equation}
H = 4\pi \int {\cal H}_{ADM} dR = 4\pi \int R^{2} \left(
\frac{1}{y} \pi_{y}^{2} + {(2R)}^{-2} ( R^{-2} - 1 ) y \right) dR
\end{equation}
We use the coordinate representation (in Lund's midisuperspace)
\begin{eqnarray}
\hat{y} &=& y \\
\hat{\pi}_{y} &=& \frac{\hbar}{i} \frac{\delta}{\delta y}
\end{eqnarray}
So the corresponding Schr\"{o}dinger equation (in a different context
we could call it the Wheeler-DeWitt equation, see sec. 2) is
\begin{equation}
i \hbar \frac{\partial \Psi[y;t]}{\partial t} = \int \left(
- \frac{\hbar^{2}}{y} \frac{\delta^{2}}{\delta y^{2}} +
\frac{1}{4} (R^{-4} - R^{-2}) y \right) \Psi[y;t] R^{2}dR
\end{equation}
where $|\Psi>$ is the quantum state, and $\Psi[y;t] = <\Psi|
y(R,t)>$ is the wave functional. In this representation it is
a functional of the field $y(r)$ and a function of time $t$.
As one can see, we use the ``$y {\pi}_{y}$-ordering"\footnote{The field
$y$ is always to the left of its conjugate momenta.} in (3.26),
but different ordering will not change our results, which anyway
are correct only semiclassicaly.
A very important feature of (3.23) is that there are no $R$-derivatives
in $H$. This meens that the infinite degrees of freedom (d.o.f.) are
{\em decoupled}. Let $R_{s}$ be the surface of the dust ball,
then we can divide $R_{s}$ to $N$ equal parts, $R_{k} = \frac{R_{s}}{
N} k ~(k=1,2,...,N)$.
The ``continuum limit" is $N \rightarrow \infty$.
The vector space $\{y(r)\}$ is now
$\{ \vec{y} = (y_{1},y_{2},...,y_{N}) \}$ where $y_{k} = y(R_{k})$,
and the Schr\"{o}dinger
equation (3.26) becomes
\begin{equation}
i \hbar \frac{\partial \Psi(\vec{y},t)}{\partial t} = \sum_{k=1}^{N}
\left[ - a_{k} \frac{\hbar^{2}}{y_{k}} \frac{\partial^{2}}{\partial
y_{k}^{2}} + b_{k} y_{k} \right] \Psi(\vec{y},t)
\end{equation}
where $ a_{k} = R_{k}^{2} $ and $ b_{k} = (R_{k}^{-2} - 1)/4 $ are
positive constants ($b_{k}$ is positive because $0 \leq R \leq 1$ [6]).
Because of the decoupling we can write $|\Psi>$ as a direct
product
\begin{equation}
|\Psi> = |\Psi_{1}> |\Psi_{2}> \cdot \cdot \cdot |\Psi_{N}>
\end{equation}
and from (3.27) we have now
\begin{equation}
i \hbar \frac{\partial \Psi_{k}(y_{k},t)}{\partial t} =
\left[ - a_{k} \frac{\hbar^{2}}{y_{k}} \frac{\partial^{2}}{\partial
y_{k}^{2}} + b_{k} y_{k} \right] \Psi_{k}(y_{k},t)
\end{equation}
where $\Psi_{k}(y_{k},t) = < \Psi_{k}|y_{k}(t) >$.
The ADM Hamiltonian is time-independent so we have ${\cal H}_{ADM}
(y_{k}) = E_{k} = const.$ and
\begin{equation}
\left[ - a_{k} \frac{\hbar^{2}}{y_{k}} \frac{\partial^{2}}{\partial
y_{k}^{2}} + b_{k} y_{k} \right] \Psi_{k}(y_{k},t) = E_{k}
\Psi_{k}(y_{k})
\end{equation}
If we define
\begin{equation}
x_{k} \equiv y_{k} - \frac{1}{2} E_{k}
\end{equation}
we get the following harmonic oscillator Schrodinger equation
\begin{equation}
\left( - \frac{\hbar^{2}}{2m_{k}} \frac{\partial^{2}}{\partial
x_{k}^{2}} + \frac{1}{2} m_{k}
\omega^{2}_{k} x_{k}^{2} \right) \Psi_{k}(x_{k}) = \epsilon_{k}
\Psi_{k}(x_{k})
\end{equation}
where $ m_{k}=1/2a_{k} ~,~ \omega_{k} = \sqrt{8a_{k}b_{k}} $ and
$\epsilon_{k}=m_{k}{(\omega_{k}E_{k})}^{2}/8$. The solutions
of (3.32) are $ |\Psi_{k}> = | n_{k} >$ , where $|n_{k}>$ is a
one-dimensional harmonic oscillator with energy
$ \epsilon_{k} = \hbar \omega_{k} ( n_{k} + 1/2 ) $ , or
\begin{equation}
\frac{1}{8} a_{k}^{1/2} E_{k}^{2} = \hbar \left( n_{k} + \frac{1}{2}
\right)
\end{equation}
So the space of quantum states describing this spherically symmetric
dust ``universe" is spanned by
\begin{equation}
\{ |\Psi_{n_{1},n_{2},...,n_{N}} > = |n_{1}>|n_{2}> \cdot \cdot
\cdot |n_{N}> \}
\end{equation}
The total energy is
\begin{equation}
E = \sum_{k=1}^{N} E_{k} = \sum_{k=1}^{N} {\left( 8 \hbar
[2(R_{k}^{-4} - R_{k}^{-2})]^{-1/2} \right)}^{1/2} {(n_{k} + 1/2)}^{1/2}
\end{equation}
In the $N \rightarrow \infty$ limit, one can have a finite
energy only if one uses the Wick order, so $n_{k} + 1/2$ must be
replaced with $n_{k}$, but we will come to that later.
\vspace{1cm}
\section{Midisuperspace for a Collapsing Star}
\subsection{The Homogeneous Case}
\setcounter{equation}{0}
In the homogeneous case (the OS case), it is convenient to
use (2.1). The 3-metric in the $(R,\theta,\phi)$-coordinates, is
\begin{equation}
h_{ij} = \hbox{diag}\left( a^{2} {\left( \frac{d \chi}{d R}
\right)}^{2} ,~ a^{2}sin^{2} \chi,~ a^{2}sin^{2}\chi sin^{2} \phi
\right)
\end{equation}
and from (3.8),(3.15) and (4.1) we get
\begin{equation}
R = \lambda' e^{\lambda - \rho} = cos \chi
\end{equation}
We see that $R$ is not a ``usual radial coordinate".
For example the origin ($r=0$) is $R=1$, and $R_{min} =
cos \chi_{0} \geq 0$. From (3.15) one can see that $ R = r_{c}/
r_{e}$ [6], where $r_{c}$ is the circumference radius, and $r_{e}$
is the extrinsic radius of curvature.
Because the horizon is a minimal area, $R(horizon)=0$.
One can get this explicitly in the OS model, because
a static black hole corresponds to
$\chi_{0}=\pi/2$, so $ R(horizon) = cos \chi_{0} = 0 $.
For asymptotically flat space $R$ goes to unity at spatial
infinity. It is convenient to use a different radial
coordinate\footnote{The coordinate $r$ grows with the usual
radial coordinate while $R$ decreases.}
\begin{equation}
r \equiv sin \chi = {(1 - R^{2})}^{1/2}
\end{equation}
Now one should replace (3.23) with
\begin{equation}
H = \int_{0}^{r_{s}} {\cal H}(r) dr =
\int_{0}^{r_{s}} \left( \frac{32 r \sqrt{1-r^{2}}}{3 \pi}
\frac{P_{y}^{2}}{y} + \frac{3 \pi r}{2\sqrt{1-r^{2}}} y \right) dr
\end{equation}
where $ y = 8e^{\lambda} = 8ar $, and $ P_{y}= 3\pi y \dot{y}/64r\sqrt{
1- r^{2}}$. It is very easy to see that (4.4) is the correct Hamiltonian,
because from (2.6) and (4.4) we have $ P_{y} = 3\pi r P_{a} / 2 \alpha_{0}
\sqrt{1-r^{2}}$, so
\begin{eqnarray}
H &=& 12 \pi \int_{0}^{sin\chi_{0}} dr \frac{r^{2}}{\sqrt{1-r^{2}}}
\left( \frac{P^{2}_{a}}{4 \alpha_{0}^{2} a} + a \right) \nonumber \\
&=& \frac{P^{2}_{a}}{4 \alpha_{0} a} + \alpha_{0} a
\end{eqnarray}
which is exactly the gravitational part of (2.6).
The Schr\"{o}dinger equation is
\begin{equation}
\hat{H} |\Psi> = E |\Psi>
\end{equation}
In Lund's formalism $E$ is an undefined constant, but in the
collapsing star case, we must impose the matching conditions:
the solution must be smoothly matched to the outside Schwarzschild
space-time. This will constrain $E$. From (2.1) and (2.4) we have
\begin{equation}
E = 32\pi^{2} \int_{0}^{\chi_{0}} sin^{2}\chi d \chi a^{3} \rho
\end{equation}
Because $E$ is time independent, we can easily calculate it at
the beginning of the collapse, when the star is at rest. Using
$ a(t=0) = a_{0} $ and $ \rho(t=0)= 3 M / 4\pi R_{0}^{3} $, we get
\begin{equation}
E = \frac{ 2 \alpha_{0} a_{0}^{3} M}{R_{0}^{3}}
\end{equation}
Now we can use the matching conditions (2.2) to get
\begin{equation}
E = \alpha_{0} a_{0}
\end{equation}
and the Schr\"{o}dinger equation (4.6) is exactly (2.7),and one
can get the results of chapter 2.
Notice that we could get (4.9) from the requirement that
the collapse start ($t=0$) at rest. We will use that in the
inhomogeneous case.
In the homogeneous case the difference between Lund's
midisuperspace and a collapsing star midisuperspace is that
in the latter the energy $E$ is constraint. In the inhomogeneous
case the constraints are more complicated, but one can
deal with them in a similar way.
\vspace{0.5cm}
\subsection{The Inhomogeneous Case}
We saw that (in the homogeneous case) one can use Lund's
formalism, impose the energy condition (4.9), and get the OS
results. This can be generalized to
the inhomogeneous case.
Let us first go beck to the homogeneous case. Using (4.4)
and (4.9), the Schrodinger equation (4.6) can be written as
\begin{equation}
\int_{0}^{r_{s}} \left[ \frac{32 r\sqrt{1-r^{2}}}{3\pi}
\frac{P_{y}^{2}}{y}+ \frac{3\pi r}{2\sqrt{1-r^{2}}} (y - y_{0})
\right] \Psi[y] dr = 0
\end{equation}
where $y_{0}=y(r,t=0)=8a_{0}r$.
Now we can generalized to the inhomogeneous case. The result
(3.23) (or (4.4)) are correct for {\em any} spherically symmetric
dust ball (not just for the homogeneous one). In particular, they
are correct in the case of a collapsing (spherically symmetric
dust) star. If we want the collapse to start ($t=0$) at
rest, which is the generalization of (4.9),
then we must have\footnote{$E=\int_{0}^{r_{s}} {\cal E}
(r) dr$, so ${\cal H}(r) = {\cal E}(r)$, and from (4.4) and (4.10)
we get (4.11).}
\begin{equation}
{\cal E}(r) = \frac{3 \pi r}{2\sqrt{1-r^{2}}}y_{0}
\end{equation}
because only in that case $\dot{y}(r,t=0) = 0$. So the form
of (4.10) is quite general: it is correct also in the inhomogeneous
case. The only
difference is that in the inhomogeneous case, the
Friedmann radius, $a$, can be a
function of $r$ too. So as a function of $(r,t)$ the field
solution is different, but it has the same form $y(r,t) =
8a(r,t)r$. The space of all field solutions that describe
a collapsing star is of course a subspace of all the field
solutions (Lund's midisuperspace), but at this stage we do
not need to know the specific restrictions; what is
important is that we can use (4.10).
After making the discretization $ r_{k}=r_{s} k / N
{}~(k=1,2,...,N) $ , we get from (4.10)
\begin{equation}
\sum_{k=1}^{N}\left( \alpha_{k}\frac{P_{y_{k}}^{2}}{y_{k}}
+ \beta_{k} (y_{k} - y_{k}^{(0)} ) \right) \Psi(\vec{y}) = 0
\end{equation}
where $ \alpha_{k} = (32 r_{k} \sqrt{1-r_{k}^{2}})/3\pi $ , $ \beta_{k} =
3 \pi r_{k} / 2 \sqrt{1-r_{k}^{2}} $ and $ y_{k}^{(0)} = y_{0}(r_{k}) $.
Using $ \Psi(\vec{y}) = \prod_{k} \Psi_{k}(y_{k}) $ we have
a set of N independent equations
\begin{equation}
\left( \alpha_{k}\frac{P_{y_{k}}^{2}}{y_{k}}
+ \beta_{k} (y_{k} - y_{k}^{(0)} ) \right) \Psi_{k}(y_{k}) = 0
\end{equation}
Defining $ x_{k} \equiv y_{k} - y_{k}^{(0)} / 2 $ , we get the
(harmonic oscillator) Schr\"{o}dinger equation
\begin{equation}
\left( -\frac{\hbar^{2}}{2m_{k}} \frac{\partial^{2}}{
\partial k_{k}^{2}} + \frac{1}{2} m_{k}
\omega^{2}_{k} x^{2}_{k} \right) \Psi_{k}(x_{k}) =
\epsilon_{k} \Psi_{k}(x_{k})
\end{equation}
where $ m_{k} = 1 / 2 \alpha_{k} $ , $ \omega_{k} = 8 r_{k} $
and $ \epsilon_{k} = m_{k} \omega_{k}^{2} {(y_{k}^{
(0)})}^{2} / 8 $. The
quantization conditions are
\begin{equation}
r_{k} m_{k} {(y_{k}^{(0)})}^{2} = \hbar \left( n_{k} + \frac{1}{2}
\right)
\end{equation}
The total energy is
\begin{equation}
E = 3 \pi \sum_{k=1}^{N} \frac{r_{k} y_{k}^{(0)}}{2
\sqrt{1-r_{k}^{2}} }
\end{equation}
and from (4.15) we get
\begin{equation}
E = \hbar \sum_{k=1}^{N} \Omega_{k} (n_{k} + 1/2)
\end{equation}
where $\Omega_{k} = 4 / a_{k}^{(0)} $, $(a_{k}^{(0)} = a(r_{k},
t=0))$.
So a quantum state describing a collapsing dust star, starting
at rest, can be written as
\begin{equation}
|\Psi(\mbox{star})> = |n_{1}>|n_{2}> \cdot \cdot \cdot |n_{N}>
\end{equation}
where $|n_{k}>$ is a one dimensional harmonic oscillator
(exited to the level $n_{k}$) with frequency $\omega_{k}=8r_{k}$.
Each $|n_{k}>$ and so $|\Psi>$ are bound states,
and we end up with quantization conditions.
In the homogeneous case $ y_{k}^{(0)} = 8 a_{0} r_{k} $ ,
($ a_{k}^{(0)} = a_{0} $) and from (4.17) we get
\begin{equation}
E_{hom} = \frac{4 \hbar}{ a_{0} } \sum_{k} (n_{k}
+ 1/2 )
\end{equation}
and using (4.9) we have
\begin{equation}
\frac{1}{4} \alpha_{0} a_{0}^{2} = \hbar \sum_{k} (n_{k} + 1/2)
\end{equation}
which is (2.10) (remember that in the homogeneous case there is
only one d.o.f. so $N=k=1$) .
In the general inhomogeneous case, $\Omega_{k}$ is not
$k$-independent, but we can use ``mean field" reasoning
and write (4.17) as
\begin{equation}
E = \hbar \sum_{k=1}^{N} \Omega_{k} (n_{k} + 1/2) =
\hbar <\Omega> \sum_{k=1}^{N} (n_{k} + 1/2)
\end{equation}
One can use (4.21) as a definition of $<\Omega>$, which is
the ``average" of $\Omega_{k}$.
The results (4.18) and (4.21) are correct for any collapsing
spheriacly symmetric dust star, which start at rest. But in
the case of a static black hole, one can relate $E$ and
$<\Omega>$ to the mass of the black hole. In the homogeneous
case we have $ a_{0} = 2M_{bh} $ so from (4.9) we have $E = 6
\pi^{2} M_{bh}$, and from (4.17) $<\Omega> = \Omega_{k} =
{(2M_{bh})}^{-1}$. In the inhomogeneous case this can be generalized
by dimensional arguments to
\begin{equation}
E \sim \frac{1}{<\Omega >} \sim M_{bh}
\end{equation}
wher $\sim$ denote equality up to a constant of order unity.
Now using (4.21) and (4.22) we get
\begin{equation}
M^{2}_{bh} \sim \hbar \sum_{k=1}^{N} (n_{k} + 1/2)
\equiv \hbar (n_{tot} + 1/2)
\end{equation}
Which is the generalization of (2.12) to the general
inhomogeneous case, in agreement with Bekenstein's result.
\vspace{1cm}
\section{Black Hole Thermodynamics}
\subsection{Entropy}
\setcounter{equation}{0}
Using the standard statistical mechanics definition,
the entropy of a (macroscopic) black hole is
\begin{equation}
S_{bh} = \mbox{ln} [{\cal N}_{bh}(M)]
\end{equation}
where we choose $k_{B}$ to be one, and ${\cal N}_{
bh}(M)$ is the number of microscopic states that correspond
to the same macroscopic state (a black hole)
with mass $M$. The microscopic states are (4.18), and for a
given $M$ (or a given $E$), ${\cal N}_{bh}(M)$ is the number
of different $|\Psi(\mbox{star})>$-states, that satisfy (4.21)
(or (4.23)).
Consider first the limit $N \rightarrow \infty$:
according to (4.21) the energy, $E$, will
be finite only if
we use the Wick order. Then
if only a finite number of d.o.f. are excited, $E$ will be finite.
In that case there are infinitely many other d.o.f. that are in their
ground states, and we face two (probably related) problems:
first, our semiclassical approximation is very bad when most of
the d.o.f. are in their ground state. Second, ${\cal N}_{bh}$
is infinite (there are infinitely many ways to choose a
finite number of exited d.o.f. from an infinite number of total
d.o.f.) so the entropy will diverge.
As a matter of fact, the limit $N \rightarrow \infty$ is
questionable. For example (2.10) is as much a radius
quantization condition as it is a mass quantization condition.
This means that one cannot simply divide $a_{0}$ infinitely many
times. This is clearly a quantum gravitational issue.
But there is a way to avoid it:
If we choose {\em not} to use the Wick order,
then if $E$ is finite, $N$ must be finite too.
We see that (2.12) or its generalization (4.23) provide us
with a natural cutoff\footnote{In our geometrical units $M_{P} =
l_{P} = \hbar^{1/2}$.}
$N_{max} \sim (M/M_{P})^{2}$, which for a classical black
hole is a big, but still finite.
Of course $ 1 \leq N \leq N_{max} $. In a sense, $N$ describe
the amount of inhomogenity. For $N=1$ we have the homogeneous
case, for $N=2$ the ``almost" homogeneous one, and so on until
$N=N_{max}$ which describe the general inhomogeneous star.
We have then
\begin{equation}
{\cal N}_{bh}(M) = \sum_{N=1}^{N_{max}} {\cal N}_{bh}
(N,M)
\end{equation}
where ${\cal N}_{bh}(N,M)$ is the corresponding number
for a specific $N$. Our semiclassical approximation is
good as long as $N$ is much smaller then $N_{max}$, but we
will see that the contribution to (5.2) from $ N > N_{max}/2 $
is the same as from $ N < N_{max}/2 $. So at least we have
a good estimate to (5.2).
It is easy to see that the $N$'s that
will contribute to (5.2) must satisfy
$N = N_{max} - 2j ~,~ j = 0,1,...,(N_{max}-1)/2 $.
Let us start from $N=N_{max}$. In that case
we have only one state (4.18), $|\Psi > = |0>_{1} |0>_{2} ...
|0>_{N_{max}}$ . Next we consider $N=N_{max}-2$, it is
easy to see that they are $N_{max}-2$ states\footnote{The
oscilators are distinguishable because they have different
frequencies, $\omega_{k} = 8 r_{k}$. Or in other words:
different $k$'s correspond to different shells
which are distinguishable because they have different radii.},
$ |\Psi> = |0>_{1}..|1>_{k}..|0>_{N_{max}-2} $. In a similar
way, we have for any $N=N_{max}-2j$
\begin{equation}
{\cal N}_{bh}(N,M) = C^{j}_{N_{max}-1-j}
\end{equation}
where $C^{k}_{m}$ are the binomial coefficients. So
\begin{equation}
{\cal N}_{bh}(M) = \sum_{j=0}^{(N_{max}-1)/2}
C^{j}_{N_{max}-1-j}
\end{equation}
Though there is no known analytic expresion for (5.4) [13], it is
elementary to show numericaly that
\begin{equation}
\sum_{j=0}^{n} C^{j}_{2n-j} \simeq \mbox{exp}(
0.962 n - 0.320 )
\end{equation}
In our case we have $n=(N_{max}-1)/2 \sim (M/M_{P})^{2}$, so
\begin{equation}
{\cal N}_{bh}(M) \sim \mbox{exp}\left( C \frac{M^{2}}{M_{P}^{2}}
\right)
\end{equation}
where $C$ is a constant of order unity.
And we find (using (5.1)) that the entropy of the black hole is
\begin{equation}
S_{bh} = C \frac{M^{2}}{M_{P}^{2}} + S_{0}
\end{equation}
The entropy is proportional to the surface area of the
black hole, or equivalently, it goes linearly with the quantum
number $n_{tot}$, see (4.23), in agreement with the
Bekenstein-Hawking entropy.
We could use a different approach to calculate the entropy:
One can use the Wick order, so (4.22) is replaced with
\begin{equation}
E = \hbar <\Omega> \sum_{k=1}^{N} n_{k}
\end{equation}
but still take a finite $N_{max}$:
{}From (2.10) and $R_{0} \sim a_{0}$ we have that
$ \Delta R_{0} \geq \hbar / R_{0} $, and from $ R_{0} = a_{0} r_{s} $ we
get $ (\Delta R_{0})_{min} = a_{0} r_{s} / N_{max} $. So in the
case of black holes, $r_{s}=1$ and $ R_{0} \sim M $ , we
have $N_{max} \sim R^{2}_{0} / \hbar \sim M^{2} / M^{2}_{P}$.
In this case $N$ can take all the integer values between $N_{max}$ and
unity; the degeneracy is
\begin{equation}
{\cal N}_{bh}(M) = \sum_{N=1}^{N_{max}-1} C^{N}_{N_{max}-1}
= 2^{N_{max} - 1}
\end{equation}
Now the entropy is
\begin{equation}
S_{bh} = ln2 \frac{M^{2}}{M^{2}_{P}} + \tilde{S}_{0}
\end{equation}
We see that (5.7) and (5.10) have the same form, the entropy is
proportional to the surface area of the black hole, but the
prefactors are different. One should get the correct prefactor
in the full exact model.
Another thing to notice is that the degeneracy (and so the
entropy) of the gravitational d.o.f. is very similar to the
degeneracy of other field d.o.f. [14], so it is tempting to
think that the gravitational d.o.f. that we use are more
appropriate for a unified scheme.
\vspace{0.5cm}
\subsection{Temperature}
Using the standard thermodynamical definition of the
temperature we have
\begin{equation}
T_{bh}^{-1} = \frac{ \partial S_{bh} }{ \partial E_{bh}}
\end{equation}
We have $E_{bh} \sim M$, and using (5.6) we get
\begin{equation}
T_{bh} \sim \frac{M_{P}^{2}}{M}
\end{equation}
in agreement with the Hawking temperature [8].
One may argue that the microscopic states (4.18)
are microscopic both to a freely falling observer
(``Kruskal observer") and to an outside observer (``Schwarzschild
observer"). This means that (5.1) should be the same for Kruskal
observers as well. Then the entropy (5.7), and temperature (5.12),
are the same for both the
Kruskal and Schwarzschild observers. This contradicts
the known results that a thermal Schwarzschild state corresponds
to a zero Kruskal temperature [7,14]. But remember
that though the microscopic states (4.18) are the same for
both observers, the {\em macroscopic states} are quite different.
For a Schwarzschild observer there is an horizon, and from the
no-hair theorems, there is only one macroscopic quantity (in
the Schwarzschild case) by which one can determine the state. This
is of course the mass $M$ of the star. In that case the degeneracy
is exactly what we get from (4.21), and indeed we have (5.7) and (5.12).
On the other hand, for a Kruskal observer, there is no horizon,
and the macroscopic state is determine by an {\em infinite} number
of macroscopic quantities. For example in our model, we have a
global hyperbolic space-time, so a classical solution is determined
by the initial data\footnote{The question of classical (and of
course quantum) observables in gravity
is an open one, but {\em in principle}
one should be able to determine those quantities.}
$(y(r,t=0),\dot{y}(r,t=0))$. In our case we
have $\dot{y}(r,t=0) = 0$, so a classical solution is determine
by $y(r,t=0)$, or equivalently by all the moments
$ P_{n} = \int_{0}^{r_{s}} r^{n} y(r,t=0) dr $, which are
macroscopic quantities. This means that all the $\Omega_{k}$'s
in (4.23) are determined, and (at least semiclassically) the state (4.18)
is determined {\em completely} by the macroscopic state. This means
that for a Kruskal observer there is no degeneracy, and the
entropy and temperature vanish.
The entropy (5.1) is sometimes called ``entangeled entropy",
but we think that (at least in the case of black holes) it
should be consistent with the thermal entropy. The way to
check this is to couple the system to other fields.
When we couple the gravitational d.o.f. to other fields, we have
the following picture: The fields are in ``equilibrium" with
the gravitational d.o.f. (the black hole). According to a
Schwarzschild observer, it is a thermal equilibrium with
the temperature (5.12), and according to a Kruskal observer it is
a zero temperature situation. This is a static scenario,
in agreement with the ``eternal black hole" picture [7],
and with the Thermo-field approach [15].
One can consider also a dynamical situation: a black hole
creation and evaporation. This will be done in the next subsection.
\vspace{0.5cm}
\subsection{Hawking Evaporation}
So far we have studied only a static picture as seen by a Schwarzschild
observer, in which a black hole is in a thermal state in equilibrium
with the outside region. But one can use our formalism to study
also the dynamical process of Hawking evaporation. Using the
semiclassical adiabatic arguments, one assumes that at each time the
star is in a state (4.18), and the Hawking radiation is the result
of a transition between a level $|\Psi(n_{tot})>$ (see (4.23)) to one of the
closest levels, $|\Psi(n_{tot}-1)>$ [3]. Using energy conservation
and (4.23), the radiation frequency satisfies
\begin{equation}
\hbar \omega_{rad} = \Delta M(n_{tot},n_{tot}-1) \sim
\frac{M_{P}^{2}}{M} ~.
\end{equation}
On the other hand the temperature is proportional to the
radiating energy (frequency), so we have
\begin{equation}
T_{H} \sim \hbar \omega_{rad} = \Delta M \sim
\frac{M^{2}_{P}}{M}
\end{equation}
in agreement with (5.12), and with Hawking results.
In this dynamical situation, one can calculate the lifetime
of the level $|\Psi(n_{tot})>$ [3]. This should be finite, because
there is an interaction with the vacuum state of the radiation fields.
Now this is not the Kruskal vacuum\footnote{Known as the
Hartle-Hawking [16], or Israel [15] vacuum.}
(like in the eternal black
hole case). The vacuum state is now the Unruh vacuum [17].
This lifetime can be estimated to be proportional to the
inverse of the imaginary part of the effective action (in the
Unruh vacuum), and one get the mass rate [3]
\begin{equation}
\frac{d M}{d t} \sim \frac{T}{\Delta M} \sim \frac{
M^{2}_{P}}{M^{2}}
\end{equation}
in agreement with Hawking results, which assume a black body
radiation rate.
If we ``extrapolate" our results to the quantum region
($n \sim 1$), we can say that there should be a ``quantum
remnant" of mass $M_{rem} \sim M_{P}$ at the end of the
Hawking evaporation [18]. But this is pure speculation
because we ignore back-reaction as well as strong
quantum effects in our model.
\vspace{1cm}
\section{Concluding Remarks}
\setcounter{equation}{0}
In this work we used the canonical quantization approach
of spherically symmetric dust matter universes, first given
by Lund, and apply it to the case of collapsing stars and
black holes. The quantum states describing those universes
are bound states and one gets a discrete spectrum.
First let us consider some of the physical consequences of
the quantized spectrum.
One may question the collapse process itself, because if the
collapsing star must satisfy (2.11) for example, then in the
space of all masses and initial radii, only a set of measure
zero satisfy it, so maybe most of the stars will not
collapse at all? This is not the case because though the
mass is a constant (by energy conservation), $R_{0}$ (or in
the general case, all the other geometrical quantities) can
fluctuate, and one must calculate $ \Delta R_{0} / R_{0} $.
If this is a very small number, then the collapse is possible
in a general situation. Using (2.11) we have
\begin{equation}
\frac{\Delta R_{0}}{R_{0}} \sim \frac{\hbar^{2} n}{
M R_{0}^{3} } \sim {\left( \frac{M_{P}}{M} \right)}^{1/2}
{\left( \frac{l_{P}}{R_{0}} \right)}^{3/2} \sim \frac{1}{n}
\end{equation}
which for astronomical objects is a very small number. For
example in the case of our sun, we have $ \Delta R_{\odot}
/ R_{\odot} \sim 10^{-100} $.
This means that for astronomical objects, the mass quantization
cannot affect the classical collapse process. On the other hand,
if we ``extrapolate" our results to Plankc size objects, then
the collapse itself may be affected by the quantization
conditions. This may be another reason to consider stable
Planck size objects (black holes)?
The effect of the mass quantization on the Hawking radiation
spectrum, will be mainly on very large wavelengths, $ \lambda
\geq M_{bh}$. The black hole cannot radiate or absorb radiation
with $\lambda > M_{bh}$ because it correspond to $\Delta M$
smaller than the distance between nearest levels. For astronomical
objects this effect will be hard to detect.
So we see that for ``classical objects" (astronomical stars),
for which $n >> 1$, the correspondence principle works,
and the quantum effects are negligible.
It is also quite easy to recover the classical laws of
black hole thermodynamics: using (4.23) and ${\cal A}
= 16 \pi M^{2}$ we get the first law of
(Schwarzschild) black hole thermodynamics
\begin{equation}
\delta M = \frac{\partial M}{\partial n} \delta n \sim
\frac{M^{2}_{P}}{M} \delta {\cal A}
\end{equation}
And because $M_{bh} \sim E_{bh}$ we have $S_{bh} \sim {\cal A}$ and
$T_{bh} \sim M^{2}_{P} / M_{bh}$.
The second law is just (5.7) with the fact that $\Delta S_{bh}
\geq 0$ for an isolated system, while the generalized second
law is $\Delta S_{tot} \geq 0$, where $ S_{tot} = S_{bh} + S $.
In the case of spherically symmetric dust matter, the infinite
number of gravitational degrees of freedom decoupled, and
each shell of dust moves independently. It is possible to
choose the coordinates and the field variables such that each
shell is an harmonic oscillator. This
is a simple generalization of the homogeneous Oppenheimer-
Snyder model.
In the case of black holes, the discrete spectrum gives
the Bekenstein area quantization: the area of the black hole
is an integer number times the Planck area.
It is very easy to calculate the degeneracy of this system
(of independent oscillators), and from it to get the entropy
of the black hole. The results agree with the known
Bekenstein-Hawking entropy: the entropy is proportional to
the surface area of the black hole. Then one can use the
standard thermodynamic definitions to get all the other
thermodynamic quantities (e.g. Hawking temperature).
It seems surprising that our simple model (of spherically
symmetric dust matter) gives ``enough" degeneracy, and the
correct Bekenstein-Hawking entropy. One might think that in
the general case (no symmetry, and a general
matter field) the degeneracy will be much bigger, and so also
the entropy, which would contradict known results.
But this is not necessarily the case. One should remember
that we have been able to quantize the dust system, because
we could fix the gauge completely, which means that we correctly
choose the coordinate system and solved the
constraints. A consistent quantization of the general
case (if it exists) may be achieved by a ``free field
representation", which will be a generalization of our
independent harmonic oscillators. If this is the case,
then the general degeneracy will be quite similar to what
we have in the dust case, as will the black hole thermodynamics.
Maybe this is one thing that we can learn from our simple
model\footnote{Though infinite-dimensional.}.
It should be interesting to study some extensions of
our model, and to see if our results will survive.
Another thing that may be interesting to study in more detail,
is the interaction between the geometrical
degrees of freedom that we use and other fields.
\vspace{1cm}
{\bf Aknowledgment}\\
I would like to thank S. Deser and G. `t Hooft for very
helpful discassions.
\newpage
{\bf REFERENCES}\\
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\end{document}
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{
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A Provençal life
Francophiles should look away now. The story you are about to read will thrill you and induce soul-churning envy all at once.
"I still walk down the centre of Aix-en-Provence and feel I should pinch myself to see if I am dreaming," muses Adi Bukman, an Aussie expat thriving in the heart of Provence. "Most days I say to myself, 'little girl from Innisfail — here I am'. I feel so lucky to be here. It's always sunny, that golden hour is the most magical I have ever seen. It's no wonder all the famous painters lived here to capture it."
To complete the dreamy scenario that is Adi's life, she lives with her Dutch husband, Frits, and their two boys in a 17th-century stone home furnished with their eclectic collection of antiques and quirky finds, and oozing French charm. "I like that it's 17th century, and the history of the home — that it was the gate-keeper's home," Adi explains. "You should see the other home a few hundred metres away. From my studio window, in autumn when the trees have lost their leaves, you can see what used to be a factory that made the old tiles in our home. During the war that factory became a transport camp to bring all the Jews together here, to transport them to the various camps around Europe. Did the Germans occupy my house with the French family that lived here? What has this house seen? The stories the walls have soaked up!"
If the walls could talk, there'd no doubt be stories of hundreds of lives lived and lost, of times of bounty and times of heartache. None of it is lost on the family who are its current custodians. "As beautiful as it is, it is a family home," Adi adds. "The kids run through it while playing with the dog or dripping wet from the pool. It's lived in and it lives."
Frits is a chemical engineer, working not far from home at an oil plant. The couple had their first stint in France as newlyweds, and it took time to settle in. "I feel after three years in a place you start to feel like its home, or have enough love from the community to feel like it's a home," Adi says. Their first son, Remy, came along during that time. "Remy spent the first 15 months of his life here, with a French nanny who spoke to him in her mother tongue," Adi says. Being a new mum far from home is never easy, and Adi had pangs to return to Australia, so they packed up and moved to Perth. Rafael was born not long afterwards, and the couple spent five happy years soaking up the West Australian life.
But when the chance to move back to Provence popped up, Adi and Frits felt they weren't quite finished with la vie Française and decided to head back. "Since moving, the boys spent the first three years saying 'I want to go home'," Adi says. "They remember home to be the beach and their school friends. It's only been the past two years they have fitted into the life of the French, conquered the language and feel settled. Heaven forbid if I offer a sandwich for lunch because here it's a three-course meal at school. Ultimately, ask them where they want to live and in a split second the answer is clear — Australia — where they speak English."
For now, another move back home again is not on the cards, and the family is savouring the slow lane of southern France. "It's about late starts, late finishes and a two-hour break in between for lunch," Adi says. "You have to stop for lunch. There is so much to see, so much to do and the town attracts some of the most amazing people." Aix, as it's known, is 30 kilometres from Marseille and boasts 300 days of sunshine a year. Its cobbled streets, cute cafes, gorgeous boutiques, myriad fountains and lively city squares attract plenty of tourists and many of Adi and Frits' relatives and friends, who often spend summer months staying at their beautiful property. "May is spring and the amazing bloom of flowers is a sight to see,'' Adi says. "It's like looking at a Renoir painting. We start living outside again and everything starts happening, a lot of soirees to enjoy the warmer evenings."
One of her favourite ways to while away the days is 'brocanting' — the quintessential French merger of antique shopping and flea market trawling. "I love the thrill of getting up early to get there before the hordes, walking idly through and imagining where the things I like could go," she says. "In spring and summer it's the season, so every weekend there are treasures screaming to be found. There are also antique fairs and those I always go to." She's turned her passion into a little business, sourcing one-off French treasures for others who may not be nearby, or nearly as savvy on the shopping circuit. "I have a 'secret man' whom I tend to selfishly keep to myself," Adi teases. "I go to him often, treasure hunting for myself and for clients. He always has a list from me and helps me tick it off."
Ever resourceful, and wanting to share all she's been lucky to glean from her time in Provence, Adi, a hairdresser by trade, has found work in various forms. "After knowing enough French to get by I started my own business as a hair stylist, and now I have a studio at home and work as a stylist two days a week," she explains. "I also hosts workshops in my home and on location in some of the most beautiful parts of the world. I do hair, make-up, photography, and collaborate with other photographers styling for them or sourcing props, locations or models. I am contacted a lot by people overseas who want to run a workshop here and need someone to organise it for them. So I tend to do a little of that too."
It's ingratiated her into a tight-knit, but friendly community. "We really feel we are accepted in the village we live in and the community of Aix," Adi says. "My friend always makes a joke with me that if she comes shopping with me she must add an hour on because every corner I turn I must stop to say hello to someone I know. So we really are locals now."
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data-ad-client="ca-pub-8496100294517982″
data-ad-slot="2200753907″>
It's a wonderful existence — the south of France has clearly stolen her heart, but it's a place much further south that will always win it back. "I have Australia in my blood," Adi says. "The people are so nice and real and raw. We lived at the beach in Scarborough and it's that feeling of sun, sea, and surf that I think of when thinking of home. Also the network of people you have around you is so great. I miss that … and fish and chips on the beach."
The complete story was originally published in Australian Country issue 21.2. Click here to subscribe to our magazine
Words Tamara Simoneau
Photography Deborah Deulofeu, Peggy Cormary, Jinky Arts & Aki Bukman
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,528
|
On the 27th of November, the creative creator of the pop-up restaurant Shelf Life launches a new project…CHALAIT!
Like the scientist / chef's unique approach to Shelf Life which focused on preservation, Susanne Tobler once again leads an unexplored path by launching a restaurant focusing on alpine cuisine with a special emphasis on dairy.
We haven't had a sneak peek yet, but our inside source says the location is modern and edgy while maintaining a cozy and welcoming vibe.
The restaurant will be in town until the 20th of January and is part of the Tastelab Project, a group of chemists, scientists and engineers who are passionate about food and have joined together to intertwine science and food to create delicious, innovative cuisine in downtown Zurich.
Sounds like a delicious excuse for a night out with the girls!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,133
|
Q: C++ Composition Constructor So I've got a program with classes Car, Route and Taxi. I'm using composition in order to obtain the data from the classes Car and Route. Each class needs to be able to enter data on it's own and the data entered to be checked. I'm having difficulties with how to make the constructor of the Taxi class the way that he uses and checks the data with the get/set methods of the classes Car and Route.
How to access the parameters in the setRouteTaxidata method?
Any suggestions?
class RouteTaxi {
private:
int id;
public:
Car car;
Route route;
RouteTaxi();
~RouteTaxi();
void setRouteTaxidata(string cbrand, string cmodel, int cyears, int cseatingCapacity, double cloadCapacity, double cfuelConsumption,string rnodes, double rrouteLength, int rtoursPerDay, int i);
};
RouteTaxi::RouteTaxi(){
setRouteTaxidata(??) ??????
}
void RouteTaxi::setRouteTaxidata(string cbrand, string cmodel, int cyears, int cseatingCapacity, double cloadCapacity, double cfuelConsumption, string rnodes, double rrouteLength, int rtoursPerDay, int i){
car.setBrand(cbrand);
}
A: Your constructor RouteTaxi() is a default constructor. It must create a valid object without any of those, so you would have to have acceptable default values.
There's no good way to make a default route, unless you have a "nowhere" route with a "nocar" car, so you can instead require those arguments in the constructor, as in this example:
#include<string>
using std::string;
class Car {
public:
Car(string cbrand, string cmodel, int cyears, int cseatingCapacity, double cloadCapacity, double cfuelConsumption){}
};
class Route {
public:
Route(string rnodes, double rrouteLength, int rtoursPerDay){}
};
class RouteTaxi
{
private:
int id;
public:
Car car;
Route route;
RouteTaxi(string cbrand, string cmodel, int cyears, int
cseatingCapacity, double cloadCapacity, double cfuelConsumption,
string rnodes, double rrouteLength, int rtoursPerDay, int i)
: car(cbrand, cmodel, cyears, cseatingCapacity, cloadCapacity, cfuelConsumption),
route(rnodes, rrouteLength, rtoursPerDay),
id(i)
{}
static RouteTaxi generate_from_console_input();
};
// this is a factory function (class static member of RouteTaxi)
RouteTaxi RouteTaxi::generate_from_console_input() {
// input from console
string cbrand, cmodel, rnodes;
int cyears, cseatingCapacity, rtoursPerDay, i;
double cloadCapacity, cfuelConsuption, rrouteLength;
// return the object
// this is an error, using uninitialized data, simply because I am not actually getting the data from the console. You will do that, so it will not be a problem.
return RouteTaxi(cbrand, cmodel, cyears, cseatingCapacity, cloadCapacity, cfuelConsuption, rnodes, rrouteLength, rtoursPerDay, i);
}
Note the colon and initializer list which the constructor uses to initialize other objects before it executes its own constructor body. http://en.cppreference.com/w/cpp/language/initializer_list
The factory generator is another way to create objects. You need all the info before you call the actual constructor, so ... write a function which gets it, before you actually call the constructor.
This is a well known design pattern. See https://en.wikipedia.org/wiki/Creational_pattern. Although those go into a lot more detail. You are not dealing with a lot of different types of RouteTaxi and subclasses.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,480
|
/**
* @author Boris V. Kuznetsov
*/
package java.security;
import org.apache.harmony.security.tests.support.MySignature1;
import org.apache.harmony.security.tests.support.MySignature2;
import junit.framework.TestCase;
/**
* Tests for <code>Signature</code> constructor and methods
*
*/
public class Signature_Impl2Test extends TestCase {
/**
* Provider
*/
Provider p;
/*
* @see TestCase#setUp()
*/
protected void setUp() throws Exception {
super.setUp();
p = new MyProvider();
Security.insertProviderAt(p, 1);
}
/*
* @see TestCase#tearDown()
*/
protected void tearDown() throws Exception {
super.tearDown();
Security.removeProvider(p.getName());
}
/*
* Class under test for Signature getInstance(String)
*/
public void testGetInstanceString1() throws Exception {
Signature sig = Signature.getInstance("ABC");
checkSig1(sig, p);
}
/*
* Class under test for Signature getInstance(String)
*/
public void testGetInstanceString2() throws Exception {
Signature sig = Signature.getInstance("CBA");
checkSig2(sig, p);
}
/*
* Class under test for Signature getInstance(String, String)
*/
public void testGetInstanceStringString1() throws Exception {
Signature sig = Signature.getInstance("ABC", "MyProvider");
checkSig1(sig, p);
}
/*
* Class under test for Signature getInstance(String, String)
*/
public void testGetInstanceStringString2() throws Exception {
Signature sig = Signature.getInstance("CBA", "MyProvider");
checkSig2(sig, p);
}
/*
* Class under test for Signature getInstance(String, Provider)
*/
public void testGetInstanceStringProvider1() throws Exception {
Provider p1 = new MyProvider();
Signature sig = Signature.getInstance("ABC", p1);
checkSig1(sig, p1);
}
/*
* Class under test for Signature getInstance(String, Provider)
*/
public void testGetInstanceStringProvider2() throws Exception {
Provider p2 = new MyProvider();
Signature sig = Signature.getInstance("CBA", p2);
checkSig2(sig, p2);
}
private void checkSig1(Signature s, Provider p) throws Exception {
byte[] b = { 1, 2, 3, 4 };
assertTrue("getInstance() failed", s instanceof MySignature1);
assertEquals("getProvider() failed", p, s.getProvider());
assertEquals("getAlgorithm() failed", "ABC", s.getAlgorithm());
try {
s.sign();
fail("No expected SignatureException");
} catch (SignatureException e) {
}
s.initVerify(new MyPublicKey());
try {
s.sign();
fail("No expected SignatureException");
} catch (SignatureException e) {
}
s.initSign(new MyPrivateKey());
s.sign();
assertEquals("Incorrect state", Signature.SIGN, ((MySignature1) s)
.getState());
assertTrue("sign() failed", ((MySignature1) s).runEngineSign);
s.initVerify(new MyPublicKey());
s.update((byte) 1);
s.initSign(new MyPrivateKey());
s.update((byte) 1);
assertEquals("Incorrect state", Signature.SIGN, ((MySignature1) s)
.getState());
assertTrue("sign() failed", ((MySignature1) s).runEngineUpdate1);
s.initSign(new MyPrivateKey());
try {
s.verify(b);
fail("No expected SignatureException");
} catch (SignatureException e) {
}
s.initVerify(new MyPublicKey());
s.verify(b);
assertEquals("Incorrect state", Signature.VERIFY, ((MySignature1) s)
.getState());
assertTrue("verify() failed", ((MySignature1) s).runEngineVerify);
}
private void checkSig2(Signature s, Provider p) throws Exception {
byte[] b = { 1, 2, 3, 4 };
assertEquals("getProvider() failed", p, s.getProvider());
assertEquals("getAlgorithm() failed", "CBA", s.getAlgorithm());
s.initVerify(new MyCertificate());
try {
s.sign(b, 0, 5);
fail("No expected IllegalArgumentException 1");
} catch (IllegalArgumentException e) {
}
s.initSign(new MyPrivateKey());
s.sign(b, 0, 3);
assertTrue("sign() failed", MySignature2.runEngineSign);
s.update(b);
s.initSign(new MyPrivateKey());
s.update(b);
assertTrue("update() failed", MySignature2.runEngineUpdate2);
s.initSign(new MyPrivateKey());
try {
s.verify(b, 0, 3);
fail("No expected SignatureException");
} catch (SignatureException e) {
}
s.initVerify(new MyPublicKey());
try {
s.verify(b, 0, 5);
fail("No expected IllegalArgumentException");
} catch (IllegalArgumentException e) {
} catch (SignatureException e) {
}
s.verify(b, 0, 3);
assertTrue("verify() failed", MySignature2.runEngineVerify);
}
private class MyProvider extends Provider {
MyProvider() {
super("MyProvider", 1.0, "Provider for testing");
put("Signature.ABC", "org.apache.harmony.security.tests.support.MySignature1");
put("Signature.CBA", "org.apache.harmony.security.tests.support.MySignature2");
}
MyProvider(String name, double version, String info) {
super(name, version, info);
}
}
private class MyKey implements Key {
public String getFormat() {
return "123";
}
public byte[] getEncoded() {
return null;
}
public String getAlgorithm() {
return "aaa";
}
}
private class MyPublicKey extends MyKey implements PublicKey {}
private class MyPrivateKey extends MyKey implements PrivateKey {}
private class MyCertificate extends java.security.cert.Certificate {
public MyCertificate() {
super("MyCertificateType");
}
public PublicKey getPublicKey() {
return new MyPublicKey();
}
public byte[] getEncoded() {
return null;
}
public void verify(PublicKey key) {}
public void verify(PublicKey key, String sigProvider) {}
public String toString() {
return "MyCertificate";
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,045
|
Pineville är en stad (city) i Rapides Parish i delstaten Louisiana i USA. Staden hade 14 384 invånare, på en yta av 33,99 km² (2020). Den ligger vid floden Red River och på andra sidan floden ligger den lite större staden Alexandria. Pineville är säte för Louisiana College.
Referenser
Externa länkar
Officiell webbplats
Orter i Louisiana
Rapides Parish
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,381
|
{"url":"http:\/\/keisan.casio.com\/exec\/system\/1180573189","text":"Normal distribution (chart) Calculator\n\nCalculates a table of the probability density function, or lower or upper cumulative distribution function of the normal distribution, and draws the chart.\n\n select function\u00a0 probability density f lower cumulative distribution P upper cumulative distribution Q mean \u03bc standard deviation \u03c3 \u03c3\uff1e0 [ initial percentile x increment repetition ]\n The default value \u03bc and \u03c3 shows the standard normal distribution.$\\normal Normal\\ distribution\\ N(x,\\mu,\\sigma)\\\\[10](1)\\qquad probability\\ density\\\\\\hspace{30}f(x,\\mu,\\sigma)={\\large\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2}}\\\\(2)\\qquad lower\\ cumulative\\ distribution\\\\\\hspace{30}P(x,\\mu,\\sigma)={\\large\\int_{\\small-\\infty}^{\\hspace{25}\\small x}}f(t,\\mu,\\sigma)dt\\\\(3)\\qquad upper\\ cumulative\\ distribution\\\\\\hspace{30}Q(x,\\mu,\\sigma)={\\large\\int_{\\small x}^{\\hspace{25}\\small\\infty}}f(t,\\mu,\\sigma)dt\\\\$\n\nSending completion\n\nTo improve this 'Normal distribution (chart) Calculator', please fill in questionnaire.\nMale or Female ?\nMale Female\nAge\nUnder 20 years old 20 years old level 30 years old level\n40 years old level 50 years old level 60 years old level or over\nOccupation\nElementary school\/ Junior high-school student\nHigh-school\/ University\/ Grad student A homemaker An office worker \/ A public employee\nSelf-employed people An engineer A teacher \/ A researcher\nA retired people Others\nUseful?\nVery Useful A little Not at All\nPurpose of use?","date":"2017-07-26 08:40:15","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\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 1, \"\/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\": 0.9066554307937622, \"perplexity\": 6911.331763704344}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-30\/segments\/1500549426086.44\/warc\/CC-MAIN-20170726082204-20170726102204-00680.warc.gz\"}"}
| null | null |
\section{Introduction: Extrasolar planetary systems}
The first "hot Jupiter" around the nearby solar-type star 51 Peg was discovered
using precise radial velocity measurements (Mayor \& Queloz 1995).
The first multiple planet system around $\upsilon$ And was discovered also using
radial velocities (Butler et al. 1999). Indeed, this search technique
continues to be the most successful (Fischer et al. 2004, Pont et al. 2004,
Konacki et al. 2004).
The total sample of extrasolar planets todate discovered using the
radial velocity technique is reaching 135 planets (see the
Extrasolar Planets Encyclopaedia by J. Schneider 2005
{\footnote{ http://www.obspm.fr/planets }}
). In this sample there are 14
multiple planet systems, containing between 2 and 4 planets. This means that
a significant fraction of the planets are in planetary systems.
A new search technique of transits has been systematically pursued by the OGLE
Collaboration, who published low amplitude ($\Delta I <0.08$ mag)
transiting planetary candidates based on extensive $I$-band photometry
of several fields spread across the Milky Way
(Udalski et al. 2002a, 2002b, 2002c, 2003). While many of these
are binaries or blends, the OGLE sample contains real planets: five of
them have been confirmed using radial velocities
(Bouchy et al. 2004, Pont et al. 2004, Konacki et al. 2003, 2004).
Due to purely geometric effects, the discovery of
short period transiting planets ($P<5$ days) is favoured.
One of the many advantages of the known transiting planets is that their
orbital plane is well defined. Using the reasonable assumption
(unproved until now) that
other planets in these stars lie in the same orbital plane, one
can search for other transits. A major
limitation is the quality of the photometry,
which demands confirmation of any other planets by radial velocities.
Here we report the search for double transiting planets in
OGLE-TR-10, OGLE-TR-56, OGLE-TR-111, OGLE-TR-113, and OGLE-TR-132,
which are the OGLE systems confirmed todate to harbor single planets
(Table 1).
The initial search for additional unnoticed eclipses in the light curves
is followed by examination of the existing radial velocity data.
We find two possible interesting candidates, OGLE-TR-10 and OGLE-TR-111,
of which OGLE-TR-111 is the most promising one because its light
curve can be phased with a $P=16$ day period in absence of the known
planet OGLE-TR-111b. We conclude that further photometric coverage of
this system is needed.
\section{Search for Double Transiting Planets}
\begin{table}[t]\tabcolsep=1pt\small
\begin{center}
\caption{Sample OGLE Stars With Confirmed Planets.}
\label{Table1}
\begin{tabular}{l@{ }l@{ }l@{ }l@{ }l@{ }}
\hline
\multicolumn{1}{l}{OGLE}&
\multicolumn{1}{l}{RA}&
\multicolumn{1}{l}{DEC}&
\multicolumn{1}{l}{P(d)}&
\multicolumn{1}{c}{Reference}\\
\hline
&\\
TR10 & 17 51 28.25 & -29 52 34.9& 3.10139&Konacki+ 2005 \\
TR56 & 17 56 35.51 & -29 32 21.2& 1.21192&Konacki+ 2003 \\
TR111& 10 53 17.91 & -61 24 20.3& 4.01610&Pont+ 2004 \\
TR113& 10 52 24.40 & -61 26 48.5& 1.43248&Bouchy+ 2004, Konacki+ 2004 \\
TR132& 10 50 34.72 & -61 57 25.9& 1.68965&Bouchy+ 2004 \\
&\\
\hline
\end{tabular}
\end{center}
\end{table}
Systematics in the OGLE light curves are explored by Udalski et al. (2002a, 2002b, 2002c, 2003),
Drake (2003), and Sirko \& Paczy\'nski (2003). They discuss the possible
contaminants and different effects that are present in the OGLE transit sample.
We have decided to search for double transiting planets only on the
stars that have been confirmed by radial velocities, where the parameters
of the planet are accurately known.
The stellar sample considered here is listed in Table 1, along with the main
properties of the transiting planets.
The most important parameter to consider when searching for additional planets
is the orbital inclination, as the range of semimajor orbital axis of
additonal transiting planets depend strongly on this.
Even though the confirmed OGLE planets are very close to their parent stars,
fortunately, they have transits with flat bottoms, implying orbital inclination angles
$i>85$ deg, allowing the search for planets more distant from their stars.
The procedure followed is very simple, and can be described into four
main steps.
First, phasing the light curve of each star with the known period of the planet,
and substracting the points around the planetary transit.
Typical OGLE light curves contain 1000-1500 data points, of which 70-140
around the transit are discarded. The light curves with the main transits
removed are called "reduced" light curves hereafter. We compute the dispersion
of these light curves by binning 10 contiguous datapoints, and check on the
stability of this dispersion (e.g. in other OGLE
light curves that turned out to be low amplitude grazing binaries
it was found that the dispersion increases).
This step was carried out for all the stars of Table 1.
Second, examination of this "reduced" light curve (Figure 1), to search for
asymmetries or excess points below the normal star baseline, beyond the
photometric errors.
This step was carried out for all the stars of Table 1.
Third, phasing the reduced light curve to search for periodicity that
may reveal an additional transiting planet. At this stage we pay particular
attention to periods which would be in mean motion resonances with the
original periods.
This step was carried out only for OGLE-TR-10 and OGLE-TR-111.
Finally, if a good candidate is found, we check the existing radial velocitiy
to confirm if they are consistent with the presence of an additional
planet, and try to derive its physical parameters from the photometry and spectroscopy.
This step was carried out only for OGLE-TR-111
We discuss here the stars OGLE-TR-10, OGLE-TR-56, OGLE-TR-113, and OGLE-TR-132, reserving the next
section for a more thorough discussion of the most promising candidate, OGLE-TR-111.
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics{minnitid_fig1.ps}}
\caption{
Magnitude distribution for
OGLE-TR-10, OGLE-TR-56, OGLE-TR-111, and OGLE-TR-113,
obtained after discarding the eclipses from the known transiting planets.
The open histograms are an expanded version of the hashed histograms
in order to illustrate possible excess of points below the mean
magnitudes. Stars OGLE-TR-10 and OGLE-TR-111 show several points
4$\sigma$ fainter than the mean magnitudes.
}
\label{Fig01}
\end{figure}
\subsection{OGLE-TR-10}
OGLE-TR-10 shows larger than normal scatter compared with other OGLE stars at
this magnitude ($I=14.93$). Its planet was suggested by
Bouchy et al. (2004), and was finally confirmed by Konacki et al. (2005).
After substraction of the 117 points (from a total of 1082 points)
next to the transit of OGLE-TR-10b,
we find $\sigma_I=0.006$ mag. There are a group of fainter points suggesting
the possibility of additional transits (Figure 1), including
3 points located $4\sigma$ below the mean magnitude.
However, the reduced light curve could not be phased adequately to reveal any
clear low amplitude transits. More accurate photometric follow up of this star is needed.
\subsection{OGLE-TR-56}
OGLE-TR-56b was the first secure
OGLE planet: it was confirmed as a planetary mass object by the
radial velocities of Konacki et al. (2003).
We substracted 137 points (out of 1115 total points) next to the transit of OGLE-TR-56b.
The scatter in the reduced light curve of OGLE-TR-56 gives $\sigma_I=0.005$,
smaller than OGLE-TR-10 in spite of being $0.4$ mag fainter.
However, the magnitude distribution of the reduced light curve is very symmetric,
and there is no evidence for points from additional transits.
\subsection{OGLE-TR-113}
OGLE-TR-113b was confirmed as a planet almost simultaneously by two different groups:
Bouchy et al. (2004) and Konacki et al. (2004).
The reduced light curve was obtained by eliminating 67 points centered on the
OGLE-TR-113b transits (out of a total of 1517 points).
The scatter in the reduced light curve of OGLE-TR-113 is very low, $\sigma_I=0.004$.
However, we find evidence for a long term periodic behaviour with $P=26.5$ d, at
very low amplitude ($0.005$ mag). If this is taken into account, the light curve
becomes even tighter. This is a good case of a star where strong limits may be put
on the absence of additional transiting giant planets with orbital semimajor
axis $a<0.2$ AU.
\subsection{OGLE-TR-132}
OGLE-TR-132b was confirmed with the radial velocities of Bouchy et al. (2004).
OGLE-TR-132 is the faintest star of this sample.
We substracted 71 points (out of 1044 total points) next to the transit of OGLE-TR-132.
The dispersion of its light curve is large, $\sigma_I=0.008$,
preventing us from putting useful limits to the presence of additional transiting planets.
\section{The Case of OGLE-TR-111}
\subsection{Stellar parameters for OGLE-TR-111}
We have previously carried out a selection of the most promising
OGLE planetary candidates using low dispersion spectroscopy in combination
with optical and near-infrared
photometry (Gallardo et al. 2004). This work identified
OGLE-TR-111 as one of the most likely candidates to host exoplanets. The
planet OGLE-TR-111b was discovered by Pont et al. (2004)
using precise velocities.
Based on the spectroscopy, Pont et al. (2004) derive the following
stellar parameters for OGLE-TR-111:
mass $M = 0.82 M\odot$,
radius $R = 0.85 R\odot$,
temperature $T_{eff} = 5070$ K,
gravity $log ~g = 4.8$, and
metallicity $[Fe/H]=0.12$ dex.
Based on a low resolution spectrum plus optical and infrared photometry,
Gallardo et al. (2004) derive the following stellar parameters:
radius $R = 0.71 R\odot$,
temperature $T_{eff} = 4460$ K,
reddening $E(B-V) = 0.16$,
absolute magnitude $M_V = 6.82$, and
distance $D = 850$ pc for OGLE-TR-111.
Even though the values from these independent studies are not identical,
both studies agree in the parameters within the uncertainties.
In order to derive planetary parametes hereafter
we use for consistency the stellar parameters adopted by Pont et al. (2004),
with the cautionary remark that the major uncertainties on the planetary
parameters (mostly the radius) would arise from the uncertainties in the stellar properties.
\subsection{Photometric Evidence for OGLE-TR-111c}
The Existing planet OGLE-TR-111b is a massive planet with mass $M_p sin~i=0.53 ~M_J$,
radius $R = 1.0 R_J$, in an orbit with
period $P=4.0166$ days and semimajor orbital axis $a=0.047 ~AU$
(Pont et al. 2004). They call this planet OGLE-TR-111b the "missing link".
The transit signature of this planet is clearly seen in the
phased OGLE lightcurve (Figure 2).
Aside from the
transit of OGLE-TR-111b, additional points well below the
baseline are observed, clumped at a single phase. Specifically,
there are 7 points located $4\sigma$ below the mean magnitude. This clumping hinted
at the presence of additional transits with
a period multiple of $P=4.011$ dat, the OGLE-TR-111b period.
Because the additional feature is 180 degrees out of phase we have to
make sure that it is not a noisy secondary eclipse caused by an
eclipsing binary system.
This possibility is discarded because of (a) the new transits are not always
present, and (b) the radial velocities of Pont et al. (2004) show
low amplitudes.
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics{minnitid_fig2.ps}}
\caption{
OGLE lightcurve for OGLE-TR-111 phased for $P=4.0166$ days
(1176 photometric points), showing the transit of OGLE-TR-111b
at phase 0.6. Several points clumped at phase 0.1 hint at the presence
of additional eclipses.}
\label{Fig01}
\end{figure}
There are nine transits of OGLE-TR-111b observed by OGLE.
Looking at the unphased light curve (Figure 3), it became clear
that there might be some possible additional transits interleaved with
some of the transits of OGLE-TR-111b. Two of these transits are
indicated with the arrows in Figure 3.
Figure 3 also clearly illustrates the fortunate fact that the period of
OGLE-TR-111b is a multiple of the Earth's rotation period, as discussed
by Pont et al. (2004), resulting in
a favorable transit observing condition during the OGLE observations.
We then proceeded to phase the OGLE data in search for additional transits.
For this, 68 points around the OGLE-TR-111b
transits were discarded, phasing only the remaining 1108 baseline points.
A new phased possible transit was found with a period of $P=16.0644$ days,
as shown in Figure 4.
This period would in 1:4 mean motion resonance with
the previously known planet of this system OGLE-TR-111b, with $P = 4.0161 ~d$.
Caution should be taken with this period, because it is largely based on
two transit like features. The aspect of these two individual
transits is not different from the other nine transits of OGLE-TR-111b,
but a larger number of individual transits must be observed in order
to confirm this periodic nature.
Pending confirmation of this possible period, we note that a
1:4 mean motion resonance should not be totally unexpected.
Let us consider
a simple example from our own neighborhood: the Jovian system is a good
example, because it is considered to be a stable "mini Solar system".
The Galilean satellites
Io and Ganimede share a 1:4 mean motion resonance.
But also Europa has a 1:2 mean motion
resonance with Io (and 2:1 with Ganimede), so on a very speculative
side one could search for
another planet around OGLE-TR-111 in a similar mean
motion resonance in between the two possible planets, with a period of $P=8.03$
days. In the case of the OGLE-TR-111 with an orbital inclination
of 88-89 deg, planets located as far out as 0.3 AU might be detected. This would
correspond to the 3:7 resonance
with the outer planet (e.g. Callisto is in 3:7 mean motion resonance with Ganimede).
With the more accurate periods to be determined by this years OGLE campaign,
one can tune the searches for transits from
additional planets to specific mean motion resonances.
The possible transit appears to be symmetric, with a duration of
$t_T=4$ hours. The transit also appears to
have a flat portion lasting about $2$ hours, ruling out a grazing eclipse,
and constraining the orbital inclination angle to $88-89$ degrees.
Other transient features such as star spots cannot be ruled out with the
available data. However, it is very suggestive that the two
additional transits alternate with the main transits of OGLE-TR-111b
(Figure 4).
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics{minnitid_fig3.ps}}
\caption{
Unphased OGLE data for OGLE-TR-111, showing the individual
transits of OGLE-TR-111b indicated by the vertical lines.
Two additional eclipses can be identified at the locations of the arrows.
}
\label{Fig02}
\end{figure}
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics{minnitid_fig4.ps}}
\caption{
Lightcurve of OGLE-TR-111 phased with $P=16.0644$ days
after discarding 68 points around the transits of OGLE-TR-111b.
The possible 4 hour long transit of OGLE-TR-111c is seen.
}
\label{Fig03}
\end{figure}
\subsection{Spectroscopic Evidence for OGLE-TR-111c}
The photometry of OGLE-TR-111 suggests the presence of an additional
giant planet with $P=16.0664$ days, but it is not conclusive.
Here we check if the existing
spectroscopic data are consistent with this suggestion, and try to
derive the parameters of the possible additional planet.
We would call this possible extrasolar transiting planet OGLE-TR-111c, {\it a.k.a.}
the "missing link's sister".
Based on the radial velocities of the star, it is straightforward to fit the
eight individual velocity measurements of Pont et al. (2004)
with a double Keplerian model instead of a single one
(Figure 4). Given the original parameters for OGLE-TR-111b,
this fit is not free, however: the period and the phase of
OGLE-TR-111c are fixed by
the photometric data. The amplitude is a free parameter, which is clearly
poorly constrained due to the limited orbital coverage:
only one half of the putative second planet orbit is covered.
With this caveat, if there is an additional planet,
we obtain the parameters listed in Table 2. This Table presents the
photometric results and the final parameters for the
possible OGLE-TR-111c planet as well.
Different fits changing the original parameters for OGLE-TR-111b
are allowed, but are not significantly better.
The amplitude of the radial velocities induced by the additional planet are
scaled by the factor $M_P/M_*$. In the case of
OGLE-TR-111c, the fit shown in Figure 4 has
velocity semiamplitude $V_r=60$ m/s, and $M_P=0.75 ~M_J$.
For comparison, the O-C residuals of the single planet fit are
$24$ m/s, which are reduced to $14$ m/s with the double Keplerian model.
The major difficulties for better constraining the parameters of the
two possible planets
circling the star OGLE-TR-111 are the uncertainty in the photometry and
the limited spectroscopic coverage.
Clearly, further photometric and spectroscopic observations over
an extended period of time are needed
to confirm the existence of OGLE-TR-111c, and to refine the measured parameters
of the planets.
Finally, the planetary radius listed in Table 2 is also uncertain
because it depends on the adopted stellar size.
In particular, adoption of a different stellar radius, $R = 0.71 R\odot$
from Gallardo et al. (2004), leads to a smaller planet radius $R = 0.7 ~R_J$ for
OGLE-TR-111c, which would translate into a higher density.
\begin{figure}[h]
\resizebox{\hsize}{!}{\includegraphics{minnitid_fig5.ps}}
\caption{
Radial velocity measurements for OGLE-TR-111 from Pont et al. (2004).
The dotted line shows their fit for OGLE-TR-111b, the dashed line shows the
effect of OGLE-TR-111c, and the solid curve the combined effect of the
two planets.
}
\label{Fig01}
\end{figure}
\begin{table}[t]\tabcolsep=1pt\small
\begin{center}
\caption{Possible OGLE-TR-111c planetary parameters.}
\label{Table2}
\begin{tabular}{l@{ }l@{ }}
\hline
\hline
&\\
Orbital period & $16.0644 \pm 0.0050 ~d$ \\
Semimajor axis & $0.12 \pm 0.01 ~AU $ \\
Orbital eccentricity & $ 0 $ (assumed) \\
Transit epoch & $JD~2453064.73\pm 0.01 $ \\
Transit duration & $4 \pm 1 ~h$ \\
Transit depth & $0.01 \pm 0.005 ~mag$ \\
Orbital inclination & $88-89 ~deg$ \\
Systemic velocity & $25.40 \pm 0.05~km/s$ \\
RV semiamplitude & $60 \pm 20 ~km/s$ \\
Planet mass & $M = 0.7 \pm 0.2 ~M_J$ \\
Planet radius & $R = 0.85 \pm 0.15 ~R_J$ \\
Planet density & $\rho = 1.4 \pm 0.3 ~g/cm^3$ \\
&\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Discussion of an Optimistic Scenario}
We have argued that the evidence is not conclusive to distinguish
between a second transiting planet around OGLE-TR-111 and a false positive detection.
However, here we briefly discuss the implications of a positive confirmation
of the existence of OGLE-TR-111c. If confirmed, this new planet would be unique and hold
several interesting records:
$\bullet$ OGLE-TR-111 would be the first extrasolar system with multiple
planets detected by transits, and the first multiple system for which
planetary masses and radii are measured.
With a distance of 850 pc, OGLE-TR-111 it would also be the most distant
extrasolar planetary system discovered todate.
$\bullet$ The transit of OGLE-TR-111c allows to constrain further the
inclination of the orbital plane of the system to $i = ~88-89 deg$.
This would prove for the first time that the orbits of two planets of
an extrasolar planetary system are coplanar to within 2 degrees.
The most important implication of coplanarity is that careful photometric
monitoring may reveal rocky planets in this and other systems
by timing the transits of giant planets Holman \& Murray (2004), or by directly
detecting their transits.
$\bullet$ With radius $R = 0.85 ~R_J$, mass $M = 0.7 ~M_J$,
and density $\rho = 1.4 ~g/cm^3$, the planet OGLE-TR-111c would be
smallest and densest extrasolar planet found. It could be called a true
Jovian planet, albeit with shorter period,
because it has size like Saturn, density like Jupiter, and mass
intermediate between them. Its radius, however, could be as small as $R = 0.7 ~R_J$.
$\bullet$ With $P = ~16.0644$ d,
OGLE-TR-111c would also the transiting planet with the longest orbital period,
lying at the arbitrary boundary between "hot" and
"normal" giant extrasolar planets.
In that sense it could also be considered a
"missing link" on its own, justifying the "missing link's sister"
nickname given here.
$\bullet$ Comparison with the properties of OGLE-TR-111b would directly
confirm the effects of inflation due to stellar
irradiation, as the less massive planet closer to the star would be bigger than
the more massive OGLE-TR-111c.
In fact, planet OGLE-TR-111b would be 1.22 times bigger than OGLE-TR-111c.
This result is independent of the uncertainties in the stellar mass and radius.
However, reducing the uncertainties in the radii is very important in
order to test the models.
$\bullet$ If the period of $P = ~16.0644$ d
is confirmed,
the OGLE-TR-111 system would an interesting case for stability studies because of the
presence of a 1:4 mean motion resonance.
The confirmation of the orbital resonances coupled with the low inclination
of the system the exiting possibility of the search
for transiting smaller (rocky?) planets in OGLE-TR-111.
\section{Conclusions}
We have started a search for second planets around existing
transiting systems. This search has been applied
to the specific case of the stars
OGLE-TR-10, OGLE-TR-56, OGLE-TR-111, OGLE-TR-113, and OGLE-TR-132.
Even though OGLE-TR-56, OGLE-TR-113, and OGLE-TR-132 do not show
evidence for scatter above what is expected, there are
several data points of OGLE-TR-10 and OGLE-TR-111 below their mean
magnitudes that suggest the possibility of additional transits.
While it was not possible to phase the OGLE-TR-10 reduced light curve
in order to find the orbital period for another planet, the
OGLE-TR-111 reduced light curve was successfully phased with $P\approx 16$ days.
Thus we explore the possibility of a putative additional planet in this system.
Based on published photometry and radial velocities, we
tentatively derive the following parameters for OGLE-TR-111c:
orbital period
$P = ~16.0644$ d,
semimajor axis $a = 0.12 ~AU$,
mass $M = 0.75 ~M_J$,
radius $R = 0.85 ~R_J$, and
density $\rho = 1.5 ~g/cm^3$.
We stress that the possibility of a false positive detection is not ruled out,
and that this must be confirmed with additional data.
The major difficulty for securing the present claim of another planet
transiting around OGLE-TR-111 is the uncertainty in the photometry and
the limited spectroscopic coverage. Both difficulties would
be easily overcome with more observations. We thus stress the need
for the photometric and spectroscopic follow up of this system.
However, we speculate that extensive transit searches by space missions (KEPLER,
COROT, MPS) will find numerous multiple systems, as predicted by Holman \& Murray (2004).
The present search for double transit planetary systems suggests a number of
interesting follow-up studies.
Theoretical modeling of formation and stability
of such systems including different migration scenarios can be pursued.
These systems would provide also a good opportunity to test various
models of the effects of stellar radiation in planetary atmospheres.
Further observation of multiple transit systems would refine the
planetary parameters and reveal similarities and differences between
the structures of "hot" and "normal" extrasolar giant planets.
\begin{acknowledgements}
DM is supported by Fondap Center for Astrophysics 15010003.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Built on water: Floating Houses
By Ambista
Architects from New York to Shanghai are increasingly being confronted with the same problem: Too little space for too many people. The challenge of developing new habitable spaces within the city is not easy. Many architects, contractors and urban planners are tackling this situation with floating architecture.
The architects of Waterstudio.NL not only design floating houses in the luxury segment in the IJburg district of Amsterdam but also in the rest of the world. © Koen Olthuis – Waterstudio.NL
New living space on the water
Whether it's Asia, the US or Europe, living space is becoming an important resource in the major cities of the world. Most cities have little room to grow in the central urban area and increasing rents are symptomatic of this crisis. Metropolitan regions in the immediate vicinity of water are trying to develop new living spaces with floating houses in response to the housing shortage.
Floating houses take care of two problems at once: They meet the demand for living space in large cities and also serve as flood protection. Coastal cities in particular are extremely affected by climate change and the resulting rise in sea level. They are therefore looking for new strategies to cope with the water and turn the disadvantage into an advantage.
Floating houses in Amsterdam
It is no wonder that the Netherlands is considered a pioneer when it comes to floating houses. Around a quarter of the country lies below sea level. For the Dutch, water has long been an important element of urban planning. Amsterdam is a major European city known worldwide for the many houseboats that create additional living space in the canals.
However, not only do the residents of Amsterdam live on the water in the city centre but also in the eastern part of the city. The new IJburg district was created here on artificially raised sand islands. In the first construction phase, a total of 18,000 apartments with living space for 45,000 people were created. The Waterbuurt district in the western section of IJburg was also planned at the same time – the Floating Houses IJburg project by Amsterdam-based Marlies Rohmer Architects & Urbanists.
Lacking a firm subsoil, the neighbourhood functions primarily with bridges and jetties, which provide access to the residences. Gardens are not allowed, but living close to the water makes up for it. A lock ensures that the inland sea on which the houses float is separated from the IJMeer. This prevents the apartments from drifting out to sea. The project was completed in 2011 and included both social housing and condos.
Architecturally, however, IJburg is still a long way from being fully developed. To the east, the "Water District" continues to grow. By 2020, the Dutch architectural firm Waterstudio.NL wants to complete around 380 additional apartments, offices, floating gardens and a restaurant. Everything is possible for the architects – from a bungalow to a three-storey residential building.
Amphibious houses on the Thames
Other countries, such as Great Britain, are also discovering water as additional living space. This is how the amphibious houses near Marlow on the Thames in Buckinghamshire came to be. The homes were designed by Baca Architects in London. When the tide is low, the house rests on the ground like a conventional building and can also float in the event of flooding.
This is made possible by a kind of dry dock made of reinforced concrete, which serves as the base of the house. As the floodwaters fill the trough, the house is buoyed up to the surface of the water. An anchoring system keeps it in position and buoyancy is ensured by air chambers under the floor.
Living on the water: The future is now
In Hong Kong and Macau, people have been living on the water for a long time – in jungle settlements consisting of old sailboats that have fallen into disuse. In the US, water communities also have a long tradition. Seattle has one of the largest collections of floating houses in Portage Bay and Lake Union. And Germans are also finding life on the water more and more attractive.
In Hamburg, for example, additional moorings for houseboats and floating houses are being built. The idea of floating architecture is no longer a vision of the future, it is a reality. People learn to live with water and use it for urban development. And not only in Europe or Asia, but throughout the entire world.
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Floating Homes Architectural Solutions to Sea Level Rise
By Lukasz Stepnik
Translated by Aga Zano
2019.April.29
As we all know, when Yahweh unleashed a great flood upon humanity, he saved Noah and his family by ordering Noah to build an ark. This story was recorded not only by Jews, but also by other nations, such as the Sumerians and Mesopotamians (except their God was called Ea or Enki, and those saved from the deluge were known as Utnapishtim or Ziusudra). Clearly, they were all onto something. No wonder people of reason are preparing for another flood to come – even in Poland, too.
In the 1995 film Waterworld, Kevin Costner's character – a fish-like mutated human sailing the endless ocean – takes his beloved Helen for an underwater journey. There, he shows her the drowned ruins of New York City. Corals cover reinforced concrete structures that look like the haunted graveyard of a once-glorious civilization that brought doom upon itself. Scarce dialogues make it difficult to deduce the exact cataclysm that resulted in the world's submergence under water, but we can assume it was to do with melting ice caps and the immense amounts of carbon dioxide emissions that had been seeping into the atmosphere for decades.
Is the deluge coming for us?
Most scientific prognoses for the Earth's future look not unlike the Kevin Costner film. This makes the screenwriters for Waterworld pioneers of environmental enlightenment, who educated the masses on the dangers of global warming. Realistic calculations estimate that by the year 2100, ocean waters will rise by 1.5 metres. The Maldives and Bahamas will disappear from maps, just like many European coastal cities and towns that offer luxury seaside vacations. Such a rise in the water level will threaten the lives of over 150 million people living in flooded areas, causing mass migration, and a radical reshuffling of the global economy and politics.
Some of these changes are now beyond remediation. Therefore, the question of time is essential. How long will humans need to adapt to this new reality? Most of the infrastructure created today, such as homes and public buildings, will most likely still exist several decades from now. Thus they should be designed with contingency planning in mind for the worst climate-change scenarios. In the meantime, it appears obvious that this is the last thing politicians and developers want to consider; they are usually interested in little else than their own short-term investment return. Even one of Poland's most famous modern urban projects – the Wilanów suburb in Warsaw – was built entirely on a flood zone. One day, it might become a financially-attractive alternative to Venice.
Polders in Jakarta
The fastest-drowning city today is Jakarta, located by the Java Sea, which is part of the Pacific Ocean. The northern part of the city has to deal with an annual water level increase of 25 centimetres. This, however, isn't due solely to global warming – local laws allow all inhabitants to dig their own wells, which leads to a gradual collapse of the ground. Indonesia is trying to save its capital with the aid of Dutch engineers, who have suggested building a network of enormous polders along the coastline. Polders are low-lying areas that fall below the sea-level, but are enclosed by barriers. This solution would help to separate the northern parts of Jakarta from the sea, and double as emergency reservoirs that would contain excess water during a crisis.
The government of Jakarta backed out of its initial idea of creating a new business district on an artificial island shaped like the mystical Garuda bird (a man-eagle of sorts), which would have isolated the coast from the sea. This concept, inspired by Dubai's urban planning solutions, was eventually abandoned in favour of a less spectacular but more innovative strategy to deal with the consequences of climate change. It involves filling polders with floating homes that could remain safely on the water surface during floods. This solution will also allow commercial use of an area that is utterly useless for traditional construction techniques.
Jakarta is the fastest-drowning city today. Photo: Adobe Stock
It is no coincidence that Jakarta hired the Dutch to oversee the project of preparing the city for changes in water levels. Apart from the colonial ties linking the two nations, the Dutch have been experts in reclaiming flooded areas for new projects for centuries. They have perfected the art of building on water, too. Amsterdam invests billions of euros in preventing the city from drowning, and it promotes living in floating homes.
The sea won't see it coming
The IJburg district, located south of Amsterdam's city centre, was built on four artificial islands connected by bridges. Between the islands, dozens of floating homes are bobbing along the shores, moored to the islands and using the city's infrastructure. The continually growing community of IJburg is known for its liberal world view (even by Dutch standards), and at its core are young, well-educated, middle-class people. They consider inhabiting the sea a chance to realize their dreams of living close to nature. Life on a barge or a floating platform today is a hippie extravagance. Soon, it might become a necessity.
Villa IJburg in Amsterdam. Photo: Architect Koen Olthuis, Waterstudio.NL
"The Netherlands has long been a pioneer in reclaiming land from water, spending centuries drying out the sea to build. That may have been a mistake," says Koen Olthuis, an architect and the founder of Waterstudio (an architectural firm that designs floating buildings). When so much land is threatened by floods, perhaps it's better to invest in a house that can double as a boat, should crisis strike? This is an intriguing alternative for traditional urban planning, but it's not a new idea. For centuries, Cambodians, Indians and Nigerians have built many towns and villages on areas flooded during rainy season, or just covered in water all year long. Some have created stilt houses, while others have preferred floating platforms that allow the settlements to move during immediate dangers, but also to acclimatize to changing weather conditions.
Such dynamically changing urban development principles could have a significant impact on the ways we all use our cities. It could lead to removing the attachment to particular patches of land. In many densely-urbanized areas all over the world, the amount of dry land available for new building projects is continuously shrinking. This means that throughout the years, there will be more people willing to move their homes to water. It can already be observed in data from the Netherlands, Denmark, Britain and France, where serious consideration is given to creating whole districts made entirely of floating homes.
Koen Olthuis is convinced that this type of building is the best solution for the architecture of the future. His projects are made not only for residential housing; he also designs floating hospitals, schools and theatres. They can easily travel between cities or districts, enriching the infrastructure and reaching places where they are most needed at the moment. However, his strategy raises many questions – most people don't view living on water as a guarantee of permanence. This opinion is further cemented by news of disasters, like the recent collapse of the most famous floating construction, a school in Makoko, Lagos. The school, designed by the Dutch architectural firm NLÉ, drowned after heavy rain. Many ecologists also highlight the fact that too many floating buildings could affect riverbeds and shorelines, damaging the local fauna and flora. On the other hand, several units could have a surprisingly positive impact on the environment. Floating homes are a fantastic base for the development of underwater ecosystems, because they promote the growth of plants that significantly improve the biodiversity of the habitats of various fish and birds.
This has happened underneath several residential barges, whose impact on the environment was recently analysed by a team of London scientists. As it transpired, the barges are now home to more tenants than just the handful of humans who live inside.
Vistula tenants
Prototype water estates are also appearing in Poland. In the Czerniakowski Port in Warsaw, there are already several residential barges with permanent tenants. The only problem is the lack of space in the port, which makes it impossible to add more vessels to the neighbourhood. And the queue is long, as such floating homes are a tempting alternative to land property. Living on a barge is a bargain. The vessels are prefabricated and assembled of ready-made elements, which means a brand-new home could cost as little as 4000 PLN (around £800) per square metre. A 70–80 square-metre home could therefore cost less than a studio apartment in a big city. There are manufacturers offering steel or wooden skeletons on floats, often made of barrels, or styrofoam covered in concrete. The only trouble with such homes is ensuring an even spread of weight. If we invite 20 or 30 guests and let them all stand in one part of the room, the party would most likely end in a spectacular disaster. But a little caution and a pinch of common sense don't seem to be a high price to pay for the chance to live in such an unusual and interesting environment.
The human ability to adapt to living on the water is actually quite astonishing. In today's Indonesia, there are still tribes who spend over 60% of their daily lives on boats, diving for fish. Scientists were recently baffled by research confirming that the Bajau people have adapted to centuries-long evolutionary processes by developing spleens 50% larger than those in people who live on land. This unusual change allows the Bajau people to last up to 13 minutes underwater on just one breath. Moreover, some of them regularly experience land sickness after leaving their boats. What today seems to us curious trivia might one day become the new normal. If ocean waters continue rising at their current pace, it could be that our grand- and great-grandchildren's genes will have to start making similar changes.
These solar-powered floating homes are built to withstand floods and hurricanes
By Nicole Jewell
Inhahitat
April.1.2019
Photo Credits: ARKUP & Waterstudio
As many coastal cities struggle to come up with resiliency plans in the face of rising sea levels, Dutch architect Koen Olthuis with Waterstudio is creating sustainable, solar-powered floating residences that could offer the perfect solution. Already well-known for its high-end floating homes, Waterstudio and Miami-based Arkup are now teaming up with Artefacto, an environmentally friendly Brazilian furnishing brand, to create stylish floating houses that are not only resilient to storms and sea levels, but also represent the luxury style for which Miami is known.
Waterstudio has long been recognized for creating sustainable and attractive floating homes that can provide discerning homeowners with an "avant-garde life on water." The residences are modern, cube-like structures that are completely self-sufficient, operating 100 percent off-grid thanks to solar power generation, eco-friendly waste management features, rainwater harvesting and water purification systems. Additionally, the homes are equipped with unique self-elevating systems that help the structures withstand high winds, floods and hurricanes.
In addition to the ultra sustainable and resilient features, the two-story floating homes boast interiors with a 775-square-foot living room, bedroom, kitchen and dining space, as well as an open-air rooftop lounge. Sliding glass doors, which almost make up the entirety of the front facade, lead out to a beautiful terrace.
Although the company has been working on its floating homes for some time, it recently announced a new partnership with Artefacto, a Brazilian furnishing company with a strong commitment to sustainability that is known for combining luxurious furniture made of raw materials with cutting-edge smart automation technologies. The floating residences will now be outfitted with eco-friendly furnishings, including high-end pieces made out of timber approved for use by the Brazilian Environment Department.
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Floating homes that can withstand Category 4 hurricanes will soon become a reality
By Aria Bendix
September.20.2018
Photo Credits Waterstudio
As Hurricane Florence makes it way across the Carolinas, millions of coastal residents have reason to be concerned about the structural integrity of their homes. Already, nearly 300,000 homes and businesses have lost power, and officials are reporting damage to property in Onslow County, North Carolina.
When Hurricane Harvey swept Texas last September, it damaged more than 204,000 homes and apartment buildings. Around the same time, Hurricane Irma destroyed a quarter of the homes in the Florida Keys, according to federal officials.
While the idea of a hurricane-proof home may sound far-fetched, a housing startup called Arkup has created a residence that can withstand rising sea levels and Category 4 hurricanes. The key lies in its hydraulic while lifting it 40 feet above the ocean floor.
Arkup calls the residences "livable yachts" due to their buoyant nature, which allows them to bob with the water. After debuting the designs in 2017, the company teamed up with The Advantaged Yacht Charters & Sales, the oldest yacht charter company in Miami, to make the structures available for rent and purchase. In August, The Advantaged announced that it isaccepting charter reservations online.
The residences were designed by architect Koen Olthuis, who has pioneered the concept of the floating home.
Each 4,350-square-foot unit contains four bedrooms and four-and-a-half bathrooms.
The retail price for each home is $5 million.
The residences provide 360-degree views of the water.
They also have zero emissions and are powered by solar panels on the roof.
Guests can disconnect from sewage lines, thanks to a system that collects, stores, and purifies rainwater.
The units are just as mobile as a typical yacht.
Even as coastal residents become more fearful of rising sea levels, Olthuis wants cities to see water as an asset, not a challenge, to new construction.
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Waterstudio at BBC
Are floating homes the next frontier for urban design?
Architecture that works with water, rising with floods and sitting upon unused city space, may be the future of urban planning, say these innovative designers.
Buildings and communities that can float on water may be the next step in the evolution of cities, according to some avant-garde housing designers.
Many of the world's largest cities sit next to, or are built around, large bodies of water. In the light of unprecedented population growth, climate change, flooding and rising sea levels, are floating homes the next frontier in urban living?
Watch the video to see two leading architecture firms describe their innovative concepts for life on water.
Are These Dutch Floating Homes a Solution for Rising Seas?
By Olga Mecking
August.23.2017
Houseboats have long been a common sight near Amsterdam, but a new community may signal a premise that could work elsewhere, too.
Not far from Amsterdam's Central Station lies IJburg. Hidden in plain view, the city's newest district is somewhat of an undiscovered secret. In fact, IJburg is known better to the people outside of the country rather than the ones who actually live in the Netherlands.
Moriam Hassan Balogun, who is originally from the United Kingdom, moved there in 2009 and now considers herself an "international local." She loves IJburg's family friendly atmosphere, the space, and the many cafes and possibilities for work and leisure. It also attracts many business owners and mostly people with liberal political views.
IJburg is built on four artificial islands that are connected to each other and the rest of the city via bridges. It has around 21,000 inhabitants, the first of whom moved there in early 2002. But the district still isn't completely built. Though the goal was to finish building IJburg by 2012, that has not happened due to environmental concerns and slow uptake of houses. When finished, it will offer 18,000 homes for 45,000 people and create around 12,000 jobs.
Of late, the islands have been of particular interest to climate change researchers; in particular, the area of Waterbuurt West. There, 120 floating homesteads have been built to deal with Amsterdam's housing shortage and to prevent the citizens of Amsterdam from moving farther away, to Purmerend or Almere—a phenomenon known as urban sprawl. Living on water is not that surprising in a country that's surrounded by it. All over the Netherlands, people live on barges or houseboats. But these new houses in IJburg are different because they are very visibly not boats. They are houses.A Dutch saying goes, "God created the world but the Dutch created Holland." The Netherlands has long been a pioneer in reclaiming land from water, spending centuries drying out the sea to build. That may have been a mistake, says Koen Olthuis, the founder of the Waterstudio in Rijswijk, an architectural bureau specifically devoted to designing buildings on water.
"The Dutch are crazy, that's fun about the Dutch. We are here now in a part of Holland where we shouldn't be. It's man-made," Olthuis says. A much better solution would be to simply build floating houses, or even whole floating neighborhoods instead.The technology used to build houses on water is not really new. Whatever can be built on land can also be built on water. The only difference between a house on land and a floating house is that the houses on water have concrete "tubs" on the bottom, which are submerged by half a story and act as counter-weight. To prevent them from floating out to sea, they are anchored to the lakebed by mooring poles.
As sea levels are rising globally, many cities around the world are under threat from water. Some areas are projected to disappear completely in the next few decades. Therefore, designing houses to float may, in some instances, be safer than building on land and risking frequent floods. "In a country that's threatened by water, I'd rather be in a floating house; when the water comes, [it] moves up with the flood and floats," Olthuis says. He believes that water shouldn't be considered an obstacle, but rather a new ingredient in the recipe for the city.
Floating houses are not only safer and cheaper, but more sustainable as well. Because such a house could more readily be adapted to existing needs by changing function, or even moving to a whole new location where it can serve as something else, the durability of the building is much improved. Olthuis compares this to a second-hand car: "By having floating buildings, you're no longer fixed to one location. You can move within the city, or you can move to another city, and let them be used and used again."
Houses built on land are very static, while on water it's possible to add, take away, or easily change parts. And communities built on water can be constructed more densely, which would allow for more efficient energy use. Water allows houses (and even whole cities) more flexibility, and, for Olthuis, it's this characteristic that makes it such a fascinating element.
"In a country that's threatened by water, I'd rather be in a floating house."
He sees the use and incorporation of water as the next logical step in the evolution of cities. Cities are not unlike brands, and the ones with a lot of water would be the most flexible, and therefore the most desirable. This branding is already visible in many regions around the world: Think of Los Angeles as the city of movies, New York as the city for writers. Blue cities, or cities that can utilize the water, would also be the cities that would attract residents.
But Olthuis goes one step further. He imagines cities that can quickly change, depending, for example, on the season. In the summer, they could be open to allow the collection of sun energy, and in the winter they could huddle closer together for warmth and energy preservation. He also prefers to talk about functions, or modules, rather than actual buildings.
"In the next city, it's no longer about what you have; it's about what you can load. You're going to load functions to your neighborhood on the water, and if you need new functions, you take them out and you reload them with other profiles," he imagines. Cities of the future will share certain functions, like, for example, museums, stadiums, or other facilities. "It will be a completely new way of thinking about these [establishments]."
Incorporating water into the cities will also introduce more equality, says Olthuisk, referring to a principle known as "the democracy of water." In fact, something similar is already happening not just in IJburg but in the whole of the Netherlands, where house owners and social housing recipients share neighborhoods. In IJburg itself, around 30 percent of the houses are earmarked for this very form of government assistance. People of various nations, races, religions, and ethnicities live on the island. "There's no group that's more than the other," Hassan Balogun says. However, the people who move to IJburg tend to be politically similar. "There's quite a lot of liberal thinkers, very open-minded people here. I think that like seeks like," she says. The residents of IJburg often vote for D66 and Groenlinks parties, both known for their liberal views and a focus on sustainability.
A floating home in Ijburg. (Margriet Faber/AP)
The inhabitants of IJburg don't really have the need to leave the island unless they want to. There are plenty of options in that part of the city, including cafes, gyms, yoga studios, and parks. There are also 10 schools. The whole area has an atmosphere of newness, of opportunity.
And this opportunity—the concept of floating houses—could spread to other areas around the world. Due to very strict regulations in the Netherlands, Olthuis is often exporting his ideas abroad, including to China, the United Arab Emirates, India, and the Ukraine. He recognizes that American cities face the same threat towns in the Netherlands did: urban sprawl. "So we have to bring the cities back, make them more compact," he says. "That's what I hope that people in the States will learn."
Floating houses are an idea American cities should consider not just to combat sprawl. Many major cities—like New York, Washington, or Miami—could soon find themselves under water. Olthuis does, however, caution against a Waterworld-like future.He doesn't believe in cities existing solely on water due to high costs of maintenance and constant energy consumption. He thinks the future lies in already existing cities that use naturally existing water to expand and improve. His hope is that, one day, 10 percent of the Netherlands could become a blue city. But it doesn't have to stop there. "It's not only about architecture, it's not just about having fun in IJburg. It's about rethinking how we, as communities, want to live in cities."The city may not be fully built yet, but, given its multi-faceted approach to sustainable urban design, IJburg could be seen as a first step in that direction. At the very least, it is an already existing example for how to successfully integrate water into our cities.
This story originally appeared as "Are the Floating Houses of the Netherlands A Solution Against the Rising Seas" on Pacific Standard, an editorial partner site. Subscribe to the magazine in print and follow Pacific Standard on Twitter to support journalism in the public interest.
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'Floating Homes' technology has demonstrated usefulness
By Fane Lozman
It is an unfortunate reality that those who currently live on Eastern Shores in Maule Lake will have to abandon their homes in the next 30 years because of rising tides. The only residents around Maule Lake will be the Amirillah floating islands, and any new developments built after Miami 21-like building codes are enacted.
The current Eastern Shores residents should be making plans now for where they will be moving once North Miami Beach condemns their residences for sea water intrusion. Instead, their fears are focused on the floating islands and reflect a total lack of knowledge about floating technology that has been proven over the last century.
Just like floating oil rigs moored to the ocean floor survive Category 5 hurricanes without being torn off their moorings, floating islands use similar technology. The land-based houses around Maule Lake would be swept clean off their concrete pads as the eye wall of a hurricane similar to Andrew made a direct hit, while the Maule Lake floating islands wouldn't slide an inch off their permanent moorings.
Even more impressive is that these foam-cored, reinforced concrete islands are unsinkable, even after being pelted with 200-mph, windswept debris from the destroyed houses on shore.
The West Coast of the United States has thousands of floating homes that are a welcome addition to their communities in Washington, Oregon and California. The Maule Lake floating-island residences will introduce a new generation of floating homes to the East Coast. They will be completely self-sustaining and have the "greenest" footprint of any dwelling in South Florida.
The landlubbers whose attitude is that "I got to Maule Lake first and no one else should ever join me" forget one thing. The actual lake bottom is privately owned, and the submerged lands do not belong to those who are fortunate to live on its borders. Perhaps a 50-foot-high floating privacy screen running on the east side of Maule Lake would be soothing to these residents so they would not have to be jealous of their floating neighbors?
The floating islands will also help solve a simple reality that the political leaders of North Miami Beach can no longer ignore: New sources of tax revenue will be desperately needed to supplement the hidden pension demands of civil employees (i.e. police) over the coming years.
The 29 floating islands that will be assessed at $12.5 million each will bring in a staggering $363 million in new property assessments. This windfall for the city will be further magnified by the increased tax assessments for the Eastern Shores residents as their droopy neighborhood wakes up to become part of South Florida's most unique residential community.
Like any new technology, whether it was the Wright brothers' first airplane flight, or the $3 billon Perdido oil rig anchored in 2,438 meters of water in the Gulf of Mexico, there are "talking heads" that will refuse to accept the inevitable march of technology. It makes one wonder: How many Eastern Shores residents still have horse and buggies in their back yards?
FANE LOZMAN WON A PRECEDENT-SETTING VICTORY LAST YEAR AGAINST RIVIERA BEACH WHEN THE U.S. SUPREME COURT AGREED WITH HIM THAT HIS FLOATING HOME — WHICH THE CITY HAD SEIZED AND DESTROYED UNDER LAWS GOVERNING SHIPS AT SEA — WAS A HOUSE, NOT A VESSEL COVERED BY MARITIME LAW.
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Architects' Answer to Rising Seas: Floating Homes, abc NEWS
abc NEWS, AP Associated press, Denis D.Gray, Apr 2012
BANGKOK (AP) — A floating mosque and golf course for the submerging Maldives islands. Amphibious homes in the Netherlands lifted to safety as waters surge beneath them. A hospital perched on 400 stilts to protect patients from Thailand's devastating floods and the encroaching sea.
Around the world, architects and city planners are exploring ways mankind and water may be able to coexist as oceans rise and other phenomenon induced by climate change, including extreme, erratic floods, threaten land-rooted living.
With the Dutch at the helm, projects in the cutting-edge field of aqua-architecture are already in place, including a maritime housing estate, floating prison and greenhouses in the Netherlands. An increasing number are coming on stream, and while earlier blueprints appeared to be the stuff of science fiction, advocates say leaps of imagination are still needed given the magnitude of the danger.
"The focus on floating solutions has grown enormously. It has shifted from freak architecture to more sustainable, flexible alternatives," says Dutch architect Koen Olthuis, citing growing support by governments and interest among private investors in Asia and Russia.
"We will have to live with a more watery environment. There is no choice," says Danai Thaitakoo, a Thai landscape architect whose own Bangkok house was swamped last year as the country suffered its worst floods of modern times.
The Thai capital is also among the mega coastal cities projected by the end of this century to lie totally or partially under water as global warming boosts sea levels, according to the U.N. Intergovernmental Panel on Climate Change. Others include Tokyo, London, Jakarta, Sydney and Shanghai — an apocalyptic prospect of mass migrations and economic crises.
While in earlier decades architects and planners, particularly Japanese and Americans, dreamed of entire marine cities housing millions, most today are proposing a mix of defending communities with barriers and building on water using floating platforms, raised or amphibious structures and solutions still being devised.
"Climate change will require a radical shift within design practice from the solid-state view of landscape urbanism to the more dynamic, liquid-state view of waterscape urbanism," says Danai, who is involved in several projects based on this principle. "Instead of embodying permanence, solidity and longevity, liquid perception will emphasize change, adaptation."
In a study for low-lying New York, Olthuis says he envisioned Manhattan ringed by a sea wall with outlying boroughs allowing water to enter and adapting. The world's Londons and Bangkoks, he says, may become "hydro-cities," their historic hearts and concentrated core development waterproofed and other areas "going with the flow."
The Netherlands, a third of which lies below sea level, has been managing water since the Middle Ages and is thus a pioneer in the field. It has exported its expertise to Indonesia, China, Thailand, Dubai and the Republic of the Maldives, an Indian Ocean archipelago that with a maximum elevation of about 2 meters (8 feet) is the world's lowest country. The sea-battered city of New Orleans has also sought advice from Olthuis's Waterstudio.
In the Maldives, Waterstudio has designed a network of floating islands, the first to be put in place next year, to accommodate hotels, a convention center, yacht club and villas. The "islands," secured by steel cables, are made up of pontoons with a foam core encased in concrete that can be joined together like Lego blocks. An 18-hole golf course will also be set on such platforms, each with two to three holes, connected by underwater tunnels. The $500 million project, paid for by the Maldivian government and private investors, is slated for completion in 2015.
A floating mosque, originally destined for Dubai before an economic downturn hit, is also part of the master plan, Olthuis said in an interview.
Following the principles of "water will always find its way" and "collaborating with nature," the Dutch have reversed some of their earlier strategy of tightly defending their land with dikes by allowing the sea to penetrate some areas on which housing has been constructed.
One pioneering effort was the placement of amphibious and floating homes on the River Maas in 2005. All survived major 2011 floods that forced the evacuation of villages along rain-swollen rivers.
Construction recently began on the Olthuis-designed New Water estate, 600 homes and a luxury apartment complex on land purposely inundated. Interest in water-based living and work space has accelerated over the past decade, he says, and Waterstudio's drawing boards are stacked with plans for local and international projects.
Typical amphibious houses, like the two-story ones on the Maas, consist of a structure that slides into a steel framework over a hollow foundation which, like the hull of a ship, buoys up the building when water enters.
The Maas houses sell from $310,000, about 25 percent more than equivalent homes, in part due to the cost of connecting them to utilities and drainage. But Olthuis says such linkages are simple and present no inconvenience to owners.
"Just proven technology of plug-and-play systems. All tested and used for years in Holland," he says.
"The only time you will see a difference between a floating house and the traditional one is during floods — when your house rises above the water and your neighbor's stays put," Olthuis says.
Along similar lines will be Britain's first amphibious house, recently granted planning permission along the banks of the Thames River in Buckinghamshire. The 225-square-meter (2,421-square-foot) home will be able to rise to 2.5 meters (8.2 feet) in the event of flooding.
Thai architect Chutayaves Sinthuphan, who will be unveiling a pilot amphibious house for the Thai government in September, says interest in such projects has grown since last year's floods, which killed more than 600 people and affected more than a fifth of the country's 64 million people.
"We have had proposals out for some time, but nobody paid much attention to them until the floods came," he says.
His Site-Specific Company has already built such houses for private clients, using modern techniques and materials but like other architects in Asia looking to a past when communities adapted well to annual monsoon season inundations.
They point to a riverside village in the southern province of Surat Thani, where everyone lived on homes atop bamboo rafts until all but three families moved on land. Those three homes were the only ones that survived last year's floods.
In the mid-19th century, almost all of Bangkok lived on houses built atop stilts or rafts. Since then, most canals have been paved over and the stilt houses replaced by a concrete urbanscape that holds back water instead of allowing it to flow through.
Architect Prisdha Jumsai has borrowed from traditional methods to design Thailand's first hospital for the aged. Work has begun on the 300-bed hospital over a permanently flooded area near Bangkok that is also subject to tides from the nearby Gulf of Thailand. Concrete stilts will raise its first floor about 4 meters (13 feet) above average water levels.
"We hope this will influence people not to just fill in land but to build on water. I think it will open up new ideas for Thais who can look to traditional architecture and make it more up-to-date in design," Prisdha says.
But this still appears to be a minority view.
"Most Thais look to Western, land-based models and most architects still don't talk about environmental concerns. They talk about how a house will look and make you feel good," says Danai. "But this will have to change. It's about survival."
Associated Press writer Mike Corder in The Hague contributed to this story.
Why the Dutch are dismantling their famous dislikes
Architectural Record, Tracy Metz
01-06-2006, Architecural Record
Complete woonwijk op het water
TED Talk of Koen in Hasselt great success
Devorar el mar
De Volkskrant about Waterstudio.NL:
Building a floating house, how does it work
Tecture Mag – Arkup
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With plenty of loungers and seating, the seasonal outdoor pool and hot tub is a great place to chill out and begin planning tomorrow's adventures whilst the hotel's fitness suite is equipped with a good range of cardio machines and weight stations to help you keep in shape. The resort also features a 464 seat theatre for live performances and movie screenings.
The resort is only a short stroll from Graceland and the Graceland complex, the home of Elvis Presley, take the tour and share in the unforgettable journey of the King, from humble roots to worldwide superstar and American icon.
Of course there's more to Memphis than just Graceland, Memphis is a city steeped in music' the Stax Museum of American Soul Music, Sun Studio, the Blues hall of fame and the Memphis Rock 'n' Soul Museum will appeal to music lovers of all genres. Spend the day afloat with a Memphis Riverboat Cruise or head downtown to sample some of the fantastic shopping dining and nightlife that Memphis has to offer.
To maximise your enjoyment the Nashville Music Attraction Pass is included, your passport to the music history of Nashville. Enjoy admission to 4 famous attractions including the Country Music Hall of Fame with $15 for food and beverages to be used at the Hard Rock Cafe. Tour the historic RCA Studio B, the Ryman Auditorium home to the Grand Ole Opry and the Johnny Cash Museum, officially authorized by the Estate of Johnny Cash.The Millennium Maxwell House Nashville offers everything you could wish for to immerse yourself in this wonderful city and with a brilliant range of amenities every need is catered for providing a spectacular holiday to remember.
Call one of our travel experts to book this fantastic twin centre holiday today.
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Taylor Gilbert
#11 Midfielder - UC Santa Barbara
5' 10" Height
180 lbs Weight
Senior Year
Hometown: Encinitas, CA
High School: Sante Fe Chrisitan
Major: Unknown
Eligibility: Senior
Statistics By Season
2012 UC Santa Barbara 11 7 20 12 0 0 0 - 0
2011 UC Santa Barbara 11 27 16 8 0 0 0 - 0
2012 Player Statistics By Game
02/11/2012 Stanford W 7-6 0 0 0 0 0
02/18/2012 @ Chico State W 16-12 2 4 2 0 0
02/20/2012 @ UC Davis W 12-7 2 2 2 0 0
02/25/2012 Claremont W 14-3 0 0 2 0 0
02/29/2012 Cal Poly W 10-9 1 1 1 0 0
03/05/2012 Boston College W 13-4 0 1 2 0 0
03/29/2012 @ Colorado State L 3-11 0 0 0 0 0
03/31/2012 @ Colorado W 7-2 1 1 0 0 0
04/07/2012 Loyola Marymount W 9-6 0 3 0 0 0
04/14/2012 @ San Diego State W 19-4 0 1 2 0 0
04/21/2012 Chapman L 4-9 0 0 0 0 0
04/28/2012 UNLV W 11-8 0 1 0 0 0
05/04/2012 @ Arizona State L 8-10 1 2 1 0 0
05/14/2012 Pittsburgh W 13-9 0 4 0 0 0
05/15/2012 Colorado State L 3-6 0 0 0 0 0
02/04/2011 @ California W 11-10 4 2 0 0 0
02/11/2011 @ Cal Poly L 11-12 5 2 0 0 0
02/17/2011 Utah W 13-10 4 1 1 0 0
02/19/2011 Santa Clara W 8-5 0 1 2 0 0
02/27/2011 Colorado State L 7-13 0 1 0 0 0
03/05/2011 San Diego State W 15-9 3 0 0 0 0
03/06/2011 @ Claremont W 15-9 1 1 0 0 0
03/24/2011 @ Michigan State L 10-11 1 1 0 0 0
04/02/2011 @ Loyola Marymount W 11-10 3 0 1 0 0
04/09/2011 UCLA W 15-5 1 2 0 0 0
04/23/2011 @ Chapman L 5-16 0 0 1 0 0
05/17/2011 Brigham Young L 10-13 0 2 0 0 0
Total Assists
Total Saves
Total Goals Against
2012 Captain
2012 1st Team All-Conference Midfield
2012 All-Tournament Midfield
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| 8,060
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Voice disorders are medical conditions involving abnormal pitch, loudness or quality of the sound produced by the larynx and thereby affecting speech production. These include:
Vocal fold nodules
Vocal fold cysts
Vocal cord paresis
Reinke's edema
Spasmodic dysphonia
Foreign accent syndrome
Bogart–Bacall syndrome
Laryngeal papillomatosis
Laryngitis
See also
Aphasia
Dysphonia
Human voice
Laryngectomy
Parkinson's disease
Speech disorder
Vocology
Voice changes during puberty
References
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\section{Introduction}
In the last few years, algorithms based on convolutional neural networks (CNNs) have led to dramatic advances in the state of the art for fundamental problems in computer vision, such as object detection, object localization, semantic segmentation, and object instance segmentation. \cite{Krizhevsky:2012:ICD:2999134.2999257}, \cite{DBLP:journals/corr/SimonyanZ14a}, \cite{resnets}, \cite{DBLP:journals/corr/ZagoruykoLLPGCD16}. This has led to increased interest in the applicability of convolutional neural network-based methods for problems in medical image analysis. Recent work has shown promising results on tasks as diverse as automated diagnosis diabetic retinopathy, automatic diagnosis of melanoma, precise measurement of a patient's cardiovascular ejection fraction, segmentation of liver and tumor 3D volumes, segmentation of mammogram images, and 3D knee cartilage segmentation \cite{Pratt2016200}, \cite{esteva_skin_cancer}, \cite{Prasoon2013}, \cite{DBLP:journals/corr/Tran16}, \cite{liver_segmentation}, \cite{mammogram}, \cite{ejection_fraction}.
Semantic segmentation of natural images is a long-standing and not fully solved computer vision problem, and in the past few years progress in this area has been almost exclusively driven by CNN-based models. Notable developments in recent years include the development of RCNN in 2014, fully convolutional neural networks in 2015, and the development in 2015 of the Fast-RCNN followed by the Faster-RCNN models \cite{girshick14CVPR}, \cite{fully_convolutional}, \cite{fast_rcnn}, \cite{faster_rcnn}. These algorithms were designed with semantic segmentation, object localization, and object instance segmentation of natural images in mind. Recently there has been an explosion of development in this area, with convolutional neural network based architectures such as Feature Pyramid Networks, SegNets, RefineNets, DilatedNets, and Retinanets developed all pushing benchmarks for this task forward \cite{DBLP:journals/corr/LinDGHHB16}, \cite{badrinarayanan2015segnet}, \cite{Lin:2017:RefineNet}, \cite{DBLP:journals/corr/abs-1708-02002}, \cite{dilatednet}. In addition, single-shot models such as YOLO and SSD have enabled object detection to occur at speeds up to 100-1000 times faster than region proposal based algorithms \cite{yolov3}, \cite{ssd}.
In 2015, the U-Net architecture was developed explicitly with the segmentation of medical images in mind, and used to produce state-of-the-art results on on the ISBI challenge for segmentation of neuronal structures in electron microscopic stacks as well as the ISBI cell tracking challenge 2015 \cite{unet}. U-Net architectures have since been adapted and used for a wide range of tasks in medical image analysis including volumetric segmentation of 3D structures and sparse-view CT reconstructions \cite{Han2017FramingUV}, \cite{3d_unet}.
\subsection{The Mask-RCNN Model}
\begin{figure}[t]\label{mask_rcnn_image}
\begin{center}
\includegraphics[scale=0.5]{teaser.pdf}
\end{center}
\caption{The Mask-RCNN model. Image from \cite{mask_rcnn}. Used with permission.}
\end{figure}
The Mask-RCNN model was developed in 2017 and extends the Faster-RCNN model for semantic segmentation, object localization, and object instance segmentation of natural images \cite{mask_rcnn}. Mask-RCNN is described by the authors as providing a `simple, flexible and general framework for object instance segmentation'. Mask-RCNN was used to outperform all existing single-model entries on every task in the 2016 COCO Challenge, a large-scale object detection, segmentation, and captioning challenge \cite{mscoco}.
Many modern algorithms for image segmentation fall into one of two classes: those that rely on a region proposal algorithm and those that do not. U-Net, for instance, is an example of a segmentation algorithm that does not rely on a region proposal algorithm; rather, U-Net uses an encoder-decoder framework in which a convolutional neural network learns, or encodes, a representation of the content of the image and a second network, such as a deconvolutional neural network, constructs the desired segmention mask from the learned representation produced by the encoder (note that a deconvolutional neural network may also be referred to as a fully convolutional neural network, a transposed convolutional neural network or a fractionally-strided convolutional neural network in the literature) \cite{fully_convolutional}, \cite{johnson2016perceptual}. Encoder-decoder architectures have been used in machine learning for a variety of tasks outside of object detection or segmentation for some time, such as denoising images or generating images \cite{kingma_vae}, \cite{denoising_autoencoders}.
Mask-RCNN, in contrast, relies on a region proposals which are generated via a region proposal network. Mask-RCNN follows the Faster-RCNN model of a feature extractor followed by this region proposal network, followed by an operation known as ROI-Pooling to produce standard-sized outputs suitable for input to a classifier, with three important modifications. First, Mask-RCNN replaces the somewhat imprecise ROI-Pooling operation used in Faster-RCNN with an operation called ROI-Align that allows very accurate instance segmentation masks to be constructed; and second, Mask-RCNN adds a network head (a small fully convolutional neural network) to produce the desired instance segmentations; c.f. Figure \ref{mask_rcnn_image}. Finally, mask and class predictions are decoupled; the mask network head predicts the mask independently from the network head predicting the class. This entails the use of a multitask loss function $L = L_{cls} + L_{bbox} + L_{mask}$. For additional details, we refer interested readers to \cite{mask_rcnn}.
Mask-RCNN is built on a backbone convolutional neural network architecture for feature extraction \cite{girshick14CVPR}, \cite{faster_rcnn}. In principle, the backbone network could be any convolutional neural network designed for images analysis, such as ResNet-50 or ResNet-101 \cite{resnets}; however, it has been shown that using a feature pyramid network (FPN) based on a network such as ResNet-50 or ResNet-101 as the Mask-RCNN backbone gives gains in both accuracy and speed \cite{mask_rcnn}. A feature pyramid network takes advantage of the inherent hierarchical and multi-scale nature of convolutional neural networks to derive useful features for object detection, semantic segmentation, and instance segmentation at many different scales. Feature pyramid network models all require a `backbone` network themselves in order for the feature pyramid to be constructed. Again, the backbone model is typically chosen to be a convolutional neural network known for high performance at object detection, and may be pretrained \cite{DBLP:journals/corr/LinDGHHB16}.
Although the properties of natural images will in general differ significantly from medical images, given the effectiveness of Mask-RCNN at general-purpose object instance segmentation, it is a reasonable candidate for use in automated segmentation of medical images. Here, we investigate the efficacy of a Mask-RCNN model at detecting nuclei in microscopy images.
\begin{figure}[tp]\label{sample_images}
\centering
\label{figure}
\begin{tabular}{ccc}
\includegraphics[width=30mm]{8c3ef7aa7ed29b62a65b1c394d2b4a24aa3da25aebfdf3d29dbfc8ad1b08e95a.png} &
\includegraphics[width=30mm]{d1ba6089cae2f90cb7275ece10ca393c25f60ea17e5c9c3cea2399d31fd41869.png} &
\includegraphics[width=30mm]{930f246a8e4ff273a72a6e4b3cf8e8caff94fca4eaf1dbe6f93ba37b8195c0a0.png} \\
\includegraphics[width=30mm]{8c3ef7aa7ed29b62a65b1c394d2b4a24aa3da25aebfdf3d29dbfc8ad1b08e95a_full_mask.png} &
\includegraphics[width=30mm]{d1ba6089cae2f90cb7275ece10ca393c25f60ea17e5c9c3cea2399d31fd41869_full_mask.png} &
\includegraphics[width=30mm]{930f246a8e4ff273a72a6e4b3cf8e8caff94fca4eaf1dbe6f93ba37b8195c0a0_full_mask.png} \\
\end{tabular}
\caption{Sample Nuclei Images and Masks. For each image, an individual mask is provided for each nucleus detected. To generate the masks in the bottom row, all of the masks provided for each image have been merged into a single mask. Note that images vary widely, including in size.}
\end{figure}
\section{The Data}
The data used for these experiments is image set \href{https://data.broadinstitute.org/bbbc/BBBC038/}{BBBC038v1}, available from the Broad Bioimage Benchmark Collection \cite{nature_broad}. These data were used for Stage 1 of the the 2018 Data Science Bowl, an annual competition sponsored by Booz Allen Hamilton and hosted on the data science website \href{http://www.kaggle.com}{kaggle.com}. The data consist of 729 microscopy images and corresponding annotations for each individual nucleus detected by an expert in each image; c.f. Figure \ref{sample_images}. The nuclei in the images are derived from a wide range of organisms including humans, mice, and flies. Furthermore, the nuclei in the images have been imaged and treated in variety of conditions and appear in a variety of contexts and states, including tissues and embryos, and cell division and genotoxic stress. This presents a significant additional challenge as convolutional neural networks can be expected to perform best in general when the input data is as uniform and standardized as possible. This includes standardization in terms of color, contrast, scale, and class balance.
Of these 729 images in the dataset, 664 images were used for training and validating the model and 65 images were held out for testing.
\section{Methodology}
For all experiments described here, we use a Mask-RCNN model with a feature pyramid network backbone. The implementation used is based on an existing implementation by Matterport Inc. released under an MIT License, and which is itself based on the open-source libraries Keras and Tensorflow \cite{matterport}, \cite{tensorflow}, \cite{keras}. This implementation is well-documented and easy to extend. For these experiments, we tried both a ResNet-50 feature pyramid network model and a ResNet-101 feature pyramid network model as a backbone. We note that the model with ResNet-50-FPN backbone has a somewhat lower computational load than that with a ResNet-101 backbone, but the ResNet-101-FPN gives significantly improved results with no other changes to the model or training procedure. Rather than training the network end-to-end from the start, we initialize the model using weights obtained from pretraining on the MSCOCO dataset \cite{mscoco} and proceed to train the layers in three stages: first, training only the network heads, which are randomly initialized, then training the upper layers of the network (from stage 4 and up in the ResNet model), and then reducing the learning rate by a factor of 10 and training end to end. In total we train for 100 epochs using stochastic gradient descent with momentum of 0.9, starting with a learning rate of 0.001 and ending with a learning rate of 0.0001. Although we experimented with longer and shorter training times, additional training did not lead to noticable improvement and few epochs led to underfit. We use a batch size of 6 on a single NVIDIA Titan Xp GPU. Gradients are clipped to 5.0 and weights are decayed by 0.0001 each epoch. We also conducted additional experiments with other learning rate schedules, but they gave no additional improvements and aren't reported here.
For all of these experiments, image preprocessing is kept to a minimum. Images are upsampled by a factor of two and the channel means are normalized. We conducted multiple experiments on mirroring the image edges by various numbers of pixels to improve detection of small nuclei at the edges as described in \cite{unet}, but we saw no improvement from doing so and omitted this step from the final algorithm.To help avoid overfitting, the dataset was augmented using random crops, random rotations, gaussian blurring, and random horizontal and vertical flips.
\begin{table}[t]
\begin{tabular}{l|l|c}
Backbone & AP & Mask Average IoU \\ \hline
ResNet-50 FPN & 56.06 & 66.98 \\
ResNet-100 FPN & 59.40 & 70.54
\end{tabular}\vspace{2mm}
\caption{\textbf{Instance segmentation} mask results on validation data. All results are single-model results.}
\label{results_table}\vspace{-3mm}
\end{table}
\subsection{Results}
The model with ResNet-50 backbone and parameters as described above obtains an average mask intersection over union (IoU) of 66.98\% on the validation dataset. The mean average precision at thresholds 0.5 to 0.95 by steps of size 0.05 as defined as the primary metric for the MSCOCO challenge (AP) is 56.06\% for this model \cite{mscoco}. The mode with ResNet-101 backbone and the same parameters and training procedures as described above obtains an average mask IoU of 70.54\% and a mean average precision as defined for the MSCOCO challenge of 59.40\%. These results are summarized in table \ref{results_table}. Several sample detections are illustrated in Figure \ref{sample_detections}.
\begin{figure}[tp]\label{sample_detections}
\centering
\label{figure}
\begin{tabular}{ccc}
\includegraphics[width=40mm]{detection_406b4c2e9610ef28f5167e7d995be83785a36b6b3323a2a9f82a23c90a5e68e8.png} &
\includegraphics[width=40mm]{detection_f3d6640db1e6479bd3f1bb03e21706b32b4380aa1bc068498b054ffbca47d73a.png} &
\includegraphics[width=40mm]{detection_99922b3630e35a017210ab0573745b7d2ad58f8d2c1bdf1628df265f33f940ba.png}
\end{tabular}
\caption{Sample detections from the ResNet-50-FPN model.}
\end{figure}
\section{Conclusion \& Future Work}
In this paper we demonstrate that the Mask-RCNN model, while primarily designed with object detection, object localization, and instance segmentation of natural images in mind, can be used to produce high quality results for the challenging task of segmentation of nuclei in widely varying microscopy images with very little modification. There are several similar tasks in medical image analysis for which it is likely that a Mask-RCNN based model could easily be adapted to improve performance without extensive modification or customization. Examples of this include the task of segmentation of the left ventricle of the heart, where accurate segmentations can be used to estimate a cardiac patient's ejection fraction and improve their outcomes, or liver and tumor segmentation as described in \cite{liver_segmentation}. Future work will explore the efficacy and performance of Mask-RCNN-based models for a range of such tasks.
\subsubsection*{Acknowledgments}
The author would like to thank NVIDIA Corp. for GPU donation to support this research.
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Hannah Sowerby
Based in Newcastle/Cumbria
Visit Website Email @Hannahsowerby1 Facebook
Hannah is a Cumbrian writer, performer and comedian who is based in Newcastle. She has been writing plays since she was sixteen and is particularly interested in writing comedy and about her home county of Cumbria. Her play "Bridges Apart", set in Cumbria during storm Desmond, was performed at Kendal Yarn's festival of new plays. It was chosen to be part of Best of the Fest (the top six plays) in village Staveley. It was also chosen by a London theatre producer as one in seven out of over one thousand entries, seven short plays ran at the Union Theatre for a week, and also for Playing Up at Northern Stage.
She's had four plays at the Southwark Playhouse with Full Disclosure Theatre Company, Director's Cut Theatre Company and Slackline Stories and a play picked to be part of The Twilight Hour at the Canal Café Theatre. She wrote, directed and performed in a sixteen-night run at the Edinburgh Fringe.
She has been writing and performing comedy sketches for BBC Newcastle's comedy radio show "It's Grin Up North" for over a year and a half. She is half of comedy double act Sowersprouts and has played venues such as The Stand and Alphabetti Theatre. In a bizarre turn of events she has also performed solo stand-up comedy in the shop Poundland.
During Lockdown she has produced a series of six comedy monologues "Sex, Gardening and Contact Lenses" that are available to watch on her YouTube".
Recent work:
SWIPE WRONG - FILM
Send a message to Hannah Sowerby
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Le château de Barjon est un château moderne situé à Barjon (Côte-d'Or) en Bourgogne-Franche-Comté .
Localisation
Le château est situé à la limite sud du village en bordure de plateau d'où il surplombe le cours du Volgrain et la RD 19.
Historique
En 1368, Étienne de Musigny tient la forteresse de Barjon de Jacques de Gransdson, seigneur de Pesmes, En 1413, le château appartient à De Bulligneville et vers 1450 à Nicolas Bouesseau, secrétaire du duc de Bourgogne. En 1500, il y reçoit l'assemblée de la chambre des comptes, pendant l'épidémie de peste à Dijon. Le 10 février 1505, il lègue à son fils Bénigne le fief de Barjon. Le château est ensuite divisé jusqu'en 1609 où Georges Martin de Choisey rachète à sa sœur sa part de propriété. A la fin du , le château est réputé en ruines et on relève un devis de restauration de 1787. Olivier Berger, l'actuel propriétaire, est un passionné des vieilles pierres.
Architecture
Le château actuel est constitué d'un corps de bâtiment dont la façade sud, ouverte sur la vallée, au sud, est bordée d'une terrasse. La façade nord est garnie de deux tours et le pignon oriental d'une tour rectangulaire coiffée d'un toit en pavillon. Elle est flanquée d'une tour semi-circulaire hors-œuvre, garnie de huit canonnières réparties sur trois niveaux.
Notes et références
Annexes
Articles connexes
Liste des châteaux de la Côte-d'Or
Liens externes
Barjon
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A GeekyMomma's Blog: San Jose; Here I Come!
San Jose; Here I Come!
The Innovative Learning Conference (ILC) 2008, will be held October 14-16 at the San Jose Convention Center in San Jose, Calif., and is devoted to sharing innovative educational strategies for advancing K-12 student achievement. Registration is taking place NOW, with an early-bird discount available through Oct. 6. In addition, readers of this blog can get an extra discount of $40 by registering online by Oct. 6 at www.ilc2008.org and typing in the promotional code ORG40. This conference is a "must-attend" professional development opportunity, offering educators, administrators and technology leaders a chance to experience first-hand the latest innovations in classroom technology. More than 150 concurrent sessions and over 60 workshops will provide thought-provoking strategies, applications and practices. ILC 2008 will also feature an expansive exhibit hall where attendees can view the latest products and services from some of the nation's leading technology companies, with more than 100 top solution providers in attendance. For more information on the conference, please visit www.ilc2008.org.
I wish I could be there to cheer you on, Lee - I'm sure you'll do a fantastic job.
Thanks for always being so willing to help others learn. It can be a rough road for some and I'm glad to be fighting the fight with so many quality people.
Yes, I'm a little behind on my reader. Do you have this presentation online anywhere? I shared delicious with some of my staff members a few weeks ago. I don't know why I didn't pick your brain ahead of time.
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New Update to "Enforcement: An International Litigation Guide"
The United Nations Declaration on the Rights of Indigenous Peoples specifically includes the right of indigenous peoples to "maintain, control, protect and develop their intellectual property over their cultural heritage, traditional knowledge, and traditional cultural expressions." Art. 31. Last year marked the ten-year anniversary of that declaration and the end of the first full term of INTA's inaugural Indigenous Rights Committee. Previous iterations of the committee have existed for over a decade, including task forces and subcommittees, as part of other general committees, such as the Emerging Issues and Related Rights Committees. The transformation to a full committee is testimony to INTA's engagement in this important area.
Assisting WIPO
As part of that commitment, in November 2017, INTA provided pro bono assistance to the Traditional Knowledge Division of the World Intellectual Property Organization (WIPO) in its holding of a capacity-building workshop in the Philippines. The workshop was aimed primarily at helping the indigenous peoples and local communities understand the IP system so that they can learn to make more strategic and effective use of IP tools, particularly trademarks.
The WIPO Traditional Knowledge Division supports indigenous and local communities by providing various platforms and tools to facilitate a better understanding of IP options so that these communities can make decisions regarding the proactive protection of their brands on goods and services. It also raises awareness of the defensive action that can be taken to protect names and symbols from misappropriation.
On behalf of INTA, Leslie Anne Cruz (Cesar C. Cruz & Partners Law Offices, the Philippines) participated in a seminar that provided practical legal advice to the indigenous T'Boli community, in response to the challenges they are currently facing in protecting their traditional cultural practices and producing their traditional garments.
The T'Boli Example
The Philippines is home to more than 40 different ethnic groups. Lake Sebu is home to one of the most well-known tribes, the T'Boli or Tagabili. The T'Boli continue to live traditionally, are instantly recognizable by their garments and adornments, and are famous for their complex woven fabrics and intricate beadwork. A traditional, sacred textile known as T'nalak is exchanged during marriages and is used as a covering during childbirth. The unique pattern of this cloth is inspired by the "dreams" of the weavers and is based on traditional knowledge and practices passed down through generations. It is reflective of a rich and diverse cultural heritage interlaced with collective imagination and cultural meaning. The tradition associated with T'nalak is considered to be the most characteristic of the T'Boli people and is synonymous with their natural history and cultural home of Lake Sebu.
Notwithstanding the registration of a collective mark, T'NALAK TAU SEBU, the T'Boli face a number of challenges, including sacrilegious use of their cloth by fashion designers, lack of attribution when authentic cloth is obtained, use and exploitative pricing without any remuneration to the T'Boli people, and the commercial manufacture of T'nalak by others from areas other than Lake Sebu, where the cloth should originate.
The workshop participants included the T'Boli peoples and government and nongovernmental representatives from around the ASEAN region, including various IP offices. Ms. Cruz was able to provide some practical recommendations and answer questions associated with the T'NALAK TAU SEBU collective mark. Her experience highlighted the important role INTA has played in enabling peoples like that T'Boli to effectively exercise their IP rights.
The Challenges for Brands
Just as important as educating indigenous and local communities in the effective use of available IP tools is ensuring brand owner awareness associated with the appropriation of traditional cultural expressions, as exemplified by the T'Boli's concerns.
The challenge for many brand owners is that traditional cultural expressions (TCEs), including names, signs, and symbols, are often accessible and known to the public. It is, therefore, often assumed that these artistic or cultural expressions of indigenous and local communities are available for use without regard to their source. This may lead consumers to mistakenly assume a connection between the business concerned and the relevant indigenous community whose TCEs are being used. Without understanding an indigenous community's traditional ownership of TCEs and focusing only on trademark rights, businesses may end up inadvertently appropriating TCEs, and in some instances, causing potential offense and negative publicity for the brand because of misuse.
Coming Soon: Tips for Avoiding TCE Appropriation
To help brand owners navigate the pitfalls of possible TCE appropriation, the Indigenous Rights Committee, under the guidance of Jern Ern Chuah (Advanz Fidelis Sdn Bhd, Malaysia), has developed some helpful tips for INTA members on a range of matters to be considered when developing brands that may include TCEs. This list will be published under separate cover in an upcoming issue of the INTA Bulletin.
By respecting the intrinsic identity of these connections in developing their brands, brand owners can contribute to the ongoing sustainability of communities and their cultures and positively benefit from the United Nations Declaration on the Rights of Indigenous Peoples.
Although every effort has been made to verify the accuracy of items in the INTA Bulletin, readers are urged to check independently on matters of specific concern or interest. Law & Practice updates are published without comment from INTA except where it has taken an official position.
© 2018 International Trademark Association
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Coreglia Ligure es una localidad y comune italiana de la provincia de Génova, región de Liguria, con 275 habitantes.
Evolución demográfica
Referencias
Enlaces externos
Página web oficial de Coreglia Ligure (en italiano)
Localidades de la provincia de Génova
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{"url":"https:\/\/proofwiki.org\/wiki\/Number_of_Twin_Primes_less_than_Powers_of_10","text":"# Number of Twin Primes less than Powers of 10\n\nFor each $n \\in \\N$, the number of pairs of twin primes below $10^n$ is given by the following sequence:\n$2, 8, 35, 205, 1224, 8169, 58 \\, 980, 440 \\, 312, 3 \\, 424 \\, 506, 27 \\, 412 \\, 679, \\ldots$\n\u2022 1976: Tables Concerning Irregularities in the Distribution of Primes and Twin Primes to $10^{11}$","date":"2019-12-15 13:38:56","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\": 0.9975049495697021, \"perplexity\": 194.60327509503762}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575541308149.76\/warc\/CC-MAIN-20191215122056-20191215150056-00158.warc.gz\"}"}
| null | null |
Impotencija ili erektivna disfunkcija je u širem smislu nesposobnost, nemoć u duhovnom, stvaralačkom, ili nekom drugom smislu. U užem smislu, impotencija je spolna nemoć muškarca za obavljanje spolnog čina zbog organskih ili psiholoških uzroka.
Impotencija se može manifestirati kao potpuna nesposobnost postizanja erekcije (ako ona traje duže od tri mjeseca), ili je njena nekonzistentnost ili sposobnost održavanja erekcije samo kratkotrajna (prolazna).
Uzroci potencije mogu biti: stres na radnom mjestu ili doma, uživanje alkohola, pomanjkanje vitamina, pomanjkanje želje za spolnošću, psihološki razlozi. Češće se javlja kod starijih muškaraca. Najvažniji organski uzroci nemoći su: kardiovaskularna bolest i dijabetes, neurološki problemi, hormonska insuficijencija (hipogonadizam) i nuspojave lijekova. Psihološka impotencija je zbog misli ili osjećaja (psiholoških razloga), a ne od fizičke nemogućnosti; to je nešto rjeđe, ali se često može pomoći. U psihološkoj impotenciji postoji snažan odgovor na placebo liječenje.
Osim liječenja temeljnih uzroka kao što je nedostatak kalija ili zagađenje pitke vode arsenom, prva linija liječenja erektilne disfunkcije sastoji se od tretmana PDE5 inhibitora (kao što je sildenafil). U nekim slučajevima liječenje može uključivati prostaglandinske tablete u uretru, injekcije ili rekonstruktivnu kirurgiju.
Izvori
Seksualnost
Psihijatrija
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In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia
Amy Sheppard, Leon N. Davies
Ophthalmic Research Group
Aston Research Centre for Healthy Ageing
Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia.
Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length.
Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side.
Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass.
Investigative Ophthalmology and Visual Science
https://doi.org/10.1167/iovs.10-5787
Creative Commons Attribution Non-Commercial No Derivatives License
ocular accommodation
biometry
ophthalmological diagnostic techniques
smooth muscle
pilot projects
refractive errors
10.1167/iovs.10-5787
In Vivo Analysis of Ciliary Muscle Morphologic Changes
Final published version, 635 KBLicence: CC BY-NC-ND 3.0
http://www.iovs.org/content/51/12/6882
Dive into the research topics of 'In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia'. Together they form a unique fingerprint.
Refractive Errors Medicine & Life Sciences 100%
Temporal Muscle Medicine & Life Sciences 17%
Optical Coherence Tomography Medicine & Life Sciences 12%
Statistical Factor Analysis Medicine & Life Sciences 10%
Volunteers Medicine & Life Sciences 10%
Nose Medicine & Life Sciences 9%
Analysis of Variance Medicine & Life Sciences 9%
Sheppard, A., & Davies, L. N. (2010). In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia. Investigative Ophthalmology and Visual Science, 51(12), 6882-6889. https://doi.org/10.1167/iovs.10-5787
Sheppard, Amy ; Davies, Leon N. / In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia. In: Investigative Ophthalmology and Visual Science. 2010 ; Vol. 51, No. 12. pp. 6882-6889.
@article{e0ffecf372d24e9cb7f0c502cee5dedb,
title = "In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia",
abstract = "Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia. Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length. Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side. Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass. ",
keywords = "ocular accommodation, adult, biometry, ciliary body, ophthalmological diagnostic techniques, humans, smooth muscle, pilot projects, refractive errors, reproducibility of results, optical coherence tomography, young adults",
author = "Amy Sheppard and Davies, {Leon N.}",
note = "Creative Commons Attribution Non-Commercial No Derivatives License",
doi = "10.1167/iovs.10-5787",
journal = "Investigative Ophthalmology and Visual Science",
publisher = "Association for Research in Vision and Ophthalmology Inc.",
Sheppard, A & Davies, LN 2010, 'In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia', Investigative Ophthalmology and Visual Science, vol. 51, no. 12, pp. 6882-6889. https://doi.org/10.1167/iovs.10-5787
In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia. / Sheppard, Amy; Davies, Leon N.
In: Investigative Ophthalmology and Visual Science, Vol. 51, No. 12, 12.2010, p. 6882-6889.
T1 - In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia
AU - Sheppard, Amy
AU - Davies, Leon N.
N1 - Creative Commons Attribution Non-Commercial No Derivatives License
N2 - Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia. Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length. Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side. Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass.
AB - Purpose. To use anterior segment optical coherence tomography (AS-OCT) to analyze ciliary muscle morphology and changes with accommodation and axial ametropia. Methods. Fifty prepresbyopic volunteers, aged 19 to 34 years were recruited. High-resolution images were acquired of nasal and temporal ciliary muscles in the relaxed state and at stimulus vergence levels of -4 and -8 D. Objective accommodative responses and axial lengths were also recorded. Two-way, mixed-factor analyses of variance (ANOVAs) were used to assess the changes in ciliary muscle parameters with accommodation and determine whether these changes are dependent on the nasal–temporal aspect or axial length, whereas linear regression analysis was used to analyze the relationship between axial length and ciliary muscle length. Results. The ciliary muscle was longer (r = 0.34, P = 0.02), but not significantly thicker (F = 2.84, P = 0.06), in eyes with greater axial length. With accommodation, the ciliary muscle showed a contractile shortening (F = 42.9. P < 0.001), particularly anteriorly (F = 177.2, P < 0.001), and a thickening of the anterior portion (F= 46.2, P < 0.001). The ciliary muscle was thicker (F = 17.8, P < 0.001) and showed a greater contractile response on the temporal side. Conclusions. The accommodative changes observed support an anterior, as well as centripetal, contractile shift of ciliary muscle mass.
KW - ocular accommodation
KW - adult
KW - biometry
KW - ciliary body
KW - ophthalmological diagnostic techniques
KW - humans
KW - smooth muscle
KW - pilot projects
KW - refractive errors
KW - reproducibility of results
KW - optical coherence tomography
KW - young adults
UR - http://www.iovs.org/content/51/12/6882
U2 - 10.1167/iovs.10-5787
DO - 10.1167/iovs.10-5787
JO - Investigative Ophthalmology and Visual Science
JF - Investigative Ophthalmology and Visual Science
Sheppard A, Davies LN. In vivo analysis of ciliary muscle morphologic changes with accommodation and axial ametropia. Investigative Ophthalmology and Visual Science. 2010 Dec;51(12):6882-6889. Epub 2010 Jul 29. doi: 10.1167/iovs.10-5787
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
DNA methylation is a well studied, heritable epigenetic modification which plays an important role in gene regulatory mechanisms. It is associated with a broad range of biological processes of direct clinical relevance, including X-chromosome inactivation, genomic imprinting, silencing of repetitive DNA and carcinogenesis \citep{Feinberg1983, Li1993, Baylin2011}. Methylation occurs when a methyl group is attached to a DNA nucleotide. In vertebrate genomes, methylation is observed almost exclusively on 5-methylcytosine (5-mC) residues in the context of CpG dinucleotides. Due to increased vulnerability of 5-mC to randomly deaminate into thymine, most of the genome is depleted from CpG dinucleotides, except from small CpG-rich regions, termed CpG islands (CGIs) \citep{Bird2002}. Hyper-methylation of CGIs near promoter regions is generally associated with transcriptional repression; however, outside of this well documented case, the association between DNA methylation across promoter-proximal regions and transcript abundance is considerably weaker and poorly understood \citep{Jones2012}.
Recent advances in high-throughput sequencing technology have made it possible to measure the methylation level of cytosines on a genome-wide scale with single nucleotide resolution. Sodium bisulphite treatment of DNA followed by sequencing (BS-seq) efficiently converts unmethylated cytosines to uracils (which are subsequently amplified as thymines by PCR) and leaves the 5-mCs unmodified \citep{Krueger2012}. To obtain DNA methylation levels, reads are aligned to a reference genome allowing changes of cytosines to thymines during the mapping procedure. A variant of BS-seq technology, termed Reduced Representation Bisulphite Sequencing (RRBS) \citep{Meissner2005}, uses methylation-sensitive restriction enzymes to cleave the DNA at specific loci before bisulphite treatment. This results in measuring in greater coverage and at lower cost the methylation level of CpG-rich regions genome-wide.
Despite the widespread take up of BS-seq technology, statistical modelling of such data is still challenging, yet it is crucial in order to uncover biological regulatory mechanisms. Analysis of BS-seq data has mainly focused on identifying differentially methylated regions (DMRs) across different conditions. Some notable DMR methods are BSmooth \citep{Hansen2012}, Bi-Seq \citep{Hebestreit2013} and $M^{3}D$ \citep{Mayo2015}. While DMR detection methods are often crucial ingredients in exploratory data analysis pipelines, they do not provide a clear platform to quantitatively understand the relationship between DNA methylation and gene expression. Most studies use DMR detection as a prefilter, and then simply correlate mean methylation levels across each region (often taken to be promoter-proximal) with gene expression. Adopting this simple approach, genome-wide studies \citep{Hansen2011, Bock2012a} have reported only modest correlation between average DNA-methylation and gene expression {(Pearson's correlation coefficient $r \approx$ -0.3)}.
In this paper, we argue that part of the difficulty in quantitatively associating methylation levels with gene expression resides in the simplistic encoding of DNA methylation across a region as a simple average. DNA methylation often displays reproducible, spatially correlated patterns ({\it profiles}); Figure~\ref{fig:methylation-profiles} shows two examples from an ENCODE data sets \citep{Dunham2012}. This spatial reproducibility was exploited by \cite{Mayo2015} to provide more powerful tests for DMR, and by \cite{Vanderkraats2013} to group genes with similar differential methylation patterns and corresponding expression changes. These results suggest that a precise quantification of the spatial variability in the DNA methylation mark may aid the quest to quantitatively understand the interplay between methylation and transcription. We propose a probabilistic model of methylation profiles, based on latent variable models, which allows us to associate with each region of interest a set of features capturing precisely the methylation profile across the region. We then show that, using such features, we can construct an accurate machine learning predictor of gene expression from DNA methylation, achieving test correlations twice as large as previously reported.
\begin{figure}[!pb]
\begin{center}
\begin{minipage}{0.39\textwidth}
\includegraphics[width=\textwidth]{figures/plekhh3-2}
\end{minipage
\hspace*{0.8pt}
\begin{minipage}{0.39\textwidth}
\includegraphics[width=\textwidth]{figures/ccr10-2}
\end{minipage}
\caption{\small{Methylation patterns for the PLEKHH3 and CCR10 genes from the K562 cell line over $\pm7kb$ promoter region. Each point represents the relative CpG location w.r.t. TSS and the corresponding DNA methylation level. The dashed horizontal lines show the average methylation level. The shapes of methylation profiles are very different, however the average methylation level cannot explain them. Also, note that there are no CpG measurements in the (-$6kb$, -$4kb$) region for the CCR10 gene, and the learned methylation profiles can be thought as imputing the missing values by taking into consideration the spatial co-dependence of nearby CpGs.}}
\label{fig:methylation-profiles}
\end{center}
\end{figure}
The rest of the paper is organised as follows: we start off by providing a high-level description of our approach. We then describe precisely the statistical methodology we propose. We illustrate our approach on three ENCODE data sets, showing that higher order features allow much more accurate predictions of gene expression. We also show how such features can be used to cluster regions according to their methylation profiles, and show that five prototypical methylation profiles appear to explain most variability in promoter-proximal methylation in human cell lines.
\section{Approach}
In this paper, we propose a novel probabilistic machine learning methodology to quantify the profile of DNA methylation across genomic regions from BS-seq data. Our motivation is practical: inspection of many BS-seq data sets reveals that methylation levels across promoter-proximal regions often show reproducible, spatially correlated profiles. Figure~\ref{fig:methylation-profiles} shows two example promoter-proximal regions which clearly display such spatial correlations, resulting in characteristic methylation shapes. We propose a method to quantitate such qualitative information.
The method is based on a Generalised Linear Model of basis function regression coupled with a Binomial observation likelihood, and allows us to associate each region with a set of basis function coefficients which capture the methylation profile. We show how such higher order features can then be used in downstream analysis to yield a significantly improved estimate of the correlation between methylation and gene expression, and to identify prototypical methylation profiles across promoter regions.
\section{Methods}
\subsection{Modelling DNA methylation profiles}
As in all HTS-based assays, the output of a BS-seq experiment is a set of reads aligned to the genome; the main difference is that the bisulphite treatment changes to thymine any unmethylated cytosine. Thus, the same base on the genome will appear as cytosine on some reads, and as thymine on others; the ratio of reads containing a cytosine readout to total reads gives a measurement of the sample methylation level. This measurement process at a single cytosine can be naturally modelled with a Binomial distribution, where the number of successes represents the number of reads on which the cytosine actually appears as C, and the number of attempts is the total number of reads mapping to the specific site. Let \emph{t} be the total number of reads that are mapped to a specific CpG site, and let \emph{m} of these reads to contain methylated cytosines. Then, for each CpG site we assume that $m \sim \mcal{B}inom(t,p)$, where $p$ is the unknown methylation level.
In this paper, and in many practical studies, we are interested in learning the methylation patterns of fixed-width genomic regions, e.g. promoters. Hence, each genomic region $i (i = 1, \ldots, N)$ can be represented as a vector of CpG locations $\mvec{x}_{i}$, where each entry corresponds to the location of the CpG in the genomic region, relative to a reference point such as the Transcription Start Site (TSS). It should be noted that the vector lengths $L_{i}$ may vary between different genomic regions, since they depend on the number of actual CpG dinucleotides found in each region. For each region $i$, we also have a vector of observations $\mvec{y}_{i}$, containing the methylation levels of the corresponding CpG sites; each entry consists of the tuple $y_{il} = (m_{il}, t_{il})$, where, $m_{il}$ is the number of 5-mC reads mapped to the $l$-th CpG site in region $i$, and $t_{il}$ corresponds to the total number of reads.
Direct comparison of the observation vectors $\mvec{y}_{i}$ for different regions is complicated due to the variability in the vector lengths. To enable comparisons between these regions, we formulate our problem as a regression problem, where the methylation profile of each genomic region is modelled as a linear combination of a set of latent basis functions. Let $f(\mvec{x}_{i})$ be a latent function representing the methylation profile for genomic region $i$. Since the observed methylation data contain the proportion of methylated reads out of the total reads for each CpG site, each entry of the vector $\mvec{y}_{i}$ takes values in the $[0, 1]$ interval. Thus, we introduce an unconstrained latent function $g(\mvec{x}_{i})$ defined so that $f(\mvec{x}_{i})$ is the probit transformation of $g(\mvec{x}_{i})$: $f(\mvec{x}_{i}) = \Phi \big(g(\mvec{x}_{i})\big)$, where $\Phi(\cdot)$ denotes the cumulative distribution function (cdf) of the standard normal distribution. Let $\mvec{f}_{i} = f(\mvec{x}_{i})$ and $\mvec{g}_{i} = g(\mvec{x}_{i})$ be shorthand for the values of the latent functions.
Given the values of the latent function $\mvec{f}_{i}$ for region $i$, the observations $y_{il}$ for each CpG site are independent and identically distributed Binomial variables, so we can define the joint log-likelihood for region $i$ in factorised form:
\begin{equation}\label{eq:bpr-likelihood}
\log p(\mvec{y}_{i} | \mvec{f}_{i}) = \sum\limits_{l = 1}^{L_{i}} \log \bigg\lbrace \mcal{B}inom\big(m_{il} | t_{il}, \Phi(g_{il})\big) \bigg\rbrace
\end{equation}
From its final form, we refer to this observation model as the Binomial distributed Probit Regression (BPR) likelihood function. Notice that the BPR model explicitly accounts for the coverage variability across CpG sites through the use of the Binomial observation model: as the variance of a binomial distribution decreases rapidly with the number of attempts, the model will be very strongly constrained by highly covered sites. Hence, it handles in a principled way the uncertainty present in low coverage reads during the analysis of BS-seq data.
\subsection{Feature Extraction}
To constrain the latent function $\mvec{g}_{i}$ we assume it is given as a linear combination of fixed non-linear basis functions $h_{j}(\cdot)$ of the input space $\mvec{x}_{i}$, of the form:
\begin{equation}\label{eq:linear-comb-basis-func}
\mvec{g}_{i}(\mvec{x}_{i}, \mvec{w}_{i}) = \sum\limits_{j = 0}^{M-1} w_{j} h_{j}(\mvec{x}_{i}) = \mvec{H}_{i} \mvec{w}_{i}
\end{equation}
where $\mvec{w}_{i} = (w_{i,0}, \ldots, w_{i,M-1})^T$, $\mvec{H}_{i}$ is the $L_{i} \times M$ design matrix, whose elements are given by $\mvec{H}_{ilj} = h_{j}(x_{il})$, and $M$ denotes the total number of basis functions. Hence, its probit transformation $\mvec{f}_{i}$ is given by:
\begin{equation}
\mvec{f}_{i}(\mvec{x}_{i}, \mvec{w}_{i}) = \Phi\big(\mvec{g}_{i}(\mvec{x}_{i}, \mvec{w}_{i})) = \Phi\big(\mvec{H}_{i} \mvec{w}_{i}\big)
\end{equation}
One should note that even though the function $\mvec{g}_{i}$ is linear with respect to the parameters $\mvec{w}_{i}$, the latent function $\mvec{f}_{i}$ is non linear due to the presence of the probit transformation. In this study, we consider Radial Basis Functions (RBFs); for a single input variable $x$, the RBF takes the form $h_{j}(x) = exp(-\gamma || x - \mu_{j} ||^2)$, where $\mu_{j}$ denotes the location of the $j^{th}$ basis function in the input space and $\gamma$ controls the spatial scale.
Learning the methylation profiles $\mvec{f}_{i}$ for each genomic region, is equivalent to optimising the model parameters $\mvec{w}_{i}$. The parameters $\mvec{w}_{i}$ can be considered as the extracted features which quantitate precisely notions of shape of a methylation profile. Optimising $\mvec{w}_{i}$ involves maximising Equation~(\ref{eq:bpr-likelihood}) for each genomic region; however, by increasing the number of basis functions, we also increase the resolution for the shape of the methylation profiles, which might lead to overfitting. To ameliorate this issue, we maximise a penalised version of Equation~(\ref{eq:bpr-likelihood}), by adding a regularisation term $\mcal{E}(\mvec{w}_{i})$ to the log-likelihood function which will encourage the weights to decay to zero:
\vspace*{3pt}
\begin{equation}\label{eq:bpr-lik-penalized}
J(\mvec{w}_{i}) = \log p(\mvec{y}_{i} | \mvec{f}_{i}, \mvec{w}_{i}) - \mcal{E}(\mvec{w}_{i})
\end{equation}
where $\mcal{E}(\mvec{w}_{i}) = \frac{1}{2} \mvec{w}^{T}_{i}\mvec{w}_{i}$ is the squared two-norm.
This approach is known as ridge regression or weight decay.
Direct maximisation of $J(\mvec{w}_{i})$ w.r.t parameters $\mvec{w}_{i}$ is intractable due to presence of the probit transformation. We use gradient-based numerical optimisation techniques, such as Conjugate Gradient (CG) and BFGS, to perform the optimisation.
\subsection{Predicting gene expression}
The extracted higher-order methylation features across promoter-proximal regions can be used for downstream analysis, such as predicting transcript abundance, or performing clustering in order to learn prototypical methylation patterns that occur at promoters across different cell lines.
To quantitatively predict expression at each promoter region, we construct a regression model by taking as input the higher-order methylation features extracted from each promoter-proximal region. The performance of the regression model is evaluated by computing the root-mean squared error (RMSE) and the Pearson's correlation coefficient (r) between the predicted and the measured (log-transformed) gene expression levels. We compare our proposed model's performance with the standard approach \citep{Hansen2011, Bock2012a} which uses the average methylation level across a region as input feature (this approach can be thought of as fitting a constant function across each genomic region). We have tested both a linear regression model and a variety of non-linear models, such as SVM regression, Random Forests and Multivariate Adaptive Regression Splines (MARS) \citep{Friedman1991}. The SVM regression is consistently better than the other regression models, hence, we choose this model for the rest of our analysis.
In addition to the methylation profile features, we consider two supplementary sources of information which could plausibly act as confounders in the predictions. The first feature accounts for the goodness of fit of each methylation profile to the observed methylation data using the RMSE as error measure, intuitively quantitating the noisiness in the methylation profile. The second feature considers the number of CpG dinucleotides present in each promoter region. It is thought that CpG density may play a functional role in controlling gene expression, with the main evidence being the existence of CpG islands \citep{Deaton2011}.
\subsection{Clustering methylation profiles}
To cluster methylation profiles we consider a mixture modelling approach \citep{McLachlan2004}. We assume that the methylation profiles $\mvec{f}$ can be partitioned into at most K clusters, and each cluster $k$ can be modelled separately using the BPR likelihood as our observation model. The log-likelihood for the mixture model is defined as:
\begin{equation}\label{eq:mixture-model}
p(\mvec{y} | \msvec{\Theta}) = \sum\limits_{i = 1}^{N} \log \bigg\lbrace\sum_{k = 1}^{K} \pi_{k} p(\mvec{y}_{i} | \mvec{f}_{i}, \mvec{w}_{k}, z_{i} = k)\bigg\rbrace
\end{equation}
where $\msvec{\Theta} = (\pi_{1}, \ldots , \pi_{k}, \mvec{w}_{1}, \ldots, \mvec{w}_{k})$, $\pi_{k}$ are the mixing proportions (with $\pi_{k} \in (0,1) \; \forall k$ and $\sum_{k}\pi_{k} = 1$), $\mvec{w}_{k}$ are the methylation profile parameters and $z_{i}$ are the latent variables denoting to which cluster each genomic region belongs. To avoid cluttering the notation, we will omit the dependence of the observation model on the latent variables $z_{i}$.
\subsubsection*{Parameter Estimation}
To estimate the model parameters $\msvec{\Theta} = (\pi_{1}, \ldots , \pi_{k}, \mvec{w}_{1}, \ldots, \mvec{w}_{k})$, the Expectation Maximization (EM) algorithm \citep{Dempster1977} is considered. EM is a general iterative algorithm for computing maximum likelihood estimates when there are missing or latent variables, as in the case of mixture models. EM alternates between inferring the latent variables given the parameters (E-step), and optimising the parameters given the posterior statistics of the latent variables (M-step). Formally, during the E-step we compute the responsibility that component $k$ takes for explaining observations $\mvec{y}_{i}$:
\begin{equation} \label{eq:compute-resp}
\gamma(z_{ik}) = \myfrac[4pt]{\pi_{k} p(\mvec{y}_{i} | \mvec{f}_{i}, \mvec{w}_{k})}{\sum_{j = 1}^{K} \pi_{j} p(\mvec{y}_{i} | \mvec{f}_{i}, \mvec{w}_{j})}
\end{equation}
The M-step consists of updating the model parameters so as to maximise the expected complete data log-likelihood. The mixing proportions $\pi_{k}$ are updated as follows:
\begin{equation} \label{eq:update-mix-prop}
\pi_{k} = \frac{1}{N}\sum\limits_{i = 1}^{N} \gamma(z_{ik})
\end{equation}
To re-estimate the observation model parameters $\mvec{w}_{k}$ we need to optimise the following quantity:
\begin{equation} \label{eq-em-opt}
\ell(\mvec{w}_{k}) = \sum_{i} \gamma(z_{ik}) \sum_{l} \log \bigg\lbrace \mathcal{B}inom \big(m_{il} | t_{il}, \Phi(g_{il}; \mvec{w}_{k})\big) \bigg \rbrace
\end{equation}
However, direct optimisation of $\ell(\mvec{w}_{k})$ w.r.t parameters $\mvec{w}_{k}$ is intractable, thus, we resort again to numerical optimisation strategies. This variant of EM algorithm is known as Generalised EM, or GEM, and it is proved to converge to the maximum likelihood estimate \citep{Wu1983}. It should be noted that the penalised version of the BPR likelihood, given in Equation (\ref{eq:bpr-lik-penalized}), can be easily incorporated in the clustering approach.
\section{Data Sets}
To evaluate the performance of the proposed methodology we use real datasets that are publicly available from the ENCODE project consortium \citep{Dunham2012}. More specifically, the following three Tier 1 cell lines are used:
\begin{enumerate}
\item{K562 immortalized cell line, coming from a human female with chronic myelogenous leukemia.}
\item{GM12878 lymphoblastoid cell line, produced from the blood of a female donor with northern and western European ancestry by EBV transformation.}
\item{H1-hESC embryonic stem cells, coming from a human male.}
\end{enumerate}
The RRBS data for all three cell lines are produced by the Myers Lab at HudsonAlpha Institute for Biotechnology (GEO: GSE27584). The data are already pre-processed and aligned to the \textit{hg19} human reference genome, and can be downloaded from the web accessible database at UCSC. For our analysis, we use the resulting BED files and we ignore strand information. To obtain more accurate methylation level estimates, we pool together all available replicates. To investigate the correlation between DNA methylation profiles and transcript abundance, we use the corresponding paired-end RNA-seq data produced by Caltech (GEO: GSE33480). The RNA-seq data are pre-processed and mapped to the \textit{hg19} human reference genome using TopHat and transcription quantification, in FPKM (Fragments Per Kilobase transcript per Million mapped reads), is produced using Cufflinks \citep{Trapnell2012}. The RNA-seq data are filtered in order to keep only protein-coding genes.
To define promoter regions, we extract the TSS from the corresponding RNA-seq data, which are annotated based on both versions v3c and v4 of GENCODE GRCh37. Then, we consider N base pairs upstream and downstream from each TSS, resulting in promoter regions of length 2N base pairs. Since the cell lines are coming from different genders, the sex chromosomes are discarded from further analysis.
\section{Results}
\subsection{Methylation profiles are highly correlated with gene expression}
Initially, we examine whether gene expression levels might be predictable from DNA methylation patterns alone. We therefore extract higher-order features from promoter regions of $\pm 7kb$ around the TSS by learning the corresponding methylation profiles using the BPR observation model. To ensure that the promoter-proximal regions will have enough data to learn reasonable methylation profiles, we discard regions with less than $15$ CpGs, and restrict our attention to regions which exhibit spatial variability in methylation levels. We applied the same pre-processing steps for the three ENCODE cell lines, which resulted in $7093$ promoters for K562, $6022$ for GM12878 and $5753$ for H1-hESC cell line.
We model the methylation profiles using nine RBFs, which results in ten extracted features including the bias term. In addition to these features, we use the goodness of fit in RMSE and the CpG density across each region. We then train the SVM model on the resulting 12 features using a random subset of $70\%$ of the promoter-proximal regions. We test the model's ability to quantitatively predict expression levels on the remaining $30\%$ of the data. Our results show a striking improvement in prediction accuracy when compared to using the mean methylation level as input feature.
\begin{figure}[!tpb]
\centerline{\includegraphics[width=0.78\textwidth]{figures/k562-scatter}}
\caption{\small{Quantitative relationship between DNA methylation patterns and expression. (\textbf{A}) Scatter plot of predicted gene expression using the BPR model on the x-axis versus the measured (log-transformed) gene expression values for the K562 cell line on the y-axis. Each methylation profile is modelled using nine RBFs. In addition to these features, the SVM regression model uses as input the goodness of fit in RMSE and the CpG density. Each shaded blue dot represents a different gene and the darker the colour, the higher the density of points. The red dashed line indicates the linear fit between the predicted and measured expression values, which are highly correlated (Pearson's $r = 0.7$, p-value $<$ 2.2e-16), indicating a quantitative relationship between methylation profiles across promoter-proximal regions and transcript abundance. The model performance is also assessed by RMSE, which is 2.63. (\textbf{B}) Scatter plot of predicted and measured gene expression values when using the average methylation level as input feature in the SVM model; correlation has decreased substantially ($r$ = 0.31 and RMSE = 3.52).}}
\label{fig:k562-scatter}
\end{figure}
Figure~\ref{fig:k562-scatter}A shows a scatter plot of the predicted and measured expression values for the K562 cell line, with Pearson's $r$ = 0.7 (p-value of t-test $<$ 2.2e-16) and RMSE = 2.63, demonstrating that the shape of methylation patterns across promoter-proximal regions is well correlated to mRNA abundance. Figure~\ref{fig:k562-scatter}B shows the performance of the regression model when using the mean methylation level as input feature. It is evident that this approach cannot capture the diverse patterns present across the promoter regions, leading to poor prediction accuracy ($r$ = 0.31 and RMSE = 3.52). Notice that the mean methylation approach erroneously predicts gene expression values only in the (-2, 4) interval, whereas the BPR model captures more accurately the dynamic range of expression. Interestingly, the mean approach erroneously predicts the majority of genes to have expression value around -1, clearly indicating that summarising DNA methylation by a single average is insufficient to capture the complex relationship with expression. Finally, one should observe the horizontal stripe around -3 on both figures: these are genes whose lack of expression cannot be attributed to DNA methylation patterns, possibly implicating other regulating mechanisms (e.g. histone marks, binding of transcription factors, etc.), or difficulties in the measurement process of RNA-seq experiments (e.g. due to genes having relatively non-unique transcript sequences or multiple promoters).
We then consider the relative importance of the various features in predicting gene expression: in particular, we are interested in determining whether including goodness of fit or CpG density as covariates has any impact on predictive performance. For each cell line, we learn five SVM regression models, each having a different number of input features. The first four models consider as input the extracted higher-order methylation features with a combination of the two additional features we described in the previous section, whereas the last model takes the average methylation level as input feature. To statistically assess our results, we perform 20 random splits in training and test sets and evaluate the model performance on the corresponding test sets. Figure~\ref{fig:model-performance} shows boxplots of correlation coefficients for the three ENCODE cell lines, where each boxplot indicates the performance of the prediction model on the 20 random splits of the data. The results demonstrate that by considering higher-order features we can build powerful predictive models of gene expression; and in the case of K562 and GM12878 we have more than 2-fold increase in correlation.
\begin{table}[!b]
\begin{center}
{\begin{tabular}{@{}c|cccccccc@{}}\toprule
Cell Line & {$\pm2kb$} & {$\pm3kb$} & {$\pm4kb$} & {$\pm5kb$} & {$\pm6kb$} & {$\pm7kb$} & {$\pm8kb$} & {$\pm9kb$}\\\midrule
K562 & 0.63 & 0.69 & 0.69 & 0.67 & 0.67 & \textbf{0.70} & 0.67 & 0.67 \\
GM12878 & 0.62 & 0.62 & 0.64 & 0.61 & 0.62 & \textbf{0.61} & 0.61 & 0.61 \\
H1-hESC & 0.46 & 0.49 & 0.48 & 0.43 & 0.49 & \textbf{0.50} & 0.47 & 0.49 \\\bottomrule
\hline
\end{tabular}}
\caption{Pearson's $r$ when considering different promoter region windows. \small{For various length promoter-proximal regions, we show the performance (in Pearson's $r$) of methylation profiles in accurately predicting gene expression. The BPR model has high correlation across all different-length regions for all cell lines considered in this study.}}
\label{Tab:01}
\end{center}
\end{table}
Concentrating on the importance of the additional features for the prediction process, we observe that the addition of CpG density does not have a significant prediction improvement compared to using only the shape of methylation profiles as input features (paired Wilcoxon test p-value = 0.22, 0.18 and 0.02 for K562, GM12878 and H1-hESC, respectively). On the other hand, the goodness of fit of the methylation profile in RMSE has a positive impact on the prediction performance (paired Wilcoxon test p-value = 4.8e-05, 4.8e-05 and 0.0001 for K562, GM12878 and H1-hESC, respectively). Finally, we explore the importance of considering different promoter region windows. Table~\ref{Tab:01} shows Pearson's $r$ when considering various length promoter-proximal regions around the TSS. In general, the BPR model maintains its high predictive power across all cell lines for all different-length regions.
\begin{figure}[!tpb]
\centerline{\includegraphics[width=0.79\textwidth]{figures/model-corr-boxplot}}
\caption{\small{Boxplot of correlation coefficients for the three ENCODE cell lines (K562, GM12878 and H1-hESC) with different input features for the SVM regression. The 'Profile full' model corresponds to the extracted BPR features plus the two additional features. Each boxplot indicates the performance using 20 random splits of the data in training and test sets. Paired Wilcoxon test shows that the high quantitative relationship between the shape of DNA methylation and expression exists in various cell lines, and is significantly better predictor than using the average methylation level (p-value = 8.4e-12). Regarding the two additional features, we observe that the goodness of fit measured in RMSE has a positive impact in correlation, whereas the CpG density does not improve the prediction performance. Paired Wilcoxon tests between K562 and other cell lines, show that K562 has significantly higher prediction accuracy (p-value = 4.8e-05 for both GM12878 and H1-hESC).}}
\label{fig:model-performance}
\end{figure}
\subsection{Methylation profiles are predictive of gene expression across different ENCODE cell lines}
We showed that gene expression is effectively predicted from the BPR model by using higher-order methylation features among various cell lines. Next, we further explore if the proposed model maintains predictive power across different cell lines. That is, we apply the regression model trained on one cell line to predict expression levels in another cell line, by using the learned methylation profiles in those cell lines as input features to the regression model. Figures~\ref{fig:cross-cell-line}A-B show confusion matrices of correlation coefficients for the cross-cell line prediction process, using the BPR model and the mean methylation level approach, respectively. Figure~\ref{fig:cross-cell-line}C shows an example of applying the model learned from GM12878 methylation patterns to predict expression levels of the K562 cell line. The BPR model effectively predicts gene expression ($r$ = 065 and 0.49 for predicting K562 and H1-hESC, respectively), while, the mean methylation approach provides a poor estimate of correlation ($r$ = 0.28 and 0.22 for predicting K562 and H1-hESC, respectively).
\begin{figure}[!tp]
\centerline{\includegraphics[width=0.75\textwidth]{figures/corr-across-cell-lines}}
\caption{\small{Prediction accuracy across different cell lines. (\textbf{A}) Confusion matrix of correlation coefficients across cell-lines when using the BPR model with nine RBFs as input features to the regression model. Each $(i,j)$ entry of the confusion matrix, corresponds to training a regression model from $i^{th}$ cell line and predicting gene expression levels for the $j^{th}$ cell line. The colour of the confusion matrix corresponds to Pearson's $r$ value, the darker the colour the higher the correlation. (\textbf{B}) The corresponding correlation coefficients when using the mean methylation level as input feature to the regression model. Comparing both confusion matrices, it is evident that the methylation profile approach is more powerful in predicting expression levels across different cell lines. (\textbf{C}) Application of the model learned from GM12878 cell line to predict expression levels of the K562 cell line, using methylation profiles (top) and mean methylation levels (bottom) as input features.}}\label{fig:cross-cell-line}
\end{figure}
The results indicate that the quantitative relationship between DNA methylation profiles and mRNA abundance is not cell line specific, but that the model captures patterns of association between methylation and expression which hold across different cell lines. Although the proposed models have high prediction accuracy across all cell lines, the H1-hESC cell line shows consistently weaker correlations. This finding is in line with recent studies, reporting weaker correlations of gene expression and chromatin features for the H1-hESC cell line \citep{Dong2012}, and with observations that mRNA-encoding genes in stem cells are transcriptionally paused during cell differentiation \citep{Min2011}.
\subsection{Clustering DNA methylation profiles across promoter - proximal regions}
We next use the higher order methylation features to cluster DNA methylation patterns across promoter-proximal regions and examine whether distinct methylation patterns across different cell lines are associated to gene expression levels. We apply the same pre-processing steps described in the previous sections and we consider genomic regions of $\pm 7$kb around the TSS. We use the Bayesian Information Criterion (BIC) to set the number of clusters to five. We model the methylation profiles at a slightly lower spatial resolution, using four RBFs, as we are interested in capturing broader similarities between profiles, rather than fine details. Figure~\ref{fig:clusters}A shows the five distinct methylation profiles that were learned from each cell line after applying the EM algorithm. To investigate the association of promoter methylation profiles and transcription, in Figure~\ref{fig:clusters}B we show boxplots with the corresponding mRNA expression values that are assigned to each cluster for each cell line. From the resulting methylation profile clusters, we seek to characterize the common features that are responsible for the corresponding mRNA abundance.
As expected, clusters corresponding to hyper-methylated regions (Cluster 4, green) are associated with repressed genes across all cell lines, confirming the known relationship of DNA methylation around TSSs with gene repression. Also, two distinct patterns emerge: an S-shape profile (Cluster 5, yellow) with hypo-methylated CpGs upstream of TSS, which become gradually methylated at the gene body, and the reverse S-shape pattern (Cluster 3, orange). Genes associated with these profiles have intermediate expression levels for K562 and GM12878, and relatively high expression for H1-hESC. The most interesting pattern is the U-shape methylation profile (Cluster 2, blue), with a hypo-methylated region around the TSS surrounded by hyper-methylated domains. These profiles are associated with high transcriptional activity at their associated genes across all cell lines (t-test p-value $<$ 2.2e-16 for all paired cluster comparisons across cell lines). Surprisingly, uniformly low-methylated domains (Cluster 1, red) seem in general to be repressed, except from the H1-hESC cell line, suggesting a different type of relationship between DNA methylation and expression in embryonic stem cells. The clustering analysis confirms, in a complementary way, that DNA methylation profiles and transcriptional process are tightly connected to each other, and this relationship can be generalized across all cell lines considered in this study.
\begin{figure*}[!tpb]
\centerline{\includegraphics[width=0.99\textwidth]{figures/cluster-profiles}}
\caption{\small{Clustering DNA methylation profiles across promoter-proximal regions. (\textbf{A}) Five clustered methylation profiles over $\pm 7$kb promoter region w.r.t. TSS in the direction of transcription for the three ENCODE cell lines (K562, GM12878 and H1-hESC). Each methylation profile is modelled using four RBFs. Comparing the clustered profiles it is evident that there are five prototypical methylation shapes across the cell lines. (\textbf{B}) Boxplots with the corresponding expression levels of the protein-coding genes assigned to each cluster for each of the three cell lines. The colours match with the clustered methylation profiles shown above. The numbers below each boxplot correspond to the total number of genes asssigned to each cluster. T-test shows that the U-shape methylation profiles (Cluster 2, blue) correspond to significantly higher expression (p-value $<$ 2.2e-16) compared to the expression of genes assigned to the remaining methylation profiles.}} \label{fig:clusters}
\end{figure*}
To provide a biological insight on the potential methylation mechanisms that regulate transcription, we consider the purity of the clustering across different cell lines, i.e., which fraction of genes assigned to a certain cluster in a certain cell line are assigned to the same cluster in the other cell lines. Surprisingly, around 68\% of the genes assigned to the U-shape profile are present in all three cell lines, while the intersection of genes assigned to the other clusters ranges between 20\% to 40\%. Interestingly, the promoter-proximal regions clustered to the U-shape methylation profile are dominated by CGIs. Of all common promoters assigned to the U-shape profiles, 95.6\% are CGI associated. Not surprisingly, hyper-methylated promoters are only 35.7\% CGI associated, however uniformly low-methylated promoters are 65.9\% CGI associated. This suggests, that promoters associated with totally unmethylated CGIs surrounded by hyper-methylated domains are transcriptionally active across cell lines. Indeed, we find that 35\% of the U-shape profile genes are associated with a curated set of housekeeping genes \citep{Eisenberg2013}. On the contrary, only a small fraction of genes assigned to hyper-methylated domains or uniformly low-methylated domains are housekeeping genes (1.4\% and 17.7\% respectively). Finally, around 22\% of the genes assigned to the S-shape and reverse S-shape profiles are associated with housekeeping genes.
\section{Discussion}
Alterations in DNA methylation are associated with regulatory roles and are involved in many diseases, most notably cancer \citep{Baylin2011}. Therefore, unravelling the function of DNA methylation and its relationship to transcription, is essential for understanding biological processes and for developing biomarkers for disease diagnostics \citep{Laird2003}.
Our results demonstrate that representing methylation patterns by their average level is insufficient to understand the link between DNA methylation and expression, and one should consider the shape of the methylation profiles at the vicinity of the promoters. The contributions of this paper are twofold. First, we introduced a generic modelling approach to quantitate spatially distributed methylation profiles via the BPR model. The BPR features enabled us to build a powerful predictive model for gene expression in various cell lines which more than doubled the predictive accuracy of current methods based on average methylation levels.
Second, we have shown how the BPR features can be used in downstream analyses by clustering spatially similar methylation profiles. We revealed five distinct groups of methylation patterns across promoter regions that are well correlated with gene expression and are well reproducible across different cell lines. Some of these patterns recapitulate existing biological knowledge. The U-shape methylation profile, consisting of hypo-methylated CGIs followed by hyper-methylated CGI shores, has been identified in different studies, and is termed as 'canyon' \citep{Jeong2014} or 'ravine' \citep{Edgar2014}. Our findings are in line with \cite{Edgar2014}, where ravines are in general positively correlated with mRNA abundance. Since, the main difference of the U-shape methylation profile and the uniformly low-methylated profile is the CGI shore methylation, our results support the hypothesis that hyper-methylation on the edges of CGIs enhances transcriptional activity.
The existence of U-shape methylation profiles may help to explain observations that the methylation of gene body was sometimes positively correlated with transcript abundance \citep{Varley2013, Lou2014}. We hypothesize that these regions may correspond to U-shape methylation profiles, or a mixture of U-shape and S-shape methylation profiles. Another relevant study, showed that hyper-methylation of CGI shores on the mouse genome was associated with increased DNMT3A activity, which resulted in positive correlation with transcriptional activity; indicating that methylation outside of CGIs may by used for maintaining active chromatin states for specific genes \citep{Wu2010}.
In this study, we focused on RRBS data, however, given the considerable robustness of the BPR model to low coverage, we expect that it may also be well suited for Whole Genome Bisulphite Sequencing data, which have the advantage of providing a more comprehensive coverage of CpG sites genome-wide. As an extension of this analysis, further work could include building a model to relate differential methylation profiles with differential gene expression levels, and evaluate the importance of profile changes in regulation of gene expression across different cell types. More generally, it is increasingly clear that transcriptional activity is regulated by a complex and still incompletely understood interaction network of molecular players, including DNA methylation, histone modifications and transcription factor binding. Several recent computational studies have highlighted the dependencies between these players \citep{Dong2012, Benveniste2014}. The BPR model provides an effective way of recapitulating DNA methylation patterns using higher order features, and may therefore play an important role in building more effective integrative models of high-throughput data.
\section*{Acknowledgements}
We thank Duncan Sproul for valuable comments and discussion. \emph{Funding}: CAK is supported in part by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh. GS is funded by the European Research Council through grant MLCS306999.
\bibliographystyle{natbib}
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"redpajama_set_name": "RedPajamaArXiv"
}
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In this series of articles and podcasting we will take alook at interesting information about lots of things related to Java, PDF and HTML5, and also take a look behind the scenes at IDR solutions and interview some key members of the Development team about the JPedal Java PDF Library and the PDF to HTML5 Converter.
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Download the Podcast (5.12 minutes) to listen offline, when your busy or when your on the move.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Blues central defender Andreas Christensen has provided an update on his Chelsea future, saying he hopes to sign a new deal 'soon'.
The 25-year-old's current contract runs out at the end of the season but insists he is not in a 'bad situation' with the club and hopes to be in a position to commit his future to Chelsea.
'Hopefully soon,' Christensen said when asked if he is close to committing his long-term future with the Blues. 'I feel valued, I'm happy but I do not get into the contract situation.
'I do not know why everyone thinks it is a bad situation. Because I'm still happy and I enjoy playing football.'
Christensen, who is currently away on international duty with Denmark, believes he is valued highly by the club and head coach Thomas Tuchel.
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The Academy graduate recently scored his first ever Chelsea goal in our 4-0 win over Malmo and has gone from strength to strength under our German head coach's stewardship.
'I do not think about it at all,' Andreas reiterated about his contract situation. 'I enjoy playing football, so I do not know.'
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Integralne części miejscowości w Polsce
Wg TERYT jest ich 2
Krzekowo – część miasta Szczecin
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Krzekowo – dawne osiedle administracyjne Szczecina, należące do dzielnicy Pogod
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
In recent years, deep learning has demonstrated useful capabilities and potential in many application domains~\cite{he2016deep,devlin2019bert}.
However, the performance of deep neural nets often deteriorates significantly when deployed on test data that exhibit different distributions from the training data.
This is widely recognized as the \emph{domain shift} problem~\cite{quinonero2009dataset}.
For instance, activity recognition models trained on the data from adults are likely to fail when being tested on children's activities, and the performance of natural image classification models tend to perform poorly when tested on artistic paintings.
A common technique to address the problem is domain adaptation (DA)~\cite{ben2007analysis,ganin2016domain,daume2006domain}. It learns to maximize model performance on a given target domain with the help of labeled source domains by bridging the distribution gap.
One limitation of DA is the reliance on target domains.
This makes DA less applicable in real-world scenarios that demand good generalization performance on \emph{unseen} distributions.
Domain generalization (DG)~\cite{muandet2013domain,li2018learning,wang2021generalizing} is attracting increasing attention in recent years.
DG aims to learn a generalizable model that can perform well on unseen distributions after being trained on multiple source domains.
Existing work can be categorized into three types: 1) learn domain-invariant representations~\cite{ganin2016domain,li2018domain1}, 2) meta-learning~\cite{li2018learning,balaji2018metareg,bui2021exploiting}, and 3) data augmentation-based DG~\cite{zhou2021domain,huang2021fsdr}.
In this paper, we focus on data augmentation, specifically Mixup~\cite{zhang2018mixup} which is a simple but effective approach.
Mixup generates new samples via linear interpolations between any two pairs of data.
It increases the quantity and diversity of training data to boost the generalization of deep nets~\cite{zhang2020does}.
Mixup can be used for domain generalization directly~\cite{wang2020heterogeneous}. Recent works, such as FACT~\cite{xu2021fourier} and MixStyle~\cite{zhou2021domain}, have adapted it in computer vision tasks with application-specific knowledge.
Despite the success of Mixup, an important research question remains open: \emph{ Is there any versatile Mixup learning strategy for general domain generalization problems?}
Our specific interest is to enhance the capability of Mixup for general domain generalization based on theoretical and empirical analysis of its current limitations.
First, we notice that vanilla Mixup cannot discern domain information and class information, which can negatively affect its performance due to the entangled domain-class knowledge.
Second, Mixup in DG can easily generate synthetic noisy data points when two classes are close to each other. This reduces the discrimination of the classifier.
We propose the domain-invariant Feature MIXup with Enhanced Discrimination (\emph{FIXED\xspace}) approach, to address these limitations of Mixup.
It incorporates domain-invariant representation learning into Mixup, which enables the diverse data augmentation with useful information for the generalized model.
Then, FIXED\xspace introduces a large margin to reduce synthetic noisy data points in the interpolation process. It is a simple yet effective approach.
Through theoretical analysis, we present insights on the design rationale and superiority of FIXED\xspace.
Note that our FIXED\xspace is not limited to specific applications and can be applied to general classification tasks, in contrast to existing Mixup methods which are designed for computer vision tasks (e.g.,~\cite{zhou2021domain,xu2021fourier}). We have conducted extensive experiments on seven benchmarks across two modalities: 1) image classification (image data) and 2) sensor-based human activity recognition (time series data). The results demonstrate significant superiority of FIXED\xspace over nine state-of-the-art approaches, outperforming the best baseline by \textbf{6.5\%} in terms of average test accuracy on the domain time series generalization task which is still in its infancy.
To summarize, our contributions are three-fold:
\begin{itemize}
\setlength\itemsep{0em}
\item Simple yet effective algorithm: For DG, we propose FIX\xspace to enhance the diversity of useful information and implement FIXED\xspace by introducing the large margin to reduce synthetic noisy data during Mixup. FIXED\xspace remains quite simple but effective.
\item New theoretical insights: We offer theoretical insights from both the cover range and class distance perspectives to explain the rationale behind our algorithm.
\item Good Performance: We conduct comprehensive experiments on seven benchmarks across two modalities: image classification (image) and sensor-based human activity recognition (time series). Experimental results demonstrate the superiority of FIXED\xspace, especially with \textbf{6.5\%} improvements for domain generalization in time series which is still in infancy.
\end{itemize}
\section{Related Work}
\input{sec-related}
\section{Preliminaries}
We follow the definition in \cite{wang2021generalizing}.
In domain generalization, we are given $M$ labeled source domains $\mathcal{S} = \{ \mathcal{S}^i | i = 1,\cdots, M \}$ and $\mathcal{S}^i = \{(\mathbf{x}_j^i, y_j^i)\}_{j=1}^{n_i}$ denotes the $i^{th}$ domain, where $n_i$ denotes the number of data in $\mathcal{S}^i$.
The joint distributions between each pair of domains are different and denoted as $\mathbb{P}^i_{XY} \neq \mathbb{P}^j_{XY}, 1 \leq i \neq j \leq M$.
The goal of DG is to learn a robust and generalized predictive function $h: \mathcal{X} \rightarrow \mathcal{Y}$ from the $M$ training sources to achieve minimum prediction error on an unseen test domain $\mathcal{S}_{test}$ with an unknown joint distribution (i.e., $\min_{h} \mathbb{E}_{(\mathbf{x},y)\in \mathcal{S}_{test}} [\ell(h(\mathbf{x}),y)]$). $\mathbb{E}$ is the expectation and $\ell(\cdot, \cdot)$ is the loss function.
All domains, including the source domains and the unseen target domains, have the same input and output spaces (i.e., $\mathcal{X}^1 = \mathcal{X}^2 =\cdots = \mathcal{X}^M = \mathcal{X}^T \in \mathbb{R}^{m}$). $\mathcal{X}$ is the $m$-dimensional instance space and $\mathcal{Y}^1 = \mathcal{Y}^2 =\cdots =\mathcal{Y}^M = \mathcal{Y}^T=\{1,2,\cdots, K \}$. $\mathcal{Y}$ is the label space and $K$ is the number of classes.
\subsection{Background}
According to~\cite{wang2021generalizing}, data augmentation is a common technique to cope with domain generalization problems.
Among existing methods, Mixup~\cite{zhang2018mixup} is a popular data augmentation approach and has shown good performance in many fields.
It constructs synthetic training samples based on two random data points:
\begin{equation}
\label{eqa:mix}
\begin{aligned}
\lambda\sim Beta(\alpha, \alpha),\\
\tilde{\mathbf{x}} =\lambda \mathbf{x}_i + (1-\lambda) \mathbf{x}_j,\\
\tilde{y} = \lambda y_i + (1-\lambda)y_j,
\end{aligned}
\end{equation}
where $Beta(\alpha, \alpha)$ is the Beta distribution and $\alpha \in (0,\infty)$ is a hyperparameter that controls the strength of interpolation between feature-target pairs, recovering the ERM principle as $\alpha \rightarrow 0$.
Mixup extends the training distribution by incorporating the intuition that linear interpolations of feature vectors should lead to linear interpolations of the associated targets into the training set.
As a powerful data augmentation technique, Mixup has played a vital role to enhance sample diversity in domain generalization problems~\cite{wang2020heterogeneous,xu2021fourier,zhou2021domain}.
\begin{figure}[htbp]
\centering
\subfigure[Vanilla Mixup]{
\label{fig:demo1_1}
\includegraphics[width=0.45\columnwidth,height=0.15\textwidth]{./fig/demo1_1.pdf}
}
\subfigure[Domain-inv. fea. Mixup]{
\label{fig:demo1_2}
\includegraphics[width=0.45\columnwidth,height=0.15\textwidth]{./fig/demo1_2.pdf}
}
\caption{Toy examples to illustrate limitations of Mixup.
Colors and shapes denote classes and domains, respectively.
Red squares are synthetic data points generated by domain Mixup.
(a) Mixup generates unrecognizable synthetic data.
(b) Mixup with only class information can mitigate such an issue.
}
\label{fig:demo1}
\end{figure}
\begin{figure}[htbp]
\centering
\subfigure[Mixup with small margin]{
\label{fig:demo2_1}
\includegraphics[width=0.45\columnwidth]{./fig/demo2_1.pdf}
}
\subfigure[Mixup with large margin]{
\label{fig:demo2_2}
\includegraphics[width=0.45\columnwidth]{./fig/demo2_2.pdf}
}
\caption{Toy examples to illustrate the limitations of vanilla Mixup.
Colors and shapes denote classes and domains, respectively.
(a) Mixup tends to generate noisy data. (b) The large margin can reduce generations of synthetic noisy data.}
\label{fig:demo2}
\end{figure}
\subsection{Limitations of Mixup-based DG}
Although the vanilla Mixup method can enhance data diversity, it fails to discern which features are useful for training the model.
It only increases the diversity of all features equally.
In DG, it cannot distinguish domain features and classification features, which results in the entanglement of domains and classes.
It is unclear which parts of the increased diversity are useful for matching the categories.
When incorrect matching between categories and features occurs, vanilla Mixup negatively affects model performance due to the introduction of interfering information.
\figurename~\ref{fig:demo1_1} shows that vanilla Mixup directly mixes data without discerning classification and domain information.
When mixing data points with the cyan ``+''s and blue ``o''s (circled in red), the red square data points are generated.
Since the red square points are generated by two different classes, their labels should be in between the two classes, which means these points should lie between the two classes.
However, as can be observed from \figurename~\ref{fig:demo1_1}, the red squares almost completely overlap with the blue``+''s, which means they prefer to be the blue class according to the locations.
Mixed domain information interferes with the matching of synthetic data points and synthetic labels.
Not only vanilla Mixup, but also some adapted Mixup variants (e.g., manifold Mixup~\cite{verma2019manifold} which mixes in the hidden states) have the same limitation.
On the other hand, Mixup in DG is more likely to generate noisy synthetic data points~\cite{lu2022semantic}.
Even when samples from different classes in the same domain are away from each other, data points from another domain with different distributions may be close to a cluster with a different category.
When two clusters are close to each other, noisy synthetic data points are more likely to be generated.
As shown in \figurename~\ref{fig:demo2_1}, the blue cluster and the red cluster are very close.
Noisy data points (e.g., the synthetic data points generated by the red ``+''s and the blue ``+''s) are generated with a high probability by Mixup.
\section{The Proposed \textnormal{FIXED} Method}
In this paper, we propose the domain-invariant Feature MIXup with Enhanced Discrimination (FIXED\xspace) method to address the aforementioned limitations of Mixup-based DG methods.
The model architecture of FIXED\xspace is illustrated in \figurename~\ref{fig:frame}.
We introduce its two critical modules as follows.
\subsection{FIX: Domain-invariant Feature MIXup}
We first introduce \emph{domain-invariant} feature Mixup to discern the domain and class information, which is our main contribution.
As suggested in \cite{ganin2016domain}, domain-invariant features contain more informative knowledge for classification than raw data~\cite{zhang2018mixup} or the manifold Mixup~\cite{verma2019manifold}.
Such feature Mixup is general and can be embedded in many existing DG methods.
Let $\mathbf{z}$ be the domain-invariant feature. Then, our approach can be formulated as:
\begin{equation}
\label{eqa:fix}
\begin{aligned}
\lambda\sim Beta(\alpha, \alpha),\\
\tilde{\mathbf{z}} =\lambda \mathbf{z}_i + (1-\lambda) \mathbf{z}_j,\\
\tilde{y} = \lambda y_i + (1-\lambda)y_j.
\end{aligned}
\end{equation}
Note that this is \emph{not} the same as manifold Mixup~\cite{verma2019manifold} which operates on random layers and does not involve domain-invariant feature learning.
On the other hand, although domain-invariant feature learning alone brings about improvements for generalization, they usually lack diversity due to restrictions on the learning process.
Therefore, increasing diversity of these features can make classification information diverse and avoid entangling with useless domain information.
Since the diversity of class information is increased, the corresponding labels are also mixed for better matching, which is different from Mixstyle~\cite{zhou2021domain} and FACT~\cite{xu2021fourier}.
As shown in \figurename~\ref{fig:demo1_2}, when Mixup is performed on domain-invariant features that have no interference from classification information, the diversity of data is indeed enhanced with almost no unrecognizable synthetic data points being generated.
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{./fig/frame.pdf}
\caption{The network architecture of FIXED\xspace.}
\label{fig:frame}
\end{figure}
As shown in \figurename~\ref{fig:frame}, we adopt DANN~\cite{ganin2016domain} to learn domain-invariant features for its popularity and effectiveness
Nevertheless, FIX\xspace can also work with other methods for domain-invariant learning, which is shown in later experiments in Sec. \ref{sec-exp-exten}\footnote{In the following, if there is no special note, FIX\xspace denotes FIX\xspace implemented with DANN.}.
The outputs of the bottleneck layer are viewed as domain-invariant features. The Mixup operation is performed on this layer.
Correspondingly, the class labels are also mixed to increase the diversity of data while domain labels remain unchanged.
Feature Mixup is performed within each batch.
Concretely, for a batch of $\mathbf{z}$, we shuffle its indices and obtain $\hat{\mathbf{z}}$.
Then, $\mathbf{z}$ and $\hat{\mathbf{z}}$ are mixed to obtain $\tilde{\mathbf{z}}$, which is used as the inputs for the subsequent layers.
At the same time, $\tilde{y}$ is generated accordingly.
\subsection{Enhancing Discrimination}
To further enhance discrimination, we introduce a \emph{large margin} loss into Mixup just as in \cite{lu2022semantic} to complete the design of FIXED\xspace.
We follow~\cite{elsayed2018large} to derive the large margin loss as:
\begin{equation}
\begin{aligned}
\ell_{lm}(h(\mathbf{x}_i),y_i) = & \mathscr{G}_{k\neq y_i} \max\{0, \\&
\gamma+ d_{h,\mathbf{x}_i,\{k,y_i\}} \mathrm{sign}(h_k(\mathbf{x}_i) - h_{y_i}(\mathbf{x}_i) \},
\end{aligned}
\label{eqa:margin}
\end{equation}
where $\ell_{lm}$ is large margin loss, $\mathscr{G}$ is an aggregation operator for the multi-class setting, and $\mathrm{sign}(\cdot)$ adjusts the polarity of the distance.
$h_k: \mathcal{X} \rightarrow \mathbb{R}$ is a function that generates a prediction score for classifying the input vector $\mathbf{x}\in\mathcal{X}$ to class $k$.
$\gamma$ is the distance to the boundary which we expect. $d_{h,\mathbf{x},\{k_1,k_2\}}$
is the distance of a point $\mathbf{x}$ to the decision boundary of class $i$ and $j$, which can be computed as:
\begin{equation}
d_{h,\mathbf{x},\{k_1,k_2\}} = \min_{\delta} ||\delta||_p, s.t.~ h_{k_1}(\mathbf{x}+\delta) = h_{k_2}(\mathbf{x}+\delta),
\label{eqa:distbound}
\end{equation}
where $||\cdot||_p$ is $l_p$ norm.
As shown in~\cite{elsayed2018large}, Eq.~\eqref{eqa:margin} can be computed as:
\begin{equation}
\begin{aligned}
\mathscr{G}_{k \neq y_{i}} \max \{ 0, \gamma+\frac{h_{k}\left(\mathbf{x}_{i}\right)-h_{y_{i}}\left(\mathbf{x}_{i}\right)}{\left\|\nabla_{\mathbf{x}} h_{k}\left(\mathbf{x}_{i}\right)-\nabla_{\mathbf{x}} h_{y_{i}}\left(\mathbf{x}_{i}\right)\right\|_{q}} \},
\end{aligned}
\label{eqa:approx}
\end{equation}
where $q = \frac{p}{p-1}$.
As shown in \figurename~\ref{fig:demo2_2}, large margin can reduce noisy synthetic data points to enhance discrimination.
\subsection{Summary}
Combining the domain-invariant feature learning module and the large margin module, the objective function of FIXED\xspace can be formulated as:
\begin{equation}
\begin{aligned}
\min \mathbb{E}_{(\mathbf{x}_1,y_1),(\mathbf{x}_2,y_2)\sim \mathbb{P}} \mathbb{E}_{\lambda\sim Beta(\alpha,\alpha)} [\ell_{lm}(G_y( \\ \mathrm{Mix}_{\lambda}(\mathbf{z}_1,\mathbf{z}_2)),\mathrm{Mix}_{\lambda}(y_1,y_2))+
\ell_d(G_d(R_{\eta}(\mathbf{z}_1)),D)],
\end{aligned}
\end{equation}
where $\mathbb{P}$ denotes the distribution of all data. $\operatorname{Mix}(\cdot, \cdot)$ is a Mixup function, $\mathbf{z}_1=G_f(\mathbf{x}_1)$, $\mathbf{z}_2=G_f(\mathbf{x}_2)$ with
$G_f, G_y, G_d$ the feature net, classification layer, and discriminator, respectively. We perform FIXED\xspace in batches.
$\ell_d$ is the cross-entropy loss.
$R_{\eta}$ is the gradient reversal layer with hyperparameter $\eta$~\cite{ganin2016domain} and $D$ is the domain label.
Note that DANN is only one possible option for domain-invariant learning.
We show that FIXED\xspace can work with CORAL~\cite{sun2016deep} as an alternative implementation in Section \ref{sec-exp-exten}.
\section{Analytical Evaluation}
In this section, we offer theoretical analysis to shed light on the reasons behind the remarkable performance of FIXED\xspace. We perform our analysis from two aspects: 1) distribution coverage and 2) inter-class distance.
\begin{figure}[t!]
\centering
\subfigure[Implicit shrinkage]{
\label{fig:insight1}
\includegraphics[width=.4\columnwidth]{./fig/insight1.pdf}
}
\subfigure[Possible distribution range]{
\label{fig:insight2}
\includegraphics[width=.45\columnwidth]{./fig/insight2.pdf}
}
\caption{Toy examples of theoretical insights. (a) After the implicit shrinkage via Mixup, classes (denoted by different colors) mix together, bringing difficulty to classification. Different shapes denote domains. (b) FIXED\xspace enlarges the distribution cover range. Vertices represent distributions while colored areas represent possible $\mathcal{O}$ in Prop.~\ref{pro:dg1} and $\mathcal{O}'$ in Prop.~\ref{pro:dg2}.}
\label{fig:insight}
\end{figure}
\subsection{Background}
For a distribution $\mathbb{P}$ with an ideal binary labeling function $h^*$ and a hypothesis $h$, we define the error $\varepsilon_{\mathbb{P}}(h)$ in accordance with~\cite{ben2010theory} as:
\begin{equation}
\label{eqa:def-eps}
\varepsilon_{\mathbb{P}}(h) = \mathbb{E}_{\mathbf{x}\sim \mathbb{P}} |h(\mathbf{x}) - h^*(\mathbf{x})|.
\end{equation}
The error of Mixup is given as~\cite{carratino2020mixup}:
\begin{equation}
\label{eqa:def-epsmix}
\varepsilon^{\text{Mixup}}(h) = \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n
\mathbb{E}_{\lambda} \ell(h(\lambda\mathbf{x}_i +(1-\lambda)\mathbf{x}_j), \lambda y_i +(1-\lambda)y_j),
\end{equation}
where $\lambda\sim Beta(\alpha, \alpha)$ and $n$ is the number of data samples.
We also give the definition of $\mathcal{H}$-divergence in accordance with~\cite{ben2010theory}.
Given two distributions $\mathbb{P}, \mathbb{Q}$ over a space $\mathcal{X}$ and a hypothesis class $\mathcal{H}$,
\begin{equation}
\label{eqa:def-hdive}
d_{\mathcal{H}}(\mathbb{P},\mathbb{Q}) = 2\sup_{h\in \mathcal{H}} |Pr_{\mathbb{P}}(I_h)- Pr_{\mathbb{Q}}(I_h)|,
\end{equation}
where $I_h = \{\mathbf{x}\in\mathcal{X}|h(\mathbf{x})=1\}$.
We often consider the $\mathcal{H}\Delta \mathcal{H}$-divergence in~\cite{ben2010theory} where the symmetric difference
hypothesis class $\mathcal{H}\Delta \mathcal{H}$ is the set of functions characterized by disagreements between hypotheses.
For any $n\in \mathbb{N}, [n] = \{1, \cdots, n\}$ is the set of nonzero integers up to $n$.
For any $\alpha, \beta >0, [a,b]\subset [0,1]$, $Beta_{[a,b]}(\alpha, \beta)$ denotes the truncated Beta distribution on $[a,b]$.
$j \sim Unif([n])$ represents uniform random sampling.
\begin{theorem}
\label{thm:da,ben}
(Theorem 2.1 in~\cite{sicilia2021domain}, modified from Theorem 2 in~\cite{ben2010theory}). Let $\mathcal{X}$ be a space and $\mathcal{H}$ be a class of hypotheses corresponding to this space. Suppose $\mathbb{P}$ and $\mathbb{Q}$ are distributions over $\mathcal{X}$. Then, for any $h\in \mathcal{H}$, the following holds
\begin{equation}
\label{eqa:da}
\varepsilon_{\mathbb{Q}}(h) \leq \lambda'' + \varepsilon_{\mathbb{P}}(h)+ \frac{1}{2} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{Q}, \mathbb{P})
\end{equation}
with $\lambda''$ the error of an ideal joint hypothesis for $\mathbb{Q}$ and $\mathbb{P}$.
\end{theorem}
\theoremautorefname~\ref{thm:da,ben} provides an upper bound on the target-error.
$\lambda''$ is a property of the dataset and hypothesis class and is often ignored.
\theoremautorefname~\ref{thm:da,ben} demonstrates the necessity to learn domain invariant features.
\begin{proposition}
\label{pro:dg1}
(Proposition 3.1 in~\cite{sicilia2021domain}, modified from Proposition 2 in~\cite{albuquerque2020adversarial}). Let $\mathcal{X}$ be a space and $\mathcal{H}$ be a class of hypotheses corresponding to this space. Let $\mathbb{Q}$ and the collection $\{\mathbb{P}_i \}_{i=1}^M$ be distributions over $\mathcal{X}$ and let $\{\phi_i \}_{i=1}^M$ be a collection of non-negative coefficient with $\sum_i \phi_i = 1$. Let the object $\mathcal{O}$ be a set of distributions such that for every $\mathbb{S}\in \mathcal{O}$ the following holds
\begin{equation}
\label{eqa:dg1-con}
\sum_i \phi_i d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j).
\end{equation}
Then, for any $h\in \mathcal{H}$,
\begin{equation}
\label{eqa:dg1}
\begin{aligned}
\varepsilon_{\mathbb{Q}}(h)\leq& \lambda_{\phi} + \sum_i \phi_i \varepsilon_{\mathbb{P}_i}(h) + \frac{1}{2}\min_{\mathbb{S}\in\mathcal{O}} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{S}, \mathbb{Q})\\ & + \frac{1}{2}\max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{P}_j)
\end{aligned}
\end{equation}
where $\lambda_{\phi} = \sum_i \phi_i \lambda_i$ and each $\lambda_i$ is the error of an ideal joint hypothesis for $\mathbb{Q}$ and $\mathbb{P}_i$.
\end{proposition}
Proposition~\ref{pro:dg1} gives an upper bound for domain generalization.
In the right-hand side of Eq.~\ref{eqa:dg1}, the first term can be ignored and the second term is a convex combination of the source errors. The third term demonstrates the importance of diverse source distributions so that the unseen target $\mathbb{Q}$ might be near $\mathcal{O}$ while the final term is a maximum over the source-source divergences.
As shown in \figurename~\ref{fig:insight2}, the area with yellow and green is the possible $\mathcal{O}$.
\subsection{Why Mixup is not Good Enough}
\begin{proposition}
\label{pro:mix}
(modified from Theorem 1 in~\cite{carratino2020mixup}). Let $\theta \sim Beta_{[\frac{1}{2},1]}(\alpha, \alpha)$ and $j\sim Unif([n])$ be two random variables with $\alpha>0$, $n>0$, and let $\bar{\theta} = \mathbb{E}_{\theta}\theta$. For any training set $\mathcal{S}_n$, there exist two random perturbations $(\delta_i, \epsilon_i)$ with $\mathbb{E}_{\theta,j} \delta_i = \mathbb{E}_{\theta,j} \epsilon_i = 0, i\in[n].$ Denote $\varepsilon^{Mixup}$ the error of Mixup, we have
\begin{equation}
\label{eqa:pmixtr}
\begin{aligned}
&\varepsilon^{Mixup}(h)
= \frac{1}{n}\sum_{i=1}^n \mathbb{E}_{\theta,j} \ell(h(\tilde{\mathbf{x}}_i), \tilde{y}_i)\\
=&\frac{1}{n}\sum_{i=1}^n \mathbb{E}_{\theta,j} \ell(h(\bar{\mathbf{x}}+\bar{\theta}(\mathbf{x}_i - \bar{\mathbf{x}})+\delta_i), \bar{y}+\bar{\theta}(y_i - \bar{y})+\epsilon_i).
\end{aligned}
\end{equation}
\end{proposition}
Note that $\bar{\theta} \in [1/2,1]$.
From Eq.~\eqref{eqa:pmixtr}, we can see that the transformation from $(\mathbf{x}_i,y_i)$ to $(\tilde{\mathbf{x}}_i,\tilde{y}_i)$ \emph{shrinks} the inputs and the outputs towards their mean with perturbations.
When there exist spurious relations induced by redundant domain information, it may bring confusion when performing Mixup, which is demonstrated in \figurename~\ref{fig:insight1}.
Moreover, introducing the large margin can make classes far from each other and thereby reduce confusion during Mixup.
\subsection{FIXED\xspace has Larger Distribution Coverage}
We derive our theory to prove that FIXED\xspace has a larger distribution coverage.
\begin{proposition}
\label{pro:dg2}
Let $\mathcal{X}$ be a space and $\mathcal{H}$ be a class of hypotheses corresponding to this space. Let $\mathbb{Q}$ and the collection $\{\mathbb{P}_i \}_{i=1}^M$ be distributions over $\mathcal{X}$ and let $\{\phi_i \}_{i=1}^M$ be a collection of non-negative coefficient with $\sum_i \phi_i = 1$. Let the object $\mathcal{O}'$ be a set of distributions such that for every $\mathbb{S}\in \mathcal{O}'$ the following holds
\begin{equation}
\label{eqa:dg2-con}
d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j).
\end{equation}
Then, for any $h\in \mathcal{H}$,
\begin{equation}
\label{eqa:dg2}
\begin{aligned}
\varepsilon_{\mathbb{Q}}(h)\leq& \lambda' + \sum_i \phi_i \varepsilon_{\mathbb{P}_i}(h) + \frac{1}{2}\min_{\mathbb{S}\in\mathcal{O}'} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{S}, \mathbb{Q})\\& + \frac{1}{2}\max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{P}_j)
\end{aligned}
\end{equation}
where $\lambda'$ is the error of an ideal joint hypothesis.
\end{proposition}
\begin{proof}
On one hand, with \theoremautorefname~\ref{thm:da,ben}, we have
\begin{equation}
\label{eqa:p1s1}
\varepsilon_{\mathbb{Q}}(h) \leq \lambda_1 +\varepsilon_{\mathbb{S}}(h)+ \frac{1}{2}d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{S}, \mathbb{Q}), \forall h\in \mathcal{H}, \forall\mathbb{S}\in \mathcal{O}'.
\end{equation}
On the other hand, with \theoremautorefname~\ref{thm:da,ben}, we have
\begin{equation}
\label{eqa:p1s2}
\varepsilon_{\mathbb{S}}(h) \leq \lambda_2 +\varepsilon_{\sum_i \phi_i\mathbb{P}_i}(h)+ \frac{1}{2}d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S}), \forall h\in \mathcal{H}.
\end{equation}
Since $\varepsilon_{\sum_i \phi_i\mathbb{P}_i}(h) = \sum_i \phi_i\varepsilon_{\mathbb{P}_i}(h)$, and $d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j)$, we have
\begin{equation}
\label{eqa:p1s3}
\begin{aligned}
\varepsilon_{\mathbb{Q}}(h) \leq & \lambda' + \sum_i \phi_i \varepsilon_{\mathbb{P}_i}(h)+ \frac{1}{2}d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{S}, \mathbb{Q})\\ & +\frac{1}{2}\max_{i,j}d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S}), \forall h\in \mathcal{H}, \forall\mathbb{S}\in \mathcal{O}'.
\end{aligned}
\end{equation}
Eq.~\eqref{eqa:p1s3} for all $\mathbb{S}\in \mathcal{O}'$ holds. Proof ends.
\end{proof}
\begin{proposition}
\label{cor:rel}
Under the same conditions in \ref{pro:dg2},
\begin{equation}
\mathcal{O}=\{S|\sum_i \phi_i d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j)\},
\end{equation}
\begin{equation}
\mathcal{O}'=\{S|d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j)\},
\end{equation}
we have
\begin{equation}
\label{eqa:rel}
\mathcal{O}\subset \mathcal{O}'.
\end{equation}
\end{proposition}
\begin{proof}
On one hand, for any $\mathbb{S} \in \mathcal{O}$, we have
\begin{equation}
\label{eqa:p2s1}
\sum_i \phi_i d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{S}) \leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j).
\end{equation}
On the other hand, with the triangle inequality, we have
\begin{equation}
\label{eqa:p2s2}
d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S})\leq\sum_i \phi_i d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i, \mathbb{S}).
\end{equation}
Combining these two inequalities, we have
\begin{equation}
\label{eqa:p2s3}
d_{\mathcal{H}\Delta \mathcal{H}}(\sum_i \phi_i\mathbb{P}_i, \mathbb{S})\leq \max_{i,j} d_{\mathcal{H}\Delta \mathcal{H}}(\mathbb{P}_i,\mathbb{P}_j).
\end{equation}
Therefore, $\mathbb{S}\in \mathcal{O}'$. Proof completed.
\end{proof}
From Prop.~\ref{pro:dg2} and Prop.~\ref{cor:rel}, $\mathcal{O}'$ has a larger possible cover range than $\mathcal{O}$ (used in Prop.~\ref{pro:dg1}), which brings more \emph{diversity}.
As shown in the right part of \figurename~\ref{fig:insight2}, the red area is the possible increased area.
Possible areas contain distributions that may be in $\mathcal{O}(\mathcal{O}')$ while confirmed areas contain distributions that must be in $\mathcal{O}(\mathcal{O}')$.
It reveals that the area with a constant distance to purple lines and blue vertices is larger than the area with the same distance to blue vertexes, where purple lines can be viewed as Mixup of vertexes expressed as $\sum_i \phi_i\mathbb{P}_i$.
Moreover, yellow areas may have a higher possibility to satisfy Eq. \eqref{eqa:dg2-con} since the points in it have shorter distances from purple lines and blue vertices.
\subsection{Insights from Inter-class and Intra-class Distances}
Recently, it has come to researchers' attention that just learning domain invariant features maybe not enough for good generalization and discrimination, especially in the field of domain adaptation~\cite{tang2020unsupervised,bui2021exploiting}.
A common approach to enhance generalization and discrimination is to enlarge the inter-class distance and decrease the intra-class distance~\cite{blei2003latent}, which has already been utilized for domain adaptation~\cite{hu2015deep} and domain generalization~\cite{kim2021selfreg}.
From Eq.~\ref{eqa:pmixtr}, feature Mixup can be perceived as a tool to decrease the intra-class distance, while the large margin loss can enlarge the inter-class distance. This indicates that FIXED\xspace enhances generalization and discrimination.
\section{Experimental Evaluation}
While most literature on DG evaluates the algorithms on image classification datasets, we perform evaluations on \textbf{both} image classification and sensor-based human activity recognition (i.e., time series) data.
This can help study the generality of our method across multiple modalities.
\subsection{Evaluation on Image Classification Datasets}
\subsubsection{Datasets}
We adopt three popular DG benchmark datasets.
(1) \textbf{Digits-DG}~\cite{zhou2020deep}, which contains four digit datasets including MNIST~\cite{lecun1998gradient}, MNIST-M~\cite{ganin2015unsupervised}, SVHN~\cite{netzer2011reading}, SYN~\cite{lecun1998gradient}.
The four datasets differ in font style, background, and image quality.
Following~\cite{zhou2020deep}, we select 600 images per class from each dataset.
(2) \textbf{PACS}~\cite{li2017deeper}, which is an object classification benchmark with four domains (i.e., photos, art-paintings, cartoons, sketches).
There exist large discrepancies in image styles among different domains.
Each domain contains seven classes and there are 9,991 images in total.
(3) \textbf{Office-Home}~\cite{venkateswara2017deep}, which is an object classification benchmark that contains four domains (i.e., Art, Clipart, Product, Real-World).
The domain shift comes from image styles and viewpoints.
Each domain contains 65 classes and there are 15,500 images in total.
The statistical information of each dataset is presented in \tablename~\ref{tb-data-imgs}.
\begin{table}[htbp]
\centering
\caption{Information on visual datasets.}
\label{tb-data-imgs}
\resizebox{1\linewidth}{!}{%
\begin{tabular}{clcccc}
\toprule
Dataset&Domain Names& Domain&Class& Samples of each domain& Total Samples\\
\midrule
Digits-DG&(M,MM,SVN,SYHN)&4&10&(600;600;600;600)&2,400\\
PACS&(A,C,P,S)&4&7&(2,048;2,344;1,670;3,929)&9,991\\
Office-Home&(A,C,P,R)&4&65&(2,427;4,365;4,439;4,357)&15,588\\
\bottomrule
\end{tabular}%
}
\end{table}
\subsubsection{Baselines and Implementation Details}
For the experiments using ResNet-18, i.e., Office-Home and PACS datasets, and Digits-DG dataset that uses DTN as the backbone following~\cite{liang2020we}, we re-implement several recent strong comparison methods by extending the DomainBed~\cite{gulrajani2020search} codebase for fair study.
For the algorithms that are not implemented by ourselves, we copy their results from their papers when the settings are the same.
Our reproductions are marked with *.
We select the best model via results on validation datasets.
Specifically, we split each source domain with a ratio of $8:2$ for training and validation following DomainBed and report average results of three trials\footnote{The ratio $8:2$ is suggested by DomainBed and recent works for fair comparisons. We note that several methods adopted $9:1$ or $8:1$ which involves more training data that are easier to perform better. In these cases, our method still outperforms them.}.
\input{tab/tb-spcvis}
\subsubsection{Results and Discussion}
\tablename~\ref{tab:my-table-pacs}, \tablename~\ref{tab:my-table-off}, and \tablename~\ref{tab:my-table-digitsdg} show the results on PACS, Office-Home, and Digits-DG datasets respectively where PACS and Office-Home used ResNet-18.
We observe that our FIXED\xspace approach consistently outperforms all comparison methods.
For PACS, \tablename~\ref{tab:my-table-pacs} demonstrates that our method can have an over $1\%$ improvement compared to the second-best one.
In an absolute fair environment, our method can even achieve an over $3\%$ improvement compared to the methods with stars.
$3\%$ is a remarkable improvement since some methods, e.g. DANN, only have slight improvements compared to ERM.
\tablename~\ref{tab:my-table-off} illustrates that our method can also have an over $1\%$ improvement compared to the second-best one for Office-Home while \tablename~\ref{tab:my-table-digitsdg} illustrates that our method can have an over $0.3\%$ improvement compared to the latest methods for Digits-DG.
We observe more insightful conclusions.
(1) Vanilla Mixup even performs worse than ERM on some benchmarks, which illustrates that mixed domain information interfaces performances seriously.
(2) There are small performance gaps among different methods on some benchmarks, e.g. Office-Home; and even ERM can achieve acceptable results. It is caused because there are few differences among different domains (Office-Home) or some other reasons.
(3) Different domain splits are important for DG. For example, in Digits-DG, MixStyle shows a significant improvement on the second task. Even Jigen performs better than all other methods.
Hence, it is important to perform several random trials to record the averaged performance (also refer to \figurename~\ref{fig:threepacs}-\figurename~\ref{fig:threeusc} for the stability of our method).
\subsection{Evaluation on Human Activity Recognition}
\input{tab/tb-activity-all}
\begin{table}[htbp]
\centering
\caption{Information on HAR datasets.}
\label{tb-data-harss}
\begin{tabular}{crrrr}
\toprule
Dataset& Subjuects&Sensors&Classes& Samples\\
\midrule
DSADS&8&3&19&1,140,000\\
USC-HAD&14&2&12&5,441,000\\
UCI-HAR&30&2&6&1,310,000\\
PAMAP&9&3&18&3,850,505\\
\bottomrule
\end{tabular}%
\end{table}
\begin{table}[htbp]
\centering
\caption{Information on HAR in three settings.}
\label{tb-data-hars}
\resizebox{0.5\textwidth}{!}{%
\begin{tabular}{crrrrr}
\toprule
Setting& Domain&Sensor&Class&Samples of each domain& Total Samples\\
\midrule
X-Person&4&2&12&(1,401,400;1,478,000;1,522,800;1,038,800)&5,441,000\\
X-Position&5&3&19&(1,140,000)*5&5,700,000\\
X-Dataset&4&2&6&(672,000;810,550;514,950;470,850)&2,468,350\\
\bottomrule
\end{tabular}%
}
\end{table}
\subsubsection{Datasets and Settings}
We evaluate our method on several DG benchmarks with three different settings on human activity recognition.
Four datasets are used: UCI daily and sports dataset (DSADS)~\cite{barshan2014recognizing}, USC-SIPI human activity dataset (USC-HAD)~\cite{zhang2012usc}, UCI human activity recognition using smartphones data set (UCI-HAR)~\cite{anguita2012human} and PAMAP2 physical activity monitoring dataset
(PAMAP2)~\cite{reiss2012introducing}.
UCI daily and sports dataset (DSADS) consists of 19 activities collected from 8 subjects wearing body-worn sensors on 5 body parts.
USC-SIPI human activity dataset (USC-HAD) composes of 14 subjects (7 male, 7 female, aged from 21 to 49) executing 12 activities with a sensor tied on the front right hip.
UCI human activity recognition using smartphones data set (UCI-HAR) is collected by 30 subjects performing 6 daily living activities with a waist-mounted smartphone.
PAMAP2 physical activity monitoring dataset (PAMAP2) contains data of 18 different physical activities, performed by 9 subjects wearing 3 sensors.
We mainly use the sliding window technique to preprocess data.
The statistical information on each dataset is presented in \tablename~\ref{tb-data-harss}.
We constructed three settings for extensive evaluations of our method:
(1) \textbf{Cross-Person}: This setting utilizes USC-HAD dataset and 14 persons are divided into four groups. Each domain contains 12 classes.
Each sample has two sensors with six dimensions.
(2) \textbf{Cross-Position}: This setting utilizes DSADS dataset and data from each position corresponds to a different domain. There are 19 classes in total.
Each sample contains three sensors with nine dimensions.
(3) \textbf{Cross-Dataset}: This setting utilizes all four datasets, and each dataset corresponds to a different domain.
Data of six common classes, including walking, walking upstairs, walking downstairs, sitting, standing, and laying, are selected.
We choose two sensors from each dataset that belong to the same position. Data is down-sampled to ensure the dimensions of data in different datasets same.
It is easy to see that cross-dataset setting is more challenging than the other two since it contains more diversities in datasets.
For more information, please refer to \tablename~\ref{tb-data-hars}
\subsubsection{Baselines and Implementation Details}
For all three settings, we reproduce eight state-of-the-art baselines by ourselves.
In addition, for cross-person on USC-HAD, we add the results of GILE~\cite{qian2021latent} obtained from results via their code.
We used different architectures for activity recognition.
The network contains two blocks, where each has one convolution layer, one pool layer, and one batch normalization layer.
A single fully-connected layer is used as the bottleneck block while another fully-connected layer serves as the classifier.
In each step, each domain selects 32 samples.
The maximum training epoch is set to 150.
For all methods except GILE, the Adam optimizer with a learning rate $10^{-2}$ and weight decay $5 \times 10^{-4}$ is used.
We tune hyperparameters for each method and select their best results to report.
We report average results of three trials.
For each benchmark, we randomly split each source domain into $80\%$ for training and $20\%$ for validation.
\subsubsection{Results and Discussion}
The results on time series are presented in \tablename~\ref{tab:har}.
Overall, our method has an improvement of $\mathbf{6.5}\%$ average accuracy than the second-best method on average of these three settings.
This demonstrates generalization capability across different tasks of our method.
These settings represent DG scenarios of different difficulties (e.g., cross-dataset is much more difficult than cross-person), thus they can thoroughly reflect the performance of all methods in different situations.
As shown in \tablename~\ref{tab:har}, some methods deteriorate seriously for these three HAR benchmarks while our method achieves the best average accuracy on average and performs best almost on every task.
The results are consistent with image datasets.
To sum up, our method is effective in both image and time series datasets, indicating that it is a general approach for domain generalization.
\begin{figure*}[t!]
\centering
\label{cfig:img_vis}
\subfigure[ERM]{
\label{fig:cimgvis1}
\includegraphics[width=0.31\linewidth]{./fig/vis1app.png}
}
\subfigure[Vanilla Mixup]{
\label{fig:cimgvis2}
\includegraphics[width=0.31\linewidth]{./fig/vismixupapp.png}
}
\subfigure[Manifold Mixup~\cite{verma2019manifold}]{
\label{fig:cimgvis3}
\includegraphics[width=0.31\linewidth]{./fig/visFmixapp.png}
}
\subfigure[Domain-invariant feature]{
\label{fig:cimgvis5}
\includegraphics[width=0.31\linewidth]{./fig/visdann1app.png}
}
\subfigure[Large margin]{
\label{fig:cimgvis6}
\includegraphics[width=0.31\linewidth]{./fig/visdann2app.png}
}
\subfigure[Our method]{
\label{fig:cimgvis4}
\includegraphics[width=0.31\linewidth]{./fig/vis2app.png}
}
\caption{Visualization of the t-SNE embeddings of learned feature spaces for PACS with different methods. Different colors correspond to different classes and different shapes correspond to different domains. \emph{Best viewed in color and zoom in.}}
\end{figure*}
\subsection{Qualitative Analysis}
\input{tab/tb-ablat-margin}
\begin{figure}[t!]
\centering
\subfigure[PACS]{
\includegraphics[width=0.22\textwidth]{./fig/marginmix-pacs.pdf}
\label{fig:img_ablat-pacs}
}
\subfigure[Cross-Person]{
\includegraphics[width=0.22\textwidth]{./fig/marginmix-crossperson.pdf}
\label{fig:img_ablat-cross-person}
}
\caption{Ablation study of FIXED\xspace.}
\label{fig:img-ablat-m}
\end{figure}
\subsubsection{Ablation Study}
We report our ablation study in \figurename~\ref{fig:img-ablat-m}.
Compared with vanilla Mixup, Manifold Mixup shows slight improvements.
It is caused by the reason that the model trained with categories as goals is biased towards containing more classification information in the deeper layers\footnote{We try our best to perform Manifold Mixup in the deeper layers, which leads remarkable improvements compared to Vanilla Mixup, but it is still worse than ours.}.
It proves our motivation of domain-invariant Mixup from another view.
Compared with Manifold Mixup, it is obvious that directly using Mixup on domain-invariant features has a remarkable improvement, which demonstrates that increasing the diversity of useful information for model training is able to bring benefits.
In addition, we present results on large margin with existing methods in \tablename~\ref{tab:my-table-ablat-marg} After introducing large margin, there are extra improvements compared with domain-invariant Manifold Mixup.
These experiments demonstrate that two components are both effective.
\subsubsection{Visualization Study}
We present visualization results to show the rationale of our method
As shown in \figurename~\ref{fig:cimgvis1}, the same class in different domains has different distributions with ERM (data points with the same color and different shapes locate in different places), which is just like \figurename~\ref{fig:demo1_1}.
Moreover, some classes are close to each other, which is like \figurename~\ref{fig:demo2_1}.
If we do nothing for these situations, synthetic noisy points will be generated as mentioned above.
Even if vanilla Mixup or Manifold Mixup are used, the above issues cannot be solved (see \figurename~\ref{fig:cimgvis2} and \figurename~\ref{fig:cimgvis3} respectively).
With domain invariant features, FIXED\xspace can reduce the influence generated by redundant domain information (\figurename~\ref{fig:cimgvis5}) while it can bring enhance discrimination.
Overall, with both considerations, FIXED\xspace achieves the best visualization effects in \figurename~\ref{fig:cimgvis4} where two problems have been relieved to a certain degree and thus leads to the best performance in \figurename~\ref{fig:img-ablat-m}.
\begin{figure*}[t!]
\centering
\subfigure[Office-Home]{
\label{fig:fixcoraloff}
\includegraphics[width=0.45\columnwidth]{./fig/fixcoraloff.pdf}
}
\subfigure[Cross-Person HAR]{
\label{fig:fixcoralhar}
\includegraphics[width=0.45\columnwidth]{./fig/fixcoralhar.pdf}
}
\subfigure[PACS]{
\label{fig:threepacs}
\includegraphics[width=0.45\columnwidth]{./fig/pacsthreetime.pdf}
}
\subfigure[Cross-Person HAR]{
\label{fig:threeusc}
\includegraphics[width=0.45\columnwidth]{./fig/uscthreetime.pdf}
}
\caption{\figurename~\ref{fig:fixcoraloff} and \figurename~\ref{fig:fixcoralhar} are results of FIXED\xspace with CORAL extension while \figurename~\ref{fig:threepacs} and \figurename~\ref{fig:threeusc} are results on cross-person HAR and PACS with three trials to show the robustness of our algorithm.
\label{fig:fixext}
\end{figure*}
\begin{figure*}[t!]
\centering
\subfigure[$\alpha$]{
\label{fig:sensmix}
\includegraphics[width=0.22\textwidth]{./fig/sensemix.pdf}
}
\subfigure[$\eta$]{
\label{fig:senseta}
\includegraphics[width=0.22\textwidth]{./fig/senseeta.pdf}
}
\subfigure[$\gamma$]{
\label{fig:sensgamma}
\includegraphics[width=0.22\textwidth]{./fig/sensegamma.pdf}
}
\subfigure[Top $k$]{
\label{fig:sensek}
\includegraphics[width=0.22\textwidth]{./fig/sensek.pdf}
}
\caption{Parameter sensitivity analysis.}
\label{fig:img_sens}
\end{figure*}
\subsection{More Analysis}
\label{sec-exp-exten}
\subsubsection{Extensibility}
To demonstrate that FIX\xspace is an extensible DG approach, we replace the adversarial learning module in FIX\xspace with CORAL loss~\cite{sun2016deep} to learn domain-invariant features.
For better comparison, we do not use large margin and we denote it as FIX\xspace-CORAL.
We compare it with CORAL and Vanilla Mixup on two benchmarks: Office-Home and Cross-Person.
From \figurename~\ref{fig:fixcoraloff} and \figurename~\ref{fig:fixcoralhar}, we can see FIX\xspace-CORAL achieves the best results on both benchmarks compared with CORAL and vanilla Mixup, which demonstrates FIXED\xspace~is a general approach for domain generation.
In addition, in \figurename~\ref{fig:fixcoraloff}, it can be observed that FIX\xspace-CORAL without large margin loss almost achieves the same performance as FIXED\xspace with large margin loss.
It may be caused by that CORAL performs better than adversarial training on Office-Home (rf. \tablename~\ref{tab:my-table-off}), thereby obtaining improved domain-invariant features.
\subsubsection{Robustness}
Our method involves Mixup strategy and random domain splits which may introduce instabilities.
In this section, we evaluate its robustness.
\figurename~\ref{fig:threepacs} and \figurename~\ref{fig:threeusc} demonstrate that our method is robust against random seeds and different domain splits in several trials.
This implies that our method can be easily applied to real applications.
\subsubsection{Parameter Sensitivity Analysis}
There are mainly four hyperparameters in our method: $\alpha$ for Beta distribution in Mixup, $\eta$ for the weight of adversarial learning, $\gamma$ for the required distance to boundaries in Eq.~\eqref{eqa:approx}, and top $k$ for the aggregation class number in Eq.~\eqref{eqa:approx}.
We evaluate the parameter sensitivity of our method in \figurename~\ref{fig:img_sens} where we change one parameter and fix the other to record the results.
From these results, we can see that our method achieves better performance in a wide range, demonstrating that our method is not sensitive to hyperparameter choices.
We also note that $\eta$ for DANN is a bit sensitive and may need attention in real applications.
\section{Conclusions and Future Work}
In this paper, we proposed FIXED\xspace, a general approach for domain generalization.
FIXED\xspace performs Mixup on domain-invariant features to increase diversity by discerning domain and class information.
To mitigate the noisy synthetic data problem in Mixup, we introduced the large margin loss.
FIXED\xspace can be embedded in many existing DG methods, and we presented implementations based on DANN and CORAL.
We provided theoretical insights to our algorithm.
Extensive experiments on seven datasets across two modalities demonstrated that FIXED\xspace yielded SOTA results on all datasets.
In the future, we plan to incorporate our approach into representation learning and meta-learning DG methods to improve performance and deploy it on more applications.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG2-RP-2020-019); the Joint NTU-WeBank Research Centre on Fintech (Award No: NWJ-2020-008); the Nanyang Assistant Professorship (NAP); the RIE 2020 Advanced Manufacturing and Engineering (AME) Programmatic Fund (No. A20G8b0102), Singapore; and Future Communications Research \& Development Programme (FCP-NTU-RG-2021-014).
Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore.
\bibliographystyle{IEEEtranN}
{\footnotesize
\subsection{Domain Generalization}
DG aims to learn a model from a single or multiple source domains to generalize well to unseen target domains.
According to~\cite{wang2021generalizing}, existing DG methods can be divided into three groups: 1) representation learning~\cite{jin2020style,li2018deep,peng2019domain}, 2) learning strategy~\cite{dou2019domain,shi2021gradient}, and 3) data manipulation~\cite{nazari2020domain,zhou2020deep,qiao2020learning}.
Represent learning is one of the most common approaches for domain adaptation and domain generalization.
Since DG can be viewed as an extension of DA, many traditional DA methods can be applied to DG~\cite{ganin2016domain,sun2016deep,li2018domain1}.
Domain-adversarial neural network (DANN)~\cite{ganin2016domain} learns domain-invariant features via adversarial training of the generator and the discriminator.
The discriminator aims to distinguish the domains, while the generator aims to fool the discriminator to learn domain invariant feature representations.
Deep Coral~\cite{sun2016deep} learns a nonlinear transformation that aligns correlations of
layer activations in deep neural networks.
MMD-AAE~\cite{li2018domain1} imposes the Maximum Mean Discrepancy (MMD) measure to align the distributions among different domains, and matches the aligned distribution to an arbitrary prior distribution via adversarial feature learning.
These methods all attempt to learn feature representations that are supposed to be universal to the seen source domains, and are expected to generalize well on the target domain.
There are also other methods for domain-invariant learning on DG~\cite{mahajan2021domain,arjovsky2019invariant}.
Recent research works point out that just learning domain-invariant feature representation may be not enough for DA~\cite{zhao2019learning} and DG~\cite{bui2021exploiting}.
Therefore, \cite{bui2021exploiting} proposed a novel theoretically sound framework - mDSDI - to further capture the usefulness of domain-specific information.
The proposed FIXED\xspace method is another approach to exploit additional information beyond the domain-invariant feature representation.
Learning strategy is another group for DG approaches.
In this group, methods focus on exploiting the general learning strategy to enhance model generalization capability.
MLDG~\cite{li2018learning} proposed a model agnostic training procedure for DG that simulates train/test domain shift during training by synthesizing virtual testing domains within each mini-batch.
The meta-optimization objective requires that steps to improve training domain performance must also improve test domain performance. Fish~\cite{shi2021gradient} utilizes an inter-domain gradient matching objective that targets domain generalization by maximizing the inner product between gradients from different domains. By forcing the gradient direction to be invariant for different domains, it can be generalized to unseen targets.
Similar to Fish, \cite{mansilla2021domain} conjectured that conflicting gradients within each mini-batch contain information specific to the individual domains which are irrelevant to others when training with multiple domains.
It characterizes the conflicting gradients and devised novel gradient agreement strategies based on gradient surgery to alleviate such disagreements to improve generalization.
The data manipulation group of approaches is the most closely related to our work. Thus, we elaborate on it in the next subsection.
\subsection{Data Augmentation and Mixup for DG}
Data augmentation is used in DG through either domain randomization~\cite{yue2019domain}, self-supervised learning (e.g., JiGen~\cite{carlucci2019domain}), adversarial augmentation (e.g., CrossGrad~\cite{shankar2018generalizing}), or Mixup~\cite{zhang2018mixup}.
In \cite{yue2019domain}, an approach for domain randomization and pyramid consistency is proposed to learn a model with high generalizability.
In particular, it randomizes the synthetic images with the styles of real images in terms of visual appearance using auxiliary datasets.
JeGen~\cite{carlucci2019domain} learns the semantic labels in a supervised fashion, and broadens its understanding of the data by learning from self-supervised signals to solve a jigsaw puzzle on the same images. This helps the network to learn the concepts of spatial correlation while acting as a regularizer for the classification task.
CrossGrad~\cite{shankar2018generalizing} trains a label and a domain classifier in parallel on examples perturbed by loss gradients of each other's objectives.
Recently, SFA~\cite{li2021simple}, a simple feature augmentation method, attempts to perturb the feature embedding with Gaussian noise during training.
Mixup~\cite{zhang2018mixup} is a simple but effective technique to increase data diversity.
It extends the training distribution by incorporating the intuition that linear interpolations of feature vectors shall lead to linear interpolations of the associated targets.
There are several variants of Mixup.
Manifold Mixup~\cite{verma2019manifold} is designed to encourage neural networks to predict less confidently on interpolations of hidden representations, which leverages semantic interpolations as additional training signals.
Note that manifold Mixup is a common approach while both the vanilla Mixup and FIX\xspace can be regarded as specific implementations of manifold Mixup for different purposes.
CutMix~\cite{yun2019cutmix} replaces the removed regions with a patch from another image and mixed the ground truth labels proportionally to the number of pixels of combined images.
Puzzle Mix~\cite{kim2020puzzle} explicitly utilizes the saliency information and the underlying statistics of the natural examples to prevent misleading supervisory signals.
Since Mixup is a natural way for data augmentation, many variants of Mixup have been proposed for DA and DG.
Recent works~\cite{xu2020adversarial,wu2020dual} use the vanilla Mixup method for domain adaptation without modification.
DM-ADA~\cite{xu2020adversarial}, a Mixup-based method for DA, utilizes domain mixup on the pixel level and the feature level to improve model robustness.
It guarantees domain-invariance in a more continuous latent space, and guides the domain discriminator to judge sample difference between the source and target domains.
Mixstyle~\cite{zhou2021domain} increases the image style information to enhance the diversity of domains without change to classification labels.
FACT~\cite{xu2021fourier} mixes the amplitude spectrum of images after Fourier transform to force the model to capture phase information.
Wang et al.~\cite{wang2020heterogeneous} mixes up samples in multiple domains with two different sampling strategies.
Existing methods generally ignore the domain-invariant features and discrimination of Mixup.
In addition, they are designed for specific tasks and need to be modified for each application domain. FIXED\xspace can address these limitations.
|
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{"url":"http:\/\/www.comfsm.fm\/~dleeling\/physci\/psb1\/164.html","text":"# psb1 164 \u2608 Name:\n\n1. Section one measurement: If a bar of soap is floating in a liquid, then what must be true about the density of the soap relative to the density of the liquid?\n2. Section two motion: The time versus distance xy scatter graph below shows the motion of three different marbles. The marbles are labeled A, B, and C. All three marbles start from a time of 0 seconds and a distance of 0 centimeters. The following questions use this graph.\n\n_____ Which marble is slowing down?\n3. _____ Which marble is speeding up?\n4. _____ Which marble is moving at a constant speed?\n5. __________ What is the speed of marble A between five and nine seconds?\n6. __________ What is the speed of marble B between four and eight seconds?\n7. __________ What is the speed of marble C between five and eight seconds?\n8. Section three accelerated motion: ____________________ If a ball falls for two times as long a duration of time, by what multiple will the distance be increased? Will the distance covered increase by two, three, four, or eight times the original distance?\n9. Section four momentum: For marbles rolling into a line of marbles, what two rules do the marbles obey?\n10. The velocity v at the bottom of a banana leaf ramp for a marble released from a height h is given by the formula: $v\u221dh$. If the marble is rolled from a height four times as high, by what factor does the velocity v increase?\n11. Extension (cm) Force (gmf)\n0 0\n3 15\n630\n945\nSection five force: _______ Data was gathered for the extension of an elastic band using a cup and marbles to generate the force. The data in the table is from the experiment. Is the elastic band a linear elastic material?\n12. _______________ For the data above, determine Hooke's constant for the elastic band using Hooke's law [ F = \u2212kx ].\n13. _______________ Trustingleen is a 26 kg RipStik rider. In the photo a physical science instructor has convinced Trustingleen to ride a RipStik down a sloping sidewalk with an abrupt end. The plan is for the instructor to catch Trustingleen and let the board crash off the end of the sidewalk. For this to work the instructor will have to bring Trustingleen from a speed of 3 m\/s to 0 m\/s in 0.3 seconds. How much force will the instructor have to exert to bring Trustingleen to a safe stop?\n14. Section six heat: \u0394T = ____________________ \u00b0C What is the change in temperature \u0394T for metal which contained the most heat energy?\n15. ___________________ Which material holds the most heat?\n16. Section seven earth coordinates:____________________ A student walked north on a line of longitude starting at N 6\u00b0 54.467' and ending at N 6\u00b0 54.567'. The student measured a distance of 192 meters. Determine the number of meters per minute of latitude based on this data.\n17. Section eight weather: __________________ What type of cloud is shown in the image?\n18. I wake the babes from their midnight dreams, I shake the ground and rattle the beams, Before me comes a bright blue flash, While all around me the rains lash. I roll across hill and dale, I reflect off of rill and vale, The spawn of Bergeron, With a crack, bang, or rumble, I bring the fear on. What am I? __________________\n19. Section nine waves: A RipStik was ridden across a wet cloth towel soaked in water with food color. The RipStik was then swizzled across a large sheet of presentation paper. The swizzle wave can be seen in the diagram below.\n\n\u03bb = _______________ Determine the wavelength \u03bb of the RipStik swizzle wave.\n20. a = _______________ Determine the amplitude a of the RipStik swizzle wave.\n21. \u03c4 = _______________ The RipStik took a duration of 1.36 seconds to travel the 68 centimeters seen on the diagram above. Calculate the period \u03c4 for the RipStik swizzle wave.\n22. f = _______________ Calculate the RipStik swizzle wave frequency f.\n23. \u0475wave = _______________ Use the wavelength \u03bb and frequency f to calculate the velocity \u0475wave of the RipStik swizzle wave.\n24. _______________ Given a speed of sound of 350 m\/s and an echoing surface 100 meters away, how long a duration in seconds would you expect for a sound to travel out and echo back to you?\n25. Section ten spectra: Orange and brown are the same hue. Both orange and brown have a hue angle of 30\u00b0 What characteristics of light make these two colors different?\n26. Section twelve electricity: current i = ____________ In my kitchen a 780 Watt toaster, a 300 Watt rice cooker, and a 1250 Watt microwave oven are all connected to a single 120 Volt outlet. If all three are turned on, calculate the current i used by all three together.\n27. __________ An outlet can safely deliver 18 Amps. Can I safely run all three appliances in the above question at the same time?\n28. Section eleven optics: The data provided below is the apparent depth of the image of a penny under sheets of glass versus the actual depth of the penny. Plot the data on the graph.\n29. ________________ Calculate the slope of the linear best fit line. The slope is the index of refraction for glass.\n30. Section thirteen chemistry: A purple flower was boiled in water. The water turned purple. Based on the general rules for color change discovered in that laboratory, determine from the color changes listed below whether the following unknowns are likely to be an acid or a base. Put your answers in the blanks below the unknown.\nsubstancecolor changeacid or base?\ncream of tartarbright pink\nsodium hydroxide (lye)dark blue green\npotassium chloridepurple\n31. Section fifteen site swap notation:\nThe mathematical equation 4 + 5 = 9 is to the site swap equation 334233\nas\nthe mathematical equation 3 + 5 = 10 is to the site swap equation 252525\nExplain why.","date":"2018-05-21 16:46:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5004072785377502, \"perplexity\": 2497.1942012898044}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794864461.53\/warc\/CC-MAIN-20180521161639-20180521181639-00004.warc.gz\"}"}
| null | null |
Q: Automapper map source to dest with custom object How I can map:
Object: (
class CashCaseDTO { string CaseId {get;set;} double CaseTotalAmount {get;set; }
To:
public class CashCase
{
public string Id { get; set; }
}
public class CashCaseDifference
{
public decimal Amount { get; set; }
}
I have problem with mapping amount to Difference.Amount.
What should I use? Custom Value resolver or converter?
A: This conversion is supported by default in the latest version.
|
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HomeNewsFor the first time, female security officers deployed in Masjid al-Haram
For the first time, female security officers deployed in Masjid al-Haram
Staff Desk
Female security officials will make female visitors follow Corona epidemic precautionary measures in the Grand Mosque.
Makkah: For the first time in Saudi Arabia, female military officials have been deployed to provide security at the Grand Mosque (Masjid al-Haram) during Ramadan, Hajj, and Umrah.
According to the international news agency, the Saudi government had decided to deploy female officials to check and monitor the women coming to the Masjid al-Haram, which has now been implemented.
On the social networking site Twitter, the Saudi Interior Ministry shared photos of female security personnel performing their duties in the Masjid al-Haram and wrote in the caption that female military personnel is also performing security duties during Hajj and Umrah.
#من_الميدان ، "أمن الحج والعمرة"@security_gov pic.twitter.com/5j93CKcmzl
— وزارة الداخلية (@MOISaudiArabia) April 19, 2021
During the Corona epidemic, it was becoming difficult for male personnel to implement SOPs from female worshipers and devotees, on which female personnel has been deployed. In addition, they will be responsible for the security and precautionary measures of female visitors.
It should be noted that under the vision of Saudi Crown Prince Muhammad bin Salman, the policy of empowering women is being implemented in the kingdom while women are also being trained in the security sector.
Masjid Al Haram
Staff Desk of The Islamic Information covers the latest news and valuable information that Muslim readers need.
Russian Air Strikes Kills 200 in Syria
Indian Islamic Scholar Maulana Wahiduddin Khan Passes Away at 96
Tina Rahimi, First Hijabi Boxer at the 2022 Commonwealth Games Wins Medal
byMaisah
Ramadan 2018 Will Probably Start From May 17
Saudi Arabia Has Allowed Flights To Israel For The First Time!
Muslims In Norway Recited Quran On The Same Street After The Incident
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Home Tags Posts tagged with "Brittney Schulman"
Brittney Schulman
Limo driver indicted in Cutchogue crash that killed Smithtown women
by Phil Corso - March 16, 2016
Suffolk County District Attorney Tom Spota outlines the investigation on Wednesday. Photo by Alex Petroski
A makeshift memorial is erected at the scene of the fatal Cutchogue crash. Photo by Phil Corso
By Phil Corso & Alex Petroski
Story last updated 3.17.16, 8:15 a.m.
A fatal crash was the result of a limousine's dangerous U-turn at a busy intersection in Cutchogue, and on Wednesday, a special grand jury placed the blame on the driver.
Carlos F. Pino, 58, of Old Bethpage, surrendered to police Wednesday and was arraigned on four charges of criminally negligent homicide, four counts of assault, failure to yield the right of way, reckless driving and other traffic violations, Suffolk County District Attorney Tom Spota said. Pino was attempting a U-turn near the intersection of Depot Lane and County Route 48 on July 18 when Steven Romeo, 55, of Peconic, T-boned the limo, killing four and injuring six.
The crash killed Smithtown's Brittney Schulman, 23, and Lauren Baruch, 24, as well as Stephanie Belli, 23, of Kings Park, and Amy Grabina, 23, of Commack, who were all riding in a limousine in the middle of a weekend wine tour on the eastern part of the Island. The collision also injured passengers Joelle Dimonte, 25, of Elwood, Melissa Angela Crai, 23, of Scarsdale, Alicia Arundel, 24, of Setauket, and Olga Lipets, 24, of Brooklyn. Romeo, the DA said, was operating the truck under the influence of alcohol and was charged with driving while intoxicated the day of the crash.
Pino pleaded not guilty to the charges on Wednesday and was given a cash bail $50,000 and bond of $100,000. His next court date was scheduled for April 19. Romeo also pleaded not guilty to two counts of driving while intoxicated and one charge of driving while ability impaired by alcohol on Wednesday and was released with his next court date set for April 26.
"I think they may have been somewhat surprised," Spota said when family members of the victims were notified that Pino, and not Romeo, would be indicted as a result of the crash. "They either expected that it would be the other way around, that Romeo would be the party who would be completely at fault, or perhaps it was just a totally unavoidable accident. Indeed, what the grand jury has found out is that it was totally unavoidable, only as to Romeo, but not as to Pino."
At the scene, Pino had told police he did not see any oncoming traffic, Spota said. But the subsequent investigation revealed why.
The county had been investigating the crash over recent months, and on Wednesday, the DA announced that while Romeo may have been driving while impaired, the risky U-turn still made it nearly impossible for the collision to be avoided. The grand jury conducted a five-hour investigation of the crash and found that Pino had "limited sight lines looking into westbound traffic" because a Jeep Liberty was positioned in the intersection waiting to turn left onto Depot Lane, Spota said in a statement.
Spota said the Jeep Liberty "completely blocked the limo driver's view of the oncoming traffic in the main travel lanes." And despite the fact that the main westbound travel lanes were not visible, the DA said Pino failed to take any precaution to make sure he could safely enter the westbound travel lanes and he continued to make the U-turn.
"A perfectly sober Steven Romeo could not avoid this crash. An intoxicated Steven Romeo could not avoid this crash. It was simply unavoidable from Romeo's perspective," Spota said. "Romeo can be held criminally responsible for driving while intoxicated but he cannot be held criminally responsible for the crash."
Related: Vineyard visit ends in tragedy for Commack, Smithtown West grads
Southold Police Chief Martin Flatley said during the press conference that unfortunately many limo drivers exiting Vineyard 48 in Cutchogue try to make the dangerous left U-turn that ended up being fatal, because it is the fastest route to head back west.
"There are other ways to head back west, but that's the easiest way for them to do it," Flatley said.
There is now a traffic light at that intersection, Flatley said.
Spota said Romeo was heading west at about 55 miles per hour when the crash occurred. He did not see the limo enter the intersection until he was about 200 feet away, the district attorney said.
"Mr. Romeo had only 200 feet to react to the hazard he saw, and stop his vehicle," Spota said. "Traveling at 55 mph, it would have taken 1.6 seconds to perceive the limo in his path, to realize he must apply his brakes, and then to begin braking. This would leave Romeo with even less distance, 129 feet, to avoid a crash — impossible for him to do. In fact our experts tell us that at 55 mph it would have taken anyone 263 feet to stop and avoid the crash."
After investigating the crash, Spota said the incident was "unavoidable," thus keeping a grand jury from indicting Romeo for vehicular manslaughter or criminally negligent homicide.
From left, Amy Grabina, Brittany Schulman, Lauren Baruch and Stephanie Belli. Photos from Facebook
Belli, Baruch and Schulman were all decorated members of the national and language honor societies by the time they graduated from Smithtown High School West. Over the summer, schools Superintendent James Grossane said Belli, a 2010 graduate, had an infectious smile and was an enthusiastic student and member of the district's championship kick line team. Baruch, a 2009 graduate, was best known for her booming laugh and unforgettable smile, Grossane said. Schulman, he said, was another 2010 graduate and had a profound love for her family.
Grabina graduated in 2010 from Commack High School and went on to pursue accounting at Florida State University, ultimately landing a job at Ernst & Young in Tallahassee, Fla.
Tragic Cutchogue crash spurs Stony Brook into action
by Phil Corso - August 25, 2015
From left to right, Stephanie Belli's sister Diana and mother Carol receive their copy of the book with Rabbi Cohen of Chabad at Stony Brook. Photo from Chabad at Stony Brook
Four hundred acts of kindness turned out to be an underestimate.
It has been one month since a horrific Cutchogue car crash killed four North Shore women, and Chabad at Stony Brook set out to assemble a book of kind acts to show how good could come out of tragedy. But by the time that book was finished last week, it had grown into a much bigger list.
Smithtown's Brittney Schulman, 23, and Lauren Baruch, 24, as well as Stephanie Belli, 23, of Kings Park, and Amy Grabina, 23, of Commack were riding in a limousine in the middle of a weekend wine tour on the eastern part of the Island when Steven Romeo, 55, T-boned their vehicle as it made a risky U-turn, killing the girls and injuring five others.
After the crash, Romeo was arraigned at Eastern Long Island Hospital and charged with driving while intoxicated. He was initially ordered held in lieu of $500,000 cash bail, or $1 million bond, but that bail was reduced to $50,000 cash or $100,000 bond. Suffolk County District Attorney Tom Spota said Romeo had recorded a blood alcohol content of .066 percent — below the legal limit of .08 — when he was tested roughly one hour after the crash. The DWI charge, however, was not dropped, Spota said. No additional charges were filed against Romeo as the investigation continued.
Romeo's court date, which was originally set for last week, was adjourned to Sept. 18.
The tragedy sent shockwaves through the greater North Shore community, and Chabad at Stony Brook called on everyone to help.
"People came out in big numbers to post all these heartfelt things they were going to do," said Rabbi Shalom Ber Cohen of Chabad at Stony Brook, who helped launch the project in the wake of the tragic crash. "We've always encouraged to respond to darkness with light, and to evil with good."
The group launched a Facebook group called "Goodness & Kindness x 400 for our girls," and acquired thousands of page views in a matter of days, Cohen said. The goal, he said, was to remember the lives of those lost by compiling a book of names and acts of goodness committed in their honor, to show victims' families that they were not alone in their darkest hour.
"We felt we were swarming in death," Cohen said. "This was an act of goodness and kindness to bring more goodness to the world. While we can't bring the girls back, when the community comes back and shows we are there, it does bring some kind of goodness."
Good deeds included anything from committing to donate to worthy causes to something as simple as paying for succeeding cars in a Starbucks drive-thru.
Cohen, along with wife Chanie Cohen, a Chabad program coordinator, as well as Rabbi Chaim Grossbaum, Rabbi Motti Grossbaum and the rest of his staff, delivered those books to the victims' families over the last week and said they helped everyone move forward in a time of great loss.
Diana Belli, sister of Stephanie Belli, took to the "Goodness & Kindness" Facebook page to express her gratitude.
"Thank you so much! With love, my entire family," she wrote on the page. "This means a lot to us."
North Shore seeks solace in wake of deadly limo crash
by Phil Corso - July 29, 2015
Red ribbons are one way North Shore residents are remembering the fatal crash victims. Photo from Smithtown Historical Society
One week has passed, but no amount of time can ever truly heal the wounds endured by the greater North Shore community since four of its own were killed in a horrific limousine crash.
Anyone driving through the streets of Smithtown and its surrounding communities this week could notice the red ribbons wrapped around trees in memory of Smithtown's Brittney Schulman, 23, and Lauren Baruch, 24, as well as Stephanie Belli, 23, of Kings Park, and Amy Grabina, 23, of Commack. The four girls were killed when Steven Romeo, 55, T-boned their limousine with his pickup truck in Cutchogue last Saturday, injuring Romeo, along with limo driver Carlos Pino, 58, of Bethpage, Joelle Dimonte, 25, of Elwood, Melissa Angela Crai, 23, of Scarsdale, Alicia Arundel, 24, of Setauket, and Olga Lipets, 24, of Brooklyn.
After the crash, Romeo was arraigned at Eastern Long Island Hospital and charged with driving while intoxicated. He was initially ordered held in lieu of $500,000 cash bail, or $1 million bond, but that bail was reduced to $50,000 cash or $100,000 bond last Thursday, according to Suffolk County District Attorney Tom Spota. At a press conference on Friday, Spota said Romeo had recorded a blood alcohol content of .066 percent when he was tested roughly one hour after the crash. The DWI charge, however, was not dropped despite his BAC coming in below the legal limit of .08, Spota said. No additional charges were filed against Romeo as the investigation continued.
The past week saw the funerals of all four of the victims, while those injured were released from hospital care by the middle of this week. The North Shore community planned to take one of its first steps toward closure on Wednesday night at Smithtown High School West, where residents, elected officials and members of Mothers Against Drunk Driving were scheduled to meet. The event was borne out of a Facebook page titled "Candlelight Vigil for Our Girls," which was put into action in the days following the tragedy. By Wednesday, the page had collected more than 6,000 names to its roster and countless photos of mourning and support for the victims' families.
Marianne Howard, executive director with the Smithtown Historical Society, was one of the several Smithtown residents to tie red ribbons around trees in front of the society's property. She said various businesses throughout town, including Towers Flowers of Nesconset and James Cress Florist of Smithtown, helped donate the ribbons to the cause.
"We mourn the loss of four beautiful souls who were taken too early from our community," she said. "We send our deepest condolences to their families and friends. May they rest in peace."
Chabad at Stony Brook also signed onto the cause of finding good in a tragic situation, launching its own Facebook event page, "responding to dark, with light," in memory of the four girls and challenging residents to commit 400 random acts of goodness and kindness in their honor.
Chaya Klein Grossbaum of Chabad at Stony Brook said once the goal was reached, the group would print a book detailing each singular act.
Crash victims could have been any of us
by TBR Staff - July 23, 2015
Tragedy hit close to home over the weekend — countless lives were shattered when an alleged drunk driver slammed into a limousine carrying a group of eight young women, killing four who hailed from our own North Shore communities.
Saturday's Cutchogue crash captivated communities near and far. Those who knew the women, and even those who didn't, mourned, as the crash sent shock waves across the Island.
Brittney Schulman, Lauren Baruch, Stephanie Belli and Amy Grabina were friends, daughters, girlfriends, sisters and young women just starting their adult lives. Tragic doesn't even begin to explain what happened on that Cutchogue road.
But the women weren't alone, and the surviving four women, who remain hospitalized as of Monday, need our support.
At a press conference on Monday, Suffolk County District Attorney Tom Spota told a crowd of reporters, many of whom came from affiliate stations and out-of-town papers, to be reasonable, in light of a recent incident in which a member of the press entered the hospital in an attempt to see one of the survivors.
"We have four who survived, who certainly have suffered horrible, horrible trauma," Spota said. "Not only bodily trauma, but certainly mentally. And we have people — reporters — who are trying to sneak in to talk to these young women. I just think that we really should — let's all think about it and let's be reasonable here."
We find these actions disrespectful to the victims and survivors and their families and do not stand behind them. As journalists, we understand the responsibility news organizations have to inform the public about events such as this, but sneaking into a hospital room is excessive, and it is not right to serve a readership at a victim's expense.
As a community newspaper, we are protective of the neighborhoods we cover because we live here. When we get word of car crashes, many of us have to wonder if a loved one was involved. What happened on Saturday could have happened to any one of us.
To the women recovering, the families affected and the communities trying to come to terms with these losses, we will still be here to listen if and whenever you are ready to speak. Our thoughts are with you.
Governor commits millions to energy storage projects
How Suffolk Democratic leadership is looking toward the 2019 and 2020 elections
Suffolk County Police Department Lacrosse Huntington Kevin Redding Bill Landon Stony Brook University Police Kings Park Miller Place Stony Brook Rocky Point Suffolk County Port Jefferson Village Centereach Port Jefferson Desiree Keegan Port Jefferson Station Rita J. Egan Northport Ward Melville Smithtown Daniel Dunaief Setauket Huntington Station Mount Sinai
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 5,763
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Non-Alcoholic Fatty Liver Disease (NAFLD)
LITMUS Partners
Members Zone
The European IMI2-funded LITMUS consortium has reached a major milestone in the biomarker qualification process by having our letter of intent (LOI) accepted by the U.S. Food and Drug Administration (FDA). With this LOI, LITMUS aims to qualify non-invasive biomarkers to enable simpler diagnosis and better treatment for patients with non-alcoholic steatohepatitis (NASH). The FDA supports and encourages the study of non-invasive biomarkers for NASH, where better tools to diagnose and treat patients are needed. The acceptance allows LITMUS to submit a qualification plan, providing a more comprehensive description of the steps the requestor will take to provide the necessary supportive data for biomarker qualification. At the same time LITMUS has also finalized two scientific advice interactions with the European Medicines Agency (EMA).
Members of the WP7 Regulatory team involved in creating the LOI are as follows:
Elisabeth Erhardtsen (Nordic Bioscience)
Richard Torstenson (Abbvie)
Daniel Guldager Kring Rasmussen (Nordic Bioscience)
Morten Karsdal (Nordic Bioscience)
Guido Hanauer (Takeda)
Camilla Bertelsen (Novo Nordisk)
Julia Brosnan (Pfizer)
Vlad Ratziu (Hôpital Pitié-Salpêtrière, Paris)
Patrick Bossuyt (Amsterdam University Medical Centre)
Quentin M. Anstee (Newcastle University)
GROUND-BREAKING €34 MILLION PROJECT TO DEVELOP BETTER TEST FOR LIVER DISEASE
A pioneering European research project aims to lead to new diagnostic tests to assess patients with non-alcoholic fatty liver disease (NAFLD) and identify those most at risk for developing severe inflammation and liver scarring.
Liver Investigation: Testing Marker Utility in Steatohepatitis (LITMUS) funded by the European Innovative Medicines Initiative 2 Joint Undertaking, brings together clinicians and scientists from prominent academic centres across Europe with companies from the European Federation of Pharmaceutical Industries and Associations (EFPIA). Their common goals are developing, validating and qualifying better biomarkers for testing NAFLD.
The €34 million project is co-ordinated by Newcastle University, working closely with the lead EFPIA partner, Pfizer Ltd. LITMUS will include 47 international research partners based at leading international universities and some of the world's largest pharmaceutical companies.
Affecting 20 – 30 % of the population worldwide, NAFLD is caused by a build-up of fat in the liver cells, which leads to inflammation, scarring of the liver and ultimately cirrhosis. It is strongly linked to obesity and type 2 diabetes.
Although many people have NAFLD, less than one in 10 will come to harm as a result1. The challenge is to identify those people that will be most severely affected and are going to progress to liver cirrhosis or cancer so that appropriate care can be provided earlier. At present this requires a liver biopsy, which can only be done in specialist hospitals, so there is a need for better diagnostic tools.
Professor Quentin Anstee, from Newcastle University's Institute of Cellular Medicine and Consultant Hepatologist at Newcastle Hospitals NHS Foundation Trust, is co-ordinating the LITMUS consortium.
He said: "Non-alcoholic fatty liver disease is already the most common underlying cause of liver transplant in the USA and, with the obesity epidemic in Europe, we are very close behind.
"LITMUS will unite clinicians and academic experts from centres across Europe with scientists from the leading pharmaceutical companies, all working together to develop and validate new highly-accurate blood tests and imaging techniques that can diagnose the severity of liver disease, predict how each patient's disease will progress and monitor those changes, better or worse, as they occur.
"Lack of easy and accurate diagnostic tests means that many patients go undiagnosed until late in the disease process. It has also held-back efforts to develop new medical treatments for NAFLD. Availability of better diagnostic tests will help us to target care at an early stage of disease to the people who are going to be most severely affected. It will also help us to develop more effective medical treatments for NAFLD and to run the clinical trials that the regulatory agencies need so that they can licence these medicines to be prescribed by doctors."
Professor Chris Day, Vice-Chancellor and President, Newcastle University who is himself a Consultant Hepatologist with an international reputation and part of the research team, added: "Tackling Non-alcoholic fatty liver disease is a major public health challenge and the award of such a large grant from the EU, allowing us to bring together pharma and academia in this way, gives us real hope of making significant advances in the diagnosis and treatment of this increasingly common and often devastating disease."
Julia Brosnan, Senior Director of External Collaborations & Scientific Alliances, Pfizer, who also serves as the industry project lead for LITMUS, said: "This is an exciting project and we look forward to working with the other LITMUS partners to develop new diagnostic tests for NAFLD, which is too often undiagnosed in patients. We hope the results of this project will help change that."
This project has received funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No 777377. This Joint Undertaking receives support from the European Union's Horizon 2020 research and innovation programme and EFPIA.
Living with NAFLD – Yvonne's story
Yvonne Gray, 62, was diagnosed with NAFLD in 2011. A retired headteacher who lives in Fulwell, Sunderland, she is now a governor of the national adult liver patient support group, LIVErNORTH:
"I do consider myself really fortunate in that it was by chance my liver disease was eventually diagnosed, when my "mildly abnormal" liver function tests were picked up by a consultant treating me for a Vitamin D deficiency and he then referred me to a hepatologist.
"By the time I was referred to the Freeman Hospital Liver Unit in Newcastle I had Non-Alcoholic Fatty Liver Disease, specifically NASH, Stage 3 Liver Disease with significant fibrosis.
"It is interesting that even now my liver function tests are still presenting as 'mildly abnormal', thereby not appearing to trigger any major cause for concern. However, my two liver biopsies say something totally different.
"It is exciting to hear about this new collaboration. I would be delighted if there could be a simple blood test. The alternative, a liver biopsy, can involve a stay in hospital, is uncomfortable and can be painful."
NAFLD – the facts
A healthy liver should contain little or no fat, however, it is thought that 1 in every 3 people in Europe has some degree of NAFLD, where there is an excessive amount of fat accumulation in the liver. While this doesn't always cause harm, it can develop into an inflammatory form of the condition called steatohepatitis (NASH) that in turn causes fibrous scar tissue to form in the liver and leads to serious liver damage including cirrhosis in some patients. It can also increase the risk of cancer in the liver, heart attack, and stroke.
Follow LITMUS on Twitter: @LITMUS_IMI
This project has received funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No 777377.
Copyright © 2022 LITMUS Project
This project has received funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No. 777377. This Joint Undertaking receives support from the European Union's Horizon 2020 research and innovation programme and EFPIA. Visit IMI.
The information contained in this website reflects only the author's view. Neither IMI nor the European Union, EFPIA, or any Associated Partners are responsible for any use that may be made of the information it contains.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,976
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Q: Regex: Exclude character in a match Given the following characters 0100003949753>3132471430009343+ 010001628> I would like to select everything between > and + but I dont want to include the > and +.
This pattern >.*\+ results in >3132471430009343+. Can someone tell me how to exclude the two characters in order to get 3132471430009343?
A: Use lookbehind or \k,
(?<=>)[^+]*(?=\+)
DEMO
OR
>\K[^+]*(?=\+)
DEMO
OR
Use capturing groups,
>([^+]*)(?=\+)
DEMO
A: You should simply use capturing group:
>(.*)\+
to match selected part. Demo
A: It depends on the program which you using, but look for positive look behind/ahead and grouping.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,552
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FCA challenges Italy tax agency concerning the value of U.S. business
Fiat Chrysler Automobiles (FCA) has challenged a claim by Italy's tax officials over the valuing of its U.S. Chrysler business that could leave it with an unwanted tax bill just days prior to an expected key merger agreement with Peugeot owner PSA.
On Wednesday, a source knowledgeable with the matter stated the Italian tax agency believed Fiat Chrysler (FCA) had undervalued the value of its U.S. business, after its phased acquisition of Chrysler by 5.1 billion euros.
"We strongly disagree with this preliminary report," stated an FCA spokesperson.
The tax audit comes at a delicate time for the automaker which is finalizing discussions with PSA, the maker of Peugeot and Citroen, over a scheduled $50 billion merger to form the world's fourth-largest automaker.
A source close to PSA stated the details were public and came as no surprise for the French automaker.
The tax audit is "another complication" in the form of a binding merger agreement with PSA, however, one but which FCA can manage, an expert at Italian broker Equita said.
"We think it will be less relevant than the GM lawsuit, as negotiations with tax authorities are ongoing, which are expected to be closed by year-end and possibly leading to an accord on a much lower amount," Martino De Ambroggi stated.
Last month competitor General Motors submitted a racketeering lawsuit against FCA, accusing it of bribing union officials in the U.S. over many years to corrupt the bargaining process.
FCA has turned down these allegations as "groundless".
The Italian tax authority audit, which is associated with the transactions dating back to 2014, could result in FCA having to pay back taxes for $1.5 billion, the source stated.
"We are sure that we will successfully make the case for a material reduction in the assessment," the FCA spokesperson stated.
Equita FCA Fiat Chrysler Automobiles Martino De Ambroggi
Lawrence Stroll looks for a huge stake in Aston Martin
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 8,191
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Nobody, have been unable to find any. Perhaps this website will lead to a few.
Mr. Ross Coffin - he taught me the true value of the written word.
To join the Navy and yes I did. However, it was far from what I thought it would really be like.
Can't remember..... I think I was too drunk.
Steve Pike. He was my best friend and I haven't heard anything from him since about 1979.
This is a new survey submission, submitted on November 24, 2004.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,459
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Brentwood School District » News » What's New » Literacy Support Specialist
Literacy Support Specialist
BRENTWOOD SCHOOL DISTRICT WELCOMES LITERACY SUPPORT SPECIALIST
BRENTWOOD, MO – The Brentwood School District is pleased to announce that Ms. Kelsey Grammer has been approved by the Board of Education to fill the newly created role of Literacy Support Specialist. The role was created to support the district's literacy goals by working with classroom teachers and administrators on literacy curriculum, research, and evidence-based reading interventions.
"We are incredibly excited to have Kelsey take on this new role," said Dr. Brian Lane, Brentwood Superintendent. "Literacy is the foundation for our students' academic success and our students, staff, and families will greatly benefit from Kelsey's knowledge and enthusiasm." As part of her duties, Ms. Grammer will oversee the state dyslexia mandate that was introduced last year.
Ms. Grammer came to Brentwood in 2017 from Monroe Elementary in the St. Charles School District, where she was a Second and Fourth Grade Teacher. During her six years there, she served in several leadership roles related to curriculum and instruction. Ms. Grammer earned a Bachelors of Science in Education and Master of Education in Learning, Teaching, and Curriculum, both from the University of Missouri, Columbia.
In addition to her leadership experiences in education, Ms. Grammer has dedicated much of her career to work in literacy. She has attended the Reading Institute at Teachers' College in New York City as well as other conferences led by Lucy Calkins, Jennifer Serravallo and other leaders in literacy curriculum development. "I am very excited to step into the role of Brentwood's Literacy Support Specialist," Grammer said. "I have a passion for literacy instruction and am looking forward to working with all of our hard-working teachers and students."
While this is a new position for the district, the administration was able to make it cost-neutral by restructuring the science coach position. "When we have a need in the district, we will address it," Dr. Lane said. "But we will always strive to be good stewards of our taxpayer dollars in doing so," he said.
One of the first major tasks for Ms. Grammer will be to help implement and coordinate school-wide literacy programs for the district. Part of this work will be to help manage the requirements of the dyslexia mandate that was established last year. The Literacy Support Specialist will work with the reading specialist regarding dyslexia screening, assist grade-level teams on the core curriculum, and provide support for student interventions.
The introduction of this role was a direct result of the work by the Brentwood School District literacy strategic planning committee work and will help offer needed support to staff members, students, and families in areas of literacy and dyslexia.
For the latest news, information, and updates, follow the district on Twitter @BrentwoodMOSD or visit www.brentwoodmoschools.org.
Founded in 1920, the Brentwood School District is a highly rated, award winning public school district serving the Brentwood community in St. Louis County, Missouri.
www.brentwoodmoschools.org
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,818
|
ALTER TABLE ion_sessions CHANGE user_data user_data TEXT CHARACTER SET utf8 COLLATE utf8_unicode_ci NULL DEFAULT NULL ;
ALTER TABLE ion_sessions CHANGE user_agent user_agent TEXT CHARACTER SET utf8 COLLATE utf8_unicode_ci NULL DEFAULT NULL ;
-- Media lang
ALTER TABLE media_lang CHANGE alt alt VARCHAR( 255 ) CHARACTER SET utf8 COLLATE utf8_unicode_ci NULL DEFAULT NULL;
-- Extended fields modifications
ALTER TABLE extend_field CHANGE `value` `value` TEXT CHARACTER SET utf8 COLLATE utf8_unicode_ci NULL DEFAULT NULL;
ALTER TABLE extend_field CHANGE description description VARCHAR( 255 ) CHARACTER SET utf8 COLLATE utf8_unicode_ci NULL;
-- User table modification
ALTER TABLE users CHANGE username username VARCHAR(120) CHARACTER SET utf8 COLLATE utf8_unicode_ci NOT NULL;
ALTER TABLE users ADD salt VARCHAR(50) NOT NULL;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,235
|
<?php
namespace Bake\Shell\Task;
/**
* Task for creating cells.
*
* @property \Bake\Shell\Task\BakeTemplateTask $BakeTemplate
* @property \Bake\Shell\Task\TestTask $Test
*/
class CellTask extends SimpleBakeTask
{
/**
* Task name used in path generation.
*
* @var string
*/
public $pathFragment = 'View/Cell/';
/**
* {@inheritDoc}
*/
public function name()
{
return 'cell';
}
/**
* {@inheritDoc}
*/
public function fileName($name)
{
return $name . 'Cell.php';
}
/**
* {@inheritDoc}
*/
public function template()
{
return 'View/cell';
}
/**
* Bake the Cell class and template file.
*
* @param string $name The name of the cell to make.
* @return string
*/
public function bake($name)
{
$this->bakeTemplate($name);
return parent::bake($name);
}
/**
* Bake an empty file for a cell.
*
* @param string $name The name of the cell a template is needed for.
* @return void
*/
public function bakeTemplate($name)
{
$templatePath = implode(DS, ['Template', 'Cell', $name, 'display.ctp']);
$restore = $this->pathFragment;
$this->pathFragment = $templatePath;
$path = $this->getPath();
$this->pathFragment = $restore;
$this->createFile($path, '');
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,640
|
Madison Keys falls apart at Bank of the West…
Madison Keys falls apart at Bank of the West Classic
By Carl Steward |
STANFORD — For a set, rising American hopeful Madison Keys looked every bit the worthy stand-in for Serena Williams, who was supposed to play her first Bank of the West Classic match Wednesday night before her injury withdrawal just five days earlier.
Keys started her night by pummeling Australian Ajla Tomljanovic 6-1 in a 23-minute opening set. The tall 20-year-old Floridian lost just two points on her serve in the feature match, hammered four aces, won six straight games while barely breaking a sweat and made her off-court friend Tomljanovic look overmatched against her powerful ground strokes.
But a funny thing happened on the way to Keys filling Serena"s shoes at Stanford"s Taube Family Tennis Center. Tomljanovic picked up her game, and Keys completely fell apart, ultimately losing 1-6, 6-4, 6-1 to the 22-year-old who was born in Croatia.
It was a startling turnabout, and Keys" exit was a tough blow for the Bank of the West event, which not only lost Williams before the tournament even began but had two other seeded players, No. 3 Carla Suarez Navarro and No. 6 Andrea Petkovic, ousted in the afternoon session.
A decent crowd of 2,415 turned out to watch the seventh-seeded Keys — even though many of the fans in attendance had obviously bought advance tickets to see world No. 1 Williams — and they gave Keys a rousing ovation when she was introduced before the match. And for the first set, she gave them a Serena-like show. Even Tomljanovic was wowed.
"I really thought I"d be off the court in about 35 minutes," said Tomljanovic, unseeded and ranked 69th in the world.
But once Tomljanovic started finding the range, Keys didn"t react well. Her power game turned erratic, then came fully unglued in a lopsided final set. After scoring three breaks of serve in the opening set, Keys didn"t break Tomljanovic once over the final two sets while being broken four times.
"She really raised her level, and I think my level dropped," Keys said. "It"s never easy feeling like you were playing so well in the first set and then you let things get away. It"s tough to keep yourself in the moment and get yourself to calm down, and I wasn"t able to. I just have to take it, learn from it and move on."
In short, it was a very disappointing effort from a player many believe to be the American heir apparent to Williams. For the most part, Keys has made progress toward that end, reaching the semifinals of this year"s Australian Open and the quarterfinals at Wimbledon. She came into the Bank of the West event 18th in the world, just two spots off her highest ranking.
On the flip side, Keys has yet to win a WTA event on American soil and has made just two WTA finals since turning pro in 2009.
Williams, who pulled out with an elbow injury, will be looking to complete the Grand Slam at the U.S. Open in September and is also seeking to tie Steffi Graf for the most Grand Slam singles titles in the Open era at 22.
American Alison Riske scored the first major upset of the tournament when she knocked off Spain"s Suarez Navarro 6-4, 6-4. The 25-year-old Riske, unseeded and ranked 59th, won five of the last six games in the first set, then jumped out to a 5-2 lead in the second set but had to hold on to turn back a rally by Suarez Navarro, ranked 10th in the world.
Petkovic fell to fellow German Mona Barthel 5-7, 6-2, 7-6 (4). Petkovic knocked off Venus Williams at last year"s Bank of the West in the quarterfinals before losing to Serena in the semis.
In the quarterfinals, Barthel could meet top-seeded Caroline Wozniacki, who makes her tournament debut Thursday night against American Varvara Lepchenko.
In the late match, eighth-seeded Elina Svitolina outlasted former Stanford star Nicole Gibbs 6-3, 7-6 (5) to reach the quarterfinals.
Follow Carl Steward on Twitter at twitter.com/stewardsfolly.
View a Bank of the West photo gallery at photos.mercurynews.com.
Arraignment delayed for teen charged in 2018 Fairfield mall shooting
Carl Steward
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,785
|
{"url":"http:\/\/mathonline.wikidot.com\/higher-order-directional-derivatives","text":"Higher Order Directional Derivatives\n\n# Higher Order Directional Derivatives\n\nRecall from the Directional Derivatives page that if $z = f(x, y)$ is a two variable real-valued function and $\\vec{u} = (a, b)$ is a unit vector, then the directional derivative of $f$ in the direction of $\\vec{u}$ is given by:\n\n(1)\n\\begin{align} \\quad D_{\\vec{u}} \\: f(x, y) = \\lim_{h \\to 0} \\frac{f(x + ha, y + hb) - f(x, y)}{h} \\end{align}\n\nWe saw that we could compute directional derivatives of $f$ with the following formula:\n\n(2)\n\\begin{align} \\quad D_{\\vec{u}} \\: f(x, y) = \\left ( a, b \\right ) \\cdot \\left ( \\frac{\\partial z}{\\partial x} , \\frac{\\partial z}{\\partial y} \\right ) = a \\frac{\\partial z}{\\partial x} + b \\frac{\\partial z}{\\partial y} \\end{align}\n\nWe will now look at computing higher order directional derivatives. The process is much the same as computing higher order partial derivatives. Suppose that we have a function $z = f(x, y)$ and a unit vector $\\vec{u} = (a, b)$. Then the first directional derivative of $f$ in the direction of $(a, b)$ is $D_{\\vec{u}} \\: f(x, y) = a \\frac{\\partial z}{\\partial x} + b \\frac{\\partial z}{\\partial y}$ as noted above. The directional derivative in the direction of $\\vec{u}$ will define a function, say:\n\n(3)\n\\begin{align} \\quad z^* = D_{\\vec{u}} \\: f(x, y) \\end{align}\n\nIf we want to take the second directional derivative of $f$ in the direction of $(a, b)$, then we have that:\n\n(4)\n\\begin{align} \\quad D^2_{\\vec{u}} \\: f(x, y) = (a, b) \\cdot \\left ( \\frac{\\partial z^*}{\\partial x} , \\frac{\\partial z^*}{\\partial y} \\right ) \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = a \\frac{\\partial z^*}{\\partial x} + b \\frac{\\partial z^*}{\\partial y} \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = a \\frac{\\partial}{\\partial x} \\left ( D_{\\vec{u}} \\: f(x, y) \\right ) + b \\frac{\\partial}{\\partial y} \\left ( D_{\\vec{u}} \\: f(x, y) \\right ) \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = a \\left ( a \\frac{\\partial^2 z}{\\partial x^2} + b \\frac{\\partial^2 z}{\\partial x \\partial y} \\right ) + b \\left ( a \\frac{\\partial^2 z}{\\partial y \\partial x} + b \\frac{\\partial^2 z}{\\partial y^2} \\right ) \\end{align}\n\nIf the second partial derivatives of $f$ are continuous, then by Clairaut's theorem we have that:\n\n(5)\n\\begin{align} \\quad D^2_{\\vec{u}} \\: f(x, y) = a^2 \\frac{\\partial^2 z}{\\partial x^2} + 2ab \\frac{\\partial^2 z}{\\partial y \\partial x} + b^2 \\frac{\\partial^2 z}{\\partial y^2} \\end{align}\n\nA formula for the second directional derivative of a three variable real-valued function $w = f(x, y, z)$ can be obtained in a similar manner.\n\nOf course, we can take successively higher order directional derivatives if we so choose. It's not practical to remember the formulas for computing higher order direction derivatives of a function of several variables though.\n\nLet's look at an example of finding a higher order directional derivative.\n\n## Example 1\n\nFind the second order and third order directional derivative of the function $f(x, y) = 2xy^2$ in the direction of $(1, 2)$.\n\nThe vector $(1, 2)$ is not a unit vector. We have that the unit vector in the direction of $(1, 2)$ is given by:\n\n(6)\n\\begin{align} \\quad \\vec{u} = \\frac{(1, 2)}{\\| (1, 2) \\|} = \\frac{(1, 2)}{\\sqrt{5}} = \\left ( \\frac{1}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}} \\right ) \\end{align}\n\nTherefore the first directional derivative in the direction of $(1, 2)$ is given by:\n\n(7)\n\\begin{align} \\quad D_{\\vec{u}} \\: f(x, y) = \\frac{1}{\\sqrt{5}} 2y^2 + \\frac{2}{\\sqrt{5}} 4xy \\\\ \\quad D_{\\vec{u}} \\: f(x, y) = \\frac{2}{\\sqrt{5}} y^2 + \\frac{8}{\\sqrt{5}} xy \\end{align}\n\nWe will now take the second directional derivative in the direction of $(1, 2)$:\n\n(8)\n\\begin{align} \\quad D^2_{\\vec{u}} \\: f(x, y) = D_{\\vec{u}} \\left ( \\frac{2}{\\sqrt{5}} y^2 + \\frac{8}{\\sqrt{5}} xy \\right ) \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = \\left ( \\frac{1}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}} \\right ) \\cdot \\left (\\frac{\\partial}{\\partial x} \\left ( \\frac{2}{\\sqrt{5}} y^2 + \\frac{8}{\\sqrt{5}} xy \\right ), \\frac{\\partial}{\\partial y} \\left ( \\frac{2}{\\sqrt{5}} y^2 + \\frac{8}{\\sqrt{5}} xy \\right ) \\right ) \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = \\left ( \\frac{1}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}} \\right ) \\cdot \\left ( \\frac{8}{\\sqrt{5}} y, \\frac{4}{\\sqrt{5}} y + \\frac{8}{\\sqrt{5}} x \\right ) \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = \\frac{1}{\\sqrt{5}} \\frac{8}{\\sqrt{5}} y + \\frac{2}{\\sqrt{5}} \\frac{4}{\\sqrt{5}} y + \\frac{2}{\\sqrt{5}} \\frac{8}{\\sqrt{5}} x \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = \\frac{8}{5} y + \\frac{8}{5} y + \\frac{16}{5} x \\\\ \\quad D^2_{\\vec{u}} \\: f(x, y) = \\frac{16}{5} (x + y) \\end{align}\n\nLastly, we find the third order directional derivative as follows:\n\n(9)\n\\begin{align} \\quad D^3_{\\vec{u}} \\: f(x, y) = \\left ( \\frac{1}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}} \\right ) \\cdot \\frac{16}{5} \\left ( \\frac{\\partial}{\\partial x} (x + y) , \\frac{\\partial}{\\partial y} (x + y) \\right ) \\\\ \\quad D^3_{\\vec{u}} \\: f(x, y) = \\left ( \\frac{1}{\\sqrt{5}}, \\frac{2}{\\sqrt{5}} \\right ) \\cdot \\frac{16}{5} (1, 1) \\\\ \\quad D^3_{\\vec{u}} \\: f(x, y) = \\frac{16}{5\\sqrt{5}} + \\frac{32}{5\\sqrt{5}} \\\\ \\quad D^3_{\\vec{u}} \\: f(x, y) = \\frac{48}{5\\sqrt{5}} \\end{align}","date":"2019-02-23 19:43:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 9, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 1.0000085830688477, \"perplexity\": 584.8680169435921}, \"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-2019-09\/segments\/1550249530087.75\/warc\/CC-MAIN-20190223183059-20190223205059-00529.warc.gz\"}"}
| null | null |
#include <algorithm>
#include <cmath>
#include <cstdlib>
#include <functional>
#include <iterator>
#include <limits>
#include <random>
#include <string>
#include <vector>
#include <gtest/gtest.h>
#include "tensorflow/lite/kernels/internal/optimized/integer_ops/pooling.h"
#include "tensorflow/lite/kernels/internal/reference/integer_ops/pooling.h"
#include "tensorflow/lite/kernels/internal/test_util.h"
namespace tflite {
namespace {
// Runs the reference and optimized MaxPool functions and asserts the values
// are the same.
void RunOneMaxPoolTest(const PoolParams& params,
const RuntimeShape& input_shape, const int8* input_data,
const RuntimeShape& output_shape) {
const int buffer_size = output_shape.FlatSize();
std::vector<int8> optimized_maxpool_output(buffer_size);
std::vector<int8> reference_maxpool_output(buffer_size);
reference_integer_ops::MaxPool(params, input_shape, input_data, output_shape,
reference_maxpool_output.data());
optimized_integer_ops::MaxPool(params, input_shape, input_data, output_shape,
optimized_maxpool_output.data());
for (int i = 0; i < buffer_size; i++) {
ASSERT_TRUE(reference_maxpool_output[i] == optimized_maxpool_output[i]);
}
}
// Creates random input shape (batch, height, width, depth), then computes
// output shape based on value of `padding_same`:
// `padding_same` == true, calculate output with padding == "SAME"
// `padding_same` == false, calculate output with padding == "VALID"
// With input/output shapes computed, fills the input data and calls the
// test function.
void CreateDataAndRunMaxPool(bool padding_same) {
const int batch = UniformRandomInt(1, 20);
const int input_depth = UniformRandomInt(32, 700);
const int output_depth = input_depth;
const int input_width = UniformRandomInt(64, 128);
const int input_height = UniformRandomInt(64, 128);
const int stride_width = UniformRandomInt(1, 10);
const int stride_height = UniformRandomInt(1, 10);
const int filter_width = UniformRandomInt(1, 10);
const int filter_height = UniformRandomInt(1, 10);
const int output_width =
padding_same ? (input_width + stride_width - 1) / stride_width
: (input_width - filter_width + stride_width) / stride_width;
const int output_height =
padding_same
? (input_height + stride_height - 1) / stride_height
: (input_height - filter_height + stride_height) / stride_height;
auto input_shape =
RuntimeShape({batch, input_height, input_width, input_depth});
auto output_shape =
RuntimeShape({batch, output_height, output_width, output_depth});
const int buffer_size = input_shape.FlatSize();
std::vector<int8> input_data(buffer_size);
FillRandom(&input_data);
PoolParams params;
params.stride_height = stride_height;
params.stride_width = stride_width;
params.filter_height = filter_height;
params.filter_width = filter_width;
params.quantized_activation_min =
static_cast<int8_t>(std::numeric_limits<int8_t>::lowest());
params.quantized_activation_max =
static_cast<int8_t>(std::numeric_limits<int8_t>::max());
auto compute_padding = [](int stride, int in_size, int filter_size,
int out_size) {
int padding = ((out_size - 1) * stride + filter_size - in_size) / 2;
return padding > 0 ? padding : 0;
};
params.padding_values.width =
compute_padding(stride_width, input_width, filter_width, output_width);
params.padding_values.height = compute_padding(stride_height, input_height,
filter_height, output_height);
RunOneMaxPoolTest(params, input_shape, input_data.data(), output_shape);
}
TEST(TestMaxPool, SymmetricQuantMaxPool) {
const int kTestsToRun = 10;
for (int i = 0; i < kTestsToRun; i++) {
CreateDataAndRunMaxPool(/*padding_same=*/true);
CreateDataAndRunMaxPool(/*padding_same=*/false);
}
}
} // namespace
} // namespace tflite
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,155
|
{"url":"http:\/\/existentialtype.wordpress.com\/tag\/functional-programming\/","text":"## Old neglected theorems are still\u00a0theorems\n\nMarch 20, 2014\n\nI have very recently been thinking about the question of partiality\u00a0vs totality in programming languages, a perennial topic in PL\u2019s that every generation thinks it discovers for itself. \u00a0And this got me to remembering an old theorem that, it seems, hardly anyone knows ever existed in the first place. \u00a0What I like about the theorem is that it says something specific and technically accurate about the sizes of programs in total languages compared to those in partial languages. \u00a0The theorem provides some context for discussion that does not just amount to opinion or attitude (and attitude alway seems to abound when this topic arises).\n\nThe advantage of a total programming language such as Goedel\u2019s T is that it\u00a0ensures, by type checking, that every program terminates, and that every\u00a0function is total. There is simply no way to have a well-typed program that\u00a0goes into an infinite loop. This may seem appealing, until one considers that\u00a0the upper bound on the time to termination can be quite large, so large that some terminating programs\u00a0might just as well diverge as far as we humans are concerned. But never mind that,\u00a0let us grant that it is a virtue of \u00a0T that it precludes divergence.\n\nWhy, then, bother with a language such as PCF that does not rule out\u00a0divergence? After all, infinite loops are invariably bugs,\u00a0so why not rule them out by type checking? (Don\u2019t be\u00a0fooled by glib arguments about useful programs, such as operating systems,\u00a0that \u201crun forever\u201d. After all, infinite streams are programmable in the\u00a0language M of inductive and coinductive types in which all functions terminate. Computing infinitely does\u00a0not mean running forever, it just means \u201cfor as long as one wishes, without\u00a0bound.\u201d) \u00a0The notion does seem appealing until one actually tries to write a program in a language such as T.\n\nConsider computing the greatest common divisor (GCD) of two natural\u00a0numbers. This can be easily programmed in PCF by solving the following\u00a0equations using general recursion:\n\n$\\begin{array}{rcl} \\textit{gcd}(m,0) & = & m \\\\ \\textit{gcd}(0,m) & = & m \\\\ \\textit{gcd}(m,n) & = & \\textit{gcd}(m-n,n) \\quad \\text{if}\\ m>n \\\\ \\textit{gcd}(m,n) & = & \\textit{gcd}(m,n-m) \\quad \\text{if}\\ m\n\nThe type of $\\textit{gcd}$ defined in this manner\u00a0has\u00a0partial function type\u00a0$(\\mathbb{N}\\times \\mathbb{N})\\rightharpoonup \\mathbb{N}$, which\u00a0suggests that it may not terminate for some inputs. But we may prove by\u00a0induction on the sum of the pair of arguments that it is, in fact, a total\u00a0function.\n\nNow consider programming this function in T. It is, in fact,\u00a0programmable using only primitive recursion, but the code to do it is rather painful (try it!). One way to see the problem is that in T the only form\u00a0of looping is one that reduces a natural number by one on each recursive call;\u00a0it is not (directly) possible to make a recursive call on a smaller number other\u00a0than the immediate predecessor. In fact one may code up more general patterns\u00a0of terminating recursion using only primitive recursion as a primitive, but if\u00a0you examine the details, you will see that doing so comes at a significant price\u00a0in performance and program complexity. Program complexity can be mitigated by\u00a0building libraries that codify standard patterns of reasoning whose cost of\u00a0development should be amortized over all programs, not just one in particular.\u00a0But there is still the problem of performance. Indeed, the encoding of more\u00a0general forms of recursion into primitive recursion means that, deep within the\u00a0encoding, there must be \u201ctimer\u201d that \u201cgoes down by ones\u201d to ensure that the\u00a0program terminates. The result will be that programs written with such\u00a0libraries will not be nearly as fast as they ought to be. \u00a0(It is actually quite fun to derive \u201ccourse of values\u201d recursion from primitive recursion, and then to observe with horror what is actually going on, computationally, when using this derived notion.)\n\nBut, one may argue, T is simply not a serious language. A more serious\u00a0total programming language would admit sophisticated patterns of control without\u00a0performance penalty. Indeed, one could easily envision representing the natural\u00a0numbers in binary, rather than unary, and allowing recursive calls to be made by\u00a0halving to achieve logarithmic complexity. This is surely possible, as are\u00a0numerous other such techniques. Could we not then have a practical language\u00a0that rules out divergence?\n\nWe can, but at a cost. \u00a0One limitation of total programming\u00a0languages is that they are not universal: you cannot write an interpreter for T\u00a0within T (see Chapter 9 of PFPL for a proof). \u00a0More importantly,\u00a0this limitation extends to any total language whatever. \u00a0If this limitation does not seem important, then consider the Blum Size Theorem (BST)\u00a0(from 1967), which places a very different\u00a0limitation on total languages. \u00a0Fix any total\u00a0language, L, that permits writing functions on the natural numbers. Pick\u00a0any blowup factor, say $2^{2^n}$, or however expansive you wish to be. \u00a0The BST\u00a0states that there is a total function on the natural numbers that is\u00a0programmable in L, but whose shortest program in L is larger by the given blowup factor than its shortest program in PCF!\n\nThe underlying idea of the proof is that in a total language the proof of\u00a0termination of a program must be baked into the code itself, whereas in\u00a0a partial language the termination proof is an external verification condition\u00a0left to the programmer. Roughly speaking, there are, and always will be,\u00a0programs whose termination proof is rather complicated to express, if you fix in\u00a0advance the means by which it may be proved total. (In T it was primitive\u00a0recursion, but one can be more ambitious, yet still get caught by the BST.) \u00a0But\u00a0if you leave room for ingenuity, then programs can be short, precisely because\u00a0they do not have to embed the proof of their termination in their own running\u00a0code.\n\nThere are ways around the BST, of course, and I am not saying otherwise. \u00a0For example, the BST merely guarantees the existence of a bad case, so one can always argue that such a case will never arise in practice. \u00a0Could be, but I did mention the GCD in T problem for a reason: there are natural problems that are difficult to express in a language such as T. \u00a0By fixing the possible termination arguments in advance, one is tempting fate, for there are many problems, such as the Collatz Conjecture, for which the termination proof of a very simple piece of code has been an open problem for decades, and has resisted at least some serious attempts on it. \u00a0One could argue that such a function is of no practical use. \u00a0I agree, but I point out the example not to say that it is useful, but to say that it is likely that its eventual termination proof will be quite nasty, and that this will have to be reflected in the program itself if you are limited to a T-like language (rendering it, once again, useless). \u00a0For another example, there is no inherent reason why termination need be assured by means similar to that used in T. \u00a0We got around this issue in NuPRL\u00a0by separating the code from the proof, using a type theory based on a partial programming language, not a total one. \u00a0The proof of termination is still required for typing in the core theory (but not in the theory with \u201cbar types\u201d for embracing partiality). \u00a0But it\u2019s not baked into the code itself, affecting its run-time; it is \u201coff to the side\u201d, large though it may be).\n\nUpdates:\u00a0word smithing,\u00a0fixed bad link, corrected gcd, removed erroneous parenthetical reference to Coq, fixed LaTeX problems.\n\n## What, If Anything, Is A Declarative\u00a0Language?\n\nJuly 18, 2013\n\nBack in the 1980\u2019s it was very fashionable to talk about \u201cdeclarative\u201d programming languages. \u00a0But to my mind\u00a0there was never a clear definition of a \u201cdeclarative language\u201d, and hence no way to tell what is declarative and what is not. \u00a0Lacking any clear meaning, the term came to refer to the arbitrary conflation of functional with logic programming to such an extent that \u201cfunctional-and-logic-programming\u201d almost became a Germanic thing-in-itself (ding an sich). \u00a0Later, as the logic programming wave subsided, the term \u201cdeclarative\u201d, \u00a0like \u201cobject-oriented\u201d, came to be an expression of approval, and then, mercifully, died out.\n\nOr so I had thought. \u00a0Earlier this week I attended a thriller of an NSF-sponsored workshop on high-level programming models for parallelism, where I was surprised by the declarative zombie once again coming to eat our brains. \u00a0This got me to thinking, again, about whether the term has any useful meaning. \u00a0For what it\u2019s worth, and perhaps to generate useful debate, here\u2019re some things that I think people mean, and why I don\u2019t think they mean very much.\n\n1. \u201cDeclarative\u201d means \u201chigh-level\u201d. \u00a0This just seems to replace one vague term by another.\n2. \u201cDeclarative\u201d means \u201cnot imperative\u201d. \u00a0But this flies in the face of reality. \u00a0Functional languages embrace and encompass imperative programming as a special case, and even Prolog has imperative features, such as assert and retract, that have imperative meaning.\n3. \u201cDeclarative\u201d means \u201cfunctional\u201d. \u00a0OK, but then we don\u2019t really need another word.\n4. \u201cDeclarative\u201d means \u201cwhat, not how\u201d. \u00a0But every language has an operational semantics that defines how to execute its programs, and you must be aware of that semantics to understand programs written in it. \u00a0Haskell has a definite evaluation order, just as much as ML has a different one, and even Prolog execution is defined by a clear operational semantics that determines the clause order and that can be influenced by \u201ccut\u201d.\n5. \u201cDeclarative\u201d means \u201cequational\u201d. \u00a0This does not distinguish anything, because there is a well-defined notion of equivalence for\u00a0any programming language, namely observational equivalence. \u00a0Different languages induce different equivalences, of course, but how does one say that one equivalence is \u201cbetter\u201d than another? \u00a0At any rate, I know of no stress on equational properties of logic programs, so either logic programs are not \u201cdeclarative\u201d or \u201cequational reasoning\u201d is not their defining characteristic.\n6. \u201cDeclarative\u201d means \u201creferentially transparent\u201d. \u00a0The misappropriation of Quine\u2019s terminology only confuses matters. \u00a0All I\u2019ve been able to make of this is that \u201creferentially transparent\u201d means that beta-equivalence is valid. \u00a0But beta equivalence is not a property of an arbitrary programming language, nor in any case is it clear why this equivalence is first among equals. \u00a0In any case why\u00a0you would decide\u00a0a priori\u00a0on what equivalences you want before you even know what it means to run a program?\n7. \u201cDeclarative\u201d means \u201chas a denotation\u201d. \u00a0This gets closer to the point, I think, because we might well say that a declarative\u00a0semantics is one that gives meaning to programs as some kind of mapping between some sort of spaces. \u00a0In other words, it would be a synonym for \u201cdenotational semantics\u201d. \u00a0But every language has a denotational semantics (perhaps by interpretation into a Kripke structure to impose sequencing), so having one does not seem to distinguish a useful class of languages. \u00a0Moreover, even in the case of purely functional programs, the usual denotational semantics (as continuous maps) is not fully abstract, and the fully abstract semantics (as games) is highly operational. \u00a0Perhaps a language is declarative in proportion to being able to give it semantics in some \u201cfamiliar\u201d mathematical setting?\n8. \u201cDeclarative\u201d means \u201cimplicitly parallelizable\u201c. \u00a0This was certainly the sense intended at the NSF meeting, but no two \u201cdeclarative\u201d languages seemed to have much in common. \u00a0Charlie Garrod proposes just \u201cimplicit\u201d, which is pretty much synonymous with \u201chigh level\u201d, and may be the most precise sense there is to the term.\n\nNo doubt this list is not exhaustive, but I think it covers many of the most common interpretations. \u00a0It seems to me that none of them have a clear meaning or distinguish a well-defined class of languages. \u00a0Which leads me to ask, is there any such thing as a declarative programming language?\n\n[Thanks to the late Steve Gould for inspiring the title of this post.]\n\n[Update: wordsmithing.]\n\n## Introductory FP Course\u00a0Materials\n\nSeptember 15, 2012\n\nThe course materials for our first-semester introductory programming course (which I\u2019ve discussed elsewhere on this blog) are now available here.\n\nThe course materials for our second-semester data structures and algorithms course are available here.\n\nThanks to Dan Licata and Guy Blelloch for helping make these available. \u00a0Comments are most welcome.\n\n## Intro Curriculum Update\n\nAugust 17, 2012\n\nIn previous posts I have talked about the new introductory CS curriculum under development at Carnegie Mellon. After a year or so of planning, we began to roll out the new curriculum in the Spring of 2011, and have by now completed the transition. As mentioned previously, the main purpose is to bring the introductory sequence up to date, with particular emphasis on introducing parallelism and verification. A secondary purpose was to restore the focus on computing fundamentals, and correct the drift towards complex application frameworks that offer the students little sense of what is really going on. (The poster child was a star student who admitted that, although she had built a web crawler the previous semester, she in fact has no idea how to build a web crawler.) A particular problem is that what should have been a grounding in the fundamentals of algorithms and data structures turned into an exercise in object-oriented programming, swamping the core content with piles of methodology of dubious value to beginning students. (There is a new, separate, upper-division course on oo methodology for students interested in this topic.) A third purpose was to level the playing field, so that students who had learned about programming on the street were equally as challenged, if not more so, than students without much or any such experience. One consequence would be to reduce the concomitant bias against women entering CS, many fewer of whom having prior computing experience than the men.\n\nThe solution was a complete do-over, jettisoning the traditional course completely, and starting from scratch. The most important decision was to emphasize functional programming right from the start, and to build on this foundation for teaching data structures and algorithms. Not only does FP provide a much more natural starting point for teaching programming, it is infinitely more amenable to rigorous verification, and provides a natural model for parallel computation. Every student comes to university knowing some algebra, and they are therefore familiar with the idea of computing by calculation (after all, the word algebra derives from the Arabic al jabr, meaning system of calculation). Functional programming is a generalization of algebra, with a richer variety of data structures and a richer set of primitives, so we can build on that foundation. It is critically important that variables in FP are, in fact, mathematical variables, and not some distorted facsimile thereof, so all of their mathematical intuitions are directly applicable. So we can immediately begin discussing verification as a natural part of programming, using principles such as mathematical induction and equational reasoning to guide their thinking. Moreover, there are natural concepts of sequential time complexity, given by the number of steps required to calculate an answer, and parallel time complexity, given by the data dependencies in a computation (often made manifest by the occurrences of variables). These central concepts are introduced in the first week, and amplified throughout the semester.\n\nTwo major homework exercises embody the high points of the first-semester course, one to do with co-development of code with proof, the other to do with parallelism. Co-development of program and proof is illustrated by an online regular expression matcher. The problem is a gem for several reasons. One is that it is essentially impossible for anyone to solve by debugging a blank screen. This sets us up nicely for explaining the importance of specification and proof as part of the development process. Another is that it is very easy, almost inevitable, for students to make mistakes that are not easily caught or diagnosed by testing. We require the students to carry out a proof of the correctness of the matcher, and in the process discover a point where the proof breaks down, which then leads to a proper solution. (See my JFP paper \u201cProof-Directed Debugging\u201d for a detailed development of the main ideas.) The matcher also provides a very nice lesson in the use of higher-order functions to capture patterns of control, resulting in an extremely clean and simple matcher whose correctness proof is highly non-trivial.\n\nThe main parallelism example is the Barnes-Hut algorithm for solving the n-body problem in physics. Barnes-Hut is an example of a \u201ctree-based\u201d method, invented by Andrew Appel, for solving this well-known problem. At a high level the main idea is to decompose space into octants (or quadrants if you\u2019re working in the plane), recursively solving the problem for each octant and then combining the solutions to make an overall solution. The key idea is to use an approximation for bodies that are far enough away\u2014a distant constellation can be regarded as an atomic body for the purposes of calculating the effects of its stars on the sun, say. The problem is naturally parallelizable, because of the recursive decomposition. Moreover, it provides a very nice link to their high school physics. Since FP is just an extension of mathematics, the equations specifying the force law and Newton\u2019s Law carry over directly to the code. This is an important sense in which FP builds on and amplifies their prior mathematical experience, and shows how one may connect computer science with other sciences in a rather direct manner.\n\nThe introductory FP course establishes the basis for the new data structures and algorithms course that most students take in either the third or fourth semester. This course represents a radical departure from tradition in several ways. First, it is a highly rigorous course in algorithms that rivals the upper-division course taught at most universities (including our own) for the depth and breadth of ideas it develops. Second, it takes the stance that all algorithms are parallel algorithms, with sequential being but a special case of parallel. Of course some algorithms have a better parallelizability ratio (a precise technical characterization of the ability to make use of parallelism), and this can be greatly affected by data structure selection, among other considerations. Third, the emphasis is on persistent abstract types, which are indispensable for parallel computing. No more linked lists, no more null pointer nonsense, no more mutation. Instead we consider abstract types of graphs, trees, heaps, and so forth, all with a persistent semantics (operations create \u201cnew\u201d ones, rather than modify \u201cold\u201d ones). Fourth, we build on the reasoning methods introduced in the first semester course to prove the correctness and the efficiency of algorithms. Functional programming makes all of this possible. Programs are concise and amenable to proof, they are naturally parallel, and they enjoy powerful typing properties that enforce abstraction in a mathematically rigorous manner. Fifth, there is a strong emphasis on problems of practical interest. For example, homework 1 is the shotgun method for genome sequencing, a parallel algorithm of considerable practical importance and renown.\n\nThere is a third introductory course in imperative programming, taken in either the first or second semester (alternating with the functional programming course at the student\u2019s discretion), that teaches the \u201cold ways\u201d of doing things using a \u201csafe\u201d subset of C. Personally, I think this style of programming is obsolete, but there are many good reasons to continue to teach it, the biggest probably being the legacy problem. The emphasis is on verification, using simple assertions that are checked at run-time and which may be verified statically in some situations. It is here that students learn how to do things \u201cthe hard way\u201d using low-level representations. This course is very much in the style of the 1970\u2019s era data structures course, the main difference being that the current incarnation of Pascal has curly braces, rather than begin-end.\n\nFor the sake of those readers who may not be up to date on such topics, it seems important to emphasize that functional programming subsumes imperative programming. Every functional language is capable of expressing the old-fashioned sequential pointer-hacking implementation of data structures. You can even reproduce Tony Hoare\u2019s self-described \u201cbillion dollar mistake\u201d of the cursed \u201cnull pointer\u201d if you\u2019d like! But the whole point is that it is rarely useful, and almost never elegant, to work that way. (Curiously, the \u201cmonad mania\u201d in the Haskell community stresses an imperative, sequential style of programming that is incompatible with parallelism, but this is not forced on you as it is in the imperative world.) From this point of view there no loss, and considerable gain, in teaching functional programming from the start. It puts a proper emphasis on mathematically sane programming methods, but allows for mutation-based programming where appropriate (for example, in engendering \u201cside effects\u201d on users).\n\nI encourage readers to review the syllabi and course materials. There is quite a large body of material in place that we plan to expand into textbook-level treatments in the near future. We also plan to write a journal article documenting our experiences with these courses.\n\nI am very grateful to my colleagues Guy Blelloch, Dan Licata, and Frank Pfenning for their efforts in helping to develop the new introductory curriculum at Carnegie Mellon, and to the teaching assistants whose hard work and dedication put the ideas into practice.\n\nUpdate: Unfortunately, the homework assignments for these courses are not publicly available, because we want to minimize the temptation for students to make use of old assignments and solutions in preparing their own work. \u00a0I am working with my colleagues to find some way in which we can promote the ideas without sacrificing too much of the integrity of the course materials. \u00a0I apologize for the inconvenience.\n\n## Words matter\n\nFebruary 1, 2012\n\nYesterday, during a very nice presentation by Ohad Kammar at Carnegie Mellon, the discussion got derailed, in part, because of a standard, and completely needless, terminological confusion involving the word \u201cvariable\u201d. \u00a0I\u2019m foolish enough to try to correct it.\n\nThe problem is that we\u2019ve all been taught to confuse variables with variables\u2014that is, program variables with mathematical variables. \u00a0The distinction is basic. \u00a0Since time immemorial (well, at least since al Khwarizmi) we have had the notion of a variable, properly so-called, which is given meaning by substitution. \u00a0A variable is an unknown, or indeterminate, quantity that can be replaced by any value of its type (a type being, at least since Russell, the range of significance of a variable). \u00a0Frege gave the first systematic study of the quantifiers, and Church exploited the crucial concept of a variable to give the most sharply original and broadly applicable model of computation, the $\\lambda$-calculus.\n\nSince the dawn of Fortran\u00a0something that is not a variable has come to be called a variable. \u00a0A program variable, in the sense of Fortran and every imperative language since, is not given meaning by substitution. \u00a0Rather, it is given meaning by (at least) two operations associated with it, one to get\u00a0its contents and one to put\u00a0new contents into it. \u00a0(And, maybe, an operation to form a reference\u00a0to it, as in C or even Algol.) \u00a0Now as many of you know, I think that the concept of a program variable in this sense is by and large a bad idea, or at any rate not nearly as important as it has been made out to be in conventional (including object-oriented) languages, but that\u2019s an argument for another occasion.\n\nInstead, I\u2019m making a plea. \u00a0Let\u2019s continue to call variables variables. \u00a0It\u2019s a perfectly good name, and refers to what is perhaps one of the greatest achievements of the human mind, the fundamental concept of algebra, the variable. \u00a0But let\u2019s stop calling those other things variables! \u00a0In my Practical Foundations for Programming Languages I coined (as far as I know) a word that seems perfectly serviceable, namely an assignable. \u00a0The things called variables in imperative languages should, rather, be called assignables. \u00a0The word is only a tad longer than variable, and rolls off the tongue just as easily, and has the advantage of being an accurate description of what it really is. \u00a0What\u2019s not to like?\n\nWhy bother? \u00a0For one thing, some languages have both\u00a0concepts, a necessity if you want your language to be mathematically civilized (and you do). \u00a0For another, in the increasingly important world of program verification, the specification formalisms, being mathematical in nature, make use of variables, which most definitely are not assignables! \u00a0But the real reason to make the distinction is, after all, because words matter. \u00a0Two different things deserve to have two different names, and it only confuses matters to use the same word for both. \u00a0This week\u2019s confusion was only one example of many that I have seen over the years.\n\nSo, my suggestion: let\u2019s call variables variables, and let\u2019s call those other things assignables. \u00a0In the fullnesss of time (i.e., once the scourge of imperative programming has been lifted) we may not need the distinction any longer. \u00a0But until then, why not draw the distinction properly?\n\nAddendum:\u00a0It seems worth mentioning that in PFPL I have a novel (afaik) treatment of the concept of a reference, which is clarified in a subsequent post.\n\n## The semester\u2019s over\n\nMay 4, 2011\n\nOne reason that I started this blog was to record my experiences with a new course on functional programming for freshmen at Carnegie Mellon. \u00a0Classes have ended, the final exam is graded, and we are in the process of taking stock of what we accomplished, and what we could do better. \u00a0This was the first instance of the course, taught to 83 first-year computer science students who volunteered to be part of the new curriculum. \u00a0(This represents about 2\/3 of the freshman class.) \u00a0All had taken a new course on imperative programming taught by Frank Pfenning the previous semester, and all were simultaneously taking 251, the legendary theory foundation course developed here over the last dozen or so years.\n\nStarting this fall the course will be part of the standard curriculum, and will be taught to a broader range of students, including the electrical engineers and the full class of CS students. \u00a0We\u2019re still working out whether the standard order will be imperative, then functional, or functional, then imperative. \u00a0Much depends on the mathematical maturity of the incoming students. \u00a0Those with weak mathematical skills (the vast majority), but some programming experience, will likely start with imperative programming, which they will take simultaneously with a standard discrete math course taught in the mathematics department. \u00a0Those with strong mathematical skills, or high ambitions, will start with functional programming. \u00a0A small number will park for one semester, taking discrete math and a programming course primarily intended for non-majors, to bring themselves up to speed.\n\nOur new data structures and algorithms course, being developed by Guy Blelloch, will be offered in the fall for students who have already have the IP and FP classes. \u00a0Guy plans to make heavy use of functional programming, and will be stressing parallel algorithms on persistent data structures as the common case, leaving the traditional sequential, ephemeral case to the imperative programming class. \u00a0Guy\u2019s course is essentially a continuation (ahem) of the FP class that emphasizes more sophisticated data structures and algorithms, and more complex programming problems.\n\nFor those who might be interested, I\u2019m attaching the final exam for the course. \u00a0The students did exceptionally well (average score of 88%), even though we feared it was too long and difficult beforehand. \u00a0The spread was very narrow, perhaps because the exam was too easy, or because this being a class of volunteers, their skills were unusually strong and uniform. \u00a0We\u2019ll know more after the fall once we\u2019ve taught the class to a broader range of students. \u00a0I do think the exam is representative of what we expected them to be able to do at the end, and they showed us that they can do it. \u00a0A clear success!\n\n15-150 Final Exam (Spring 2011)\n\n## Of Course ML Has\u00a0Monads!\n\nMay 1, 2011\n\nExamined from the point of view of ML, monads are but a particular of use of modules.\u00a0\u00a0The signature of monads is given by the definition\n\nsignature MONAD = sig\nval ret : 'a -> 'a monad\nend\n\nThere are many, many, many structures that satisfy this signature; I needn\u2019t (and, in any case, can\u2019t) rehearse them all here. \u00a0One particularly simple example should suffice to give the general idea:\n\nstructure Option : MONAD = struct\ntype 'a monad = 'a option\nfun ret x = SOME x\nfun bnd (SOME x) k = k x\n| bnd NONE k = NONE\nend\n\nThis is of course the option monad, which is sometimes used to model the data flow aspects of exceptions, perhaps with some elaboration of the NONE case to associate an exceptional value with a non-result. \u00a0(The control flow aspects are not properly modeled this way, however. \u00a0For that one needs, in addition, access to some sort of jump mechanism.)\n\nExamples like this one proliferate. \u00a0A monad is represented by a structure. \u00a0Any structure that provides the facilities specified by the MONAD signature gives rise to the characteristic sequentialization mechanisms codified by it. \u00a0Monad transformers are functors that transform one monad into another, with no fuss or bother, and no ad hoc\u00a0mechanisms required. \u00a0Standard modular programming techniques suffice to represent monads; moreover, the techniques involved are fully general, and are equally applicable to other signatures of interest (arrows, or quivers, or bows, or what have you). \u00a0Moreover, it is shown in my paper with Chakravarty and Dreyer how to integrate modules into the type inference mechanism of ML so that one can get automatic functor instantiation in those limited cases where it is self-evident what is intended. \u00a0This has been implemented by Karl Crary in a prototype compiler for an extension of Standard ML, and it would be good to see this supported in more broadly available compilers for the language.\n\nThe bulk of the mania about monads is therefore accounted for by modules. \u00a0I have no doubt, however, that you are wondering about the infamous IO monad in Haskell (and it\u2019s associated work-around, unsafePerformIO). \u00a0Isn\u2019t that a fundamental feature of the language that cannot be replicated in ML? \u00a0Hardly! \u00a0It\u2019s entirely a matter of designing the signatures of the standard basis library modules, and nothing more. \u00a0The default basis library does not attempt to segregate effects into a monad, but it is perfectly straightforward to do this yourself, by providing your own layer over the standard basis, or to reorganize the standard basis to enforce the separation.\u00a0\u00a0For example, the signature of reference cells might look like this:\n\nsignature REF = sig\ntype 'a ref\nval ref : 'a -> 'a ref IO.monad\nval ! : 'a ref -> 'a IO.monad\nval := : 'a ref -> 'a -> unit IO.monad\nend\n\nHere we are presuming that we have a fixed declaration\n\nstructure IO : MONAD = ...\n\nthat packages up the basic IO primitives that are already implemented in the run-time system of ML, more or less like in Haskell. \u00a0The other signatures, such as those for mutable arrays or for performing input and output, would be modified in a similar manner to push effects into the IO monad. \u00a0Et voila, you have monadic effects, just like in Haskell.\n\nThere\u2019s really nothing to it. \u00a0In fact, the whole exercise was carried out by a Carnegie Mellon student, Phillippe Ajoux, a couple of years ago. \u00a0He also wrote a number of programs in this style just to see how it all goes: swimmingly. \u00a0He also devised syntactic extensions to the Moscow ML compiler that provide a nicer notation for programming with monads, much as in Haskell, but better aligned with ML\u2019s conventions. \u00a0(Ideally it should be possible to provide syntactic support for any\u00a0signature, not just monads, but I\u2019m not aware of a worked-out design for the general case, involving as it would an intermixing of parsing and elaboration.)\n\nMy point is that the ML module system can be deployed by you to impose the sorts of effect segregation imposed on you by default in Haskell. \u00a0There is nothing special about Haskell that makes this possible, and nothing special about ML that inhibits it. \u00a0It\u2019s all a mode of use of modules.\n\nSo why don\u2019t we do this by default? \u00a0Because it\u2019s not such a great idea.\u00a0\u00a0Yes, I know it sounds wonderful at first, but then you realize that it\u2019s pretty horrible. \u00a0Once you\u2019re in the IO monad, you\u2019re stuck there forever, and are reduced to Algol-style imperative programming. \u00a0You cannot easily convert between functional and monadic style without a radical restructuring of code. \u00a0And you inevitably need unsafePerformIO to get anything serious done. \u00a0In practical terms, you are deprived of the useful concept of a benign effect, and that just stinks!\n\nThe moral of the story is that of course ML \u201chas monads\u201d, just like Haskell. \u00a0Whether you want to use them is up to you; they are just as useful, and just as annoying, in ML as they are in Haskell. \u00a0But they are not forced on you by the language designers!\n\nUpdate: This post should\u2019ve been called \u201cML Has Monads, Why Not?\u201d, or \u201cOf Course ML Has Comonads!\u201d, but then no one was wondering about that.\n\nUpdate: I now think that the last sentence is excessive. \u00a0My main point is simply that it\u2019s very simple to go one way or the other with effects, if you have modules to structure things; it\u2019s all a matter of library design. \u00a0A variant of ML that enforced the separation of effects is very easily constructed; the question is whether it is useful or not. \u00a0I\u2019ve suggested that the monadic separation is beguiling, but not clearly a great idea. \u00a0Alternatively, one can say that we\u2019re not that far away from eliminating laziness from Haskell, at least in this respect: just re-do the standard basis library in ML, and you\u2019re a good ways there. \u00a0Plus you have modules, and we understand how to integrate type classes with modules, so the gap is rather small.\n\nFollow","date":"2014-09-01 07:26:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"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\": 0.5915453433990479, \"perplexity\": 1476.869977893187}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1409535917463.1\/warc\/CC-MAIN-20140901014517-00157-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
| null | null |
ARLA > NEWS > NOVEMBER 2019 > HOUSING MINISTER NOT IN FAVOUR OF RENT CONTROLS
Housing Minister not in favour of rent controls
London's Labour Mayor, Sadiq Khan, has returned to his campaign to win powers which would allow him to impose rent controls despite the Housing Minister speaking out against.
The Government brought up rent controls in the A New Deal for Renting: Resetting the balance of rights and responsibilities between Landlords and Tenants consultation due to concerns that landlords would use rent hikes to get rid of tenants if they lost 'no-fault eviction' powers.
ARLA Propertymark has long made the point political parties advocating rent control systems are failing to learn the lessons of history. The last time rent controls existed in this country, the private rented sector (PRS) shrunk from 90 per cent to seven per cent.
The Rt Hon Robert Jenrick, Secretary of State for Housing, Communities and Local Government said: "I am not in favour of rent controls. As I said, that has proven to be very negative for both landlords and tenants in the past, and I do not want to see any move in that direction."
Khan has been an advocate of controls for some time, but currently has no authority to introduce them in the capital.
In his published landmark report, which sets out how the private rented sector in London should be transformed, he said: "It is high time for private renting in London to be transformed - Londoners need fundamental change that is long overdue.
"Unlike other Mayors around the world, I have no powers over the private rented sector. That's why this landmark report sets out a detailed blueprint of what the Government must do to overhaul tenancy laws, and what powers City Hall needs from them to bring rents down.
"We have made important progress over the last three years by working closely with councils and renters - from 'naming and shaming' rogue landlords and banning letting agents' fees for tenants, to being part of the successful campaign to scrap 'section 21'.
"But now we need the Government to play their part by making tenancy laws fit for purpose, and by enabling us to bring in the rent control Londoners so urgently need."
David Cox, Chief Executive, ARLA Propertymark said: "Rent controls do not work; it hits hardest those it's designed to help the most.
"The last time rent controls existed in this country, the private rented sector (PRS) shrunk to the lowest levels ever recorded. At a time of demand for PRS homes massively outstripping supply, rent controls will cause the sector to shrink. In turn, this means professional landlords will only take the very best tenants, and the vulnerable and low-income people that rent controls are designed to help, will be forced into the hands of rogue and criminal operators, who may exploit them."
ARLA Propertymark responded to the Government's A New Deal for Renting: Resetting the balance of rights and responsibilities between Landlords and Tenants consultation stressing these issues. Read the full response below.
Consultation response
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,708
|
import morepath
class app(morepath.App):
pass
@app.path(path="")
class Root:
pass
@app.html(model=Root)
def index(self, request):
return "the root"
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,686
|
{"url":"https:\/\/manual.gromacs.org\/documentation\/2019\/onlinehelp\/gmx-make_ndx.html","text":"# gmx make_ndx\u00b6\n\n## Synopsis\u00b6\n\ngmx make_ndx [-f [<.gro\/.g96\/...>]] [-n [<.ndx> [...]]] [-o [<.ndx>]]\n[-natoms <int>] [-[no]twin]\n\n\n## Description\u00b6\n\nIndex groups are necessary for almost every GROMACS program. All these programs can generate default index groups. You ONLY have to use gmx make_ndx when you need SPECIAL index groups. There is a default index group for the whole system, 9 default index groups for proteins, and a default index group is generated for every other residue name.\n\nWhen no index file is supplied, also gmx make_ndx will generate the default groups. With the index editor you can select on atom, residue and chain names and numbers. When a run input file is supplied you can also select on atom type. You can use boolean operations, you can split groups into chains, residues or atoms. You can delete and rename groups. Type \u2018h\u2019 in the editor for more details.\n\nThe atom numbering in the editor and the index file starts at 1.\n\nThe -twin switch duplicates all index groups with an offset of -natoms, which is useful for Computational Electrophysiology double-layer membrane setups.\n\nSee also gmx select -on, which provides an alternative way for constructing index groups. It covers nearly all of gmx make_ndx functionality, and in many cases much more.\n\n## Options\u00b6\n\nOptions to specify input files:\n\n-f [<.gro\/.g96\/\u2026>] (conf.gro) (Optional)\nStructure file: gro g96 pdb brk ent esp tpr\n-n [<.ndx> [\u2026]] (index.ndx) (Optional)\nIndex file\n\nOptions to specify output files:\n\n-o [<.ndx>] (index.ndx)\nIndex file\n\nOther options:\n\n-natoms <int> (0)\nset number of atoms (default: read from coordinate or index file)\n-[no]twin (no)\nDuplicate all index groups with an offset of -natoms","date":"2021-01-26 22:35:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28151679039001465, \"perplexity\": 12500.411635679373}, \"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-2021-04\/segments\/1610704803737.78\/warc\/CC-MAIN-20210126202017-20210126232017-00541.warc.gz\"}"}
| null | null |
import random
# TODO:
# Q: What quantity of middle bandwidth do you need to kill guards?
# A: Intuitively, you need the disable rate % of bandwidth, but you
# might have some edge cases to exploit with min_circs.
PATH_BIAS_PCT = 70
# XXX: Min_circs only actives the "notice" level logs
PATH_BIAS_MIN_CIRCS = 20
# XXX: An int divisor was wrong here. Fix that in Tor. We might
# even want a weighted moving average, but that will be trickier
# to analyze.
PATH_BIAS_SCALE_FACTOR = 50
PATH_BIAS_SCALE_THRESHOLD = 250
# XXX: We should only emit warnings if we are above the scaling threshhold..
PATH_BIAS_WARN_CIRCS = PATH_BIAS_SCALE_THRESHOLD*(PATH_BIAS_SCALE_FACTOR/100.0)
#############################################################
# FIXME: haxxx. Who cares, though?
def reset_globals():
global PATH_BIAS_PCT
global PATH_BIAS_MIN_CIRCS
global PATH_BIAS_SCALE_FACTOR
global PATH_BIAS_SCALE_THRESHOLD
global PATH_BIAS_WARN_CIRCS
PATH_BIAS_PCT = 70
PATH_BIAS_MIN_CIRCS = 20
PATH_BIAS_SCALE_FACTOR = 50
PATH_BIAS_SCALE_THRESHOLD = 250
PATH_BIAS_WARN_CIRCS = PATH_BIAS_SCALE_THRESHOLD*(PATH_BIAS_SCALE_FACTOR/100.0)
####################### Guard Types #########################
# Normal Guard experiences the average circuit failure rate
# of the network as a whole
class Guard:
def __init__(self, succeed_rate):
self.first_hops_total = 0
self.success_total = 0
self._first_hops = 0
self._success = 0
self.succeed_rate = succeed_rate
self.rejected_count = 0
def reset(self):
self._success = 0
self._first_hops = 0
def reject_if_bad(self):
if self.is_bad():
self.reset()
self.rejected_count += 1
def reject_rate(self):
return self.rejected_count/float(self.first_hops_total)
def _get_rate(self):
return self._success/float(self._first_hops)
def is_bad(self):
return self._first_hops >= PATH_BIAS_MIN_CIRCS and \
(self._get_rate() < (PATH_BIAS_PCT/100.0))
def build_circuit(self):
self._inc_first_hop()
if random.random() < self.succeed_rate:
self._inc_success()
# Client may give up on us after this circuit
self.reject_if_bad()
def circ_fail_count(self):
return self._first_hops - self._success
def _inc_first_hop(self):
self._first_hops += 1
self.first_hops_total += 1
if self._first_hops > PATH_BIAS_SCALE_THRESHOLD:
self._first_hops *= PATH_BIAS_SCALE_FACTOR/100.0
self._success *= PATH_BIAS_SCALE_FACTOR/100.0
def _inc_success(self):
self._success += 1
self.success_total += 1
# EvilGuard collects statistics on how evil he is, but doesn't
# actually implement any evilness
class EvilGuard(Guard):
def __init__(self, succeed_rate, adversary_capacity):
Guard.__init__(self, succeed_rate)
self.adversary_capacity = adversary_capacity # c/n probability of malicious exit
self.capture_count = 0
def pwnt_per_client(self):
return self.capture_count/float(self.rejected_count+1)
def capture_rate(self):
return self.capture_count/float(self.first_hops_total)
def compromise_rate(self):
return self.capture_count/float(self.success_total)
# PassiveEvilGuard uses a non-destructive long-term timing-based
# tagging attack to fully correlate circuits end-to-end with 100%
# accuracy. PassiveEvilGuard does not kill any circuits.
class PassiveEvilGuard(EvilGuard):
def __init__(self, succeed_rate, adversary_capacity):
EvilGuard.__init__(self, succeed_rate, adversary_capacity)
def build_circuit(self):
self._inc_first_hop()
# The presence of a malicious exit is a prior probability governed by the
# client. Decide it now.
got_malicious_exit = False
if random.random() < self.adversary_capacity:
got_malicious_exit = True
if random.random() < self.succeed_rate:
if got_malicious_exit: # via timing-based tagging attack
self._inc_success()
self.capture_count += 1
else:
self._inc_success() # "Better luck next time :/"
# Client may give up on us after this circuit
self.reject_if_bad()
# UnrepentantEvilGuard uses a destructive tagging attack to
# fully correlate circuits end-to-end with 100%
# accuracy, as well as to kill uncorrelated circuits.
#
# UnrepentantEvilGuard doesn't care if there is a defense or
# not.
class UnrepentantEvilGuard(EvilGuard):
def __init__(self, succeed_rate, adversary_capacity):
EvilGuard.__init__(self, succeed_rate, adversary_capacity)
def build_circuit(self):
self._inc_first_hop()
# The presence of a malicious exit is a prior probability governed by the
# client. Decide it now.
got_malicious_exit = False
if random.random() < self.adversary_capacity:
got_malicious_exit = True
if random.random() < self.succeed_rate:
if got_malicious_exit: # via tagging attack
self._inc_success()
self.capture_count += 1
else:
pass # "We can't deanon it? Who cares then?"
# Client may give up on us after this circuit
self.reject_if_bad()
# OmniscientEvilGuard is the worst-case adversary against
# the path bias counters implemented in Tor 0.2.3.17.
#
# OmniscientEvilGuard knows client path counts, when they are about to
# think it's bad, and when they scale, and tries to use all of these
# to fail what it can to bias client paths without appearing bad to
# them.
#
# Further in favor of the adversary, we assume that their circuit
# failure rate is actually less than the network average by
# the fraction of the network that they control (because the rest
# of the network experiences this circuit failure as part of the
# average failure).
#
# Further still, OmnscientEvilGuard is *so* omnsicient, it even knows
# when circuits will fail due to ambient noise, so it never gets
# killed by chance. (It is debatable how much this helps.. a
# smart adversary could play the stats close enough to the line
# to approach this omniscience asymptotically).
#
# Note: These omniscience assumptions all favor the attacker,
# but they also simplify analysis to get worst-case bounds easily.
#
# XXX: Introducing some fuzz into our scaling count and/or rate might
# help remove this exact omniscience in practice?
class OmniscientEvilGuard(EvilGuard):
def __init__(self, succeed_rate, adversary_capacity):
EvilGuard.__init__(self, succeed_rate, adversary_capacity)
def look_ahead(self, n):
self.prev_first_hops = self._first_hops
self.prev_success = self._success
self.prev_first_hops_total = self.first_hops_total
self.prev_success_total = self.success_total
for i in xrange(n):
self._inc_first_hop()
def stop_looking(self):
self._first_hops = self.prev_first_hops
self._success = self.prev_success
self.first_hops_total = self.prev_first_hops_total
self.success_total = self.prev_success_total
# This guard should never get caught
def reject_if_bad(self):
assert not self.is_bad()
def build_circuit(self):
self._inc_first_hop()
# The presence of a malicious exit is a prior probability governed by the
# client. Decide it now.
got_malicious_exit = False
if random.random() < self.adversary_capacity:
got_malicious_exit = True
# In reality, OmniscientEvilGuard sees less failure because some
# of the failure in the network is due to other colluding nodes.
#if random.random() < self.succeed_rate + self.adversary_capacity:
#
# Note: We cut this out, because it favors the attacker to do so.
# It removes the risk of elimination by chance (which they could mitigate
# for an unknown but possibly small cost).
if True:
if got_malicious_exit: # via tagging attack
self.capture_count += 1
self._inc_success() # "We built a circuit! Yay!"
else:
# Look-ahead only needs to be non-zero to mitigate risk of random rejection
self.look_ahead(0)
if (self._get_rate() <= (PATH_BIAS_PCT/100.0)):
self.stop_looking()
self._inc_success() # "I better be good! don't want to get caught.."
else:
pass # Fail the circuit by doing nothing. It's not useful
# Client may give up on us after this circuit
self.reject_if_bad()
# ProbabalisticEvilGuard only fails untagged circuits pct_below_path_bias
# below the warning rate
class ProbabalisticEvilGuard(EvilGuard):
def __init__(self, succeed_rate, adversary_capacity, pct_below_path_bias):
EvilGuard.__init__(self, succeed_rate, adversary_capacity)
# FIXME: There may be an optimal point where pct_below_path_bias
# is the lowest possible value that the adversary expects to control?
# Doesn't seem to be worth probing, though
self.path_bias_rate = (PATH_BIAS_PCT - pct_below_path_bias)/100.0
assert self.path_bias_rate <= 1.0
def build_circuit(self):
self._inc_first_hop()
# The presence of a malicious exit is a prior probability governed by the
# client. Decide it now.
got_malicious_exit = False
if random.random() < self.adversary_capacity:
got_malicious_exit = True
# ProbabalisticGamingGuard sees less failure because some
# of the failure in the network is due to other colluding nodes.
if random.random() < self.succeed_rate + self.adversary_capacity:
if got_malicious_exit: # via tagging attack
self._inc_success()
self.capture_count += 1
elif not self.success_total or \
self.success_total/float(self.first_hops_total) <= self.path_bias_rate:
# "Uh oh, we're failing too much, better let some through"
self._inc_success()
else:
pass # Fail the circuit by doing nothing. It's not useful
# Client may give up on us after this circuit
self.reject_if_bad()
####################### Testing and Simulation #########################
def simulate_circs_until(g, circ_count, say_when):
for i in xrange(circ_count):
g.build_circuit()
if say_when(g):
return True
return say_when(g)
# Variables:
# success_rate
# PATH_BIAS_MIN_CIRCS = 20
# PATH_BIAS_PCT = 70
def startup_false_positive_test(trials, success_rate, min_circs, path_bias_pct):
# FIXME: Look it's just easier this way, ok? Get off my back already
global PATH_BIAS_MIN_CIRCS
global PATH_BIAS_PCT
PATH_BIAS_MIN_CIRCS = min_circs
PATH_BIAS_PCT = path_bias_pct
g = Guard(success_rate)
for i in xrange(1+trials/min_circs):
simulate_circs_until(g, PATH_BIAS_SCALE_THRESHOLD, lambda g: False)
g.reset()
#print g._get_rate()
return g.rejected_count
def reject_false_positive_test(trials, success_rate, scale_circs, path_bias_pct):
# FIXME: Look it's just easier this way, ok? Get off my back already
global PATH_BIAS_MIN_CIRCS
global PATH_BIAS_SCALE_THRESHOLD
global PATH_BIAS_PCT
PATH_BIAS_SCALE_THRESHOLD = scale_circs
PATH_BIAS_PCT = path_bias_pct
g = Guard(success_rate)
# Ignore startup. We don't reject then.
simulate_circs_until(g, PATH_BIAS_SCALE_THRESHOLD, lambda g: False)
g.rejected_count = 0
simulate_circs_until(g, trials, lambda g: False)
return g.rejected_count
def generic_rate_test(g, trials, success_rate, adversary_capacity, path_bias_pct, rate_fcn):
# FIXME: Look it's just easier this way, ok? Get off my back already
global PATH_BIAS_PCT
PATH_BIAS_PCT = path_bias_pct
simulate_circs_until(g, trials, lambda g: False)
if not isinstance(g, UnrepentantEvilGuard):
assert not g.is_bad()
return rate_fcn(g)
def dos_attack_test(success_rate, dos_success_rate, path_bias_pct, scale_thresh):
global PATH_BIAS_PCT
global PATH_BIAS_SCALE_THRESHOLD
PATH_BIAS_PCT = path_bias_pct
PATH_BIAS_SCALE_THRESHOLD = scale_thresh
g = Guard(success_rate)
simulate_circs_until(g, PATH_BIAS_SCALE_THRESHOLD, lambda g: False)
g.rejected_count = 0
g.succeed_rate = dos_success_rate
simulate_circs_until(g, 10000, lambda g: g.rejected_count > 0)
return g.first_hops_total - PATH_BIAS_SCALE_THRESHOLD
################ Multi-Dementianal Analysis #####################
# If brute force doesn't work, you're not using enough
def brute_force(cmptr, functor, ranges, increment):
testpoint = map(lambda p: p[0], ranges)
maxpoint = testpoint
maxval = functor(*testpoint)
print "New extrema at "+str(maxpoint)+": "+str(maxval)
for dementia in xrange(len(ranges)):
if increment[dementia] > 0:
cmpr = lambda x, y: x<y
else:
cmpr = lambda x, y: x>y
value = ranges[dementia][0]
while cmpr(value, ranges[dementia][1]):
value += increment[dementia]
testpoint[dementia] = value
val = functor(*testpoint)
if cmptr(val, maxval):
maxval = val
maxpoint = testpoint
print "New extrema at "+str(maxpoint)+": "+str(maxval)
# FIXME: Haxx
reset_globals()
return maxpoint
def surface_plot(functor, startpoint, ranges, increment):
pass
def gradient_descent(functor, startpoint, ranges, increment):
# Warning, mentat: If brute force doesn't work, you're not using enough
# It might be wise to try to get a 3d color plot/heatmap/some other
# visualization before attempting this?
pass
def main():
#random.seed(23)
if True:
print "==================== P(Compromise|Guard) =========================="
print "\nPassiveEvilGuard compromise rate at [success_rate, adversary_capacity, path_bias_pct]:"
print "(As expected, P(CompromisedExit|PassiveEvilGuard) ~= c/n)"
print brute_force(lambda x,y: x>y,
lambda t, a,b,c:
generic_rate_test(PassiveEvilGuard(a,b), t, a,b,c,
lambda g:
g.compromise_rate()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(10000,10000), (0.75,0.75), (0.05,0.85), (70, 70)],
[0, 0, 0.2, 5])
print "\nUnrepentantEvilGuard compromise rate at [success_rate, adversary_capacity, path_bias_pct]:"
print "(As expected, P(CompromisedExit|UnrepentantEvilGuard) = 1.0)"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(UnrepentantEvilGuard(a,b), t,a,b,c,
lambda g:
g.compromise_rate()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(10000,10000), (0.75,0.75), (0.05,0.85), (70, 70)],
[0, 0, 0.2, 5])
print "\nProbabalisticEvilGuard compromise rate at [success_rate, adversary_capacity, path_bias_pct]:"
print "P(CompromisedExit|ProbabalisticEvilGuard) <= (c/n)*(100/PATH_BIAS_PCT)"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(ProbabalisticEvilGuard(a,b,5),
t,a,b,c,
lambda g:
g.compromise_rate()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(10000,10000), (0.75,0.75), (0.05,0.85), (70, 70)],
[0, 0, 0.2, 5])
print "\nOmniscientEvilGuard compromise rate at [success_rate, adversary_capacity, path_bias_pct]:"
print "P(CompromisedExit|OmniscientEvilGuard) <= (c/n)*(100/PATH_BIAS_PCT)"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(OmniscientEvilGuard(a,b), t,a,b,c,
lambda g:
g.compromise_rate()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(10000,10000), (0.75,0.75), (0.05,0.85), (70, 70)],
[0, 0, 0.2, 5])
print "\nOmniscientEvilGuard compromise at [success_rate, adversary_capacity, path_bias_pct]:"
print "P(CompromisedExit|OmniscientEvilGuard) <= (c/n)*(100/PATH_BIAS_PCT)"
print brute_force(lambda x,y: x<y,
lambda t,a,b,c:
generic_rate_test(OmniscientEvilGuard(a,b), t,a,b,c,
lambda g:
g.compromise_rate()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(10000,10000), (0.75,0.75), (0.20,0.20), (20, 80)],
[0, 0, 0.05, 20])
if True:
print "\n\n==================== Circuits pwnt per client ========================="
print "\nUnrepentantEvilGuard compromised circs at [success_rate, adversary_capacity, path_bias_pct]:"
print "circs_per_client ~= success_rate*c/n*MIN_CIRCS for c/n < PATH_BIAS_PCT || c/n < success_rate"
print " ~= success_rate*circ_attempts*c/n for c/n > PATH_BIAS_PCT && c/n > success_rate"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(UnrepentantEvilGuard(a,b), t,a,b,c,
lambda g:
g.pwnt_per_client()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(100000,100000), (0.75,0.75), (0.05,0.85), (50, 50)],
[0, 0, 0.2, 5])
print "\nPassiveEvilGuard compromised circs at [success_rate, adversary_capacity, path_bias_pct]:"
print "circs_per_client ~= success_rate * circ_attempts * c/n"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(PassiveEvilGuard(a,b),
t,a,b,c,
lambda g:
g.pwnt_per_client()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(100000,100000), (0.75,0.75), (0.05,0.85), (50, 50)],
[0, 0, 0.2, 5])
print "\nProbabalisticEvilGuard compromised circs at [success_rate, adversary_capacity, path_bias_pct]:"
print "circs_per_client ~= success_rate * circ_attempts * c/n"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(ProbabalisticEvilGuard(a,b,5),
t,a,b,c,
lambda g:
g.pwnt_per_client()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(100000,100000), (0.75,0.75), (0.05,0.85), (50, 50)],
[0, 0, 0.2, 5])
print "\nOmniscientEvilGuard compromised circs at [success_rate, adversary_capacity, path_bias_pct]:"
print "circs_per_client ~= circ_attempts * c/n"
print brute_force(lambda x,y: x>y,
lambda t,a,b,c:
generic_rate_test(OmniscientEvilGuard(a,b), t,a,b,c,
lambda g:
g.pwnt_per_client()),
#generic_rate_test(trials, success_rate, adversary_capacity, path_bias_pct):
[(100000,100000), (0.75,0.75), (0.05,0.85), (50, 50)],
[0, 0, 0.2, 5])
if True:
print "\n\n===================== False Positives ============================"
print "\nStartup false positive counts at [num_circs, success_rate, min_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs min_circs)"
print brute_force(lambda x,y: x<y,
startup_false_positive_test,
#false_positive_test(num_circs, success_rate, min_circs, path_bias_pct):
[(1000000,1000000), (0.80, 0.80), (25,250), (70, 70)],
[0, -0.1, 25, 5])
print "\nStartup false positive counts at [num_circs, success_rate, min_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs min_circs)"
print brute_force(lambda x,y: x<y,
startup_false_positive_test,
#false_positive_test(num_circs, success_rate, min_circs, path_bias_pct):
[(1000000,1000000), (0.45, 0.45), (25,250), (30, 30)],
[0, -0.1, 25, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.70, 0.70), (100,500), (70, 70)],
[0, -0.1, 50, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.75, 0.75), (100,500), (70, 70)],
[0, -0.1, 50, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.80, 0.80), (100,500), (70, 70)],
[0, -0.1, 50, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.55, 0.55), (100,500), (50, 50)],
[0, -0.1, 50, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.60, 0.60), (100,500), (50, 50)],
[0, -0.1, 50, 5])
print "\nFalse positive counts at [num_circs, success_rate, scale_circs, path_bias_pct]:"
print "(Results are some function of success_rate - path_bias_pct vs scale_circs)"
print brute_force(lambda x,y: x<y,
reject_false_positive_test,
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(1000000,1000000), (0.45, 0.45), (100,500), (30, 30)],
[0, -0.1, 50, 5])
if True:
print "\n\n===================== DoS Attack Duration ========================"
print "\nDoS attack durations (in circs) at [success_rate, dos_success_rate, path_bias_pct, scale_thresh]:"
print brute_force(lambda x,y: x<y,
dos_attack_test,
#dos_attack_test(g, num_circs, success_rate, dos_success_rate, path_bias_pct):
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(0.80, 0.80), (0.25,0.05), (30, 30), (300, 300)],
[-0.1, -0.05, 5, 100])
print "\nDoS attack durations (in circs) at [success_rate, dos_success_rate, path_bias_pct, scale_thresh]:"
print brute_force(lambda x,y: x>y,
dos_attack_test,
#dos_attack_test(g, num_circs, success_rate, dos_success_rate, path_bias_pct):
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(0.80, 0.80), (0.25,0.25), (30, 30), (200, 1000)],
[-0.1, -0.1, 5, 100])
print "\nDoS attack durations (in circs) at [success_rate, dos_success_rate, path_bias_pct, scale_thresh]:"
print brute_force(lambda x,y: x>y,
dos_attack_test,
#dos_attack_test(g, num_circs, success_rate, dos_success_rate, path_bias_pct):
#false_positive_test(num_circs, success_rate, scale_circs, path_bias_pct):
[(0.80, 0.80), (0.05,0.05), (30, 30), (200, 1000)],
[-0.1, -0.1, 5, 100])
if __name__ == "__main__":
main() #sys.argv)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,412
|
Q: Follow original author's coding style? Even if it is horrible / lazy / makes your eyes bleed? When contributing to a very old, and apparently no longer maintained, open-source or public domain project, is it suggested practice to follow the original author's coding style even if it is terrible?
More importantly, are there any reasons why one would want to follow the original author's coding style rather than clean it up significantly?
A: If you know that it's unmaintained, and you're making changes to it, then I suspect that you'll become the defacto owner/maintainer going forward.
So, that said, I'd e-mail the original author/maintainer, see what they think about the formatting or style changes, and go forward with guns blazing.
A: Follow the original coding style. It is far, far better to be consistent, even if it's not pretty to you.
If you do decide to clean up the coding style, do it separately from any other changes. Don't clutter up source control diff's with style changes. Make one (or several) checkins where the only thing you're doing is changing code style. Do not mix in real changes with meaningless changes, it makes it impossible to locate relevant changes when reviewing source control commits.
A: If you're not submitting patches, then no. Run it through a code filter and automatically format the code in your preferred style, then hack away.
A: If checking into source control, you may want to leave it alone as all the style changes will make comparing to previous versions impossible.
If you do want to change the style do it with the baseline code before you make any other changes and then check it in.
Then check it out and make your coding changes. That way it will be easier to track your changes from the point you took it over.
I personally would leave the style alone unless you are making significant changes.
A: As stated by others, ensure that any style modifications are checked in to your version control system separately from your functional modifications.
However, my suggestion is to be bold and clean up the code as you see fit; "Leave the campsite cleaner than you found it". Cheesy, but true ;)
Just make sure that you are prepared to defend your changes ;)
A: Heck no! Whatever you do don't make it worse for heavens sake!
If you're adding your own code do it in your own style. This will at least make part of the codebase easy for you to maintain and understand. If you're making minor updates to existing code then you may want to follow that style.
I'm in the same boat, I have to maintain an ancient, business critical, Access 2 based system which is a complete mishmash of styles.
A: If it's not even maintained why are you worrying about it?
I'd write using my own preferred style, and clean up any code I had to work with. I do that regularly at work even...
A: Well, I'd say if you are just making a small modification within say a function, it might be clearer to go along with the bad style.
If you create new methods, classes, properties, etc., use a clear, efficient style. You might add a comment at the start of these sections, so that others can understand what's going on. You might even add a short note at the top explaining when you started on the code, and that you were using a different style, etc.
A: What would your answer be if the question were: Should you maintain the same shoddy workmanship when you are remodeling your home?
If you're fixing a broken wall tile in the bathroom, then it's not worth re-tiling the entire wall because they didn't use the right drywall.
When you're remodeling the kitchen, then you may correct the plumbing for that room or straighten out a bad floor. That's because it makes the rest of the job easier for you. And that's what it's all about, what kind of benefit does it give you verses the cost to you/your client.
You wouldn't rewire a lamp because it needed a new bulb.
A: If you're coding to a company-defined standard (or even just a team-defined standard), and the code that you're working on does not match the standard (or is so old that it doesn't match the current standard), then definitely clean up the code if you're going to be making major changes anyway. (As others have mentioned, do that to the baseline code, and check it in. Then check it out again and make the changes that you have to make to fix bugs, add features, etc.)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,502
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\section{Introduction} \label{sec:intro}
One of the most fundamental ways to characterize the steady state of a system is through the statistical properties of currents. These have been studied both in and out of equilibrium and in both classical~\cite{Derrida2007} and quantum systems~\cite{pilgram_stochastic_2003,Esposito2009,genway_trajectory_2014}. Recently, much progress has been achieved in understanding the statistics of time-averaged currents, which are encoded in a corresponding large deviation functions (LDF), of boundary-driven diffusive systems in one dimension~\cite{Derrida2004,Bodineau2004,Bertini2005a,Bertini2006,Harris2005,Imparato2009,Lecomte2010,Shpielberg2015} as well as in other geometries~\cite{bodineau_vortices_2008,Akkermans2013}. Whereas exact microscopic solutions are often available for bulk-driven systems~\cite{derrida_exact_1998,prolhac_cumulants_2009,lazarescu_exact_2011,mallick_exact_2011,lazarescu_matrix_2013,lazarescu_physicists_2015,ayyer_full_2015},
the results for boundary-driven systems largely rest on the application of a hydrodynamic approach termed the macroscopic fluctuation theory (MFT)~\cite{spohn_large_1991,Bertini2002,jordan_fluctuation_2004,Bertini2015}, with the notable exception of~\cite{lazarescu_matrix_2013}.
Being a hydrodynamic theory, the MFT is naively expected to yield the correct statistics of currents only when the current fluctuations are small enough for the hydrodynamic description to be valid. For example, consider a single-species diffusive system on the line $0 \leq x \leq \ell$, where $\ell$ denotes the length of the system. After coarse-graining and a diffusive rescaling ($x \to x/\ell$ and $t \to t/\ell^2$~\footnote{These notations indicate that the rescaled variables are defined as $\tilde{x} \equiv x/\ell$ and $\tilde{t} \equiv t/\ell^2$, and then renamed as $x$ and $t$, respectively. Other notations for rescaling schemes should be interpreted similarly.}), the hydrodynamic equation takes the form
\begin{equation}
\label{eq:hydro}
\partial_t \rho(x) = -\partial_x J(x),
\end{equation}
with $\rho(x)$ the coarse-grained density and $J(x)$ the coarse-grained current. Since we are interested in the $\ell \to \infty$ limit, this equation is not well defined for $J(x)$ which before the rescaling is not of the order of $1/\ell$. Thus, the statistics of currents obtained by the MFT are reliable only for current fluctuations of the order of $1/\ell$. The same conclusion can be reached by another argument more directly based on the MFT, which is discussed in Appendix~\ref{app:hydro_limit}.
In this paper we study the validity of the hydrodynamic approach in regions where it is expected to fail. Quite surprisingly, we find that there are classes of models where the hydrodynamic approach captures the statistics of currents much beyond its naive regime of validity. We give a simple explanation for this phenomena and based on it argue that this behavior is expected to be generic when the mobility diverges with the density of particles.
To obtain these results, we study current LDFs of boundary-driven systems whose lattice structure is preserved, keeping a finite number of sites $L$. Since the exact current LDFs of microscopic lattice models are difficult to obtain (with the exception of the zero-range-process~\cite{Harris2005}), we consider a little-studied class of coarse-grained models, which we term {\em large-$N$ models}. A large-$N$ model consists of a one-dimensional chain of boxes, each of which holds a macroscopically large number of particles (controlled by $N$) and which relaxes instantaneously to local equilibrium. As such, it retains the lattice structure even after coarse-graining and can be thought of as an analog of the ``boxed models'' studied in~\cite{Bunin2013,Kafri2015}. In a manner similar to models of population dynamics~\cite{Elgart2004,Meerson2011} and lattice spin models in the large-spin limit~\cite{Tailleur2007,Tailleur2008}, we rescale dynamical variables and hopping rates of the model by powers of $N$. This allows us to apply the standard saddle-point techniques in the $N \to \infty$ limit.
Thanks to simplifications arising from the assumption of a macroscopic number of particles at each box (site), the current LDFs of our large-$N$ models are exactly derivable even for a finite system with any number of sites $L$. By comparing the tail behaviors of the current LDFs in the large-$L$ limit with the predictions of the MFT approach, we can observe how and when non-hydrodynamic behaviors start to emerge. Interestingly, our formulation also shows that the same microscopic dynamics may produce {\em different macroscopic models} depending on how the microscopic variables are scaled with $N$.
We note that there were previous studies on models with multiple particles per site, such as partial exclusion processes~\cite{schutz_non-abelian_1994}, inclusion processes~\cite{giardina_duality_2007}, or both~\cite{giardina_correlation_2010,carinci_duality_2013}. These studies obtained exact expressions for particle density correlations on a finite lattice with $L$ sites. The corresponding density large deviations were studied in~\cite{Tailleur2007,Tailleur2008}, but only after a gradient expansion in the $L \to \infty$ limit that washes away the lattice structure. To our knowledge, large deviation properties of these models at finite $L$ have not been properly explored~\footnote{We note that there was a previous attempt to calculate the current LDF of a discrete system by applying a saddle-point approximation directly to the microscopic model~\cite{Imparato2009}. This approximation, however, is not well controlled.}.
This paper is organized as follows. In Sec.~\ref{sec:models}, we introduce two classes of large-$N$ models, which are the symmetric partial exclusion process (SPEP) and the symmetric inclusion process (SIP). It is shown that the latter becomes equivalent to the well-studied Kipnis--Marchioro--Presutti (KMP) model~\cite{Kipnis1982,Bertini2005b} after an appropriate rescaling by $N$. In Sec.~\ref{sec:spep_current}, we study current large deviations of the SPEP, which exhibits non-hydrodynamic behaviors for current fluctuations sufficiently far beyond the naive hydrodynamic regime expected by the argument given above. In addition, we also discuss the validity of the additivity principle. In Sec.~\ref{sec:sip_current}, we analyze current large deviations of the SIP for different large-$N$ limits, which in all cases exhibit hydrodynamic behaviors for arbitrarily large current fluctuations. Based on these results, in Sec.~\ref{sec:criterion} we propose a criterion for the persistence of hydrodynamic current fluctuations in the non-hydrodynamic regime, and confirm its validity for the symmetric zero-range process. Finally, we summarize our results and conclude in Sec.~\ref{sec:conclusions}.
\section{Large-$N$ models} \label{sec:models}
We now turn to introduce the large-$N$ versions of the SPEP and the SIP. Starting with the SPEP the microscopic model is defined and used to obtain a path-integral representation for the current cumulant generating function (CGF) along with the prescription for calculating it in the large-$N$ limit. The hydrodynamic limit of the model is then presented for completeness. The section closes by giving the corresponding results for the class of SIP models.
\subsection{Microscopic dynamics} \label{ssec:models_micro}
The models are defined on a one-dimensional chain of $L$ boxes which are in contact with two particle reservoirs denoted by $a$ and $b$ (see Fig.~\ref{fig:models} for an illustration). Each box is assumed to be in local equilibrium so that the state of box $k$ is completely specified by the number of particles $n_k$, for $k = 1,\,2,\,\ldots,\,L$. A particle hops from a box to an adjacent one with a rate (in arbitrary units) given by
\begin{align}\label{eq:micro_bulk_rates}
\text{SPEP:} &\qquad (n_k,\,n_l) \xrightarrow{n_k(N - n_l)} (n_k - 1,\, n_l + 1) \qquad \text{for $l = k \pm 1$}, \nonumber\\
\text{SIP:} &\qquad (n_k,\,n_l) \xrightarrow{n_k(N + n_l)} (n_k - 1,\, n_l + 1) \qquad \text{for $l = k \pm 1$},
\end{align}
which reflects exclusion (`attractive') interactions between particles in the SPEP (SIP). It is clear that for the SPEP the range of $n_k$ is bounded from above and below ($0 \le n_k \le N$), while for the SIP $n_k$ is only bounded from below ($n_k \ge 0$). The hopping rates at the boundaries are defined similarly as:
\begin{alignat}{3}\label{eq:micro_boundary_rates}
\text{SPEP:} &\qquad n_1 &\xrightarrow{\alpha (N - n_1)} n_1 + 1, &\qquad n_1 &\xrightarrow{\gamma n_1} n_1 - 1, \nonumber\\
&\qquad n_L &\xrightarrow{\delta (N - n_L)} n_1 + 1, &\qquad n_L &\xrightarrow{\beta n_L} n_L - 1, \nonumber\\
\text{SIP:} &\qquad n_1 &\xrightarrow{\alpha (N + n_1)} n_1 + 1, &\qquad n_1 &\xrightarrow{\gamma n_1} n_1 - 1, \nonumber\\
&\qquad n_L &\xrightarrow{\delta (N + n_L)} n_1 + 1, &\qquad n_L &\xrightarrow{\beta n_L} n_L - 1.
\end{alignat}
If the system is coupled only to reservoir $a$ (reservoir $b$), the average number of particles in each box relaxes to $\bar n_a$ ($\bar n_b$) as determined by $\alpha$ and $\gamma$ ($\beta$ and $\delta$).
In what follows, we fix the contact rates to the reservoirs through $N/(\gamma + \alpha) = 1$, $N/(\beta + \delta) = 1$ for the SPEP, and $N/(\gamma - \alpha) = 1$, $N/(\beta - \delta) = 1$ for the SIP.
The parameters $\bar{n}_a$ and $\bar{n}_b$ thus fully describe the coupling with the reservoirs:
\begin{align} \label{eq:micro_boundary_densities}
\text{SPEP:} &\qquad \alpha = \bar{n}_a, \quad \beta = N - \bar{n}_b, \quad \gamma = N - \bar{n}_a, \quad \delta = \bar{n}_b \;, \nonumber\\
\text{SIP:} &\qquad \alpha = \bar{n}_a, \quad \beta = N + \bar{n}_b, \quad \gamma = N + \bar{n}_a, \quad \delta = \bar{n}_b \;.
\end{align}
This choice provides simpler expressions in the results presented below, without affecting the large-$L$ hydrodynamic behavior.
\begin{figure}[b]
\includegraphics[width = 0.49\textwidth]{SPEP.pdf}
\includegraphics[width = 0.49\textwidth]{SIP.pdf}
\caption{\label{fig:models} Illustrations of two types of large-$N$ models. (Left) The SPEP features repulsive interactions, and each box can hold at most $N$ particles. (Right) The SIP features attractive interactions, and there is no upper bound on the number of particles in each box.}
\end{figure}
With these definitions it is natural to introduce density variables according to
\begin{equation} \label{eq:density_rescaling}
\rho_k \equiv \frac{n_k}{N}, \quad \bar\rho_a \equiv \frac{\bar n_a}{N}, \quad \bar\rho_b \equiv \frac{\bar n_b}{N}\;,
\end{equation}
and rescale time as $t \to Nt$. Then the evolution of the average density profile, taken over some initial distribution and denoted by angular brackets, satisfies
\begin{equation} \label{eq:average_dyn}
\frac{\partial \langle \rho_k \rangle}{\partial t} = \langle \rho_{k-1}\rangle - 2 \langle \rho_k \rangle + \langle \rho_{k+1} \rangle
\end{equation}
for any $k = 1,\,2,\,\ldots,\,L$ with $\rho_0 \equiv \bar\rho_a$ and $\rho_{L+1} \equiv \bar\rho_b$.
We note that the discrete diffusion equation~\eqref{eq:average_dyn} is also known to hold exactly for the standard Symmetric Simple Exclusion Process (SSEP), which corresponds to the SPEP with $N = 1$.
Under this rescaling, for the SPEP, $N$ is naturally interpreted as the capacity of each box. On the other hand, for the SIP the number of particles is not bounded from above. Therefore, $N$ does not admit a natural interpretation without specifying how both ${\bar n_a}$ and ${\bar n_b}$ scale with $N$. In fact, one can choose an alternate scaling and define densities for the SIP as
\begin{equation} \label{eq:density_rescaling_alpha}
\rho_k \equiv \frac{n_k}{N^{1+\alpha}}, \quad \bar\rho_a \equiv \frac{\bar n_a}{N^{1+\alpha}}, \quad \bar\rho_b \equiv \frac{\bar n_b}{N^{1+\alpha}}
\end{equation}
with $t$ rescaled by $N$ as above and $\alpha>0$ (the rationale behind this constraint will become clear below). It is straightforward to check that \eqref{eq:average_dyn} is then unchanged. Interestingly, these two scaling choices for the SIP, as we show below, lead to different {\em macroscopic} theories. In what follows, when we also study the SIP rescaled by \eqref{eq:density_rescaling_alpha} and refer to it as SIP(1+$\alpha$), in contrast to the SIP(1) whose scaling is defined in \eqref{eq:density_rescaling}.
\subsection{SPEP -- current CGF and hydrodynamic limit}\label{ssec:models_spep}
Our interest is in calculating the current CGF which encodes the statistics of the time-averaged density current $J$. We can obtain $J$, for example, by measuring the flux of particles from box $L$ to reservoir $b$ during an interval $t \in [0,\,T]$. The CGF is then defined through
\begin{equation} \label{eq:cgf_def}
e^{NT\psi_{N,L}(\lambda,\bar\rho_a,\bar\rho_b)} = \left\langle e^{N\lambda T J}\right\rangle \quad \text{for $T \gg 1$,}
\end{equation}
where the average, denoted by angular brackets, is taken with fixed $\bar\rho_a$ and $\bar\rho_b$, and $\lambda$ is conjugate to the current $J$. Using standard methods (see Appendix~\ref{app:path}), we can write a path-integral representation of the CGF
\begin{equation} \label{eq:macro_path_integ}
e^{NT\psi_{N,L}(\lambda)} = \int \mathcal{D}\boldsymbol{\rho} \mathcal{D}\hat{\boldsymbol{\rho}} \,
\exp\left\{-N\int_{0}^{T} \mathrm{d}t \,\left[ \hat{\boldsymbol{\rho}}\cdot\dot{\boldsymbol{\rho}}
- H_L(\lambda;\boldsymbol{\rho},\hat{\boldsymbol{\rho}})\right] + o(N)\right\}
\end{equation}
with $\boldsymbol{\rho} \equiv (\rho_1, \rho_2, \ldots, \rho_L)$ the density vector and $\hat{\boldsymbol{\rho}}\equiv (\hat\rho_1, \hat\rho_2, \ldots, \hat\rho_L)$ the auxiliary `momentum' vector. For the SPEP, the Hamiltonian $H_L$ is given by
\begin{align}\label{eq:spep_H}
H^\mathrm{SPEP}_L(\lambda;\boldsymbol{\rho},\hat{\boldsymbol{\rho}})
&= \sum_{k=1}^{L-1} \left[ \rho_k (1-\rho_{k+1}) \left(e^{\hat{\rho}_{k+1} - \hat{\rho}_k}-1\right)
+ \rho_{k+1} (1-\rho_k) \left(e^{\hat{\rho}_k - \hat{\rho}_{k+1}}-1\right)\right] \nonumber\\
&\quad +\rho_1 (1-\bar{\rho}_a) \left(e^{- \hat{\rho}_1}-1\right) + \bar{\rho}_a (1-\rho_1) \left(e^{\hat{\rho}_1}-1\right) \nonumber\\
&\quad +\rho_L (1-\bar{\rho}_b) \left(e^{- \hat{\rho}_L + \lambda}-1\right) + \bar{\rho}_b (1-\rho_L) \left(e^{\hat{\rho}_L - \lambda}-1\right).
\end{align}
When $N$ is very large (in the sense of $N \gg T \gg 1$), the large-$N$ CGF $\psi_L$ can be obtained using saddle-point asymptotics
\begin{equation} \label{eq:cgf_saddle}
\psi_L(\lambda) \equiv \lim_{N\to\infty} \psi_{N,L}(\lambda) = \lim_{T \to \infty} \frac{1}{T} \inf_{\boldsymbol{\rho},\,\hat{\boldsymbol{\rho}}}\int_{0}^{T} \mathrm{d}t \,
\left[ \hat{\boldsymbol{\rho}}\cdot\dot{\boldsymbol{\rho}} - H_L(\lambda;\boldsymbol{\rho},\hat{\boldsymbol{\rho}}) \right]
\end{equation}
with the infimum taken over trajectories of $\boldsymbol{\rho}$ and $\hat{\boldsymbol{\rho}}$. As advertised above, this approximation requires only $N$ to be a large parameter, so its predictions hold for any value of $L$.
The minimization principle~\eqref{eq:cgf_saddle} is similar to that of the MFT approach~\cite{Bertini2015} for the SSEP, with $N$, instead of $L$, playing the role of the large parameter governing the saddle-point. This allows us to keep track of the lattice structure at any finite~$L$.
Assuming that the minimizing trajectory is time-independent, the saddle-point equations are given by
\begin{equation} \label{eq:time_indep_sol}
\frac{\partial \boldsymbol{\rho}}{\partial t} = \frac{\partial H_L}{\partial \hat{\boldsymbol{\rho}}} = 0, \quad \frac{\partial \hat{\boldsymbol{\rho}}}{\partial t} = -\frac{\partial H_L}{\partial \boldsymbol{\rho}} = 0.
\end{equation}
The solutions of these equations, which we denote by $\boldsymbol{\rho}^*$ and $\hat{\boldsymbol{\rho}}^*$, are typically called the {\em optimal profiles} which support the current fluctuation $J$. Then the current CGF is obtained from \eqref{eq:cgf_saddle} as
\begin{equation} \label{eq:cgf_H}
\psi_L(\lambda) = H_L(\lambda;\boldsymbol{\rho}^*,\hat{\boldsymbol{\rho}}^*).
\end{equation}
The additivity principle, proposed in~\cite{Bodineau2004} (also independently studied in \cite{jordan_fluctuation_2004}), implies that the above assumption is applicable for any value of $\lambda$. Although counterexamples were found in periodic bulk-driven systems~\cite{Bertini2005a,bodineau_distribution_2005,Bertini2006,Hurtado2011,Espigares2013}, the principle was analytically shown to be true for any open boundary-driven diffusive system with a constant diffusion coefficient and a quadratic mobility coefficient~\cite{Imparato2009} ---~without ruling out possible discontinuous transitions, which in turn were numerically discarded in~\cite{Hurtado2009} for a specific model related to the SIP. As shown below, both the SPEP and the SIP correspond to this class of systems in the hydrodynamic limit. Thus we expect that the same principle is also applicable to our large-$N$ models, and discuss arguments supporting its validity in Sec.~\ref{ssec:spep_fin_N} and Appendix~\ref{app:finiteNquantumfluctuationsSPEP}.
Finally, we show that under appropriate assumptions our large-$N$ models are well described by hydrodynamic theories. To see this, we first apply a diffusive scaling in terms of $L$, which involves writing the position of box $k$ as $x \equiv k/(L+1)$ (with the lattice spacing set to one) and rescaling time by $t \to t/(L+1)^2$. We also assume that differences between adjacent boxes, namely $\rho_{k+1} - \rho_k$ and $\hat\rho_{k+1} - \hat\rho_k$, scale as $1/(L+1)$. Then, in the $L \to \infty$ limit, the gradients $\partial_x \rho$ and $\partial_x \hat\rho$ are well defined, and \eqref{eq:macro_path_integ} can be approximated as
\begin{equation} \label{eq:hydro_path_integ}
e^{N (L+1)^2 T\psi(\lambda)} = \int \mathcal{D} \rho \mathcal{D}\hat\rho \,
\exp\left\{-N(L+1) \int_{0}^{T} \mathrm{d}t \left[ \left( \int_0^1 \mathrm{d}x \, \hat\rho \dot{\rho} \right) - H [\rho,\hat\rho] \right]\right\}.
\end{equation}
Here the Hamiltonian $H [\rho,\hat\rho]$, which is no longer dependent on $\lambda$, is now a functional of continuous profiles $\rho(x)$ and $\hat\rho(x)$. The functional typically has the form of
\begin{equation} \label{eq:H_mft}
H[\rho,\hat\rho] = \int_0^1 \mathrm{d}x \,
\left[ -D(\rho)(\partial_x \rho)(\partial_x \hat\rho) + \frac{\sigma(\rho) (\partial_x \hat\rho)^2}{2} \right]
\end{equation}
with $D(\rho)$ the diffusion coefficient and $\sigma(\rho)$ the mobility coefficient. For the SPEP, these coefficients are given by
\begin{equation}
D(\rho) = 1, \quad \sigma(\rho) = 2\rho(1-\rho),
\end{equation}
respectively. We note that this $\sigma(\rho)$ is bounded from above, with the maximum value given by $\sigma(1/2) = 1/2$. Meanwhile, the rescaling of time speeds up the microscopic dynamics, so the leftmost ($k = 1$) and rightmost ($k = L$) boxes equilibrate with the coupled reservoirs (see \emph{e.g.}~Appendix B.2 of Ref.~\cite{Tailleur2008}). Hence, the spatial boundary conditions are given by
\begin{equation}
\rho(0) = \bar{\rho}_a, \quad \rho(1) = \bar{\rho}_b, \quad \hat\rho(0) = 0, \quad \hat\rho(1) = \lambda,
\end{equation}
whose dependence on $\lambda$ keeps $\psi$ a function of $\lambda$.
In what follows we list the corresponding sets of results for the SIP(1) and the SIP(1+$\alpha$).
\subsection{SIP(1) -- current CGF and hydrodynamic limit}\label{ssec:models_sip}
It is straightforward to repeat the above derivations for the SIP. We find, using the notation of \eqref{eq:macro_path_integ},
\begin{align} \label{eq:sip1_H}
H^{\mathrm{SIP}(1)}_L (\lambda;\boldsymbol{\rho},\hat{\boldsymbol{\rho}}) &\equiv \sum_{k=1}^{L-1} \left[ \rho_k (1 + \rho_{k+1})
\left(e^{\hat\rho_{k+1} - \hat\rho_k}-1\right) + \rho_{k+1} (1 + \rho_k) \left(e^{\hat\rho_k - \hat\rho_{k+1}}-1\right)\right] \nonumber\\
&\quad+ \left[ \rho_1 (1 + \bar{\rho}_a) \left(e^{- \hat\rho_1}-1\right) + \bar{\rho}_a (1 + \rho_1) \left(e^{\hat\rho_1}-1\right)\right] \nonumber\\
&\quad+ \left[ \rho_L (1 + \bar{\rho}_b) \left(e^{- \hat\rho_L+\lambda}-1\right) + \bar{\rho}_b (1 + \rho_L) \left(e^{\hat\rho_L-\lambda}-1\right)\right].
\end{align}
In addition, the corresponding hydrodynamic Hamiltonian in the large-$L$ limit is given by \eqref{eq:H_mft} with
\begin{equation} \label{eq:coeff_sip1}
D(\rho) = 1, \quad \sigma(\rho) = 2\rho(1+\rho).
\end{equation}
We note that $\sigma(\rho)$ in this case is not bounded from above.
\subsection{SIP(1+$\alpha$) -- current CGF and hydrodynamic limit}\label{ssec:models_sip1a}
For the SIP(1+$\alpha$) we similarly find, using the notation of \eqref{eq:macro_path_integ},
\begin{align} \label{eq:sip1a_H}
H^{\mathrm{SIP}(1+\alpha)}_L(\lambda;\boldsymbol{\rho},\hat{\boldsymbol{\rho}})
&\equiv \sum_{k=1}^{L-1} \left[-(\rho_{k+1} - \rho_k)(\hat\rho_{k+1} - \hat\rho_k) + \rho_k \rho_{k+1}(\hat\rho_{k+1} - \hat\rho_k)^2\right] \nonumber\\
&\quad + (\bar\rho_a - \rho_1)\hat\rho_1 + \bar\rho_a \rho_1 \hat\rho_1^2 + (\bar\rho_b - \rho_L)(\hat\rho_L-\lambda) + \rho_L \bar\rho_b (\hat\rho_L-\lambda)^2.
\end{align}
The hydrodynamic description of this model in the large-$L$ limit is given by \eqref{eq:H_mft} with
\begin{equation} \label{eq:coeff_sip1a}
D(\rho) = 1, \quad \sigma(\rho) = 2\rho^2,
\end{equation}
where $\sigma(\rho)$ is again not bounded from above. These transport coefficients are also shared by the Kipnis--Marchioro--Presutti (KMP) model of heat conduction~\cite{Kipnis1982,Bertini2005b}. It is notable that the same microscopic model produces different macroscopic behaviors depending on the reservoir properties.
\section{Current large deviations in the SPEP}
\label{sec:spep_current}
In what follows we first show that the scaled CGF of the time-averaged current in the SPEP in the large-$N$ limit is given by
\begin{equation} \label{eq:spep_cgf}
\psi^\mathrm{SPEP}_L(\lambda) = \begin{cases}
(L+1) \sinh^2 \left(\frac{1}{L+1} \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right) &\text{if $\omega^\mathrm{SPEP} \ge 0$}, \\
-(L+1) \sin^2 \left(\frac{1}{L+1} \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right) &\text{if $\omega^\mathrm{SPEP} < 0$}.
\end{cases}
\end{equation}
where
\begin{equation} \label{eq:spep_omega}
\omega^\mathrm{SPEP} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b - (e^\lambda - 1)\bar\rho_a\bar\rho_b\right].
\end{equation}
Note that although the result depends explicitly on the sign of $\omega^\mathrm{SPEP}$, it is straightforward to verify that it is an analytic function of $\lambda$. After deriving this result, we compare \eqref{eq:spep_cgf} to the predictions of the hydrodynamic theory. As we show, for large enough currents the two theories, as one might expect using the simple argument of the introduction, do not agree. Finally, we discuss finite-$N$ effects and their implications on the additivity principle.
\subsection{Derivation of the scaled CGF} \label{ssec:spep_cgf}
\label{ssec:ldfSPEPderivation}
As stated above, assuming additivity, the problem of calculating the CGF in the large-$N$ limit is reduced to solving \eqref{eq:time_indep_sol}. To do this it is useful to use the canonical transformation~\cite{DLS2002,Tailleur2007,Tailleur2008}
\begin{equation} \label{eq:spep_canonical}
\rho_k = F_k \left[1+(1 - F_k) \hat{F}_k\right], \qquad \hat{\rho}_k = \ln \left( 1 + \frac{\hat{F}_k}{1-F_k \hat{F}_k} \right) \;,
\end{equation}
which can also be written as
\begin{equation}
F_k = \frac{\rho_k}{e^{\hat\rho_k}(1-\rho_k) + \rho_k}\;, \qquad \hat{F}_k = (e^{\hat\rho_k}-1)(1-\rho_k) + (1-e^{-\hat\rho_k})\rho_k.
\end{equation}
Then the Hamiltonian in the new set of coordinates, $K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})$, is given by
\begin{align} \label{eq:spep_K}
K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1}) - \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right]
+ \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + e^{-\lambda}\left[\hat F_L - (e^\lambda - 1)(1-F_L\hat F_L) \right] \left[\bar{\rho}_b - F_L - F_L(1 - \bar{\rho}_b)(e^\lambda - 1)\right].
\end{align}
where $\mathbf{F}=\left(F_1,F_2,\ldots, F_L\right)$ and $\hat{\mathbf{F}}=\left(\hat{F}_1,\hat{F}_2,\ldots, \hat{F}_L\right)$.
Note that the canonical transformation also adds temporal boundary conditions to the action which can be ignored in the $T \to \infty$ limit. The scaled CGF is then given by:
\begin{align} \label{eq:spep_cgf_K}
\psi_L(\lambda,\bar\rho_a,\bar\rho_b) = K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F}^*,\hat{\mathbf{F}}^*),
\end{align}
where $(\mathbf{F}^*,\hat{\mathbf{F}}^*)$ are solutions of
\begin{equation} \label{eq:spep_K_time_indep_sol}
\frac{\partial \mathbf{F}}{\partial t} = \frac{\partial K^\mathrm{SPEP}_L}{\partial \hat{\mathbf{F}}} = 0,
\qquad \frac{\partial \hat{\mathbf{F}}}{\partial t} = -\frac{\partial K^\mathrm{SPEP}_L}{\partial \mathbf{F}} = 0.
\end{equation}
In what follows, we solve these equations using the methods used in \cite{Imparato2009}. To avoid cumbersome expressions we drop the $^*$ notation from the optimal profiles $(\mathbf{F}^*,\hat{\mathbf{F}}^*)$ and use $(\mathbf{F},\hat{\mathbf{F}})$. First, we choose the Ansatz
\begin{align} \label{eq:spep_dF_dFh_saddle}
\hat F_k = -A \sinh (kB), \qquad F_k = \bar\rho_a + \frac{1}{2A}\tanh \frac{kB}{2},
\end{align}
where $A$ and $B$ are undetermined constants. It is easy to check that this Ansatz satisfies \eqref{eq:spep_K_time_indep_sol} for $1 \le k \le L-1$.
Then the constants $A$ and $B$ are determined by the remaining saddle-point equations
\begin{equation}
\frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L} = \frac{\partial K^\mathrm{SPEP}_L}{\partial F_L} = 0.
\end{equation}
These equations imply
\begin{equation}
\frac{\partial K^\mathrm{SPEP}_L}{\partial F_L} = -4A^2\cosh \frac{LB}{2} \frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L},
\end{equation}
from which we obtain
\begin{equation} \label{eq:spep_A_saddle}
A^2 = \mathcal{A}^\mathrm{SPEP} \equiv \frac{(e^\lambda - 1) [e^\lambda(\bar\rho_b - 1) - \bar\rho_b]}
{4 [1 + (e^\lambda - 1)\bar\rho_a] [e^\lambda\bar\rho_a(\bar\rho_b-1)-\bar\rho_a\bar\rho_b+\bar\rho_b]} \;.
\end{equation}
Then one can show that $-\frac{1}{A}\frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L} = 0$ has the form of
\begin{equation} \label{eq:spep_B_epsilon}
\sinh(LB - \varepsilon) + \sinh B = 2 \sinh \frac{(L+1)B - \varepsilon}{2} \cosh \frac{(L-1)B - \varepsilon}{2} = 0,
\end{equation}
where $\varepsilon$ satisfies
\begin{equation} \label{eq:spep_omega2}
\sinh^2 \frac{\varepsilon}{2}
= \omega^\mathrm{SPEP} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b - (e^\lambda - 1)\bar\rho_a\bar\rho_b\right].
\end{equation}
Given $\varepsilon$, \eqref{eq:spep_B_epsilon} is solved by
\begin{equation}
B = \frac{\varepsilon}{L+1}.
\end{equation}
Thus we have found $A$ and $B$ up to the undetermined signs of $A$ and $\varepsilon$. These signs can be fixed by noting that the optimal density profile $\boldsymbol{\rho}^*$ must always be nonnegative and that the CGF must vanish at $\lambda = 0$. Without loss of generality, for $\bar\rho_a \ge \bar\rho_b$ the optimal profiles are given by
\begin{align} \label{eq:spep_profiles}
\hat F_k^\mathrm{SPEP} &=
\begin{cases}
-\sqrt{\mathcal{A}^\mathrm{SPEP}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda < -\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right]$,} \\
-\sqrt{-\mathcal{A}^\mathrm{SPEP}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^\mathrm{SPEP}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda \ge 0$.}
\end{cases} \nonumber \\
F_k^\mathrm{SPEP} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^\mathrm{SPEP}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda < -\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right]$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^\mathrm{SPEP}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^\mathrm{SPEP}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda \ge 0$.} \\
\end{cases} \nonumber \\
\end{align}
We note that $\mathcal{A}^\mathrm{SPEP}$ and $\omega^\mathrm{SPEP}$ are negative for the intermediate range $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] < \lambda < 0$ and nonnegative otherwise. The results for $\bar\rho_a < \bar\rho_b$ are easily obtained by a sign change $\lambda \to -\lambda$ and an exchange of $\bar\rho_a$ and $\bar\rho_b$. Using these results with \eqref{eq:spep_cgf_K}, after some algebra one obtains \eqref{eq:spep_cgf}.
\subsection{Comparison with hydrodynamic results} \label{ssec:spep_hydro}
We now compare the results of the large-$N$ limit with the predictions of the hydrodynamic theory. The latter has been derived in~\cite{Bodineau2004,Imparato2009} (for the SSEP which shares the same hydrodynamic theory) and can also be obtained by holding $\lambda$ fixed in \eqref{eq:spep_cgf} and taking the large $L$ limit. The expression is given by
\begin{equation} \label{eq:spep_cgf_hydro}
\psi^\mathrm{SPEP}(\lambda) =
\begin{cases}
\frac{1}{L+1} \mathrm{arcsinh}^2 \sqrt{\omega^\mathrm{SPEP}} &\text{if $\omega^\mathrm{SPEP} \ge 0$}, \\
-\frac{1}{L+1} \mathrm{arcsin}^2 \sqrt{-\omega^\mathrm{SPEP}} &\text{if $\omega^\mathrm{SPEP} < 0$} \;,
\end{cases}
\end{equation}
and the convergence to it is illustrated in Fig.~\ref{fig:spep_cgf_lambda_psi}. In fact, one can show analytically that
\begin{equation}
\psi^\mathrm{SPEP}_L(\lambda) - \psi^\mathrm{SPEP}(\lambda)
= \frac{\mathrm{arcsinh}^4\sqrt{\omega^\mathrm{SPEP}}}{3(L+1)^3} + O\left((L+1)^{-4}\right),
\end{equation}
The sign of the leading correction term indicates that the lattice structure increases the magnitude of the current fluctuations.
To check the validity of the hydrodynamic predictions we next increase $\lambda$ as $\lambda \sim L^\zeta$. This gives
\begin{equation}
\lim_{L \to \infty} \frac{\psi^\mathrm{SPEP}_L(\lambda)}{\psi^\mathrm{SPEP}(\lambda)} =
\begin{cases}
1 &\text{ if $\zeta < 1$,} \\
\frac{4}{\Lambda^2}\sinh^2 \frac{\Lambda}{2} &\text{ if $\zeta = 1$ with $\lambda=\Lambda L$} \\
\infty &\text{ if $\zeta > 1$.}
\end{cases}
\end{equation}
This indicates that, as one would naively expect, the hydrodynamic description fails for sufficiently large currents. The threshold separating the hydrodynamic regime from the non-hydrodynamic regime is given by $\lambda \sim L$ (see Fig.~\ref{fig:spep_hydro_breakdown}).
\begin{figure}
\includegraphics[width = 0.7\textwidth]{SPEP_CGF_lambda_psi.pdf}
\caption{\label{fig:spep_cgf_lambda_psi} The scaled CGFs of the SPEP: the large $N$ limit $\psi_L^\text{SPEP}(\lambda)$ given by \eqref{eq:spep_cgf} (solid lines) and its hydrodynamic limit $\psi^\text{SPEP}(\lambda)$ given by \eqref{eq:spep_cgf_hydro} (dashed lines). For any fixed $\lambda$, $\psi_L^\text{SPEP}(\lambda)$ is equivalent to $\psi^\text{SPEP}(\lambda)$ as $L \to \infty$. The boundary conditions are given by $\bar\rho_a = 0.8$, $\bar\rho_b = 0.2$.}
\end{figure}
\begin{figure}
\includegraphics[width = 0.7\textwidth]{SPEP_CGF_L_ratio.pdf}
\caption{\label{fig:spep_hydro_breakdown} Breakdown of the hydrodynamic limit in the SPEP for $\bar\rho_a = 0.8$ and $\bar\rho_b = 0.2$. (Left) If $\lambda \sim L^\zeta$, $\psi_L$ and $\psi$ are equivalent to each other as $L\to\infty$ only for $\zeta < 1$. (Right) The scaled CGFs of current in the large $N$ limit (solid lines) and in the hydrodynamic limit (dashed lines). If $\lambda = L$, the CGFs do not converge to each other in the $L \to \infty$ limit.}
\end{figure}
As we later show, there are other models where the predictions of the hydrodynamic theory hold well beyond the naive expectation. To this end it is useful to see in detail how the predictions of the hydrodynamic limit fail for the SPEP. To do this, we note that Hamilton's equation takes the form of
\begin{equation}
\dot{\rho_k} = \frac{\partial H^\mathrm{SPEP}_L}{\partial \hat\rho_k} = J_{k-1,k} - J_{k,k+1},
\end{equation}
where $J_{k,k+1}$ is the current from box $k$ to box $k+1$. The time-averaged current $J$ can be expressed in terms of the optimal profiles (again we drop the $^*$ notation) as
\begin{align} \label{eq:spep_J_profiles}
J &= \frac{1}{L+1}\sum_{k=0}^L\left[(\rho_k - \rho_{k+1})+\rho_k(1-\rho_{k+1})(e^{\hat\rho_{k+1}-\hat\rho_k}-1)
- \rho_{k+1}(1-\rho_k)(e^{\hat\rho_k-\hat\rho_{k+1}}-1)\right] \nonumber\\
&= \underbrace{\frac{\bar\rho_a-\bar\rho_b}{L+1}}_{= \langle J \rangle} + \underbrace{\frac{1}{L+1}
\sum_{k=0}^L\left[\rho_k(1-\rho_{k+1})(e^{\hat\rho_{k+1}-\hat\rho_k}-1) - \rho_{k+1}(1-\rho_k)(e^{\hat\rho_k-\hat\rho_{k+1}}-1)\right]}_{= \delta J}.
\end{align}
Since the mean value $\langle J \rangle$ always scales as $1/L$, large values of $J$ are always dominated by the fluctuation $\delta J$ (see \cite{Meerson2014} for a similar observation). Next, note that, as shown in Fig.~\ref{fig:spep_profile}, a large $\delta J$ is supported by a plateau of the density profile close to $\rho = 1/2$ and a slope of the momentum profile which grows with $\lambda$ (and hence with $J$). In addition, as indicated by the data collapses in Fig.~\ref{fig:spep_rhoh_collapse}, the momentum profile has the scaling form
\begin{equation}
\hat\rho_k(\lambda,L) \simeq \lambda g(k/L).
\end{equation}
This implies that
\begin{equation}
\hat\rho_{k+1}(\lambda,L) - \hat\rho_k(\lambda,L) \simeq \frac{\lambda}{L} g'(k/L) \simeq L^{\zeta - 1} \partial_x g.
\end{equation}
If $\zeta < 1$, the momentum gradient decreases with $L$. Then we can approximate $\delta J$ as
\begin{align}\label{eq:spep_J_hydro}
\delta J \simeq L^{\zeta - 1} \int_0^1 \mathrm{d} x\, 2 \rho(1-\rho) \partial_x g,
\end{align}
whose integral form suggests that the current is blind to the lattice structure for any $\zeta < 1$. In other words, the current does not feel any difference between the case $\zeta = 0$ (which can be considered as proper hydrodynamic regime) and the case $0 < \zeta < 1$. Thus its fluctuations show hydrodynamic behaviors in both cases. On the other hand, if $\zeta \ge 1$, the momentum gradient increases with $L$. Then the approximate \eqref{eq:spep_J_hydro} becomes invalid, and the current becomes sensitive to the lattice structure. Thus $\zeta = 1$, which corresponds to $J = O(L^0)$ by \eqref{eq:spep_J_profiles}, is the threshold separating the hydrodynamic regime from the non-hydrodynamic one. We note that this threshold is larger than what one would naively expect from the simple argument given in Sec.~\ref{sec:intro}, {\it i.e.,} $J = O(L^{-1})$.
\begin{figure}
\includegraphics[width = 0.45\textwidth]{SPEP_profile_k_rho.pdf}\quad
\includegraphics[width = 0.45\textwidth]{SPEP_profile_k_rhoh.pdf}
\caption{\label{fig:spep_profile} The optimal profiles of the SPEP for $L = 100$, $\bar\rho_a = 0.8$, and $\bar\rho_b = 0.2$, obtained from \eqref{eq:spep_canonical} and~\eqref{eq:spep_profiles}. (Left) As $\lambda$ is increased, the density profile forms a flat bulk profile at $\rho = 1/2$ so that the factor $\rho(1-\rho)$ in $\delta J$ is maximized. (Right) As $\lambda$ is increased, the slope of the momentum profile increases so that its contribution to the time-averaged current becomes larger (see \eqref{eq:spep_J_profiles}).}
\end{figure}
\begin{figure}
\includegraphics[width = 0.45\textwidth]{SPEP_profile_k_rhohlambda.pdf}\quad
\includegraphics[width = 0.45\textwidth]{SPEP_profile_kL_rhoh.pdf}
\caption{\label{fig:spep_rhoh_collapse} (Left) Momentum profiles at different values of $\lambda$ can be collapsed by using $\hat\rho_k/\lambda$ as the vertical axis. The other parameters are fixed at $L = 100$, $\bar\rho_a = 0.8$, and $\bar\rho_b = 0.2$. (Right) Momentum profiles at different values of $L$ can be collapsed by using $k/L$ as the horizontal axis. The other parameters are fixed at $\lambda = 1$, $\bar\rho_a = 0.8$, and $\bar\rho_b = 0.2$.}
\end{figure}
\subsection{Finite-$N$ corrections and the validity of the additivity principle} \label{ssec:spep_fin_N}
In what follows, we analyze the leading finite-$N$ correction to the scaled CGF $\psi^\mathrm{SPEP}_L$. This provides a useful tool for numerical corroboration of our analytical results, and confirms the stability of the time-independent saddle-point profiles. The latter thus supports the validity of the additivity principle for the SPEP.
As explained in Appendix~\ref{app:finiteNquantumfluctuationsSPEP}, one can integrate spatio-temporal fluctuations around the saddle-point optimal solutions. This is done by using a mapping (generalizing that of Ref.~\cite{Lecomte2010}) between the CGF of the system with reservoirs at generic densities $\bar\rho_a$, $\bar\rho_b$ and the CGF for reservoirs at densities $\frac 12$.
The resulting expression is finite and analytic, which proves that the additivity hypothesis is correct with respect to continuous phase transitions towards time-dependent profiles (which, if they had existed, would have implied an instability of $\boldsymbol{\rho}^*,\hat{\boldsymbol{\rho}}^*$, reflected in a singularity of the correction).
The saddle-point contribution $\psi_L(\lambda)$ to the CGF is complemented by a $1/N$ correction:
\begin{equation} \label{eq:spep_cgf_fin_N_series}
\psi^\mathrm{SPEP}_{N,L}(\lambda)=\psi^\mathrm{SPEP}_L(\lambda) +N^{-1} \psi_L^{1,\mathrm{SPEP}}(\lambda) + o(N^{-1})
\end{equation}
with, denoting $L'=L+1$,
\begin{align}
\psi_L^{1,\mathrm{SPEP}}(\lambda)
&=
\sum_{p=1}^{L'-1}
\bigg\{
\boldsymbol c_\lambda
-
\cos\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L'}\big)}
-
\sin\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda-\cos\frac{p\pi}{L'}\big)}
\bigg\}
\label{eq:respsiLhalfdensities_sym}
\end{align}
where we defined $\boldsymbol c_\lambda= \cosh \frac{2\operatorname{arcsinh}\sqrt{\omega^{\text{SPEP}}}}{L'}$.
We numerically confirm our theoretical predictions by implementing a finite-$N$ propagator of the SPEP conditioned on a given value of $\lambda$. The eigenvalue with the largest real part corresponds to the scaled CGF $\psi^\mathrm{SPEP}_{N,L}$. As shown in Fig.~\ref{fig:spep_fin_N}, our theory correctly predicts the leading-order behaviors of $\psi^\mathrm{SPEP}_{N,L} - \psi^\mathrm{SPEP}_L$.
\begin{figure}
\includegraphics[width = 0.6\textwidth]{{Correction_SPEP_finN_ra0.8_rb0.2_L3}.eps}
\caption{\label{fig:spep_fin_N} Finite-$N$ corrections to the scaled CGF of the SPEP at $L = 3$, $\bar\rho_a = 0.8$, and $\bar\rho_b = 0.2$. The numerics (symbols) are in good agreement with the leading-order correction (dashed lines) predicted by \eqref{eq:respsiLhalfdensities_sym}. The predicted scaling exponent is also supported by the successive slopes of finite-$N$ corrections in the log-log plot.}
\end{figure}
We now detail how the large-$L$ limit (at fixed $\lambda$) of \eqref{eq:respsiLhalfdensities_sym} matches the MFT result obtained for the SSEP~\cite{appert-rolland_universal_2008}.
The $L\to\infty$ limit behavior of \eqref{eq:respsiLhalfdensities_sym} is not immediately extractable; following a procedure described in Appendix~\ref{app:finiteNquantumfluctuationsSPEP}, one obtains
\begin{align}
\psi_L^{1,\mathrm{SPEP}}(\lambda)
&=
\frac 1{8L^2} \mathcal F\big(\!-\!\mu(\lambda)\big) \ + \
O(L^{-3}).
\label{eq:largeLpsi1L}
\end{align}
Here, with $\mu(\lambda)=\operatorname{arcsinh}^2\sqrt{\omega^{\text{SPEP}}}$, we recognize the universal scaling function
\begin{align} \label{eq:univ_scaling}
\mathcal F(u)
&= 4 \sum_{p=1}^{\infty}
\Big\{
(p\pi)^2+u-p\pi\sqrt{(p\pi)^2-2u }
\Big\}
\end{align}
as the one also arising in MFT~\cite{appert-rolland_universal_2008} and Bethe-Ansatz~\cite{appert-rolland_universal_2008,prolhac_cumulants_2009} studies of current fluctuations.
The large-$L$ limit (at fixed $\lambda$) thus yields the same correction as in the MFT approach~\cite{Imparato2009} for the SSEP.
The universal scaling function $\mathcal F(u)$ is singular at a positive value $u_\text{c}=\pi^2/2$ of its argument, but this value is never reached for any real-valued $\lambda$ in \eqref{eq:largeLpsi1L}.
This confirms, as in the MFT context, that the additivity principle holds at large $L$.
\section{Current large deviations of SIP} \label{sec:sip_current}
In this section, we derive the scaled CGF of the time-averaged current of the SIP in the large-$N$ limit. It is given by
\begin{equation} \label{eq:sip_cgf}
\psi^\mathrm{SIP}_L (\lambda,\bar\rho_a,\bar\rho_b) = \begin{cases}
(L+1) \sin^2 \left(\frac{1}{L+1} \mathrm{arcsin} \sqrt{\omega^\mathrm{SIP}}\right) &\text{if $\omega^\mathrm{SIP} \ge 0$}, \\
-(L+1) \sinh^2 \left(\frac{1}{L+1} \mathrm{arcsinh} \sqrt{-\omega^\mathrm{SIP}}\right) &\text{if $\omega^\mathrm{SIP} < 0$}
\end{cases}
\end{equation}
with the differences between $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$ encoded in
\begin{equation} \label{eq:sip_omega}
\omega^\mathrm{SIP} = \begin{cases}
\omega^{\mathrm{SIP}(1)} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b + (e^\lambda - 1)\bar\rho_a\bar\rho_b\right] &\text{ for $\mathrm{SIP}(1)$.} \\
\omega^{\mathrm{SIP}(1+\alpha)} \equiv \lambda\rho_a - \lambda\rho_b + \lambda^2 \rho_a \rho_b &\text{ for $\mathrm{SIP}(1+\alpha)$.}
\end{cases}
\end{equation}
Again, one can easily verify that $\psi^\mathrm{SIP}_L$ does not have any singularity at $\omega^\mathrm{SIP} = 0$. A comparison of this result with the hydrodynamic theory shows that for both $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$ arbitrarily large current fluctuations are still correctly captured by the hydrodynamic theory, in contrast to the SPEP. We close the section with a discussion of finite-$N$ effects.
\subsection{Derivation of the scaled CGF} \label{ssec:sip_cgf}
\subsubsection{$\mathrm{SIP}(1)$}
Similarly to the SPEP, we first transform $H^{\mathrm{SIP}(1)}_L$ into a more convenient form. This is done by the canonical transformation
\begin{equation} \label{eq:sip1_canonical}
\rho_k = F_k \left[1+(1 + F_k) \hat{F}_k\right], \qquad \hat\rho_k = \ln \left( 1 + \frac{\hat{F}_k}{1+F_k \hat{F}_k} \right),
\end{equation}
which can also be written as
\begin{equation}
F_k = \frac{\rho_k}{e^{\hat\rho_k}(1+\rho_k)-\rho_k}, \qquad \hat{F}_k = (e^{\hat\rho_k}-1)(1+\rho_k) - (1-e^{-\hat\rho_k})\rho_k.
\end{equation}
After this transformation, the Hamiltonian of the new variables is given by
\begin{align} \label{eq:sip1_K}
K^{\mathrm{SIP}(1)}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1}) + \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right]
+ \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + e^{-\lambda}\Big[\hat F_L - (e^\lambda - 1)(1+F_L\hat F_L) \Big] \Big[\bar{\rho}_b - F_L - F_L(1 + \bar{\rho}_b)(e^\lambda - 1)\Big].
\end{align}
Comparing this expression with \eqref{eq:spep_K}, we find the formal correspondence
\begin{equation} \label{eq:sip1_K_spep_K}
K^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
= -K^\mathrm{SPEP}_L (\lambda,-\bar\rho_a,-\bar\rho_b;-\mathbf{F},\hat{\mathbf{F}}).
\end{equation}
By examining the time-independent saddle-point equations derived from these Hamiltonians, we find a mapping between the optimal profiles of the SPEP and the $\mathrm{SIP}(1)$:
\begin{align} \label{eq:sip1_spep_mapping}
(\hat F^*_k)^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b) &= (\hat F^*_k)^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b), \nonumber\\
(F^*_k)^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b) &= -(F^*_k)^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b).
\end{align}
This mapping can be used to obtain the optimal profiles and the scaled CGF of the $\mathrm{SIP}(1)$ from those of the SPEP. It should be noted that, when $\bar\rho_a$ and $\bar\rho_b$ are negative, we should reconsider the proper signs of $A$ and $\varepsilon$ in the optimal profiles of the SPEP. In this case, the optimal density profile $\boldsymbol{\rho}^*$ must always be nonpositive, so that it becomes nonnegative after the mapping to the $\mathrm{SIP}(1)$. Taking this into account, for $\bar\rho_a \ge \bar\rho_b$, the optimal profiles are given by (dropping $^*$)
\begin{align} \label{eq:sip1_profiles}
\hat F_k^{\mathrm{SIP}(1)} &=
\begin{cases}
-\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left(1+\frac{1}{\rho_b}\right) < \lambda < -\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right]$,} \\
-\sqrt{\mathcal{-A}^{\mathrm{SIP}(1)}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right] \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $0 \le \lambda < \ln \left(1 + \frac{1}{\rho_a}\right)$,}
\end{cases} \nonumber \\
F_k^{\mathrm{SIP}(1)} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left(1+\frac{1}{\rho_b}\right) < \lambda < -\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right]$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^{\mathrm{SIP}(1)}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right] \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $0 \le \lambda < \ln \left(1 + \frac{1}{\rho_a}\right)$,}
\end{cases}
\end{align}
where we defined
\begin{align}
\mathcal{A}^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv -\mathcal{A}^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b) \nonumber\\
&= \frac{(e^\lambda - 1) [e^\lambda(\bar\rho_b + 1) - \bar\rho_b]}
{4 [1 - (e^\lambda - 1)\bar\rho_a] [e^\lambda\bar\rho_a(\bar\rho_b+1)-\bar\rho_a\bar\rho_b-\bar\rho_b]}, \label{eq:sip1_A_saddle} \\
\omega^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv -\omega^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b). \label{eq:sip_omega_mapping-spep}
\end{align}
Note that this definition of $\omega^{\mathrm{SIP}(1)}$ yields \eqref{eq:sip_omega}. The expressions for $\bar\rho_a < \bar\rho_b$ are obtained by $\lambda \to -\lambda$ and an exchange of $\bar\rho_a$ and $\bar\rho_b$.
Finally, due to \eqref{eq:sip1_K_spep_K} and \eqref{eq:sip1_spep_mapping}, the scaled CGFs of the SPEP and the $\mathrm{SIP}(1)$ are related by
\begin{equation}
\psi^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b) = -\psi^\mathrm{SPEP}_L (\lambda,-\bar\rho_a,-\bar\rho_b)\;,
\end{equation}
from which it is straightforward to derive \eqref{eq:sip_cgf}.
Remarkably, the $\mathrm{SIP}(1)$ has a finite range of $\lambda$, whereas the SPEP has an unbounded range of $\lambda$. This is related to the fact that the domain of $\mathrm{arcsin}$ is limited to $[-1,1]$, while that of $\mathrm{arcsinh}$ is unlimited. As will be discussed later, the limited range of $\lambda$ is closely related to the persistence of hydrodynamic behaviors for extreme current fluctuations. Meanwhile, it should be noted that the limited range of $\lambda$ does not imply a limited range of the current being considered. The time-averaged current $J$ conditioned on $\lambda$, obtained from $\partial \psi^{\mathrm{SIP}(1)}_L/\partial \lambda$, still ranges from $-\infty$ to $\infty$ for both SPEP and $\mathrm{SIP}(1)$. In fact, using standard Legendre transform arguments it is easy to check that the limited range of definition of the CGF $\psi^{\mathrm{SIP}(1)}_L(\lambda)$ corresponds to exponential tails of the current distribution function.
\subsubsection{$\mathrm{SIP}(1+\alpha)$}
We now turn to the case of $\mathrm{SIP}(1+\alpha)$. Again, the Hamiltonian, given by \eqref{eq:sip1a_H}, can be simplified by a canonical transformation
\begin{equation} \label{eq:sip1a_canonical}
\rho_k = F_k (1+F_k\hat{F}_k), \qquad \hat\rho_k = \frac{\hat{F}_k}{1+F_k \hat{F}_k} \\
\end{equation}
or
\begin{equation}
F_k = \frac{\rho_k}{1+\rho_k\hat\rho_k}, \quad \hat{F}_k = \hat\rho_k(1+\rho_k\hat\rho_k),
\end{equation}
which was also used in \cite{Tailleur2008} in the context of the equivalent KMP model. This transforms the Hamiltonian into
\begin{align}\label{eq:sip1a_K}
K^{\mathrm{SIP}(1+\alpha)}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1})
+ \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right] + \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + \Big[\hat F_L - \lambda(1+F_L\hat F_L) \Big] \Big[\bar{\rho}_b - F_L - \bar{\rho}_b\lambda F_L\Big].
\end{align}
A comparison between this expression and \eqref{eq:sip1_K} shows
\begin{equation} \label{eq:sip1a_K_sip1_K}
K^{\mathrm{SIP}(1+\alpha)}_L (\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
= \lim_{N \to \infty} K^{\mathrm{SIP}(1)}_L
(N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b;N^\alpha\mathbf{F},N^{-\alpha}\hat{\mathbf{F}}).
\end{equation}
The time-independent saddle-point equations of these Hamiltonians show that the optimal profiles of the $\mathrm{SIP}(1)$ and the $\mathrm{SIP}(1+\alpha)$ are related by (again dropping $^*$)
\begin{align} \label{eq:sip1a_sip1_mapping}
(\hat F_k)^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&= \lim_{N \to \infty} N^\alpha (\hat F_k)^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b), \nonumber\\
(F_k)^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&= \lim_{N \to \infty} N^{-\alpha} (F_k)^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b).
\end{align}
Therefore, for $\bar\rho_a \ge \bar\rho_b$ the optimal profiles are obtained as
\begin{align} \label{eq:sip1a_profiles}
\hat F_k^{\mathrm{SIP}(1+\alpha)} &=
\begin{cases}
-\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $-\frac{1}{\bar\rho_b} < \lambda < \frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b}$,} \\
-\sqrt{\mathcal{-A}^{\mathrm{SIP}(1+\alpha)}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $\frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b} \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $0 \le \lambda < \frac{1}{\bar\rho_a}$,}
\end{cases} \nonumber \\
F_k^{\mathrm{SIP}(1+\alpha)} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $-\frac{1}{\bar\rho_b} < \lambda < \frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b}$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $\frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b} \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $0 \le \lambda < \frac{1}{\bar\rho_a}$,}
\end{cases}
\end{align}
where we defined
\begin{align}
\mathcal{A}^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv \lim_{N \to \infty} N^{2\alpha}\mathcal{A}^{\mathrm{SIP}(1)}
(N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b) \nonumber\\
&= \frac{\lambda(1+\lambda\rho_b)}{4 (1-\lambda\rho_a) (\rho_a - \rho_b + \lambda \rho_a \rho_b)}, \label{eq:sip1a_A_saddle} \\
\omega^{\mathrm{SIP}(1+\alpha)}
&\equiv \lim_{N \to \infty} \omega^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b).
\end{align}
Note that this definition of $\omega^{\mathrm{SIP}(1+\alpha)}$ leads to \eqref{eq:sip_omega}.
Using \eqref{eq:sip1a_K_sip1_K} and \eqref{eq:sip1a_sip1_mapping}, the scaled CGFs of the $\mathrm{SIP}(1)$ and the $\mathrm{SIP}(1+\alpha)$ are related by
\begin{equation}
\psi^{\mathrm{SIP}(1+\alpha)}_L (\lambda,\bar\rho_a,\bar\rho_b)
= \lim_{N \to \infty} \psi^{\mathrm{SIP}(1)}_L (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b),
\end{equation}
which gives \eqref{eq:sip_cgf}. As in the case of the $\mathrm{SIP}(1)$, $\mathrm{SIP}(1+\alpha)$ also has a finite range of $\lambda$, although the range of the current $J$ is unbounded.
\subsection{Comparison with hydrodynamic results} \label{ssec:sip_hydro}
For both $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$, in the $L \to \infty$ limit, the hydrodynamic expression of the scaled CGF can be written in a similar form
\begin{equation} \label{eq:sip_cgf_hydro}
\psi^\mathrm{SIP}(\lambda) = \begin{cases}
\frac{1}{L+1} \mathrm{arcsin}^2 \sqrt{\omega^\mathrm{SIP}} &\text{if $\omega^\mathrm{SIP} \ge 0$}, \\
-\frac{1}{L+1} \mathrm{arcsinh}^2 \sqrt{-\omega^\mathrm{SIP}} &\text{if $\omega^\mathrm{SIP} < 0$},
\end{cases}
\end{equation}
where we used the superscript $\mathrm{SIP}$ to refer to both large-$N$ models. This expression is in agreement with the corresponding expression for the KMP model found in \cite{Imparato2009}. When $\lambda$ is fixed, the hydrodynamic limit is reached by
\begin{equation}
\psi^\mathrm{SIP}_L(\lambda) - \psi^\mathrm{SIP}(\lambda) = -\frac{\mathrm{arcsin}^4\sqrt{\omega^\mathrm{SIP}}}{3L^3} + O(L^{-4}),
\end{equation}
as illustrated in Fig.~\ref{fig:sip_cgf}. In contrast to the SPEP, the lattice structure decreases the magnitude of fluctuations. Since the range of $\lambda$ is bounded and the two CGFs converge to each other throughout this range, we cannot find any scaling of $\lambda$ with $L$ that induces non-hydrodynamic current fluctuations.
\begin{figure}
\includegraphics[width = 0.45\textwidth]{SIP1_CGF_lambda_psi.pdf} \quad
\includegraphics[width = 0.45\textwidth]{SIP2_CGF_lambda_psi.pdf}
\caption{\label{fig:sip_cgf} The scaled CGFs of (left) $\mathrm{SIP}(1)$ and (right) $\mathrm{SIP}(2)$ in the large $N$ limit ($\psi^\mathrm{SIP}_L (\lambda)$, \eqref{eq:sip_cgf}, solid lines) and in the hydrodynamic limit ($\psi^\mathrm{SIP} (\lambda)$, \eqref{eq:sip_cgf_hydro}, dashed lines). For any fixed $\lambda$, the CGF $\psi^\mathrm{SIP}_L (\lambda)$ is equivalent to $\psi^\mathrm{SIP} (\lambda)$ as $L \to \infty$. The boundary conditions are given by $\bar\rho_a = 0.8$, $\bar\rho_b = 0.2$.}
\end{figure}
\begin{figure}
\includegraphics[width = 0.45\textwidth]{SIP1_profile_k_rho.pdf}\quad
\includegraphics[width = 0.45\textwidth]{SIP1_profile_k_rhoh.pdf}
\caption{\label{fig:sip1_profile} The optimal profiles of the $\mathrm{SIP}(1)$ for $L = 100$, $\bar\rho_a = 0.8$, $\bar\rho_b = 0.2$. As $\Delta\lambda = \lambda - \lambda_\mathrm{min}$ approaches zero, (left) the density profile develops an arbitrarily large crest, while (right) the momentum profile becomes flatter.}
\end{figure}
\begin{figure}
\includegraphics[width = 0.45\textwidth]{SIP1_profile_k_rholambda.pdf}\quad
\includegraphics[width = 0.45\textwidth]{SIP1_profile_kL_rho.pdf}\\
\includegraphics[width = 0.45\textwidth]{SIP1_profile_k_rhohlambda.pdf}\quad
\includegraphics[width = 0.45\textwidth]{SIP1_profile_kL_rhoh.pdf}
\caption{\label{fig:sip1_profile_collapse} The data collapses of optimal profiles of the $\mathrm{SIP}(1)$. The boundary conditions are given by $\bar\rho_a = 0.8$ and $\bar\rho_b = 0.2$. We fix $L = 100$ ($\lambda = 1$) while $\lambda$ ($L$) is varied.}
\end{figure}
In order to understand why hydrodynamic behaviors are still observed for arbitrarily large current fluctuations, we examine the optimal profiles of the $\mathrm{SIP}(1)$ as was done for the SPEP. Using the same argument applied to the SPEP, the time-averaged current $J$ of $\mathrm{SIP}(1)$ can be related to the optimal profiles by
\begin{align} \label{eq:sip1_J_profiles}
J &= \frac{1}{L+1}\sum_{k=0}^L\left[(\rho_k - \rho_{k+1})+\rho_k(1+\rho_{k+1})(e^{\hat\rho_{k+1}-\hat\rho_k}-1)
- \rho_{k+1}(1+\rho_k)(e^{\hat\rho_k-\hat\rho_{k+1}}-1)\right] \nonumber\\
&= \underbrace{\frac{\bar\rho_a-\bar\rho_b}{L+1}}_{= \langle J \rangle} + \underbrace{\frac{1}{L+1}
\sum_{k=0}^L\left[\rho_k(1+\rho_{k+1})(e^{\hat\rho_{k+1}-\hat\rho_k}-1) - \rho_{k+1}(1+\rho_k)(e^{\hat\rho_k-\hat\rho_{k+1}}-1)\right]}_{= \delta J},
\end{align}
where $\delta J$ becomes dominant as $\lambda$ approaches its upper and lower bounds
\begin{equation}
\lambda_\mathrm{max} = \ln \left(1 + \frac{1}{\rho_a}\right), \quad \lambda_\mathrm{min} = -\ln \left(1+\frac{1}{\rho_b}\right).
\end{equation}
For convenience, let us denote by $\Delta \lambda$ both $|\lambda - \lambda_\mathrm{min}|$ and $|\lambda - \lambda_\mathrm{max}|$. As shown in Fig.~\ref{fig:sip1_profile}, a large $\delta J$ is supported by a growing density crest and a flattening momentum profile as $\Delta \lambda \to 0$. As the data collapses in Fig.~\ref{fig:sip1_profile_collapse} indicate, as $L \to \infty$, the optimal profiles have scaling forms
\begin{align} \label{eq:sip1_profile_scaling}
\rho_k(\Delta\lambda,L) &= \Delta\lambda^{-1/2} f(k/L), \nonumber\\
\hat\rho_k(\Delta\lambda,L) &= \Delta\lambda^{1/2} g(k/L).
\end{align}
These imply that we can approximate $\delta J$ as
\begin{align}\label{eq:sip1_J_hydro}
\delta J &\simeq \frac{1}{L+1}\sum_{k=0}^L[\rho_k(1+\rho_{k+1}) + \rho_{k+1}(1+\rho_k)](\hat\rho_{k+1}-\hat\rho_k) \nonumber\\
&\simeq \frac{1}{\Delta\lambda^{1/2}L} \int_0^1 \mathrm{d} x\, 2 \rho(1+\rho) \partial_x g,
\end{align}
which has an integral form for any small $\Delta \lambda$ corresponding to large $J$. Thus, $J$ exhibits hydrodynamic behaviors for arbitrarily large $J$. An almost identical argument also applies to the $\mathrm{SIP}(1+\alpha)$, whose optimal profiles have similar shapes and satisfy the scaling relation \eqref{eq:sip1_profile_scaling}.
\subsection{Finite-$N$ effects} \label{ssec:sip_fin_N}
\subsubsection{$\mathrm{SIP}(1)$}
From \eqref{eq:spep_H} and \eqref{eq:sip1_H}, we observe that the Hamiltonians of the SPEP and the $\mathrm{SIP}(1)$ are related by
\begin{equation}
H^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b;\boldsymbol{\rho},\hat{\boldsymbol{\rho}}) = -H^\mathrm{SPEP}_L(\lambda,-\bar\rho_a,-\bar\rho_b;-\boldsymbol{\rho},\hat{\boldsymbol{\rho}}).
\end{equation}
This suggests that the leading finite-$N$ correction to the scaled CGF $\psi^{\mathrm{SIP}(1)}_L$ can be obtained by a Gaussian approximation very similarly to the one applied to the SPEP in Sec.~\ref{ssec:spep_fin_N}. Consequently, the leading finite-$N$ correction is described by analogs of \eqref{eq:spep_cgf_fin_N_series} and \eqref{eq:respsiLhalfdensities_sym}, namely
\begin{equation}
\psi^{\mathrm{SIP}(1)}_{N,L}(\lambda) = \psi^{\mathrm{SIP}(1)}_L(\lambda) + N^{-1} \psi^{1,\mathrm{SIP}(1)}_L(\lambda) + o(N^{-1})
\end{equation}
with
\begin{align}
\psi^{1,\mathrm{SIP}(1)}_L(\lambda,\bar\rho_a,\bar\rho_b) &= -\psi^{1,\mathrm{SPEP}}_L(\lambda,-\bar\rho_a,-\bar\rho_b) \nonumber\\
&=
-\sum_{p=1}^{L'-1}
\bigg\{
\boldsymbol c_\lambda
-
\cos\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L'}\big)}
-
\sin\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda-\cos\frac{p\pi}{L'}\big)}
\bigg\},
\label{eq:respsiLhalfdensities_sym_SIP}
\end{align}
where $\boldsymbol c_\lambda = \cos \frac{2\operatorname{arcsin}\sqrt{\omega^{\text{SIP}(1)}}}{L+1}$.
The correction is again an analytic function of $\lambda$ within its domain, which proves the validity of the additivity principle with respect to continuous transitions, without ruling out possible discontinuous ones.
In the large-$L$ limit, using \eqref{eq:largeLpsi1L} and the first equality of \eqref{eq:respsiLhalfdensities_sym_SIP}, we can also write
\begin{align}
\psi_L^{1,\mathrm{SIP}(1)}(\lambda)
&=
-\frac 1{8L^2} \mathcal F\big(\nu(\lambda)\big) \ + \
O(L^{-3}),
\label{eq:sip_largeLpsi1L}
\end{align}
with $\mathcal F(u)$ the universal scaling function defined in \eqref{eq:univ_scaling} and $\nu(\lambda)=\operatorname{arcsin}^2\sqrt{\omega^{\text{SIP}(1)}}$. While $\mathcal F(u)$ is singular at $u_\text{c}=\pi^2/2$, $\nu(\lambda)$ cannot be greater than $\pi^2/4$ for any real-valued $\lambda$ in \eqref{eq:sip_largeLpsi1L}. This also confirms the validity of the additivity principle with respect to continuous transitions at large $L$.
\subsubsection{$\mathrm{SIP}(1+\alpha)$}
Unlike the previous models, the leading finite-$N$ correction to $\psi^{\mathrm{SIP}(1+\alpha)}_L$ comes from a different origin. For this model, if we keep the leading finite-$N$ correction, the path integral in \eqref{eq:macro_path_integ} can be rewritten as
\begin{equation}
e^{NT\psi^{\mathrm{SIP}(1+\alpha)}_{N,L}}
= \int \mathcal{D}\boldsymbol{\rho} \mathcal{D}\hat{\boldsymbol{\rho}} \, \exp\left\{-N\int_{0}^{T} \mathrm{d}t \,
\left[ \hat{\boldsymbol{\rho}}\cdot\dot{\boldsymbol{\rho}} - H^{\mathrm{SIP}(1+\alpha)}_L - N^{-\alpha}V^{\mathrm{SIP}(1+\alpha)}_L \right]\right\},
\end{equation}
where
\begin{equation} \label{eq:sip1a_V_rho}
V^{\mathrm{SIP}(1+\alpha)}_L(\lambda,\bar\rho_a,\bar\rho_b;\boldsymbol{\rho},\hat{\boldsymbol{\rho}})
= \sum_{k=1}^{L-1} \frac{\rho_k + \rho_{k+1}}{2}\,(\hat\rho_{k+1} - \hat\rho_k)^2
+ \frac{\bar\rho_a + \rho_1}{2}\,\hat\rho_1^2 + \frac{\bar\rho_b + \rho_L}{2}\,(\hat\rho_L - \lambda)^2.
\end{equation}
Applying a saddle-point approximation as before, we obtain
\begin{equation}
\psi^{\mathrm{SIP}(1+\alpha)}_{N,L}(\lambda) = \psi^{\mathrm{SIP}(1+\alpha)}_L(\lambda) + N^{-\alpha} \psi^{1,\mathrm{SIP}(1+\alpha)}_L(\lambda) + o(N^{-\alpha})
\end{equation}
with
\begin{equation} \label{eq:sip1a_saddle_rho}
\psi^{1,\mathrm{SIP}(1+\alpha)}_L(\lambda) = V^{\mathrm{SIP}(1+\alpha)}_L(\lambda;\boldsymbol{\rho}^*,\hat{\boldsymbol{\rho}}^*),
\end{equation}
where $\boldsymbol{\rho}^*$ and $\hat{\boldsymbol{\rho}}^*$ are the optimal profiles determined in Sec.~\ref{ssec:sip_cgf}.
\subsubsection{Numerical results}
We numerically confirm our theoretical predictions by constructing a matrix representation of the SIP conditioned on $\lambda$. Since it is impossible to implement the unbounded configuration space of this model, we introduce an artificial upper bound $M$ on the number of particles in each site. The matrix representation is such that any transition that violates this upper bound is forbidden, while the other transitions occur with the same rates as the original dynamics. We expect that if $M$ is sufficiently large, the effects of $M$ become irrelevant. The results for $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1.5)$ shown in Fig.~\ref{fig:sip_finN} are both in agreement with our predictions.
\begin{figure}
\includegraphics[width = 0.49\textwidth]{{Correction_SIP1_finN_ra0.08_rb0.02_L3_M16}.eps}
\includegraphics[width = 0.49\textwidth]{{Correction_SIP1.5_finN_ra0.08_rb0.02_L3_M16}.eps}
\caption{\label{fig:sip_finN} Finite-$N$ corrections to the scaled CGF of (left) the $\mathrm{SIP}(1)$ and (right) the $\mathrm{SIP}(1.5)$ at $L = 3$, $\bar\rho_a = 0.08$, and $\bar\rho_b = 0.02$. The upper cutoff $M$ is set equal to $16$. The numerics (symbols) are in good agreement with the leading-order corrections (dashed lines) given by \eqref{eq:respsiLhalfdensities_sym_SIP} and \eqref{eq:sip1a_saddle_rho}. The predicted scaling exponents are also supported by the successive slopes of finite-$N$ corrections in the log-log plots.}
\end{figure}
\section{Criterion for persistent hydrodynamic behaviors} \label{sec:criterion}
We have shown that current fluctuations of the SPEP have a non-hydrodynamic regime, while those of the SIP always behave according to the predictions of the hydrodynamic equations. As noted in Sec.~\ref{sec:models}, one important difference between the SPEP and the SIP lies in whether the mobility coefficient $\sigma(\rho)$ is bounded from above. This suggests a connection between the presence of an upper bound on $\sigma(\rho)$ and hydrodynamic behaviors of current fluctuations. In order to investigate this connection, we examine how the optimal profiles depend on the time-averaged current $J$ within the naive hydrodynamic regime given by $J = O(L^{-1})$. An extrapolation of this dependence beyond the regime ({\em i.e.}, $J$ larger than $O(L^{-1})$) reveals whether non-hydrodynamic behaviors appear for sufficiently large $J$.
In the hydrodynamic limit, from Hamilton's equations we have
\begin{equation} \label{eq:hamilton_hydro}
\frac{\partial \rho}{\partial t} = \frac{\delta H}{\delta \rho} = \partial_x \left[ D(\rho)\partial_x \rho - \sigma(\rho)\partial_x \hat\rho \right]
\end{equation}
with $H$ given by \eqref{eq:H_mft}. This gives a relation between $J$ and the optimal profiles through
\begin{align} \label{eq:J_profiles_hydro}
J = \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[-D(\rho^*)\partial_x \rho^* + \sigma(\rho^*)\partial_x \hat\rho^*\right]
= \frac{G(\bar\rho_a) - G(\bar\rho_b)}{L+1} + \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[\sigma(\rho^*)\partial_x \hat\rho^*\right],
\end{align}
where $D(\rho) = G'(\rho)$. Then, as long as $G(\bar\rho_a)$ and $G(\bar\rho_b)$ are finite, $J$ beyond the naive hydrodynamic regime satisfies
\begin{align} \label{eq:J_profiles_hydro_approx}
J \simeq \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[\sigma(\rho^*)\partial_x \hat\rho^*\right].
\end{align}
In other words, in this regime $J$ is sensitive only to $\sigma(\rho^*)$ and $\partial_x \hat\rho^*$.
Note that when $\sigma(\rho)$ is bounded from above, an arbitrarily large $J$ can only be supported by an arbitrarily large $\partial_x \hat\rho^*$. This means that $\partial_x \hat\rho^*$ can no longer be expressed as a proper gradient for a sufficiently large $J$, in which case $J$ exhibits non-hydrodynamic behaviors, as was the case for the SPEP. Hence, the absence of an upper bound on $\sigma(\rho)$ is clearly a necessary condition for the persistence of hydrodynamic behaviors.
When $\sigma(\rho)$ is not bounded from above, a large $J$ can be supported by a large $\sigma(\rho^*)$ while $\partial_x \hat\rho^*$ remains well defined, so that $J$ is still blind to the lattice structure. Based on this possible scenario, we conjecture that the absence of an upper bound on $\sigma(\rho)$ is also a sufficient condition for the persistence of hydrodynamic behaviors. Although there is no rigorous proof yet, we can confirm this conjecture for the one-dimensional symmetric zero-range process, which provides a simple example of boundary-driven systems with non-constant $D(\rho)$ and unbounded $\sigma(\rho)$. An interested reader is referred to Appendix~\ref{app:zrp} for more discussions on this model.
\section{Conclusions} \label{sec:conclusions}
In this paper we introduced a class of large-$N$ models for one-dimensional boundary-driven diffusive systems. Using $N$ as a large parameter, we were able to obtain exact expressions for current large deviations on a finite lattice, without relying on a hydrodynamic approach. This allowed us to look at regimes where the hydrodynamic theory is naively expected to break down. Surprisingly, we found that there are classes of models, which we conjecture to be those with an unbounded $\sigma(\rho)$ as a function of $\rho$, where the predictions of the hydrodynamic theory always hold. It will be interesting to see if similar considerations also hold for models with a bulk bias and/or for large deviations of other additive observables, such as the activity.
In addition, we examined the finite-$N$ corrections and used them to argue that the additivity principle, assumed throughout the paper, is likely to hold for the models considered.
{\it Acknowledgments:} We are grateful for discussions with B.~Derrida, M.~R.~Evans, B.~Meerson, T.~Sadhu, and H.~Spohn. YB and YK were supported by an Israeli-Science-Foundation grant. YB is supported in part at the Technion by a fellowship from the Lady Davis Foundation. VL wishes to thank the hospitality of the Physics Department of Technion, Haifa, where part of the research was performed, and acknowledges support from LAABS Inphyniti CNRS project.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction and main result}
In this note, we establish a sharp local boundedness result for local minimizers of integral functionals
\begin{equation}\label{eq:int}
\mathcal F[u,\Omega]:=\int_\Omega f(x,\nabla u)\,dx,
\end{equation}
where $\Omega\subset\R^n$, $n\geq2$, is a bounded domain and the integrand $f(x,\nabla u)$ satisfies $(p,q)$-growth of the form
\begin{equation}\label{pqsim}
|z|^p\lesssim f(x,z)\lesssim |z|^q+1,
\end{equation}
see Assumption~\ref{ass} below. Local boundedness and H\"older continuity of local minimizer of \eqref{eq:int} in the case $1<p=q$ are classical, see the original reference \cite{GG82} or the textbook \cite{Giu}. Giaquinta \cite{G87} provided an example of an autonomous convex integrand satisfying \eqref{pqsim} with $p=2$ and $q=4$ that admits unbounded minimizer in dimension $n\geq6$. Similar examples can be found in \cite{Mar91,H92}, in particular it follows from \cite[Section~6]{Mar91} that if
\begin{equation}\label{pq:unbounded}
q>\frac{(n-1)p}{n-1-p}=:p_{n-1}^*\quad\mbox{and}\quad 1<p<n-1,
\end{equation}
then one cannot expect local boundedness for minimizers of \eqref{eq:int} in general. In this paper we show that condition \eqref{pq:unbounded} is sharp. Before we state our main result, we recall a standard notion of local minimizer and quasi-minimizer in the context of functionals with $(p,q)$-growth
\begin{definition}
Given $Q\geq1$, we call $u\in W_{\rm loc}^{1,1}(\Omega)$ a \emph{$Q$-minimizer} of \eqref{eq:int} if and only if
\begin{equation*}
\mathcal F(u,{\rm supp}\,\varphi)<\infty\quad\mbox{and}\quad \mathcal F(u,{\rm supp}\,\varphi)\leq Q\mathcal F(u+\varphi,{\rm supp}\,\varphi)
\end{equation*}
for any $\varphi\in W^{1,1}(\Omega)$ satisfying ${\rm supp}\;\varphi\Subset \Omega$. If $Q=1$, then $u$ is a \emph{local minimizer} of \eqref{eq:int}. Moreover, we call $u$ a \emph{quasi-minimizer} if and only if there exists $Q\geq1$ such that $u$ is a $Q$-minimizer.
\end{definition}
\begin{assumption}\label{ass} Let $f:\Omega\times \R^n\to\R$ be a Caratheodory function and suppose that $z\mapsto f(x,z)$ is convex for almost every $x\in\Omega$. Moreover, there exist $1\leq L<\infty$, $\mu\geq0$ such that for all $z\in \R^n$ and almost every $x\in\Omega$
\begin{align}
|z|^p\leq f(x,z)\leq L|z|^q+1,&\label{ass1}\\
f(x,2z)\leq \mu+Lf(x,z),\label{ass2}
\end{align}
\end{assumption}
Now we are in position to state the main result of the present paper
\begin{theorem}\label{T}
Let $\Omega\subset \R^n$, $n\geq2$ and suppose Assumption~\ref{ass} is satisfied with $1\leq p\leq q<\infty$ such that
\begin{equation}\label{eq:assq:intro}
\frac1q\geq \frac1p-\frac1{n-1}
\end{equation}
Let $u\in W_{\rm loc}^{1,1}(\Omega)$ be a quasi-minimizer of the functional $\mathcal F$ given in \eqref{eq:int}. Then, $u\in L_{\rm loc}^{\infty}(\Omega)$.
\end{theorem}
\begin{remark}
We provide the proof of Theorem~\ref{T} in Section~\ref{sec:proof}. We establish a slightly more general results in which the growth condition \eqref{ass1} is replaced by
%
$$
|z|^p-g(x)^\frac{p}{p-1}\leq f(x,z)\leq L(|z|^q+g(x)^\frac{p}{p-1})
$$
and optimal assumptions (in the Lorentz-scale) on $g$ are imposed.
\end{remark}
Let us now relate Theorem~\ref{T} to previous results in the literature. To the best of our knowledge, the best previously known relation between $p$ and $q$ that ensures local boundedness under Assumption~\ref{ass} can be found in the paper by Fusco \& Sbordone \cite{FS90} and read
\begin{equation}\label{pqwrong}
\frac1q\geq\frac1p-\frac1n
\end{equation}
see \cite[Theorem~2]{FS90} (see also the more recent result \cite[Theorem~2.3]{CMM15}) which also implies local boundedness with condition \eqref{pqwrong}). Obviously, relation \eqref{eq:assq:intro} is less restrictive than \eqref{pqwrong} and in view of the discussion above optimal for local boundedness (compare \eqref{eq:assq:intro} and \eqref{pq:unbounded}). However, we want to emphasize that \cite{CMM15,FS90} (and similarly \cite{BMS90,FS93}) contain sharp local boundedness results under additional structural assumptions on the growth of $f$, namely \textit{anisotropic growth} of the form
\begin{equation}\label{growth:aniso}
\sum_{i=1}^n|z_i|^{p_i}\lesssim f(x,z)\lesssim \sum_{i=1}^nM(1+|z_i|^{p_i}).
\end{equation}
In this case, local boundedness is proven under the condition $q\leq \overline p^*$, where $\frac1{\overline p}=\frac1n\sum_{i=1}^n\frac1{p_i}$ and $\overline p^*=\frac{n \overline p}{n-\overline p}$. This condition is optimal for local boundedness in view of the above mentioned counterexamples (the integrands in \cite{G87,Mar91,H92} satisfy growth of the form \eqref{growth:aniso})
\bigskip
The systematic study of higher regularity of minimizers of functionals with $(p,q)$-growth was initiated by Marcellini \cite{Mar89,Mar91}. By now there is a large and quickly growing literature on regularity results for minimizers of functionals with $(p,q)$-growth, and more general non-standard growth \cite{L93,Mar93}. We refer to \cite{Min06} for an overview. A currently quite active field of research is the regularity theory for so-called double phase problems where the model functional is given by
\begin{equation}\label{int:doublephase}
\int_\Omega |\nabla u(x)|^p+a(x)|\nabla u(x)|^q\,dx,
\end{equation}
where $0\leq a\in C^{0,\alpha}$ with $\alpha\in(0,1]$ see e.g.\ \cite{BCM18,CM15,CM15b,DM19,ELM04}) and \cite{Zhikov,JKO94} for some motivation for functionals of the form \eqref{int:doublephase}. For this kind of functionals rather sharp conditions for higher ($C^{1,\beta}$-) regularity are known, where $\alpha$ has to be balanced with $p,q$, and $n$. In \cite{CM15b} it was observed that the conditions on the data can be relaxed if one a priori knows that the minimizer is bounded. Obviously by Theorem~\ref{T} the results of \cite{CM15b} can be applied without any a priori assumption whenever $\frac1q\geq\frac1p-\frac1n$ and in particular can be used to improve \cite[Theorem~5.3]{CM15b}. Similarly, Theorem~\ref{T} improves the applicability of some results in \cite{BB18,CKP11} where also higher regularity results are proven \textit{assuming} a priori boundedness of the minimizer.
\smallskip
Let us very briefly explain the strategy of the proof of Theorem~\ref{T} and the origin of our improvement. In principle, we use a variation of the De-Giorgi type iteration similar to e.g.\cite{FS90,FS93,CMM15}. Recall that De-Giorgi iteration is based on a Caccioppoli inequality (which yield a reverse Poincar\'e inequality) and Sobolev inequality. The main new ingredient here is to use in the Caccioppoli inequality cut-off functions that are optimized with respect to the minimizer $u$ (instead of using affine cut-offs). This enables us to use Sobolev inequality on ($(n-1)$-dimensional) spheres instead of ($n$-dimensional) balls and thus get the desired improvement. This idea, combined with a variation of Moser-iteration, was recently used by the second author and Bella in the analysis of linear non-uniformly elliptic equations \cite{BS19a} (improving in an essentially optimal way classic results of Trudinger~\cite{T71}) and for higher regularity for integral functionals with $(p,q)$-growth \cite{BS19c} (see also \cite{BS19b} for an application in stochastic homogenization).
\smallskip
The paper is organized as follows: In Section~\ref{sec:prelim}, we recall some definitions and useful results regarding Lorentz spaces and present a technical lemma which is used to derive an improved version of Caccioppoli inequality which plays a prominent role in the proof of Theorem~\ref{T}. In Section~\ref{sec:proof}, we prove a slightly more general version of Theorem~\ref{T} which in particular includes some a~priori estimates.
\section{Preliminary results}\label{sec:prelim}
\subsection{Preliminary lemmata}
A key ingredient in the prove of Theorem~\ref{T} is the following lemma which is a variation of \cite[Lemma 3]{BS19c}
\begin{lemma}\label{L:optimcutoff}
Fix $n\geq2$. For given $0<\rho<\sigma<\infty$, $v\in L^1(B_\sigma)$ and $s>1$, we consider consider
\begin{equation*}
J(\rho,\sigma,v):=\inf\left\{\int_{B_\sigma}|v||\nabla \eta|^s\,dx \;|\;\eta\in C_0^1(B_\sigma),\,\eta\geq0,\,\eta=1\mbox{ in $B_\rho$}\right\}.
\end{equation*}
Then for every $\delta\in(0,1]$
\begin{equation}\label{1dmin}
J(\rho,\sigma,v)\leq (\sigma-\rho)^{-(s-1+\frac1\delta)} \biggl(\int_{\rho}^\sigma \left(\int_{S_r} |v|\right)^\delta\,dr\biggr)^\frac1\delta.
\end{equation}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{L:optimcutoff}]
Estimate \eqref{1dmin} follows directly by minimizing among radial symmetric cut-off functions. Indeed, we obviously have for every $\varepsilon\geq0$
\begin{equation*}
J(\rho,\sigma,v)\leq \inf\left\{\int_{\rho}^\sigma |\eta'(r)|^s\left(\int_{S_r}|v|+\varepsilon\right)\,dr \;|\;\eta\in C^1(\rho,\sigma),\,\eta(\rho)=1,\,\eta(\sigma)=0\right\}=:J_{{\rm 1d},\varepsilon}.
\end{equation*}
For $\varepsilon>0$, the one-dimensional minimization problem $J_{{\rm 1d},\varepsilon}$ can be solved explicitly and we obtain
\begin{equation}\label{1dmin:2}
J_{{\rm 1d},\varepsilon}=\biggl(\int_{\rho}^\sigma \biggl(\int_{S_r}|v|+\varepsilon\biggr)^{-\frac1{s-1}}\,dr\biggr)^{-(s-1)}.
\end{equation}
Let us give an argument for \eqref{1dmin:2}. First we observe that using the assumption $v\in L^1(B_\sigma)$ and a simple approximation argument we can replace $\eta\in C^1(\rho,\sigma)$ with $\eta\in W^{1,\infty}(\rho,\sigma)$ in the definition of $J_{{\rm 1d},\varepsilon}$. Let $\widetilde\eta:[\rho,\sigma]\to[0,\infty)$ be given by
$$\widetilde\eta(r):=1-\biggl(\int_\rho^\sigma b(r)^{-\frac{1}{s-1}}\,dr\biggr)^{-1}\int_{\rho}^rb(r)^{-\frac{1}{s-1}}\,dr,\quad\mbox{where $b(r):=\int_{S_r}|v|+\varepsilon$}.$$
Clearly, $\widetilde \eta\in W^{1,\infty}(\rho,\sigma)$ (since $b\geq\varepsilon>0$), $\widetilde \eta(\rho)=1$, $\widetilde \eta(\sigma)=0$, and thus
\begin{equation*}
J_{{\rm 1d},\varepsilon}\leq\int_{\rho}^\sigma |\widetilde\eta'(r)|^sb(r)\,dr=\biggl(\int_{\rho}^\sigma b(r)^{-\frac{1}{s-1}}\,dr\biggr)^{-(s-1)}.
\end{equation*}
The reverse inequality follows by H\"older's inequality: For every $\eta\in W^{1,\infty}(\rho,\sigma)$ satisfying $\eta(\rho)=1$ and $\eta(\sigma)=0$, we have
\begin{equation*}
1=\left(\int_\rho^\sigma \eta'(r)\,dr\right)^s\leq \int_{\rho}^\sigma|\eta'(r)|^sb(r)\,dr\biggl(\int_{\rho}^\sigma b(r)^{-\frac1{s-1}}\,dr\biggr)^{s-1}.
\end{equation*}
Clearly, the last two displayed formulas imply \eqref{1dmin:2}.
Due to the monotonicity of $(-\infty,\infty)\ni m\mapsto (\fint_{\rho}^\sigma v^m(r)\,dr)^\frac1m$, we deduce from \eqref{1dmin:2} for every $\delta>0$
\begin{equation*}
J_{{\rm 1d},\varepsilon}\leq (\sigma-\rho)^{-(s-1+\frac1\delta)}\biggl(\int_{\rho}^\sigma \left(\int_{S_r}|v|+\varepsilon\right)^{\delta}\,dr\biggr)^{\frac{1}{\delta}}.
\end{equation*}
Sending $\varepsilon$ to zero, we obtain \eqref{1dmin}.
\end{proof}
In order to derive a suitable Cacciopolli type inequality in the proof of Theorem~\ref{T}, we make use of the so-called 'hole-filling' trick combined with the following useful (and well-kown) lemma
\begin{lemma}[Lemma~6.1, \cite{Giu}]\label{L:holefilling}
Let $Z(t)$ be a bounded non-negative function in the interval $[\rho,\sigma]$. Assume that for every $\rho\leq s<t\leq \sigma$ it holds
\begin{equation*}
Z(s)\leq \theta Z(t)+(t-s)^{-\alpha} A+B,
\end{equation*}
with $A,B\geq0$, $\alpha>0$ and $\theta\in[0,1)$. Then, there exists $c=c(\alpha,\theta)\in[1,\infty)$ such that
\begin{equation*}
Z(s)\leq c((t-s)^{-\alpha} A+B).
\end{equation*}
\end{lemma}
\subsection{Non-increasing rearrangement and Lorentz-spaces}
We recall the definition and useful properties of the non-increasing rearrangement $f^*$ of a measurable function $f$ and Lorentz spaces, see e.g.\ \cite[Section~22]{TatarBook}. For a measurable function $f:\R^n\to\R$, the non-increasing rearrangement is defined by
\begin{equation*}
f^*(t):=\inf\{\sigma\in(0,\infty)\,:\,|\{x\in\R^n\,:\,|f(x)|>\sigma\}|\leq t\}.
\end{equation*}
Let $f:\R^n\to\R$ be a measurable function with ${\rm supp}f\subset \Omega$, then it holds for all $p\in[1,\infty)$
\begin{equation}\label{eq:ff*}
\int_\Omega |f(x)|^p\,dx=\int_0^{|\Omega|}(f^*(t))^p\,dt.
\end{equation}
A simple consequence of \eqref{eq:ff*} and the fact $f\leq g$ implies $f^*\leq g^*$ is the following inequality
\begin{equation}\label{est:omegat}
\sup_{|A|\leq t\atop A\subset\Omega}\int_A|f(x)|^p\leq \int_0^t(f_\Omega^*(t))^p\,dt,
\end{equation}
where $f_\Omega^*$ denotes the non-increasing rearrangement of $f\chi_\Omega$ (inequality \eqref{est:omegat} is in fact an \textit{equality} but for our purpose the upper bound suffices).
The Lorentz space $L^{n,1}(\R^d)$ can be defined as the space of measurable functions $f:\R^d\to\R$ satisfying
\begin{equation*}
\|f\|_{L^{n,1}(\R^d)}:=\int_0^\infty t^\frac1n f^*(t)\,\frac{dt}t<\infty.
\end{equation*}
Moreover, for $\Omega\subset\R^d$ and a measurable function $f:\R^d\to\R$, we set
\begin{equation*}
\|f\|_{L^{n,1}(\Omega)}:=\int_0^{|\Omega|} t^\frac1n f_\Omega^*(t)\,\frac{dt}t<\infty,
\end{equation*}
where $f_\Omega$ defined as above. Let us recall that $L^{n+\varepsilon}(\Omega)\subset L^{n,1}(\Omega)\subset L^n(\Omega)$ for every $\varepsilon>0$, where $L^{n,1}(\Omega)$ is the space of all measurable functions $f:\Omega\to\R$ satisfying $ \|f\|_{L^{n,1}(\Omega)}<\infty$ (here we identify $f$ with its extension by zero to $\R^n\setminus \Omega$). Following \cite[Section 9]{Kufner}, we define for given $\alpha>0$ the Lorentz-Zygmund space $L^{n,1}(\log L)^\alpha(\R^d)$ as the space of all measurable functions $f:\R^d\to\R$ satisfying%
\begin{equation*}
\|f\|_{L^{n,1}(\log L)^\alpha(\R^d)}:=\int_0^\infty t^\frac1n (1+|\log(t)|)^\alpha f^*(t)\,\frac{dt}t<\infty.
\end{equation*}
As above, for $\Omega\subset\R^d$ and a measurable function $f:\R^d\to\R$, we set
\begin{equation*}
\|f\|_{L^{n,1}(\log L)^\alpha(\Omega)}:=\int_0^{|\Omega|} t^\frac1n (1+|\log(t)|)^\alpha f_\Omega^*(t)\,\frac{dt}t<\infty,
\end{equation*}
and denote by $L^{n,1}(\log L)^\alpha(\Omega)$ the space of all measurable functions $f:\Omega\to\R$ satisfying $\|f\|_{L^{n,1}(\log L)^\alpha(\Omega)}<\infty$. Obviously, we have for every bounded domain $\Omega$ that $L^{n+\varepsilon}(\Omega)\subset L^{n,1}(\log L)^\alpha(\Omega)\subset L^{n,1}(\Omega)$ for every $\varepsilon>0$.
\color{black}
\section{Proof of Theorem~\ref{T}}\label{sec:proof}
In this section, we provide a proof of Theorem~\ref{T}. As mentioned in the introduction, we establish a slightly stronger statement where the growth condition \eqref{ass1} is relaxed in order to introduce a right-hand side (see Remark~\ref{rem:g} below).
\begin{assumption}\label{assgeneral} Let $f:\Omega\times \R^n\to\R$ be a Caratheodory function and suppose that $z\mapsto f(x,z)$ is convex for almost every $x\in\Omega$. Moreover, there exist $1\leq L<\infty$, $\mu\geq0$ such that for all $z\in \R^n$ and almost every $x\in\Omega$
\begin{align}
|z|^p-g(x)^{\frac{p}{p-1}}\leq f(x,z)\leq L|z|^q+g(x)^\frac{p}{p-1},&\label{ass1gen}\\
f(x,2z)\leq \mu+Lf(x,z)),\label{ass2gen}
\end{align}
where $g$ is a non-negative function satisfying $g\in L^{\frac{p}{p-1}}(\Omega)$.
\end{assumption}
In order to state an a priori estimate it is convenient to introduce suitable scale invariant versions of Soblev and $L^p$ norms. For any bounded domain $\Omega\subset\R^n$, we set
\begin{equation*}
\|v\|_{\underline W^{1,p}(\Omega)}:=|\Omega|^{-\frac1n}\|v\|_{\underline L^p(\Omega)}+\|\nabla v\|_{\underline L^p(\Omega)},
\end{equation*}
where
\begin{equation*}
\|v\|_{\underline L^p(\Omega)}:=|\Omega|^{-\frac1p}\|v\|_{L^p(\Omega)}.
\end{equation*}
Note that by definition of $\|\cdot\|_{\underline W^{1,p}(\Omega)}$, it holds
\begin{equation}\label{eq:rescaling}
\forall v\in W^{1,p}(B_R),\,R>0:\qquad \|v\|_{\underline W^{1,p}(B_R)}= \|v_R\|_{\underline W^{1,p}(B_1)}\quad\mbox{where $v_R:=\frac1Rv(R\cdot)\in W^{1,p}(B_1)$.}
\end{equation}
\begin{theorem}\label{T1}
Let $\Omega\subset \R^n$, $n\geq2$ and suppose Assumption~\ref{assgeneral} is satisfied with $1\leq p<q<\infty$ satisfying
\begin{equation}\label{eq:assq}
\varepsilon:=\varepsilon(n,p,q):=\min\biggl\{\frac1q+\frac1{n-1},1\biggr\}-\frac1p\geq0
\end{equation}
and suppose that
\begin{equation*}
g^\frac1{p-1}\in L^{n,1}(\Omega)\quad\mbox{if $p<n$ and}\qquad g^\frac1{n-1}\in L^{n,1}(\log L)^\frac{n-1}{n}(\Omega)\quad\mbox{if $p=n$}.
\end{equation*}
\color{black}Let $u\in W_{\rm loc}^{1,1}(\Omega)$ be a quasi-minimizer of the functional $\mathcal F$ given in \eqref{eq:int}. Then, $u\in L_{\rm loc}^{\infty}(\Omega)$. Moreover, if
\begin{equation}\label{eq:assq1}
\varepsilon(n,p,q)>0\quad\mbox{and}\quad 1\leq p<n
\end{equation}
there exists $c=c(L,n,p,q,Q)\in[1,\infty)$ such that every $Q$-minimizer of \eqref{eq:int} satisfies for every $x_0\in \Omega$ with $B_R:=B_R(x_0)\Subset\Omega$ the estimate
\begin{align}\label{est:T1}
\|u\|_{L^\infty(B_\frac{R}2)}\leq c(R\|u\|_{\underline W^{1,p}(B_R)}+R\|u\|_{\underline W^{1,p}(B_R)}^{1+\frac1\varepsilon(\frac1p-\frac1n)(1-\frac{p}q)}+\|g^\frac{1}{p-1}\|_{L^{n,1}(B_R)}).
\end{align}
\end{theorem}
\begin{remark}\label{rem:g}
As mentioned above, Theorem~\ref{T1} is optimal with respect to the relation between the exponents $p$ and $q$. Moreover, it is also optimal with respect to the assumption on $g$ (at least for $p<n$). Indeed, for $p>1$ consider
\begin{equation}\label{f:laplaceG}
f(x,z):= \tfrac{p+1}p |z|^p+G\cdot z,
\end{equation}
where $G\in L^\frac{p}{p-1}(\Omega,\R^n)$. Clearly $f$ satisfies Assumption~\ref{assgeneral} with $1<p=q$, $g=\frac{p-1}p|G|$ and $L=\frac{p+2}{p}$. Note that $u\in W_{\rm loc}^{1,1}(\Omega)$ is a local minimizer of the functional $\mathcal F$ given in \eqref{eq:int} and $f$ given as in \eqref{f:laplaceG} if and only if it solves locally
\begin{equation}\label{eq:rem:g}
-\Delta_p u:=-{\rm div}(|\nabla u|^{p-2}\nabla u)=\tfrac1{p+1}{\rm div}\, G.
\end{equation}
Hence, Theorem~\ref{T1} yields local boundedness for every weak solution of \eqref{eq:rem:g} provided $|G|^\frac1{p-1}\in L^{n,1}(\Omega)$. On the contrary, the (unbounded) function $u(x)=\log(-\log(|x|))$ solves (trivially) \eqref{eq:rem:g} on $B_\frac12$ with right-hand side $G=-(p+1)|\nabla u|^{p-2}\nabla u$
satisfying $|G|^\frac1{p-1}=c(p)|\nabla u|$ and thus $|G|^\frac1{p-1}\in L^{n,1+\delta}(B_\frac12)$ for every $\delta>0$ (in particular $|G|^\frac1{p-1}\in L^{n,n}(B_\frac12)=L^n(B_\frac12)$) but $|G|^\frac1{p-1}\notin L^{n,1}(B_\frac12)$.
In the interesting recent paper \cite{BM18}, a related result is proven on the Lipschitz-scale. More precisely it is proven that local minimizer of $\int_\Omega f(\nabla u)-gu\,dx$ are locally Lipschitz if $f$ satisfies (controlled) $(p,q)$-growth i.e.\
\begin{equation}\label{pq:growthd2}
(1+|z|^2)^{\frac{p-2}{2}}|\lambda|^2\lesssim \langle D^2 f(z)\lambda,\lambda\rangle\lesssim (1+|z|^2)^\frac{q-2}2|\lambda|^2
\end{equation}
with $\frac{q}{p}<1+\frac2n$ and $g$ is in the optimal Lorentz space $L^{n,1}(\Omega)$ (provided $n\geq3$). Very recently, Lipschitz-regularity of minimizers for integrands satisfying \eqref{pq:growthd2} is proven in \cite{BS19c} under the less restrictive relation $\frac{q}{p}<1+\frac2{n-1}$ in the case $g\equiv0$. It would interesting if the methods of \cite{BS19c} and \cite{BM18} can be combined to obtain Lipschitz estimates under the assumption $\frac{q}p<1+\frac2{n-1}$ and $g\in L^{n,1}$ provided $n\geq3$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{T1}]
By standard scaling and translation arguments it suffices to suppose that $B_1\Subset \Omega$ and prove that $u$ is locally bounded in $B_\frac12$. Hence, we suppose from now on that $B_1\Subset\Omega$. In Step~1--Step~3 below, we consider the case $p\in[1,n)$. We first derive a suitable Caccioppoli-type inequality (Step~1) and perform a De Giorgi-type iteration (Step~2 and 3) to prove boundedness from above for a $Q$-miniminzer. In Step~4, we discuss how this implies the claim of the theorem in the case $p\in[1,n)$. In Step~5, we sketch the adjustments for the remaining non-trivial case $p=n$.
\step 1 Basic energy estimate.
We claim that there exists $c=c(n,p,q,Q)\in[1,\infty)$ such that for every $k\geq0$ and every $\frac12\leq \rho<\sigma\leq1$ it holds
\begin{align}\label{est:substep11}
\|\nabla (u-k)_+\|_{L^p(B_\rho)}^p\leq& c\biggl(\omega(|A_{k,\sigma}|)+\frac{L|A_{k,\sigma}|^{q\varepsilon}}{(\sigma-\rho)^{\gamma}} \|(u-k)_+\|_{W^{1,p}(B_\sigma)}^q\biggr),
\end{align}
where $\gamma=\gamma(n,q):=q-1+q\min\{\frac1q+\frac1{n-1},1\}$, $\varepsilon$ as in \eqref{eq:assq},
\begin{equation}
A_{l,r}:=B_r\cap \{x\in\Omega\, :\,u(x)>l\}\qquad\mbox{for all $r>0$ and $l>0$,}
\end{equation}
and $\omega:[0,|B_1|]\to[0,\infty)$ is defined by
\begin{equation}\label{def:omegat}
\omega(t):=\int_0^t ((g^\frac1{p-1}\chi_{B_1})^*(t))^p\,dt.
\end{equation}
Fix $M>k$ and let $\eta\in C_c^1(B_1,[0,1])$ be such that $\eta=1$ in $B_\rho$ and $\mathop{\mathrm{supp}} \eta\subset B_\sigma$. Define $w:=\max\{u_M-k,0\}$ where $u_M:=\min\{u,M\}$ and set $\varphi:=-\eta^qw$. Since $u$ is a quasi-minimizer, we obtain with help of convexity of $f$ that
\begin{align*}
\int_{A_{k,\sigma}}f(x,\nabla u(x))\,dx\leq& Q\int_{A_{k,\sigma}}f(x,\nabla (u+\varphi)(x))\,dx\\
=&Q\int_{A_{k,\sigma}\cap \{u\leq M\}}f(x,(1-\eta^q)\nabla u-q\eta^{q-1}\nabla \eta(u_M-k)_+)\,dx\\
&+Q\int_{A_{k,\sigma}\cap \{u> M\}}f(x,\nabla u+q\eta^{q-1}\nabla \eta(-(u_M-k)_+)\,dx\\
\leq&Q\int_{A_{k,\sigma}\cap \{u\leq M\}}(1-\eta^q)f(x,\nabla u)+\eta^qf(x,-\frac{q\nabla \eta}{\eta}(u_M-k)_+)\,dx\\
&+\frac{Q}2\int_{A_{k,\sigma}\cap \{u> M\}}f(x,2\nabla u)+f(x,-2q\eta^{q-1}\nabla \eta(u_M-k)_+)\,dx
\end{align*}
and thus, using \eqref{ass1gen}, \eqref{ass2gen} and $|\eta|\leq1$,
\begin{align}\label{est:cacc1}
\int_{A_{k,\sigma}}f(x,\nabla u(x))\,dx\leq&Q\int_{A_{k,\sigma}\setminus B_\rho}f(x,\nabla u)\,dx+\frac{Q}2\int_{A_{k,\sigma}\cap \{u> M\}}\mu+Lf(x,\nabla u)\,dx\notag\\
&+ Q \int_{A_{k,\sigma}}g^\frac{p}{p-1}+Lq^q(1+2^{q})|\nabla \eta|^q|(u_M-k)_+|^q\,dx.
\end{align}
We claim that there exists $c=c(n,q)\in[1,\infty)$ such that
\begin{align}\label{est:infcutoff}
\inf_{\eta\in \mathcal A(\rho,\sigma)}\int_{A_{k,\sigma}}|\nabla \eta|^q|(u_M-k)_+|^q\leq& c(\sigma-\rho)^{-\gamma}\|(u-k)_+\|_{W^{1,p}(B_\sigma\setminus B_\rho)}^q|A_{k,\sigma}|^{q\varepsilon},
\end{align}
%
where $\mathcal A(\rho,\sigma):=\{\eta\in C_c^1(B_\sigma),\, \eta\equiv1\mbox{ on $B_\rho$}\}$. To show \eqref{est:infcutoff}, we use the Sobolev inequality on spheres, i.e\ there exists $c=c(n,q)\in[1,\infty)$ such that for every $r>0$
\begin{equation}\label{ineq:sobd3}
\biggl(\int_{S_r}|\varphi|^{q}\biggr)^\frac1{q}\leq c\biggl(\biggl(\int_{S_r}|\nabla \varphi|^{q_*}\biggr)^\frac1{q_*}+\frac1r\biggl(\int_{S_r}|\varphi|^{q_*}\biggr)^\frac1{q_*}\biggr),
\end{equation}
where $q_*\geq1$ is given by $\frac1{q_*}=\min\{\frac1q+\frac1{n-1},1\}$. Combining \eqref{ineq:sobd3} applied to $\varphi=(u-k)_+$ and Lemma~\ref{L:optimcutoff} with $\delta:=\frac{q_*}q>0$ yield
\begin{align*
&\inf_{\eta\in \mathcal A(\rho,\sigma)}\int_{A_{k,\sigma}}|\nabla \eta|^q|(u_M-k)_+|^q\\
\leq& (\sigma-\rho)^{-(q-1+\frac{q}{q_*})}\biggl(\int_{\rho}^\sigma \biggl(\int_{S_r}|(u-k)_+|^{q}\biggr)^\frac{q_*}{q}\biggr)^\frac{q}{q_*}\\
\leq& c(\sigma-\rho)^{-\gamma}\biggl(\int_{\rho}^\sigma \biggl[\left(\int_{S_r} |\nabla(u-k)_+|^{q_*}\right)+\left(\int_{S_r} |(u-k)_+|^{q_*}\right)\biggr]\,dr\biggr)^\frac{q}{q_*}
\end{align*}
(note that we ignored the factor $\frac1r$ in \eqref{ineq:sobd3} in view of $\frac12\leq\rho<\sigma\leq1$). Finally, we observe that $\varepsilon\geq0$ implies that $q_*\leq p$ and we obtain with help of H\"older inequality
\begin{align*
\inf_{\eta\in \mathcal A(\rho,\sigma)}\int_{A_{k,\sigma}}|\nabla \eta|^q|(u_M-k)_+|^q\leq&c(\sigma-\rho)^{-\gamma}\|(u-k)_+\|_{W^{1,q_*}(B_\sigma\setminus B_\rho)}^q\\
\leq& c(\sigma-\rho)^{-\gamma}\|(u-k)_+\|_{W^{1,p}(B_\sigma\setminus B_\rho)}^q|A_{k,\sigma}|^{q\varepsilon},
\end{align*}
and \eqref{est:infcutoff} is proven.
Since \eqref{est:cacc1} is valid for all $\eta\in \mathcal A(\rho,\sigma)$, we deduce from \eqref{est:infcutoff}, \eqref{est:omegat} and $f(x,z)\geq -g(x)^\frac{p}{p-1}$
\begin{align}\label{est:cacc2}
\int_{A_{k,\rho}}f(x,\nabla u)\,dx \leq&Q\int_{A_{k,\sigma}\setminus B_\rho}f(x,\nabla u)\,dx+\frac{Q}2\int_{A_{k,\sigma}\cap \{u> M\}}\mu+Lf(x,\nabla u)\,dx\notag\\
&+ (Q+1)\omega(|A_{k,\sigma}|)+\frac{cLQ}{(\sigma-\rho)^{\gamma}}\|(u-k)_+\|_{W^{1,p}(B_\sigma)}^q|A_{k,\sigma}|^{q\varepsilon},
\end{align}
where $c=c(n,p,q)\in[1,\infty)$ and $\omega$ is defined in \eqref{def:omegat}. Since $u$ is a quasi-minimizer and we assume $B_1\Subset\Omega$, we have that $f(\cdot,\nabla u)\in L^1(B_1)$ and $u\in W^{1,1}(B_1)$. Thus, we can send $M\to\infty$ in \eqref{est:cacc2} and the second term on the right-hand side in \eqref{est:cacc2} vanishes. Hence, we obtain with help of the hole-filling trick (namely adding $Q\int_{A_{k,\rho}}f(x,\nabla u)\,dx $ to both sides of inequality \eqref{est:cacc2})
\begin{align*}
\int_{A_{k,\rho}}f(x,\nabla u(x))\,dx\leq \theta \int_{A_{k,\sigma}}f(x,\nabla u(x))\,dx+c\biggl(\omega(|A_{k,\sigma}|)+\frac{L|A_{k,\sigma}|^{q\varepsilon}}{(\sigma-\rho)^{\gamma}}\|(u-k)_+\|_{W^{1,p}(B_\sigma)}^q\biggr),
\end{align*}
with $\theta=\frac{Q}{Q+1}\in[0,1)$ and $c=c(n,p,q)\in[1,\infty)$. Estimate \eqref{est:substep11} follows by Lemma~\ref{L:holefilling} and \eqref{ass1gen}.
\step 2 One-step improvement.
We claim that there exist $c_1=c_1(n,p,q,Q)\in[1,\infty)$ and $c_2=c_2(n,p)\in[1,\infty)$ such that for every $k>h\geq 0$ and every $\frac12\leq \rho<\sigma<1$ it holds
\begin{align}\label{est:onestep}
J(k,\rho)\leq& c_1\biggl(\omega\biggl(\frac{c_2 J(h,\sigma)^\frac{p_n^*}p}{(k-h)^{p_n^*}}\biggr)+L\biggl(\frac{J(h,\sigma)^\frac1p}{k-h}\biggr)^{p_n^*q\varepsilon}\frac{J(h,\sigma)^\frac{q}p}{(\sigma-\rho)^{\gamma}}+\biggl(\frac{J(h,\sigma)^{\frac1p}}{(k-h)}\biggr)^{p_n^*\frac{p}n}J(h,\sigma)\biggr),
\end{align}
where $p_n^*:=\frac{p n}{n-p}$ and for any $l\geq0$ and $r>0$
\begin{equation*}
J(l,r):=\|(u-l)_+\|_{W^{1,p}(B_r)}^p
\end{equation*}
Note that $k-h<u-h$ on $A_{k,r}$ for every $r>0$ and thus with help of Sobolev inequality
\begin{equation}\label{est:Aksigma}
|A_{k,\sigma}|\leq \int_{A_{k,\sigma}}\biggl(\frac{u(x)-h}{k-h}\biggr)^{p_n^*}\leq \frac{\|(u-h)_+\|_{L^{p_n^*}(B_\sigma)}^{p_n^*}}{(k-h)^{p_n^*}}\leq c \frac{J(h,\sigma)^\frac{p_n^*}p}{(k-h)^{p_n^*}}
\end{equation}
where $c=c(n,p)\in[1,\infty)$. Combining the above estimate with \eqref{est:substep11}, we obtain
\begin{equation}\label{est:onestep1}
\|\nabla (u-k)_+\|_{L^p(B_\rho)}^p\leq c_1\biggl(\omega\biggl(\frac{c_2 J(h,\sigma)^\frac{p_n^*}p}{(k-h)^{p_n^*}}\biggr)+L\biggl(\frac{J(h,\sigma)^\frac1p}{k-h}\biggr)^{p_n^*q\varepsilon}\frac{J(h,\sigma)^\frac{q}p}{(\sigma-\rho)^{\gamma}}\biggr),
\end{equation}
where $c_1=c_1(n,p,q,Q)\in[1,\infty)$ and $c_2=c_2(n,p)\in[1,\infty)$. It is left to estimate $\|(u-k)_+\|_{L^p(B_\rho)}$. A combination of H\"older inequality, Sobolev inequality and estimate \eqref{est:Aksigma} yield
\begin{align}\label{est:onestep2}
\| (u-k)_+\|_{L^p(B_\rho)}^p\leq \|(u-h)_+\|_{L^{p_n^*}(B_\sigma)}^p|A_{k,\sigma}|^{\frac{p}{n}}\leq c \biggl(\frac{J(h,\sigma)^{\frac1p}}{(k-h)}\biggr)^{p_n^*\frac{p}n}J(h,\sigma)\biggr)
\end{align}
Combining \eqref{est:onestep1} and \eqref{est:onestep2}, we obtain \eqref{est:onestep}.
\step 3 Iteration.
For $k_0\geq0$ and a sequence $(\Delta_\ell)_{\ell\in\mathbb N}\subset [0,\infty)$ specified below, we set
\begin{equation}
k_\ell:=k_0+\Delta_\ell,\quad \sigma_\ell=\frac12+\frac1{2^{\ell+1}}.
\end{equation}
For every $\ell\in\mathbb N\cup\{0\}$, we set $J_\ell:=J(k_\ell,\sigma_\ell)$. From \eqref{est:onestep}, we deduce for every $\ell\in\mathbb N$
\begin{align}\label{est:iteration}
J_\ell\leq c_1\biggl(\omega\biggl(\frac{c_2 J_{\ell-1}^\frac{p_n^*}p}{(\Delta_\ell)^{p_n^*}}\biggr)+L2^{(\ell+1)\gamma}\biggl(\frac{J_{\ell-1}^\frac1p}{\Delta_\ell}\biggr)^{p_n^*q\varepsilon}J_{\ell-1}^\frac{q}p+\biggl(\frac{J_{\ell-1}^{\frac1p}}{\Delta_\ell}\biggr)^{p_n^*\frac{p}n}J_{\ell-1}\biggr),
\end{align}
where $c_1$ and $c_2$ are as in Step~2. Fix $\tau=\tau(n,p,q)\in(0,1)$ such that
\begin{equation}\label{def:tau}
2^\gamma\tau^{\frac{q}p(1+p_n^{*}\varepsilon)-1}=\frac12.
\end{equation}
We claim that we can choose $\{\Delta_\ell\}_{\ell\in\mathbb N}$ satisfying
\begin{equation}\label{est:sumdeltaell}
\sum_{\ell\in\mathbb N}\Delta_\ell<\infty
\end{equation}
and $k_0$ (in the borderline case $\varepsilon=0$) in such a way that
\begin{equation}\label{ass:iterationJell}
J_\ell\leq \tau^\ell J_0\qquad\mbox{for all $\ell\in\mathbb N\cup\{0\}$}.
\end{equation}
\substep{3.1} Suppose that $\varepsilon>0$. Set $k_0=0$ and choose $\Delta_\ell$ to be the smallest number such that
\begin{align}\label{def:deltaell1}
c_1\omega\biggl(\frac{c_2 (\tau^{\ell-1}J_0)^\frac{p_n^*}p}{(\Delta_\ell)^{p_n^*}}\biggr)\leq \frac13\tau^\ell J_0,\qquad c_1\tau^{-(\frac{p_n^*}p+1)}J_0^{\frac{p_n^*}n}\tau^{\ell\frac{p_n}p}\leq \frac13\Delta_\ell^{p_n^*\frac{p}n}
\end{align}
and
\begin{equation}\label{def:deltaell2}
c_1L2^\gamma\tau^{-\frac{q}p(1+p_n^*\varepsilon)}J_0^{\frac{q}p(1+p_n^*\varepsilon)-1}2^{-\ell}\leq \frac13\Delta_\ell^{p_n^*q\varepsilon}
\end{equation}
is valid. The choice of $\tau$ (see \eqref{def:tau}), $\Delta_\ell$ and estimate \eqref{est:iteration} combined with a straightforward induction argument yield \eqref{ass:iterationJell}. Using $\sum_{\ell\in\mathbb N}(2^{-\alpha}+\tau^\beta)<\infty$ for any $\alpha,\beta>0$, we deduce from \eqref{def:deltaell1} and \eqref{def:deltaell2}
\begin{align}\label{sum:deltaell}
\sum_{\ell\in\mathbb N}\Delta_\ell \leq \sum_{\ell\in\mathbb N}\frac{c_2^{\frac1p-\frac1n}(\tau^{\ell-1}J_0)^\frac1p)}{(\omega^{-1}(\frac{\tau^\ell J_0}{3c_1}))^{\frac1p-\frac1n}}+c (J_0^\frac1p+J_0^{\frac1p+\frac1p(\frac1p-\frac1n)(1-\frac{p}q)\frac1\varepsilon}),
\end{align}
where $c=c(L,n,p,q,Q)\in[1,\infty)$. Next, we show that $g^\frac1{p-1}\in L^{n,1}(B_1)$ ensures that the first term on the right-hand side of \eqref{sum:deltaell} is bounded and thus \eqref{est:sumdeltaell} is valid. Indeed,
\begin{align}\label{sum:omega1}
\sum_{\ell\in\mathbb N}\frac{(\tau^{\ell}J_0)^\frac1p)}{(\omega^{-1}(\frac{\tau^\ell J_0}{3c_1}))^{\frac1p-\frac1n}}\lesssim& \int_1^\infty \frac{(\tau^x J_0)^\frac1p}{(\omega^{-1}(\frac{\tau^xJ_0}{3c_1}))^{\frac1p-\frac1n}}\,dx\notag\\
=&\frac1{|\log \tau|}\int_0^\tau\frac{(tJ_0)^\frac1p}{(\omega^{-1}(\frac{tJ_0}{3c_1}))^{\frac1p-\frac1n}}\,\frac{dt}t\notag\\
\leq&\frac{(3c_1)^\frac1p}{|\log\tau|}\int_0^{\omega^{-1}({\frac{\tau J_0}{3c_1}})} \frac{( \omega(s))^\frac1p}{s^{\frac1p-\frac1n}}\frac{\omega'(s)}{\omega(s)}\,ds.
\end{align}
Recall $\omega(t)=\int_0^t((g^\frac1{p-1}\chi_{B_1})^*(s))^p\,ds$ and $(g^\frac1{p-1}\chi_{B_1})^*$ is non-increasing, thus $\omega(t)\geq t(g^\frac{1}{p-1}\chi_{B_1})^*(t)^p$ and
\begin{align}\label{sum:omega2}
\int_0^{\omega^{-1}({\frac{\tau J_0}{3c_1}})} \frac{( \omega(s))^\frac1p}{s^{\frac1p-\frac1n}}\frac{\omega'(s)}{\omega(s)}\,ds\leq&\int_0^{\infty}s^{\frac1n-\frac1p}(s((g^\frac1{p-1}\chi_{B_1})^*(s))^p)^{-(1-\frac1p)}((g^\frac1{p-1}\chi_{B_1})^*(s))^p\,ds\notag\\
=&\int_0^{\infty}s^{\frac1n}(g^\frac1{p-1}\chi_{B_1})^*(s)\,\frac{ds}s=\|g^\frac1{p-1}\|_{L^{n,1}(B_1)}.
\end{align}
Notice that \eqref{ass:iterationJell} and $k_0=0$ implies
\begin{equation*}
\|(u-\sum_{\ell\in\mathbb N}\Delta_\ell)_+\|_{L^p(B_\frac12)}=0\quad \Rightarrow\quad \sup_{B_\frac12}u\leq \sum_{\ell\in\mathbb N}\Delta_\ell
\end{equation*}
and thus
$$
\sup_{B_\frac12}u\leq \sum_{\ell\in\mathbb N}\Delta_\ell.
$$
Hence, appealing to \eqref{sum:deltaell}-\eqref{sum:omega2}, we find $c=c(L,n,p,q,Q)\in[1,\infty)$ such that
\begin{equation}\label{est:supu}
\sup_{B_\frac12}u\leq c(\|(u)_+\|_{W^{1,p}(B_1)}+\|(u)_+\|_{W^{1,p}(B_1)}^{1+(\frac1p-\frac1n)(1-\frac{p}q)\frac1\varepsilon}+\|g\|_{L^{n,1}(B_1)}).
\end{equation}
\substep{3.1} Suppose that $\varepsilon=0$. We claim that
\begin{equation}\label{lim:J00}
\lim_{k_0\to\infty} J_0=0.
\end{equation}
Before, we give the argument for \eqref{lim:J00} we explain how \eqref{lim:J00} implies the desired claim \eqref{ass:iterationJell} in the case $\varepsilon=0$. Choose $\Delta_\ell$ to be the smallest number such that \eqref{def:deltaell1} is satisfied and choose $k_0$ sufficiently large such that
\begin{equation*
c_1L2^\gamma \tau^{-\frac{q}p}J_0^{\frac{q}p-1}\leq\frac13.
\end{equation*}
It is now easy to see that the choice of $\tau$, $\Delta_\ell$, $k_0$ and estimate \eqref{est:iteration} yield \eqref{ass:iterationJell}. In view of Substep~3.1 we also have $\sum_{\ell\in\mathbb N}\Delta_\ell<\infty$ and we have
\begin{equation*}
\sup_{B_\frac12}u\leq k_0+c(\|(u)_+\|_{W^{1,p}(B_1)}+\|g\|_{L^{n,1}(B_1)})<\infty,
\end{equation*}
where $c=c(L,n,p,q,Q)\in[1,\infty)$.
Let us now show \eqref{lim:J00}. For $k\geq 2^\frac1p|B_1|^{-\frac1p}\|u\|_{L^p(B_1)}$ we have $|A_{k,1}|\leq\frac12 |B_1|$ and thus a suitable version of Poincare inequality (see e.g.\ \cite[Proposition~3.15]{GM12}) yields
\begin{equation*}
\int_{B_1}|(u-k)_+|^p\,dx\leq c\int_{B_1}|\nabla (u-k)_+|^p\,dx,
\end{equation*}
where $c=c(n,p)\in[1,\infty)$. Hence, it suffices to show $\lim_{k\to\infty}\|\nabla (u-k)_+\|_{L^p(B_1)}^p=0$. By \eqref{ass1gen}, we have for every $k\geq0$
\begin{align}\label{lim:J001}
\int_{B_1}|\nabla (u-k)_+|^p=\int_{A_{k,1}}|\nabla u|^p\leq \int_{A_{k,1}}f(x,\nabla u)+g^\frac{p}{p-1}(x)\,dx.
\end{align}
Since $B_1\Subset \Omega$ and $f(x,\nabla u),g^\frac{p}{p-1}\in L^1_{\rm loc}(\Omega)$, the right-hand side in \eqref{lim:J001} tends to zero as $k$ tends to infinity and thus \eqref{lim:J00} is proven.
\step 4 Conclusion in the case $p<n$
In view of Step~1--Step~3, we have that if $B_1\Subset \Omega$ than $u$ is locally bounded from above in $B_\frac12$ and in the case $\varepsilon>0$, we have the estimate \eqref{est:supu}. Moreover, if $u$ is a $Q$-minimizer of $\mathcal F$, then $-u$ is a $Q$-minimizer of the functional $\widetilde{\mathcal F}(v):=\int_{\Omega} \tilde f(x,\nabla v(x))\,dx$ with $\tilde f(x,z):=f(x,-z)$. Clearly, $\tilde f$ is convex in the second component and satisfies the same growth conditions as $f$. Hence, we obtain that $u$ is locally bounded in $B_\frac12$. Moreover, if $\varepsilon>0$ there exists $c=c(L,n,p,q,Q)\in[1,\infty)$ such that
\begin{equation*}
\|u\|_{L^\infty(B_\frac12)}\leq c(\|u\|_{W^{1,p}(B_1)}+\|u\|_{W^{1,p}(B_1)}^{1+(\frac1p-\frac1n)(1-\frac{p}q)\frac1\varepsilon}+\|g\|_{L^{n,1}(B_1)}).
\end{equation*}
The conclusion of the theorem in the case $p\in[1,n)$ now follows by standard scaling, translation and covering arguments (here we use \eqref{eq:rescaling}).
\step 5 The case $p=n$.
We use the same notation as in the previous steps and sketch the necessary adjustments. Note that for $p=n$ we cannot use Sobolev inequality in the form \eqref{est:Aksigma}. In the parts not involving $\omega$ it suffices to replace $p_n^*$ by any $\tilde p\in[q,\infty)$ (recall $q>p=n$) and we leave the details to the reader. Using this replacement for the estimates related to $\omega$, we obtain local boundedness under slightly stronger assumptions on $g$, namely $g^\frac1{n-1}\in L^{n+\delta}(\Omega)$ for some $\delta>0$ (in fact this statement is already contained in \cite{FS93}). Thus we may appeal to the Moser-Trudinger inequality, which gives for some dimensional constant $c>0$, $0\le h <k , \frac12<\sigma <1$
\begin{equation}\label{est:Aksigmapn}
|A_{k,\sigma}|\leq c\exp\biggl(-\frac1c\left(\frac{k-h}{J(h,\sigma)^\frac1n}\right)^{\frac{n}{n-1}}\biggr).
\end{equation}
Let us first conclude and present the the derivation of the above inequality below.
In view of Step 3 and \eqref{est:Aksigmapn} it suffices to show that the sequence $\{\Delta_\ell\}_{\ell\in\mathbb N}$ defined by the identity
\[\omega\biggl(c\exp\biggl(-\frac1c\frac{\Delta^{\frac{n}{n-1}}_\ell}{(\tau^{\ell-1} J_0)^\frac{1}{n-1}}\biggr)\biggr)= \bar c\tau^\ell J_0,\]
for some $\bar c>0$ and $\tau\in(0,1)$ is summable. Indeed, we have
\begin{align*}
\sum_{\ell\in\mathbb N}\Delta_\ell\lesssim& \sum_{\ell\in\mathbb N}(\tau^{\ell-1} J_0)^\frac1n\left(1 + |\log(\tfrac1{c}\omega^{-1}(\bar c\tau^\ell J_0))|\right)^{\frac{n-1}{n}}\\
\lesssim& \frac1{\tau^\frac1n|\log(\tau)|}\int_0^\tau(tJ_0)^\frac1n(1+|\log(\omega^{-1}(\bar ctJ_0))|)^{\frac{n-1}{n}}\,\frac{dt}t\\
=& \frac{1}{(\tau\bar c)^\frac1n|\log(\tau)|}\int_0^{\omega^{-1}(\bar c\tau J_0)}(\omega(s))^\frac1n(1+|\log(s)|)^{\frac{n-1}{n}}\frac{\omega'(s)}{\omega(s)}\,ds.
\end{align*}
Now we can continue as before, i.e. using $\omega(s)\geq s(g^\frac{1}{n-1}\chi_{B_1})^*(s)^n$ and $\omega'(s)=(g^\frac{1}{n-1}\chi_{B_1})^*(s)^n$, we obtain
\begin{align*
\int_0^{\omega^{-1}(\bar c\tau J_0)}(\omega(s))^\frac1n(1+|\log(s)|)^{\frac{n-1}{n}}\frac{\omega'(s)}{\omega(s)}\,ds\leq&\int_0^{\infty}s^{\frac1n}(1+|\log(s)|)^{\frac{n-1}{n}}((g^\frac1{n-1}\chi_{B_1})^*(s))\,\frac{ds}s\notag\\
=&\|g^\frac1{n-1}\|_{L^{n,1}(\log L)^{\frac{n-1}{n}}(B_1)}<\infty.
\end{align*}
Finally, we present the argument for \eqref{est:Aksigmapn}. For this we recall the Moser-Trudinger inequality in the following form: there exists $c_i=c_i(n)>0$, $i=1,2$ such that for every ball $B\subset\R^d$ and every $v\in W^{1,n}(B)$
\begin{equation*}
\fint_{B}\exp\biggl(\biggl(\frac{|v-\fint_Bv|}{c_1\|\nabla v\|_{L^n(B)}}\biggr)^\frac{n}{n-1}\biggr)\leq c_2
\end{equation*}
(see e.g.\ \cite[Chapter 7]{GT}).
Since
\[A_{k, \sigma} \subset B_\sigma \cap \{ x \colon (u-h)_+ \ge k-h\}=:E_{h,k,\sigma}\]
Chebychev's inequality combined with Moser-Trudinger inequality gives \eqref{est:Aksigmapn}:
\begin{align*}
|E_{h,k,\sigma}|
\lesssim& \exp\biggl(-\left(\frac{k-h}{2^{\frac{n}{n-1}}c_1 J(h,\sigma)^\frac1n}\right)^\frac{n}{n-1}\biggr) \int_{B_\sigma} \exp\biggl(\biggl(\frac{(u-h)_+}{ 2^{\frac{n}{n-1}}c_1 J(h,\sigma)^\frac1n}\biggr)^\frac{n}{n-1}\biggr)\,dx\\
\leq&\exp\biggl(-\left(\frac{k-h}{2^{\frac{n}{n-1}}c_1 J(h,\sigma)^\frac1n}\right)^\frac{n}{n-1}\biggr) \exp\biggl(\biggl(\frac{\fint_{B_\sigma}(u-h)_+}{ c_1 J(h,\sigma)^\frac1n}\biggr)^\frac{n}{n-1}\biggr)c_2|B_\sigma|\\
\lesssim&\exp\biggl(-\left(\frac{k-h}{2^{\frac{n}{n-1}}c_1 J(h,\sigma)^\frac1n}\right)^\frac{n}{n-1}\biggr)
\end{align*}
where we use in the last estimate the Poincar\'e inequality the assumption $\sigma\leq1$.
\end{proof}
\section*{Acknowledgments}
M.S. was supported by the German Science Foundation DFG in context of the Emmy Noether Junior Research Group BE 5922/1-1. J.H. was supported by the German Science Foundation DFG in context of the Priority Program SPP 2026 "Geometry at Infinity".
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Подгајци Подравски су насељено место у саставу града Доњег Михољца у Осјечко-барањској жупанији, Република Хрватска.
Историја
До територијалне реорганизације у Хрватској налазили су се у саставу старе општине Доњи Михољац.
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На попису становништва 1991. године, насељено место Подгајци Подравски је имало 750 становника, следећег националног састава:
Референце
Спољашње везе
Доњи Михољац
Насељена места у Хрватској
Насељена места у Осјечко-барањској жупанији
Википројект географија/Насеља у Хрватској
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This past year, ERA Real Estate's objective was to better support our affiliated brokers in the form of new strategic partnerships, resources and programs that help position the agent at the center of the homeownership lifecycle.
A perfect example of this is the brand's new relationship with HomeAdvisor, a digital marketplace that connects homeowners with pre-screened, local service professionals to carry out home improvement, maintenance and remodeling projects. Now, clients working with an ERA broker have access to HomeAdvisor's network of top-rated and reviewed service professionals in their area through a self-service website and dedicated call center. ERA was the first real estate partner to offer this concierge service network-wide, giving participating brokers another client touchpoint, while providing a value-added experience for buyers and sellers.
Additionally, ERA Real Estate recently launched ERA Moves, a strategic client outreach program. This tool offers ERA-affiliated, independent agents two ways to stay top of mind with their clients, positioning them as local market experts. One of its components provides applicable homeowner discounts through agent-branded, automated emails. It also provides clients with a concierge service that reduces the stress, time and expense associated with the moving process. ERA was the first to offer this service nationwide, all at zero cost to the agent and broker. More importantly, however, the program creates another way for affiliated agents to remain in contact with their clients throughout the homeownership lifecycle. And the best part? It also creates an additional stream of revenue for affiliated brokerages through referral fees. (More: Keeping Agents Top of Mind With ERA Moves)
Now more than ever, the industry is recognizing that profitability is a competitive advantage. At ERA, we encourage our affiliated brokers to "Grow Your Way"—offering them the flexibility to choose what makes sense for their business goals, whether it's through the addition of ancillary business, customized branding or providing business consulting services. And when it comes to empowering people to make pioneering changes to their business, the strength of the ERA network is formidable. A consistent story I hear from ERA brokers is how they love to inspire, support and work with their colleagues across the network, including sharing their listing presentations, compensation strategies or blueprints for creating additional revenue through ancillary businesses.
This is what organically happens when you have a brand with such an influential culture of collaboration.
Ron Restaino, founder and owner of Restaino & Associates ERA Powered, recently shared that they were courted by many franchisors over the years and would listen politely, but ultimately declined each offer because they all required the brokerage to swallow their brand—their identity—and essentially give up their autonomy. All of that changed when he learned about the ERA Powered concept and realized it was the solution they had been looking for all along—a franchise that gives the tools, service and support needed, while allowing them to remain who they ultimately wanted to be.
ERA's "Grow Your Way" value proposition will be instrumental as we look at how the brand will continue to grow in the next decade and beyond.
We will continue to build upon the brand's founding principle that each customer, independent sales agent and brokerage is unique and should be free to develop their business as they see fit. Companies who affiliate with ERA get the best of both worlds: the support, technology and leverage that comes with joining a national brand with nearly 50 years of success, as well as the ability to retain their own local brand identity.
A new decade brings a host of exciting prospects, including the opportunity to get closer to the consumer and help them manage the changes in their own lives. "The only thing constant is change" is an appropriate expression for all of us in this industry. At ERA, adapting to change is not enough. We want to create change. We're excited by the opportunities that lie ahead and look forward to TEAM ERA being a changemaker in 2020 and beyond.
Sherry Chris is president and CEO of ERA Real Estate. For more information, please visit www.era.com.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,966
|
Q: Задание по указателям - C (СИ) С клавиатуры вводится динамическая строка. Проверить, входит ли в нее цифры 5 и 7. При доступе к элементам использовать указатели.
Что не так?
#include <stdio.h>
#include <stdlib.h>
#include <conio.h>
#include <string.h>
void main(void)
{
char *str = (char*)malloc(50 * sizeof(char));
gets(str);
char*Ykaz[strlen(str)];
printf("Vhodyat 7 ta 5?");
for (int i = 0; i < strlen(str); i++)
{
if (*str[i] == 5 || *str[i] == 7)
printf("Da, vhodyat");
}
getch();
}
A: #include <stdio.h>
int main(void)
{
#define MAXLEN 200
char *str = malloc(MAXLEN * sizeof(char));
gets_s(str,MAXLEN);
int has5 = 0, has7 = 0;
for (char * c = str; *c; ++c)
{
switch(*c)
{
case '5': has5 = 1; break;
case '7': has7 = 1; break;
}
if (has5 && has7) break;
}
if (has5) puts("String has '5'");
if (has7) puts("String has '7'");
}
Примерно так.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,326
|
Kraszewo-Rory is a village in the administrative district of Gmina Raciąż, within Płońsk County, Masovian Voivodeship, in east-central Poland.
References
Kraszewo-Rory
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,588
|
I Want to Be a Clone: How Steve Taylor Helped Me Find Myself
By John Barber • December 11, 2013
Ah hah! Steve Taylor and John Barber. I knew it. Those dudes would have it. This is the right place, but what was the word…what was the word. . .
There's no doubt that the Rabbit Room is full of my people, but I haven't always known who my people were.
The first time I found my people, I was in eighth grade. I didn't know they were my people at first, I just knew I'd finally found human beings who said out loud the things I was thinking. Looking back now, I can see that those people were more important than I ever realized back then.
In the back of the room where our church youth group met, there was a smaller room. And in that smaller room, the arcane was made real. It was my wardrobe, but inside, instead of fur coats, I found a soundboard and lighting controls. Instead of snowy woods, I found the buttons that controlled everything—the music, the microphones, the lights. The characters who occupied that room were like deities to me because they knew what the buttons did when you pushed them and because they got to choose which music played as everyone else walked into the larger room. These guys would fit so well in the Rabbit Room. One that sticks out to me was Chip, one of the adult leaders. Chip was fond of dressing in knee-high leather boots and a kilt, and he would sometimes have a Highlander sword slung on his back. For a 13-year-old, these things make an impression. In that tiny room, guys like Chip introduced me to bands like The Seventy-Sevens, and singers like Mark Heard and Steve Taylor.
For a kid who grew up exclusively on Contemporary Christian Music with a little bit of the oldies station for good measure, I wasn't used to this. In the late '80s and early '90s, when names like Michael W. Smith and Sandi Patty ruled the Southern Baptist music dial, I had no idea this other musical world existed. Oh, I knew there were bands like Petra—Christian music's answer to Bon Jovi—but I wasn't allowed to listen to them. That was too "hard." But mostly, I needed music that asked difficult questions: "What about social justice?" "What if I'm not sure about what I believe?" "What about sin?" I'm thankful for the CCM artists that filled my cassette rack back then. They gave me a vocabulary for my faith. But I really needed to move past musical Christianity 101 and look toward something that would help me know how to live; something to convince me that I wasn't alone.
Enter Steve Taylor, a Christian music iconoclast. Steve Taylor's music is a mash-up of pop, rock, and punk sensibilities (in attitude, if not in sound). It's a hard sound to categorize because it's not unusual to find reggae, rap, and hard rock on the same album. What's unique about Steve Taylor (this was especially true in the '80s and '90s) is that you never knew what he would do next.
I was a little late to Steve Taylor's game, but that just made the findings richer indeed. Instead of having to wait for new releases, I had the pleasure of digging through the shelves at Long's Christian Bookstore in Orlando and buying everything my allowance would pay for. And as I talked with people about Steve Taylor, I began to hear the stories:
Have you heard 'I Blew Up the Clinic Real Good?' The satirical song about people who blow up abortion clinics?
Have you heard his song about racism at Bob Jones University? He's banned from that campus for life!
Have you seen the video for Meltdown? It's got Blair from The Facts of Life in it!
This stuff was mind-blowing for me. But beyond the controversial topics, what always impacted me about Steve Taylor's music was how he talked about things that mattered to me. Things I'd never heard put to music before. He criticized the idea that all Christians have to look, think, and feel exactly the same in "I Want to Be a Clone" ("Cloneliness is next to godliness, right?"). He made me look differently at the public figures I was idolizing in "Hero." He made me see that "Jesus is for Losers." And mostly, he made me feel all right about the doubt that I've always carried.
Belief was never easy for me (it still isn't), but judging by the other Christian music I'd been hearing, this was my own personal flaw. I was an oddball. Everyone else got this belief thing with no trouble. My favorite Steve Taylor song was and is "Harder to Believe Than Not To" because it gives voice to what I was thinking (watch the video).
I held on to those songs for dear life.
Fast forward to a few weeks ago, when a Kickstarter campaign for Steve Taylor & the Perfect Foil's new album was launched. On one hand, it was **CHRISTMAS** come early for me, I've been awaiting new tunes from him for twenty years. But on the other, some things bubbled back up to the surface. Since the Kickstarter announcement, I've spent a lot of time with Now the Truth Can Be Told, the two-disc collection of Steve Taylor's work, and I can see how far I've come.
Now, instead of my people being limited to that small sound booth, they're here, in The Rabbit Room. Now I'm surrounded by people who say out loud the things I've always thought. My people have grown from five to five hundred. And if it weren't for Steve Taylor, Mark Heard, The Vigilantes of Love, The Seventy-Sevens, Daniel Amos, The Prayer Chain, Poor Old Lu, and on and on and on, I don't think I'd ever have had the guts to become a part of this place. I'm eternally grateful to all of them for teaching me that belief isn't easy for a lot of us. I'm grateful for their voices telling me that I'm not alone.
John Barber is a music lover, film nut, husband, and father. Last year he set out to watch 365 films in one year, and he lived to tell about it. That means he's seen more bad movies than we even want to think about.
One Minute Review: Steve Jobs
Thomas McKenzie
Danny Boyle, Aaron Sorkin, Michael Fassbender, and Kate Winslet turn their considerable talents to telling…
Taylor Leonhardt's River House
Chris Yokel
Every once in awhile, an artist and album comes around that takes you by happy…
Ron Block and Jeff Taylor: Trouble Go Down
Jeff Taylor and I have finished a duo record, with lyrics written by Rebecca Reynolds,…
Yes, yes, and yes again.
Wow! I didn't realize.
Thanks for sharing, John. I can certainly relate.
@pete December 11, 2013 at 6:13 pm
This project is worth backing for that sasquatch picture alone.
Chris Lovie-Tyler
Thanks for drawing attention to Steve Taylor, John. He's a great artist.
I especially like his lyrics and sense of humour.
Interestingly, he also wrote lyrics for the Newsboys. (Now I know why I like some of their lyrics so much.)
Chagall Guevara, one of his other projects, is definitely worth a listen. I still have the cassette!
Look forward to hearing the Perfect Foil!
Really enjoyed this post, John! I don't know Steve Taylor's music, but now I'm interested to try it out. While you were immersed in Christian contemporary music, I was being told that only hymns were godly. I'm still a little behind on everything…
But I love what you said about Taylor giving a voice to your inner world. I sometimes think writers and musicians don't realize the power of that particular aspect of their ministry. The emphasis is more often on enlightenment or encouragement. But, to me, when you speak the truth aloud, in whatever way is most real to you, you're inventing a new language…and tiny little miracles start happening in silent hearts all over the place. The dumb speak. 🙂
You're speaking my language!! Steve Taylor? Mark Heard?? Yes and yes!! Steve is truly a brilliant artist – not only a musician but a film director as well. His music had a huge impact on me as well. I hadn't heard about his new project – what a wonderful Christmas surprise!
Pretty much everything Helena said. I'm still behind too.
There's so much light in this room. Sometimes I don't know what to do with it all.
Man, John….I KNEW we were musical kin but that list of bands you just made truly proves we are musical brethren. I would add The Choir and Adam Again to that list to be uber complete. I am older than you, but the effect of those artists on me is similar even if the circumstance is somewhat different. As someone who had become DEEPLY entrenched in the underground music of the day (Heck,,,,I even roomed briefly with Kurt Cobain and members of The Screaming Trees and Beat Happening)) it was difficult to find Christian music that was anywhere near as authentic or honest as the stuff I was listening to. I was introduced to both Taylors, Steve and Terry around '86 and, actually wasn't all that impressed initially. I still am not a big fan of On the Fritz but eventually, deeper inspection led me to love these guys and further explore like minded artists. It was reassuring to know that there were other "misfits" out there and that people who had followed Christ for quite a while still struggled with things that I, as a newer believer, was trying to come to grips with. I LOVE that you illuminated these foundational artists here on the Rabbit Room. The "Christian Music Industry" is egregiously ambivalent toward their own history and the fact that Daniel Amos just released one of the best records of the past 25 years and can't even register in most current believer's musical vocabulary is an out and out shame.
Well John, I no longer have to write the part of my autobiography that deals with my musical awakening in the 90s. You just did it for me. Thanks.
You had me at Steve Taylor, but when you threw out Poor Old Lu, The Prayer Chain, VoL and the 77s, I knew you were a kindred spirit. Consider me one of your people.
I'd also encourage everyone to support Steve's project. His band consists of John Mark Painter (played with Ben Folds, Sixpence, etc.), Peter Furler (Newsboys) and Jimmy Abegg (played with Charlie Peacock and some dude named Rich Mullins).
Shawn – I want stories about Cobain!
Christopher Stewart
I was a scrawny twleve or thirteen year old, son of a Christian music station disc jockey. It was 1995 or thereabouts, when I looked into the glow of our smallish living room T.V. to see a tall, lanky man dressed in black and singing a rather melancholy rock ballad (something about how Jesus was for Losers) while tromping around lush, green countrysides and old world churchyards. I was mesmerized. Mystified. In all my Christian music upbringing I'd never seen (or heard) anything along these lines. Who was this scarecrow of a man with locks of stringy curls and why were his words and the melody behind them creating such a stir in my CCM'd boy's heart?!? I immediately ordered Squint on cassette from the local Christian book store-as any good Christian pre-adolescent is want to do and I waited anxiously for the still mysterious yet palpably inspiring music & man to arrive at my doorstep. This was soon followed by the order of Now The Truth Can Be Told on double cassette. Needless to say, I've never looked back. There was a Taylor-shaped hole in my young, Christian, musical soul and it was filled at just the right moment in just the right way. Thank the Lord for weird, local Christian music video T.V. channels. Thank the Lord for the divinely appointed anomaly that was and is Steve Taylor.
Shawn,
Ditto, what John said. Give us Kurt tales!
Mamie Rose
I wasn't around when Steve Taylor was making his music, but my dad introduced me to his music long ago and my siblings and I have been fans for years. I was so thrilled to hear that he is recording a new album. His voice and challenge to believers to combat sin is much needed in our day and age.
Nick Swirski
I came out of the late 70's punk scene – Ramones, Pistols… then to Bowie and Kiss…. came to Christ and someone gave me a Swaggart album and I thought I died!!!! Then… I somehow found Undercover, the Altar Boys and southern California Christian surf – punk and then came Steve Taylor! He and the Altar Boys gave my faith experience a dose of energy, life, and an awareness that Sunday morning music/worship was not the be all end all to real Christian music!! Great article… brought tears to my eyes and joy to my heart!!!
Aah, the Altar Boys. They were a favourite for me too, Nick!
"And you tell 'em who Jesus is, and they look at you like you're just so… out of it."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,375
|
Q: policy and mechanism I was going through my operating systems textbook and I came across the concept of "separating mechanism and policy". I wasn't sure of what that meant so I checked out wikipedia which I must admit, was not of much help either.
The separation of mechanism and
policy[1] is a design principle in
computer science. It states that
mechanisms (those parts of a system
implementation that control the
authorization of operations and the
allocation of resources) should not
dictate (or overly restrict) the
policies according to which decisions
are made about which operations to
authorize, and which resources to
allocate.
Could someone tone this down, and explain if possible with a few examples what separation of mechanism and policy means in the context of Operating systems?
A: In regard to *nix operating systems, the general idea is the security system is implemented by the kernel, and the authorization system is implemented by userspace.
The all-powerful root and suid binaries that so many people deride (whether justly or otherswise) are necessary for effective separations. It is possible to completely swap out the authentication mechanism while leaving the security intact (ssh does this, which is why it uses undocumented APIs on Windows).
A: Here is what this means for the X-Windows system.
X-Windows, at the very base level, provides a way of manipulating screen areas called 'windows'. It also provides a way to receive events that happen inside windows.
But X-Windows says nothing about title bars, menus, scrollbars or any of that stuff. It also doesn't say anything about the rules by which a particular application can make its window occupy the whole screen, or when a window has to be moved off the screen. It does provide a way for one application to force other applications to ask it permission before doing things with top-level windows, but doesn't provide any such application as part of the base server.
X-Windows is all about mechanism, not policy.
The policy is provided by the widget toolkit, by the window manager, and by other things added to the system later. Many widget toolkits, for example, use a set of overlapping sub-windows for scrollbars and ask for mouse events for these sub-windows so they can detect click and drag operations and make the sub-windows respond appropriately.
This is why, for example, GNOME and KDE can get along on the same display, and why really old X-Windows programs that know nothing about panels or desktops still work just fine on modern systems.
A: Difference between mechanism and policy
mechanism determines how to do something, policies decide what will be done.
The separation of policy from mechanism is very important principle, it allows maximum flexibility if policy decisions are to be changed later.
A: Although this is a very old question, I still want to share my point.
The reason why this passage is confusing is because of the two words "mechanism" and "policy". And in the context of software engineering, I think it's always ok to replace "mechanism" with "interface" and "policy" with "implementation".
As for the separation of interface and implementation, if you're programming in Java, then you must be quite familiar with these two concepts. By doing so, we can isolate "what to do" from "how to do", which helps us achieve system decoupling.
Why decoupling? Decoupling improves the extensibility and maintainability of the code, which means we can write less code when requirements change :)
Learn more techniques about decoupling from "The GoF Design Patterns".
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,691
|
Q: Finding recurrence relation and complexity Based on the number of operations, finding out the recurrence relation!
a = N;
var = 0;
while (a > 1)
{
var = var + a;
num = 2 * var;
a = a / 2;
}
I think that the the recurrence relaton that will be formed is: (Assignment operations are to be not counted)
T(n)= (from a=1 to N)Σ(3)
Am I right??
Now using this recurrence relation, how to find its complexity.
A: What you want to do is find how many times this operation is called, so consider this: after each call a is divided by 2, so if M = N/2 then T(M) = T(N) - 1.
Now, each iteration of this loop divides N again so you get the following:
T(N) = T(N/2) + 1 = ... = k + T(N/(2^k))
The stop condition is this: a>1 so you need to check when N/(2^k) <= 1
N/2^k = 1 -> log (n) = k
So T(N) = log(n) + T(1) = log(n)
This is the answer in 'big O' notation.
A: Empirical approach:
First reduce the "educational noise" from the code by simplifying it and add an iteration counter (c). Then look at the result (N,count) and after a while you see, that 2 ^ count = N for all N in [1,2,4,8,16,..].
So the complexity Compl(loop) = O(log_2(N)).
let rec loop a c =
match a with
| x when x > 1 ->
let a1 = a / 2
loop a1 (c+1)
| _ -> (a,c)
// after staring at the result of the computation we came up with this theory:
let theory n = int (System.Math.Log10(float n) / System.Math.Log10(2.0))
[1..64]
|> List.map (fun a -> a,loop a 0, theory a)
|> List.map (fun (a0,(a,c),aa) -> a0,c,aa)
Data:
[(1, 0, 0); (2, 1, 1); (3, 1, 1); (4, 2, 2); (5, 2, 2); (6, 2, 2); (7, 2, 2);
(8, 3, 3); (9, 3, 3); (10, 3, 3); (11, 3, 3); (12, 3, 3); (13, 3, 3);
(14, 3, 3); (15, 3, 3); (16, 4, 4); (17, 4, 4); (18, 4, 4); (19, 4, 4);
(20, 4, 4); (21, 4, 4); (22, 4, 4); (23, 4, 4); (24, 4, 4); (25, 4, 4);
(26, 4, 4); (27, 4, 4); (28, 4, 4); (29, 4, 4); (30, 4, 4); (31, 4, 4);
(32, 5, 5); (33, 5, 5); (34, 5, 5); (35, 5, 5); (36, 5, 5); (37, 5, 5);
(38, 5, 5); (39, 5, 5); (40, 5, 5); (41, 5, 5); (42, 5, 5); (43, 5, 5);
(44, 5, 5); (45, 5, 5); (46, 5, 5); (47, 5, 5); (48, 5, 5); (49, 5, 5);
(50, 5, 5); (51, 5, 5); (52, 5, 5); (53, 5, 5); (54, 5, 5); (55, 5, 5);
(56, 5, 5); (57, 5, 5); (58, 5, 5); (59, 5, 5); (60, 5, 5); (61, 5, 5);
(62, 5, 5); (63, 5, 5); (64, 6, 6)]
A: The recurrence relation is:
T(1) = a
T(n) = b + T(n/2)
The first part comes from the case where the loop variable equals 1, in which case only the comparison at the top of the loop executes. The second line comes from the constant amount of work done in the loop body, b, plus the time to execute the loop with the updated loop variable value.
The first few terms are:
n T
1 a
2 a + b
4 a + 2b
8 a + 3b
Based on this we can guess the general form:
T(n) = a + b log n
Proving that is left as an exercise; but you can just plug it in to the recurrence relation to see that it satisfies the requirements.
The asymptotic complexity is logarithmic.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,448
|
{"url":"https:\/\/www.wikiyy.com\/en\/Scale_(map)","text":"# Scale (map)\n\nWikipedia view on Wikipedia\n\nThe scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways. The first way is the ratio of the size of the generating globe to the size of the Earth. The generating globe is a conceptual model to which the Earth is shrunk and from which the map is projected.\n\nThe ratio of the Earth's size to the generating globe's size is called the nominal scale (= principal scale = representative fraction). Many maps state the nominal scale and may even display a bar scale (sometimes merely called a 'scale') to represent it. The second distinct concept of scale applies to the variation in scale across a map. It is the ratio of the mapped point's scale to the nominal scale. In this case 'scale' means the scale factor (= point scale = particular scale).\n\nIf the region of the map is small enough to ignore Earth's curvature\u2014a town plan, for example\u2014then a single value can be used as the scale without causing measurement errors. In maps covering larger areas, or the whole Earth, the map's scale may be less useful or even useless in measuring distances. The map projection becomes critical in understanding how scale varies throughout the map.[1][2] When scale varies noticeably, it can be accounted for as the scale factor. Tissot's indicatrix is often used to illustrate the variation of point scale across a map.\n\n## History\n\nThe foundations for quantitative map scaling goes back to ancient China with textual evidence that the idea of map scaling was understood by the second century BC. Ancient Chinese surveyors and cartographers had ample technical resources used to produce maps such as counting rods, carpenter's square's, plumb lines, compasses for drawing circles, and sighting tubes for measuring inclination. Reference frames postulating a nascent coordinate system for identifying locations were hinted by ancient Chinese astronomers that divided the sky into various sectors or lunar lodges.[3]\n\nThe Chinese cartographer and geographer Pei Xiu of the Three Kingdoms period created a set of large-area maps that were drawn to scale. He produced a set of principles that stressed the importance of consistent scaling, directional measurements, and adjustments in land measurements in the terrain that was being mapped.[3]\n\n## The terminology of scales\n\n### Representation of scale\n\nMap scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are:\n\n'one centimetre to one hundred metres' \u00a0\u00a0 or \u00a0\u00a0 1:10,000 \u00a0\u00a0or \u00a0\u00a0 1\/10,000\n'one inch to one mile' \u00a0\u00a0 or \u00a0\u00a0 1:63,360 \u00a0\u00a0 or \u00a0\u00a0 1\/63,360\n'one centimetre to one thousand kilometres' \u00a0\u00a0or\u00a0\u00a0 1:100,000,000 \u00a0\u00a0 or \u00a0\u00a0 1\/100,000,000. \u00a0(The ratio would usually be abbreviated to 1:100M)\n\n### Bar scale vs. lexical scale\n\nIn addition to the above many maps carry one or more (graphical) bar scales. For example, some modern British maps have three bar scales, one each for kilometres, miles and nautical miles.\n\nA lexical scale in a language known to the user may be easier to visualise than a ratio: if the scale is an inch to two miles and the map user can see two villages that are about two inches apart on the map, then it is easy to work out that the villages are about four miles apart on the ground.\n\nA lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. For example, a scale of one inch to a furlong (1:7920) will be understood by many older people in countries where Imperial units used to be taught in schools. But a scale of one pouce to one league may be about 1:144,000, depending on the cartographer's choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used.\n\n### Large scale, medium scale, small scale\n\nContrast to spatial scale.\n\nA map is classified as small scale or large scale or sometimes medium scale. Small scale refers to world maps or maps of large regions such as continents or large nations. In other words, they show large areas of land on a small space. They are called small scale because the representative fraction is relatively small.\n\nLarge scale maps show smaller areas in more detail, such as county maps or town plans might. Such maps are called large scale because the representative fraction is relatively large. For instance a town plan, which is a large scale map, might be on a scale of 1:10,000, whereas the world map, which is a small scale map, might be on a scale of 1:100,000,000.\n\nThe following table describes typical ranges for these scales but should not be considered authoritative because there is no standard:\n\nClassification Range Examples\nlarge scale 1:0 \u2013 1:600,000 1:0.00001 for map of virus; 1:5,000 for walking map of town\nmedium scale 1:600,000 \u2013 1:2,000,000 Map of a country\nsmall scale 1:2,000,000 \u2013 1:\u221e 1:50,000,000 for world map; 1:1021 for map of galaxy\n\nThe terms are sometimes used in the absolute sense of the table, but other times in a relative sense. For example, a map reader whose work refers solely to large-scale maps (as tabulated above) might refer to a map at 1:500,000 as small-scale.\n\nIn the English language, the word large-scale is often used to mean \"extensive\". However, as explained above, cartographers use the term \"large scale\" to refer to less extensive maps \u2013 those that show a smaller area. Maps that show an extensive area are \"small scale\" maps. This can be a cause of confusion.\n\n### Scale variation\n\nMapping large areas causes noticeable distortions because it significantly flattens the curved surface of the earth. How distortion gets distributed depends on the map projection. Scale varies across the map, and the stated map scale is only an approximation. This is discussed in detail below.\n\n## Large-scale maps with curvature neglected\n\nThe region over which the earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature of the earth is undetectable over a meridian distance of about 100 kilometres (62\u00a0mi) and over an east-west line of about 80\u00a0km (at a latitude of 45 degrees). If surveyed to the nearest 1 millimetre (0.039\u00a0in), then curvature is undetectable over a meridian distance of about 10\u00a0km and over an east-west line of about 8\u00a0km.[4] Thus a plan of New York City accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on the ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map.\n\n## Altitude reduction\n\nThe variation in altitude, from the ground level down to the sphere's or ellipsoid's surface, also changes the scale of distance measurements.[5]\n\n## Point scale (or particular scale)\n\nAs proved by Gauss\u2019s Theorema Egregium, a sphere (or ellipsoid) cannot be projected onto a plane without distortion. This is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. The only true representation of a sphere at constant scale is another sphere such as a globe.\n\nGiven the limited practical size of globes, we must use maps for detailed mapping. Maps require projections. A projection implies distortion: A constant separation on the map does not correspond to a constant separation on the ground. While a map may display a graphical bar scale, the scale must be used with the understanding that it will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.)\n\nLet P be a point at latitude ${\\displaystyle \\phi }$ and longitude ${\\displaystyle \\lambda }$ on the sphere (or ellipsoid). Let Q be a neighbouring point and let ${\\displaystyle \\alpha }$ be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing ${\\displaystyle \\beta }$. In general ${\\displaystyle \\alpha \\neq \\beta }$. Comment: this precise distinction between azimuth (on the Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably.\n\nDefinition: the point scale at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as\n\n${\\displaystyle \\mu (\\lambda ,\\,\\phi ,\\,\\alpha )=\\lim _{\\frac },}$\n\nwhere the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ.\n\nDefinition: if P and Q lie on the same meridian ${\\displaystyle (\\alpha =0)}$, the meridian scale is denoted by ${\\displaystyle h(\\lambda ,\\,\\phi )}$ .\n\nDefinition: if P and Q lie on the same parallel ${\\displaystyle (\\alpha =\\pi \/2)}$, the parallel scale is denoted by ${\\displaystyle k(\\lambda ,\\,\\phi )}$.\n\nDefinition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor ${\\displaystyle k(\\lambda ,\\phi )}$.\n\nDefinition: A map projection is said to be conformal if the angle between a pair of lines intersecting at a point P is the same as the angle between the projected lines at the projected point P', for all pairs of lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection.\n\nIsotropy of scale implies that small elements are stretched equally in all directions, that is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections. For example, the Mercator projection is conformal since it is constructed to preserve angles and its scale factor is isotopic, a function of latitude only: Mercator does preserve shape in small regions.\n\nDefinition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines. These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder[1] pages 203\u2014206.)\n\n### The representative fraction (RF) or principal scale\n\nThere are two conventions used in setting down the equations of any given projection. For example, the equirectangular cylindrical projection may be written as\n\ncartographers: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ${\\displaystyle x=a\\lambda }$ \u00a0\u00a0 \u00a0\u00a0${\\displaystyle y=a\\phi }$\nmathematicians: \u00a0\u00a0 \u00a0\u00a0 ${\\displaystyle x=\\lambda }$ \u00a0\u00a0 \u00a0\u00a0${\\displaystyle y=\\phi }$\n\nHere we shall adopt the first of these conventions (following the usage in the surveys by Snyder). Clearly the above projection equations define positions on a huge cylinder wrapped around the Earth and then unrolled. We say that these coordinates define the projection map which must be distinguished logically from the actual printed (or viewed) maps. If the definition of point scale in the previous section is in terms of the projection map then we can expect the scale factors to be close to unity. For normal tangent cylindrical projections the scale along the equator is k=1 and in general the scale changes as we move off the equator. Analysis of scale on the projection map is an investigation of the change of k away from its true value of unity.\n\nActual printed maps are produced from the projection map by a constant scaling denoted by a ratio such as 1:100M (for whole world maps) or 1:10000 (for such as town plans). To avoid confusion in the use of the word 'scale' this constant scale fraction is called the representative fraction (RF) of the printed map and it is to be identified with the ratio printed on the map. The actual printed map coordinates for the equirectangular cylindrical projection are\n\nprinted map: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ${\\displaystyle x=(RF)a\\lambda }$ \u00a0\u00a0 \u00a0\u00a0${\\displaystyle y=(RF)a\\phi }$\n\nThis convention allows a clear distinction of the intrinsic projection scaling and the reduction scaling.\n\nFrom this point we ignore the RF and work with the projection map.\n\n### Visualisation of point scale: the Tissot indicatrix\n\nThe Winkel tripel projection with Tissot's indicatrix of deformation\n\nConsider a small circle on the surface of the Earth centred at a point P at latitude ${\\displaystyle \\phi }$ and longitude ${\\displaystyle \\lambda }$. Since the point scale varies with position and direction the projection of the circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection. Superimposing these distortion ellipses on the map projection conveys the way in which the point scale is changing over the map. The distortion ellipse is known as Tissot's indicatrix. The example shown here is the Winkel tripel projection, the standard projection for world maps made by the National Geographic Society. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples[6][7]).\n\n### Point scale for normal cylindrical projections of the sphere\n\nThe key to a quantitative understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude ${\\displaystyle \\phi }$ and longitude ${\\displaystyle \\lambda }$ on the sphere. The point Q is at latitude ${\\displaystyle \\phi +\\delta \\phi }$ and longitude ${\\displaystyle \\lambda +\\delta \\lambda }$. The lines PK and MQ are arcs of meridians of length ${\\displaystyle a\\delta \\phi }$ where ${\\displaystyle a}$ is the radius of the sphere and ${\\displaystyle \\phi }$ is in radian measure. The lines PM and KQ are arcs of parallel circles of length ${\\displaystyle (a\\cos \\phi )\\delta \\lambda }$ with${\\displaystyle \\lambda }$ in radian measure. In deriving a point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.\n\nInfinitesimal elements on the sphere and a normal cylindrical projection\n\nNormal cylindrical projections of the sphere have ${\\displaystyle x=a\\lambda }$ and ${\\displaystyle y}$ equal to a function of latitude only. Therefore, the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base ${\\displaystyle \\delta x=a\\delta \\lambda }$ and height\u00a0${\\displaystyle \\delta y}$. By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (The treatment of scale in a general direction may be found below.)\n\nparallel scale factor\u00a0\u00a0 ${\\displaystyle \\quad k\\;=\\;{\\dfrac {\\delta x}}=\\,\\sec \\phi \\qquad \\qquad {}}$\nmeridian scale factor\u00a0 ${\\displaystyle \\quad h\\;=\\;{\\dfrac {\\delta y}}={\\dfrac }}$\n\nNote that the parallel scale factor ${\\displaystyle k=\\sec \\phi }$ is independent of the definition of ${\\displaystyle y(\\phi )}$ so it is the same for all normal cylindrical projections. It is useful to note that\n\nat latitude 30 degrees the parallel scale is ${\\displaystyle k=\\sec 30^{\\circ }=2\/{\\sqrt }=1.15}$\nat latitude 45 degrees the parallel scale is ${\\displaystyle k=\\sec 45^{\\circ }={\\sqrt }=1.414}$\nat latitude 60 degrees the parallel scale is ${\\displaystyle k=\\sec 60^{\\circ }=2}$\nat latitude 80 degrees the parallel scale is ${\\displaystyle k=\\sec 80^{\\circ }=5.76}$\nat latitude 85 degrees the parallel scale is ${\\displaystyle k=\\sec 85^{\\circ }=11.5}$\n\nThe following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's indicatrix.\n\n### Three examples of normal cylindrical projection\n\n#### The equirectangular projection\n\nThe equidistant projection with Tissot's indicatrix of deformation\n\nThe equirectangular projection,[1][2][4] also known as the Plate Carr\u00e9e (French for \"flat square\") or (somewhat misleadingly) the equidistant projection, is defined by\n\n${\\displaystyle x=a\\lambda ,}$\u00a0\u00a0 ${\\displaystyle y=a\\phi ,}$\n\nwhere ${\\displaystyle a}$ is the radius of the sphere, ${\\displaystyle \\lambda }$ is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at ${\\displaystyle \\lambda =0}$) and ${\\displaystyle \\phi }$ is the latitude. Note that ${\\displaystyle \\lambda }$ and ${\\displaystyle \\phi }$ are in radians (obtained by multiplying the degree measure by a factor of ${\\displaystyle \\pi }$\/180). The longitude ${\\displaystyle \\lambda }$ is in the range ${\\displaystyle [-\\pi ,\\pi ]}$ and the latitude ${\\displaystyle \\phi }$ is in the range ${\\displaystyle [-\\pi \/2,\\pi \/2]}$.\n\nSince ${\\displaystyle y'(\\phi )=1}$ the previous section gives\n\nparallel scale,\u00a0 ${\\displaystyle \\quad k\\;=\\;{\\dfrac {\\delta x}}=\\,\\sec \\phi \\qquad \\qquad {}}$\nmeridian scale ${\\displaystyle \\quad h\\;=\\;{\\dfrac {\\delta y}}=\\,1}$\n\nFor the calculation of the point scale in an arbitrary direction see addendum.\n\nThe figure illustrates the Tissot indicatrix for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of the major axis to the minor axis is ${\\displaystyle \\sec \\phi }$. Clearly the area of the ellipse increases by the same factor.\n\nIt is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of the RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is ${\\displaystyle \\sec \\phi }$ so when we transfer a separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a great circle). On a line at a bearing of say 45 degrees (${\\displaystyle \\beta =45^{\\circ }}$) the scale is continuously varying with latitude and transferring a separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum). Even if we could work out a distance along this line of constant bearing its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution.\n\n#### Mercator projection\n\nThe Mercator projection with Tissot's indicatrix of deformation. (The distortion increases without limit at higher latitudes)\n\nThe Mercator projection maps the sphere to a rectangle (of infinite extent in the ${\\displaystyle y}$-direction) by the equations[1][2][4]\n\n${\\displaystyle x=a\\lambda \\,}$\n${\\displaystyle y=a\\ln \\left[\\tan \\left({\\frac {\\pi }}+{\\frac {\\phi }}\\right)\\right]}$\n\nwhere a, ${\\displaystyle \\lambda \\,}$ and ${\\displaystyle \\phi \\,}$ are as in the previous example. Since ${\\displaystyle y'(\\phi )=a\\sec \\phi }$ the scale factors are:\n\nparallel scale \u00a0\u00a0\u00a0\u00a0${\\displaystyle k\\;=\\;{\\dfrac {\\delta x}}=\\,\\sec \\phi .}$\nmeridian scale \u00a0\u00a0${\\displaystyle h\\;=\\;{\\dfrac {\\delta y}}=\\,\\sec \\phi .}$\n\nIn the mathematical addendum it is shown that the point scale in an arbitrary direction is also equal to ${\\displaystyle \\sec \\phi }$ so the scale is isotropic (same in all directions), its magnitude increasing with latitude as ${\\displaystyle \\sec \\phi }$. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.\n\n#### Lambert's equal area projection\n\nLambert's normal cylindrical equal-area projection with Tissot's indicatrix of deformation\n\nLambert's equal area projection maps the sphere to a finite rectangle by the equations[1][2][4]\n\n${\\displaystyle x=a\\lambda \\qquad \\qquad y=a\\sin \\phi }$\n\nwhere a, ${\\displaystyle \\lambda }$ and ${\\displaystyle \\phi }$ are as in the previous example. Since ${\\displaystyle y'(\\phi )=\\cos \\phi }$ the scale factors are\n\nparallel scale\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0${\\displaystyle \\quad k\\;=\\;{\\dfrac {\\delta x}}=\\,\\sec \\phi \\qquad \\qquad {}}$\nmeridian scale \u00a0\u00a0 ${\\displaystyle \\quad h\\;=\\;{\\dfrac {\\delta y}}=\\,\\cos \\phi }$\n\nThe calculation of the point scale in an arbitrary direction is given below.\n\nThe vertical and horizontal scales now compensate each other (hk=1) and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the same area as the undistorted circles on the equator.\n\n#### Graphs of scale factors\n\nThe graph shows the variation of the scale factors for the above three examples. The top plot shows the isotropic Mercator scale function: the scale on the parallel is the same as the scale on the meridian. The other plots show the meridian scale factor for the Equirectangular projection (h=1) and for the Lambert equal area projection. These last two projections have a parallel scale identical to that of the Mercator plot. For the Lambert note that the parallel scale (as Mercator A) increases with latitude and the meridian scale (C) decreases with latitude in such a way that hk=1, guaranteeing area conservation.\n\n### Scale variation on the Mercator projection\n\nThe Mercator point scale is unity on the equator because it is such that the auxiliary cylinder used in its construction is tangential to the Earth at the equator. For this reason the usual projection should be called a tangent projection. The scale varies with latitude as ${\\displaystyle k=\\sec \\phi }$. Since ${\\displaystyle \\sec \\phi }$ tends to infinity as we approach the poles the Mercator map is grossly distorted at high latitudes and for this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and rhumb lines). However, at a latitude of about 25 degrees the value of ${\\displaystyle \\sec \\phi }$ is about 1.1 so Mercator is accurate to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees (centred on the equator) is accurate to within 1% or 1 part in 100.\n\nA standard criterion for good large-scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to ${\\displaystyle k=1.0004}$. Since ${\\displaystyle \\sec \\phi }$ attains this value at ${\\displaystyle \\phi =1.62}$ degrees (see figure below, red line). Therefore, the tangent Mercator projection is highly accurate within a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about 360\u00a0km (220\u00a0mi). Within this strip Mercator is very good, highly accurate and shape preserving because it is conformal (angle preserving). These observations prompted the development of the transverse Mercator projections in which a meridian is treated 'like an equator' of the projection so that we obtain an accurate map within a narrow distance of that meridian. Such maps are good for countries aligned nearly north-south (like Great Britain) and a set of 60 such maps is used for the Universal Transverse Mercator (UTM). Note that in both these projections (which are based on various ellipsoids) the transformation equations for x and y and the expression for the scale factor are complicated functions of both latitude and longitude.\n\nScale variation near the equator for the tangent (red) and secant (green) Mercator projections.\n\n### Secant, or modified, projections\n\nThe basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say ${\\displaystyle \\phi _}$ north and south. Clearly the scale is now true at these latitudes whereas parallels beneath these latitudes are contracted by the projection and their (parallel) scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes.\n\nAs an example, one possible secant Mercator projection is defined by\n\n${\\displaystyle x=0.9996a\\lambda \\qquad \\qquad y=0.9996a\\ln \\left(\\tan \\left({\\frac {\\pi }}+{\\frac {\\phi }}\\right)\\right).}$\n\nThe numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified:\n\nsecant Mercator scale, \u00a0\u00a0${\\displaystyle \\quad k\\;=0.9996\\sec \\phi .}$\n\nThus\n\n\u2022 the scale on the equator is 0.9996,\n\u2022 the scale is k=1 at a latitude given by ${\\displaystyle \\phi _}$ where ${\\displaystyle \\sec \\phi _=1\/0.9996=1.00004}$ so that ${\\displaystyle \\phi _=1.62}$ degrees,\n\u2022 k=1.0004 at a latitude ${\\displaystyle \\phi _}$ given by ${\\displaystyle \\sec \\phi _=1.0004\/0.9996=1.0008}$ for which ${\\displaystyle \\phi _=2.29}$ degrees. Therefore, the projection has ${\\displaystyle 1, that is an accuracy of 0.04%, over a wider strip of 4.58 degrees (compared with 3.24 degrees for the tangent form).\n\nThis is illustrated by the lower (green) curve in the figure of the previous section.\n\nSuch narrow zones of high accuracy are used in the UTM and the British OSGB projection, both of which are secant, transverse Mercator on the ellipsoid with the scale on the central meridian constant at ${\\displaystyle k_=0.9996}$. The isoscale lines with ${\\displaystyle k=1}$ are slightly curved lines approximately 180\u00a0km east and west of the central meridian. The maximum value of the scale factor is 1.001 for UTM and 1.0007 for OSGB.\n\nThe lines of unit scale at latitude ${\\displaystyle \\phi _}$ (north and south), where the cylindrical projection surface intersects the sphere, are the standard parallels of the secant projection.\n\nWhilst a narrow band with ${\\displaystyle |k-1|<0.0004}$ is important for high accuracy mapping at a large scale, for world maps much wider spaced standard parallels are used to control the scale variation. Examples are\n\n\u2022 Behrmann with standard parallels at 30N, 30S.\n\u2022 Gall equal area with standard parallels at 45N, 45S.\nScale variation for the Lambert (green) and Gall (red) equal area projections.\n\nThe scale plots for the latter are shown below compared with the Lambert equal area scale factors. In the latter the equator is a single standard parallel and the parallel scale increases from k=1 to compensate the decrease in the meridian scale. For the Gall the parallel scale is reduced at the equator (to k=0.707) whilst the meridian scale is increased (to k=1.414). This gives rise to the gross distortion of shape in the Gall-Peters projection. (On the globe Africa is about as long as it is broad). Note that the meridian and parallel scales are both unity on the standard parallels.\n\nInfinitesimal elements on the sphere and a normal cylindrical projection\n\nFor normal cylindrical projections the geometry of the infinitesimal elements gives\n\n${\\displaystyle {\\text{(a)}}\\quad \\tan \\alpha ={\\frac },}$\n${\\displaystyle {\\text{(b)}}\\quad \\tan \\beta ={\\frac {\\delta x}{\\delta y}}={\\frac {\\delta y}}.}$\n\nThe relationship between the angles ${\\displaystyle \\beta }$ and ${\\displaystyle \\alpha }$ is\n\n${\\displaystyle {\\text{(c)}}\\quad \\tan \\beta ={\\frac }\\tan \\alpha .\\,}$\n\nFor the Mercator projection ${\\displaystyle y'(\\phi )=a\\sec \\phi }$ giving ${\\displaystyle \\alpha =\\beta }$: angles are preserved. (Hardly surprising since this is the relation used to derive Mercator). For the equidistant and Lambert projections we have ${\\displaystyle y'(\\phi )=a}$ and ${\\displaystyle y'(\\phi )=a\\cos \\phi }$ respectively so the relationship between ${\\displaystyle \\alpha }$ and ${\\displaystyle \\beta }$ depends upon the latitude\u00a0${\\displaystyle \\phi }$. Denote the point scale at P when the infinitesimal element PQ makes an angle ${\\displaystyle \\alpha \\,}$ with the meridian by ${\\displaystyle \\mu _{\\alpha }.}$ It is given by the ratio of distances:\n\n${\\displaystyle \\mu _{\\alpha }=\\lim _{\\frac }=\\lim _{\\frac {\\sqrt {\\delta x^+\\delta y^}}{\\sqrt \\,\\delta \\phi ^+a^\\cos ^\\!\\phi \\,\\delta \\lambda ^}}}.}$\n\nSetting ${\\displaystyle \\delta x=a\\delta \\lambda }$ and substituting ${\\displaystyle \\delta \\phi }$ and ${\\displaystyle \\delta y}$ from equations (a) and (b) respectively gives\n\n${\\displaystyle \\mu _{\\alpha }(\\phi )=\\sec \\phi \\left[{\\frac {\\sin \\alpha }{\\sin \\beta }}\\right].}$\n\nFor the projections other than Mercator we must first calculate ${\\displaystyle \\beta }$ from ${\\displaystyle \\alpha }$ and ${\\displaystyle \\phi }$ using equation (c), before we can find ${\\displaystyle \\mu _{\\alpha }}$. For example, the equirectangular projection has ${\\displaystyle y'=a}$ so that\n\n${\\displaystyle \\tan \\beta =\\sec \\phi \\tan \\alpha .\\,}$\n\nIf we consider a line of constant slope ${\\displaystyle \\beta }$ on the projection both the corresponding value of ${\\displaystyle \\alpha }$ and the scale factor along the line are complicated functions of ${\\displaystyle \\phi }$. 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Porrentruy District (, ) is one of the three districts of the canton of Jura, Switzerland. Its capital is the town of Porrentruy. The French-speaking district has a population of (as of ).
Municipalities
Porrentruy is divided into a total of 20 municipalities:
Coat of arms
The blazon of the district coat of arms is Gules a Fess Argent, overall a Cockatrice Or volant holding in legs and beak a Crosier of the same.
Demographics
Porrentruy has a population () of . Most of the population () speaks French (22,008 or 91.8%) as their first language, German is the second most common (1,001 or 4.2%) and Italian is the third (306 or 1.3%). There are 8 people who speak Romansh.
, the population was 48.8% male and 51.2% female. The population was made up of 10,585 Swiss men (43.7% of the population) and 1,243 (5.1%) non-Swiss men. There were 11,322 Swiss women (46.7%) and 1,083 (4.5%) non-Swiss women. Of the population in the district, 9,552 or about 39.8% were born in Porrentruy and lived there in 2000. There were 7,448 or 31.1% who were born in the same canton, while 2,708 or 11.3% were born somewhere else in Switzerland, and 3,388 or 14.1% were born outside of Switzerland.
, there were 9,390 people who were single and never married in the district. There were 11,599 married individuals, 1,854 widows or widowers and 1,128 individuals who are divorced.
There were 3,245 households that consist of only one person and 747 households with five or more people.
The historical population is given in the following chart:
Mergers and name changes
On 1 January 2009, seventeen municipalities merged into four new municipalities
The former municipalities of Montenol, Montmelon, Ocourt, Saint-Ursanne and Seleute merged to form the new municipality of Clos du Doubs. ** The former municipalities of Asuel, Charmoille, Fregiécourt, Miécourt and Pleujouse merged to form the new municipality of La Baroche.
The former municipalities of Buix, Courtemaîche and Montignez merged to form the new municipality of Basse-Allaine.
The former municipalities of Chevenez, Damvant, Réclère and Roche-d'Or merged to form the new municipality of Haute-Ajoie.
On 1 January 2013 the former municipality of Bressaucourt merged into the municipality of Fontenais.
On 1 January 2018 the former municipality of Corban merged into the municipality of Haute-Ajoie.
On 1 January 2023 the former municipalities of Damphreux and Lugnez merged into the new municipality of Damphreux-Lugnez.
Politics
In the 2007 federal election the most popular party was the CVP which received 33.81% of the vote. The next three most popular parties were the SPS (31.98%), the FDP (16.2%) and the SVP (12.88%). In the federal election, a total of 8,374 votes were cast, and the voter turnout was 46.7%.
Religion
From the , 18,473 or 77.1% were Roman Catholic, while 2,423 or 10.1% belonged to the Swiss Reformed Church. Of the rest of the population, there were 79 members of an Orthodox church (or about 0.33% of the population), there were 15 individuals (or about 0.06% of the population) who belonged to the Christian Catholic Church, and there were 733 individuals (or about 3.06% of the population) who belonged to another Christian church. There were 8 individuals (or about 0.03% of the population) who were Jewish, and 287 (or about 1.20% of the population) who were Islamic. There were 30 individuals who were Buddhist, 8 individuals who were Hindu and 21 individuals who belonged to another church. 1,322 (or about 5.51% of the population) belonged to no church, are agnostic or atheist, and 933 individuals (or about 3.89% of the population) did not answer the question.
Education
In Porrentruy about 7,375 or (30.8%) of the population have completed non-mandatory upper secondary education, and 1,985 or (8.3%) have completed additional higher education (either university or a Fachhochschule). Of the 1,985 who completed tertiary schooling, 62.7% were Swiss men, 25.5% were Swiss women, 7.2% were non-Swiss men and 4.6% were non-Swiss women.
The Canton of Jura school system provides two year of non-obligatory Kindergarten, followed by six years of Primary school. This is followed by three years of obligatory lower Secondary school where the students are separated according to ability and aptitude. Following the lower Secondary students may attend a three or four year optional upper Secondary school followed by some form of Tertiary school or they may enter an apprenticeship.
During the 2009-10 school year, there were a total of 1,955 students attending 117 classes in Porrentruy. There were 24 kindergarten classes with a total of 416 students in the district. The district had 92.5 primary classes and 1,539 students.
References
Districts of the canton of Jura
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 96
|
package techreborn.client.gui;
import net.minecraft.client.util.math.MatrixStack;
import net.minecraft.entity.player.PlayerEntity;
import net.minecraft.text.Text;
import reborncore.client.gui.builder.GuiBase;
import reborncore.common.powerSystem.PowerSystem;
import reborncore.common.screen.BuiltScreenHandler;
import techreborn.blockentity.storage.energy.idsu.InterdimensionalSUBlockEntity;
public class GuiIDSU extends GuiBase<BuiltScreenHandler> {
InterdimensionalSUBlockEntity idsu;
public GuiIDSU(int syncID, PlayerEntity player, InterdimensionalSUBlockEntity blockEntityIDSU) {
super(player, blockEntityIDSU, blockEntityIDSU.createScreenHandler(syncID, player));
idsu = blockEntityIDSU;
}
@Override
protected void drawBackground(MatrixStack matrixStack, final float f, final int mouseX, final int mouseY) {
super.drawBackground(matrixStack, f, mouseX, mouseY);
final Layer layer = Layer.BACKGROUND;
drawSlot(matrixStack, 62, 45, layer);
drawSlot(matrixStack, 98, 45, layer);
drawArmourSlots(matrixStack, 8, 18, layer);
}
@Override
protected void drawForeground(MatrixStack matrixStack, final int mouseX, final int mouseY) {
super.drawForeground(matrixStack, mouseX, mouseY);
final Layer layer = Layer.FOREGROUND;
matrixStack.push();
matrixStack.scale(0.6f, 0.6f, 1.0f);
Text text = Text.literal(PowerSystem.getLocalizedPowerNoSuffix(idsu.getEnergy()))
.append("/")
.append(PowerSystem.getLocalizedPowerNoSuffix(idsu.getMaxStoredPower()))
.append(" ")
.append(PowerSystem.getDisplayPower().abbreviation);
drawCentredText(matrixStack, text, 35, 0, 58, layer);
matrixStack.pop();
builder.drawMultiEnergyBar(matrixStack, this, 81, 28, (int) idsu.getEnergy(), (int) idsu.getMaxStoredPower(), mouseX, mouseY, 0, layer);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 169
|
THE HOUSE OF DISCARDED DREAMS
Ekaterina Sedia
To Bill and Tait, who made this book possible
Copyright © 2010 by Ekaterina Sedia.
Cover art by Audrejs Pidjaas / Fotolia.
Cover design by Stephen H. Segal.
Ebook design by Neil Clarke
ISBN: 978-1-60701-269-6 (ebook)
ISBN: 978-1-60701-228-3 (trade paperback)
PRIME BOOKS
www.prime-books.com
No portion of this book may be reproduced by any means, mechanical, electronic, or otherwise, without first obtaining the permission of the copyright holder.
For more information, contact Prime Books.
# CONTENTS
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
About the Author
# Chapter 1
Vimbai knew that it was going to be one of those days the moment she shuffled downstairs, her socking feet blindly finding their way on the carpeted steps. Her eyes still half-shut from sleep but her nose already picking up the oily smell of freshly roasted coffee beans, she smiled just as the raised voice of her mother cut into her mind. Vimbai stopped smiling.
Ever since she was a child, she had not liked these days, when her parents fought first thing in the morning and the rest of the day came out all narrow-eyed and lopsided, devoid of the usual sense of balance and rightness in the world. Not that those were ever serious fights—the normal spousal squabbling, Vimbai supposed; nothing bad, and most families had it far worse. And yet these fights made her feel exposed and vulnerable, betrayed in her sanctuary and given to the mercy of strange hostile elements.
She slipped into the kitchen, her eyes wary now, looking from under the lowered eyelids.
"Don't squint," her mother said. "Do you want any breakfast?"
"Just coffee," Vimbai answered, and momentarily envied her mother's accent. The words, the familiar English words that melted and mushed in Vimbai's mouth, came out with startling sparkling edges, as if they were just born, unpolished by the world, rough and fresh and solid.
She sidled up to the table—they always ate their meals at the table, and even breakfast was a family occasion, an extra opportunity to either bond or hurt each other's feelings.
Her mother shook her head but poured Vimbai a large steaming cup. "You have to eat breakfast."
Vimbai's father made a sound in the back of his throat, a mild sound that seemed to serve only to remind them that he was also present and perhaps could offer opinions on breakfasts and other matters but was too absorbed in his thoughts to vocalize them.
Vimbai looked out of the window, at the familiar suburban street and the red leaves of maples that grew in this sandy soil through some miracle of gardening and landscaping. "How are you doing, dad?" Vimbai said. "Long day today?"
He nodded. "Double shift," he said. "You?"
Vimbai pursed her lips and blew on the surface of her coffee, wrinkling it like smooth brown silk. "Three classes today."
"You're coming to school with me?" her mother asked.
"Maybe," Vimbai answered. "If you're not working late again."
"You can go to the library," her mother suggested.
"Or I can take my car and drive home." Vimbai tried to keep her voice neutral—when her parents fought first thing in the morning, it was not wise to annoy her mother. If Vimbai was not careful, shit would go down and both her and her father would get it—not that they didn't deserve it, Vimbai admitted to herself. After all, why shouldn't she get in trouble every now and again?
Mother rose, pushing her chair away with a hair-raising squeak. "Fine. Suit yourself. Carpooling is of course too much trouble and inconvenience. Who cares about global warming anyway?"
"Mom." Vimbai cringed. "Don't be like this."
As Vimbai had grown older, she had realized that the arguments and the problems she had with her mother were not unusual—in fact, she suspected that teenage girls who did not get along with their mothers outnumbered those who did at least three to one. It did not make her feel any better, and she still wished—selfishly, she would be the first to admit—that her mother paid more attention to Vimbai than the news from abroad, or to the Africana studies and who set the agenda there. She wished that she would pay as half as much mind to Vimbai's problems and worries as she did to the white men trying to hijack her department.
Mother shrugged and left, and Vimbai and her father traded looks.
"What brought that on?" Vimbai said.
Her father shook his head. "Everything, darling. Be nice to her—she's having a rough time. Her department and all that. Stress."
"You have stress too." Vimbai drank her coffee, sizing up her father from the corner of her eye. He was always so much more subdued, so willing to make excuses and make peace and sacrifice, always minimizing his own fatigue and heartbreak. It's not important, darling. Such a slight man, his eyes so sad and kind. She did not know how to tell him.
"It's all right," he answered. "You get used to it; you get used to everything."
Vimbai shrugged and drank her coffee, considering all the things she never wanted to get used to; at the same time, the habitual guilt stirred—her parents had been through so much, it felt downright selfish for her to complain about anything at all. And yet, if the experience was all that mattered, wasn't hers just as valuable? All she knew was that she had to get out of here, before she became the same as her mother.
Her father was a nurse down at one of the Camden hospitals, and whenever she visited him or picked him up after work, she felt shamed for her sheltered life, reasonably devoid of suffering. This one wasn't a university hospital, and the emergency room always overflowed with gunshot wounds and overdoses, with beatings and burnings and other godawful things. Vimbai did not know how he could stand it, how it was possible to get used to things like that.
"You seem pensive today," her father said. "Hope we didn't upset you."
"Of course not," she said. "I was just thinking . . . am I getting too old to live at home?"
The words just poured out, mushed by habituality. Her parents never spoke like that, all their words considered, even in the heat of an argument. Even when they fought in Shona, even though she understood little of what they said then.
He put down his newspaper with the picture of Barack Obama on the page folded over. "Why would you say that? You know we don't want you living on your own."
"Just thinking." Vimbai finished her coffee in a few quick gulps. "No reason. Do you think Obama could really win?"
Her father shook his head. None of them thought that he could—the country is not ready, her mother said. He is black and not really American. He was like them, the unsaid words crowded. They would never accept people like us. We are to remain on the cultural margins of multiple worlds, abandoning one and never entering the other. Even Vimbai felt that though she had lived in New Jersey most of her life—she too was on the margins. What hope was there for her parents then, and how would they cope if she was really to move out? And yet, how could she not?
Vimbai decided to skip class. It was that sort of a day, and missing a lecture on invertebrate zoology seemed only fitting. What was there to learn that she couldn't find out by walking along the shore, the dirty hem of foam curling around her bare ankles? She stopped to crouch over a dead horseshoe crab and to stare at it for a while, then to flip it over and count its limp little legs, jointed and pale and slightly obscene. She flipped it back on its belly, as if the dead crab's dignity needed preserving.
The beach was deserted—just gulls and terns circling overhead, waiting for the tide, just sandpipers endlessly chasing after the retreating waves and then running away from them, just the surf and the sky, the tang of October bursting through the iodine smell of seaweed and the ocean to singe the back of Vimbai's throat. Just the wind and the promise of winter, when the beach will be gray and dead, a giant whale flank colonized by silent invisible life under the leaden clouds.
These beaches of the barrier islands lining the eastmost side of the continent like the crook of a mother's elbow had been so good to Vimbai—they nursed her through the first years here, they sustained her through school that had seemed so endless and was now over; they whispered the answer to her when her mother had asked if she considered a college major yet. Marine biology, Vimbai had answered and never lost her temper as her mother lectured that marine biology was not the same as swimming with dolphins or whatever other romantic garbage she thought Vimbai was imagining. Invertebrates, she said, the word that wondrously summed up all the fascinating transparent things that the tide left behind thrashing in tiny pools. I want to study invertebrates. Anything, she wanted to add, but your Africana Studies, anything but that continent you—both of you—carry inside; what was the point in ever leaving if you were going to bring it with you? Instead, she babbled about horseshoe crabs that were declining in number thanks to their use as fishbait and to the pharmaceutical companies who drained their blood to make vaccines.
Oh, the blood draining, the wazimamoto and the colonialism; as much as Vimbai resented the Africana Studies, her mind was its own little storehouse of legends and stories and memories not quite her own, she didn't think—but wazimamoto. The vampire, the white man who came on a medical truck to steal your blood. She learned the story from her Kenyan babysitter, an old woman who was so dark and shrunken she seemed to smolder. And her mother's verbal annotations—Vimbai could never get away from those. And then the books, anything that her mother could find translated, anything African. And yet, Vimbai's alliance was to the horseshoe crabs, the ones who were in real danger of having their blood stolen.
The sand under her feet—bare, her sneakers tied by their shoelaces slung over her shoulder—felt wet and solid, tamped down by the waves. Seaweed and driftwood, the usual refuse of the Atlantic, studded the solid sand surface and Vimbai wandered along, her sharp eyes looking for signs of movement of any critter left behind by the waves. She was skipping class, but let no one say that she did not study.
It was time to return to the car and drive back to school; before the inevitable, Vimbai went for a quick skip across the dunes—the signs and wire fences warned that any such behavior was illegal in the nature preserve, but Vimbai knew that the migratory birds had left already, and the rangers rarely visited the beach in October, so there was no one to witness her impropriety. She ducked under the wire slowly undulating in the wind, and staggered across the warm loose sand that sucked in her feet and stuck to her skin. The thickets of low shrubs and occasional grass patches clung to the sand with admirable, if misguided, determination; the scattering of yellow flowers surprised her—nothing was supposed to be flowering at this time of year, at least in New Jersey. Then her attention snagged on another dead horseshoe crab. It lay among the flowers, belly up, with something bright and white clutched in its stiff little legs.
It was a piece of paper with a fringed edge, of the kind usually sported by homemade ads. None of the pieces with the phone number on it was torn off, though, and Vimbai crouched over the dead crab and its white piece of paper as if it was an exotic chimera composed of animal and inorganic parts. The paper lay blank side up, and Vimbai tried to guess what was on its other side and how it got here. It could've been blown here by the wind, after being torn from whatever wall or bulletin board it had previously graced; it could've been thrown from a window of a passing car, speeding on the way to or from the town, a sleepy place after August but screaming and bustling in the summer, in contrast to the quiet nature preserve of the beach and its environs.
Vimbai wrested the piece of paper from the clasping pincers and turned it over. The ad was handwritten in a generous loopy scrawl, and it took her a moment to decipher what it said. "Roommate wanted for house in the dunes. Own bedroom and bathroom, separate entrance. Very reasonable rent plus one-third utilities. Any pets except fish." And the phone number.
It was true that Vimbai had thought about moving out—but the thought had remained soft and amorphous, hiding in the long creases of her pillow and only surfacing with any determination in that half-asleep state at nights and mornings. The ad had brought the thought into daylight, and as Vimbai walked back to her Saturn parked in the small paved lot off the only road that bisected the island, she thought, why not? House in the dunes and a very reasonable rent sounded quite appealing, and she had never owned any fish. She decided to call as soon as she found herself near a phone with decent reception and away from her parents' superior hearing.
The thought of the house in the dunes was put away as soon as she reached the campus and stopped by her mother's office to say hello and to check on the latest drama. There was always plenty in the Africana Studies, the most current being her mother's threats to complain and quit after the program appointed a white man as a department chair; the said chair busily set about redefining the agenda, and Vimbai's mother would simply not stand for it on general principle.
She was in her office, looking run down even though it wasn't even lunchtime yet.
"You okay?" Vimbai asked. "Sorry you're not having a good day."
"I'm fine." She looked up from the sheaves of paper strewn on her desk, memos and attendance reports and student essays mixed into a terrifying entangled mess that threatened to consume any mortal's sanity with its sheer size and complexity. "Another meeting, and after that I just have to grade."
"Don't work too hard," Vimbai advised.
Her mother only shrugged in response, not bothering to pretend that she would even consider such foolishness. "And you should probably go to your next class."
Vimbai left the office marveling at her mother's ability to sniff out any shirking of one's responsibilities, no matter how otherwise preoccupied she was. And she had been preoccupied—ever since the new department chair, Dr. Bouchard, was appointed, Vimbai's mother seemed to know no rest. Even late at night, she paced the hallway, sometimes muttering to herself in English and Shona; Vimbai could hear her voice through her closed door. All the more reason to move out, Vimbai thought.
She arrived to her class late and slunk to the back, to take sporadic notes of plants' inner workings and to brood. The tubes inside the plants formed neat organized patterns Vimbai enjoyed sketching; it felt almost like doodling rather than studying, and her thoughts flowed along with wavy lines and pooled in quiet oases of shading, neat little areas of cross-hatch pencil strokes.
"This is nice," the girl on Vimbai's left whispered, peering into her notebook.
Vimbai remembered the girl's name—Sarah. They were in a few classes together, and Sarah had irritated Vimbai on several occasions with her pre-med student's obsessive anxiety. "Thanks," Vimbai said with a little stingy smile.
Sarah smiled back, apparently oblivious to Vimbai's disinclination to make friends. It always puzzled Vimbai, this implied certainty some people possessed that their attention could not possibly be an imposition.
Vimbai turned the page and took more thorough notes than usual to indicate that she was not going to participate in any conversations.
Undeterred, Sarah waited for her after class. "Boring, huh?" she said by the way of striking up a chat.
Vimbai shrugged. "I like it. I like anatomy." She took a tentative step away.
Sarah followed, and there was really no good way of escaping her in the long straight hallways, made all the more desolate by the poisonous shade of their green paint. "You have any more classes today?"
Vimbai nodded. "African American Lit," she said.
"Oh," Sarah answered. White kids never knew what to say. "How is it?"
"Why don't you take it and find out?" Vimbai suggested with more vehemence than she felt.
"I don't think it's for me."
"Why not? You know all there is to know about it?"
Sarah shrugged. "I'm just not interested."
Of course she wasn't. Vimbai remembered her mother's frequent complaints that the white kids never took any classes at the Africana studies, that they always assumed that black equaled special interest. As much as Vimbai hated to agree with her mother, she had to in this case. But she didn't argue with Sarah—the fatigue was overwhelming, the sense that she had had this conversation and this argument too many times before. "Whatever," she said. "I have to go."
It wasn't true—her next class did not start until an hour later, but she was not in the mood for explaining herself. Another thing her mother complained about—the constant necessity of explaining oneself, of answering questions. "People are just trying to be nice," Vimbai used to argue when she was much younger. "They're just showing interest."
"Showing interest," her mother had replied, "would be bothering to do some research on their own rather than pestering people with questions. Don't you see? Even when they're nice, they're placing a burden on you. Just wait and see how quickly it gets on your nerves."
Vimbai sighed and headed for the library—it was usually empty during the lunchtime, and in the stacks it might be easy to avoid Sarah or any other overly talkative classmates who would be eager to burden her with their interest or socializing.
The library was located in the new building, adjacent to the science labs. It had tall narrow windows running all the way from the high ceiling to the tiled floors, and Vimbai liked the way sunlight striped the stacks, while others hid in the shadows, light and dark interspersed in regular narrow slats. She headed for the shelves draped in soft shadow, meandered between then into the unexplored library depths hiding reference materials—newspapers from the sixties and the seventies, artifacts no intrepid explorer would be likely to sift through—and sat on the floor, her back resting comfortably against the cloth-bound sheaves of papers. The air smelled of dust and air-freshener, mixed with Vimbai's own scent of warm skin and salt, and she curled up in this quiet welcoming ambience.
Unlikely to disturb anyone, she dug through her book bag and found her cell and the crumpled sheet of paper from the dunes. She dialed the number and almost chickened out and hung up when the female voice said "Hello."
"Hello," Vimbai answered, keeping her voice low out of the old library habit. "I'm calling about the house . . . in the dunes."
"We still have a room," the woman said brightly. "The rent is two hundred bucks a month, and you will share with myself and Felix—he has the third bedroom. Interested?"
"I'd like to see it first," Vimbai said.
"Come by tomorrow," the woman said, and dictated the address.
Vimbai wrote it down and promised to stop by.
That night, she dreamt of sea and whales. The whales floated on the silvery ocean surface like balloons, and water from their blowholes rose and fell like the fountains in Longwood Gardens. The whales sang in surprisingly soft voices, a rhymed children's song Vimbai could not remember when she woke up; but as the dream retreated, she kept smiling—the whales were a good omen.
# Chapter 2
No wonder the rent was so cheap—the house was in a woeful state of disrepair, its wooden siding bleached by the ocean winds to the color and consistency of driftwood; the street on which it nominally stood proved to be a cul-de-sac, almost concealed by the sand blown off the dunes that surrounded the house like waves. The surf pounded the beach nearby, and Vimbai suspected that the house wasn't condemned only because of pure oversight, since it clearly violated several zoning laws and the next good storm would likely flood it. Still, she could not deny that she was thoroughly charmed.
She lingered for a while on the porch, cracks between the boards wide and gnarled like fissures in dry clay. She thought she caught a palimpsest of motion in the shadows under the porch, a quick shift of light and a change in the quality of the cool dusk. Some wildlife was bound to nest there, and for no good reason Vimbai hoped for a den of the tiny dwarf foxes that still lingered in the barrier islands, despite the constant expansion of the tourist towns and vacation homes. The foxes who begged by the roads, their red tongues teasing and wet between their sharp teeth; a whole nest of tiny pups, Vimbai imagined, cuddled together in the somber secretive darkness under the porch.
"Do you want to see the rest of the house, or are you content with the crap under the porch?" a female voice said next to her.
Vimbai straightened, smiling. "I thought I saw something under there."
The girl who stared back at her smiled too, then laughed. "Of course you did." She was taller than Vimbai, and gave off an air of good health and clean strength. She wore a somewhat unseasonable yellow tank top and bleached cutoffs that exhibited her long strong legs to great advantage. She shook hands with Vimbai. "I'm Maya. We talked on the phone."
Vimbai nodded, her fingers trying to hold their own in Maya's strong grip. She shook hands like someone who liked to show strength from the beginning, but Vimbai did not think her threatening. If anything, Maya reminded Vimbai of herself, in her need to establish dominance from the start. So she squeezed back as hard as she could. "I'd like to see the rest of the place, if you don't mind."
Maya smiled more, released her grip, and turned away giving Vimbai a chance to wince and mouth 'ow' while shaking her hand.
Maya motioned for Vimbai to follow, and stepped through the banged-up screen door much molested by the elements into the kitchen that bore traces of recent but unthorough cleaning. A few plates dripped in the rack, and the linoleum floor shone with fresh traces of water. The windows let through the pale light, and its diluted quality testified to the fact that the panes had not been washed in a long while. Formica counters, bottles with bleach, and a patchy geriatric refrigerator sighing in the corner.
"It's modest," Maya said, noticing the trajectory of Vimbai's gaze, "but it works."
Vimbai nodded and followed Maya to the sunroom or perhaps the den—there was a TV and a concave couch, which at the moment cradled the languid form of a very young and very white man who Vimbai presumed was the second housemate.
"This is Felix," Maya said. "He's quiet, so pay him no mind."
Felix offered no opinion on the matter, and Vimbai dutifully turned her gaze to a couple of mismatched chairs that huddled by the wall, as if not quite believing their luck in having been rescued off a street corner on a garbage pick-up day, and a stern wooden table, covered in slicks shaped like pizza boxes. Good student living, familiar from the visits to Vimbai's study group buddies off campus. A sense of hastily put together and transitory space, with a modicum of effort to make it one's own and yet not to get attached. A ficus slowly dying in its way-too-small pot by the window where there wasn't enough light.
"The rest is straightforward," Maya said. "Bedrooms are upstairs, and then there're bathrooms and closets and shit. You want to see it, or do you want a beer?"
Honesty born out of living at home for all her life, under her mother's hawk-like gaze, compelled Vimbai to say, "I'm not twenty-one yet."
Maya shrugged. "I don't card. Don't worry, if you're in no shape to drive, I'll tell you."
"Okay then," Vimbai said. "I guess I want a beer," and only then realized that she had forfeited her right to the rest of the tour.
"I want a beer," Felix said from his trough on the couch, miraculously brought to life by a single phrase.
This outburst of verbosity encouraged Vimbai to give him a closer examination. First thing about Felix that she—or anyone, for that matter—noticed was his hair. It wasn't merely long or big; it undulated. The color of it was darker than black, a pure absence of light, so dense that no individual strands were visible. Occasionally this alarming hair reared up like tongues of flame, and then ebbed, calmed, and returned to its peaceful slow and hypnotic movement.
"I know," Maya said. "It's like a fucking lava lamp." She had returned from the kitchen bearing three golden long-necked bottles, and handed one to Vimbai. "It's even better with beer."
Felix sat up and extended both hands to take his, a motion that Vimbai found childlike, almost animal-like. For the first time, Vimbai got an unobscured look at his features.
Felix could've worked as a model for a Raphaelite painter specializing in cherubs—he had smooth porcelain skin and a small perfect mouth that seemed painted on—if it weren't for his eyes. Gigantic and fierce, with jaundiced whites streaked strongly with fat red capillaries, they rolled in his head with quiet fury, quite independently from one another.
Vimbai took a long swallow of her beer. She was aware that staring like this was impolite, but there was just no helping it when faced with Felix; fortunately, he seemed to neither mind nor notice—although the exact direction of his gaze was impossible to determine.
Maya pushed her gently toward one of the armchairs, and took the other. They did not speak a while—Vimbai staring, Felix preoccupied with his beverage, and Maya apparently giving Vimbai a chance to decide whether Felix was a sight she was willing to behold daily. Maya sat in the armchair that used to be burgundy, but currently hesitated between pink and gray; the original color survived only in the piping of the armrest over which she slung her legs carelessly, showing Vimbai the pink soles of her bare feet. Her black curly hair had been freed from the scrunchie that had held it together, and sprang up like a halo to rival Felix's.
Vimbai smiled and took another pull. "I like it here," she said. "It's a nice place."
Maya nodded. "And I," she said, "I would like my other roommate to be a girl, and a black girl at that. His pale ass," she motioned at Felix, "is plenty for me."
Felix grinned and bobbed his head, as if acknowledging a compliment. "There are forces in the world," he said cheerfully, and drank.
"Yeah yeah yeah." Maya waved her hand in the air. "Whatever." She turned her attention to Vimbai, a smile hiding in the plump corners of her mouth ready to come out as soon as Vimbai gave a signal. "So, what do you think?"
Vimbai shrugged and nodded, and found that her tongue had grown fat and lazy. She had had alcohol before, and half a beer never had this kind of an effect on her—no, it was this house, the languorous strangeness that colored the air despite the mundane furnishings; it was Felix and the black hole of his hair, Maya's sharp gaze and quick speech. The house in the dunes pulled her in, and she imagined herself sinking all the way, deep beneath the waves of sand where it was quiet and golden, blue shadows of the trees above dancing—or perhaps it was the ocean with its still forests of kelp and bivalve shells scattered about on the wavy sand bottom, shells empty like open hands. She imagined picking up one of these shells and whispering into it, her eyes closed, her weak hand pressing the pearly concave surface to her lips. Another shell to her ear, whispering in the susurrus of the sea, talking in monotone, come back, come back, baby, come back home.
And then her own lips, her slow tongue shaping words like stubborn clay, I'm sorry, mama, I'm so sorry. The sea between the shells a distance, pounding of the surf, the impossible separation by many tons of dense and cold water. Two continents, too far apart to ever hope for reconciliation.
Vimbai pressed the phone receiver to her ear, the voice of her mother so far away, so defeated and alone. "Mom?" Vimbai whispered.
Only the static of the ocean answered, the empty static in the shell of the phone like a small ghost trapped in the wires.
It came over Vimbai whenever she stayed in the house long enough. Being trapped in amber, in ocean water, in time, in distance, suspended and separated finally from everything in the world. She used to dread her mother's reaction, what she would say if Vimbai decided to move away from home; more than that, she feared her father's resigned and unconditional support. Now, she could only whisper faintly into the phone, her lips salty and barely moving, I'm sorry, mama, I'm so sorry. She did not let them help her move; her separation, this carving herself off from the rest of her family, had begun.
Maya blamed the strange effects of the house on Felix, on the gravitational pull of his hair; Felix did not argue. Vimbai thought that it was the dunes, the underwater singing of the horseshoe crabs buried in the sand for the winter; the shifting of sand, the lapping of the waves, the eroding processes that ate away at everything, that made land part of the sea and carried the sea over land, the same forces that pulled Vimbai away from everything her parents were. At night, she listened to the whistle of the wind in the rigging of the old house and its creaking moans, the lapping of the tides, unable to sleep. And so it went.
Vimbai liked to sit in the kitchen in the morning—she made coffee and waited for Maya to come downstairs. Maya, always fascinating and evasive, a strange thing in herself, something that needed to be puzzled out and unraveled. Even though Vimbai was not sure why she felt that it was her job to unravel this enigma wrapped in a striped bath robe, she looked forward to the moment when Maya stumbled downstairs, her eyes half-closed and her nostrils flared in anticipation of the hot, clear coffee; there seemed to be few things in life Maya enjoyed more than that first cup of coffee in the morning.
"Good morning," Maya said and poured herself a cup. "Thanks for making coffee—before you moved in, I was the one making it. Felix always sleeps late."
"Sure thing," Vimbai said. "I enjoy making it—I get up early anyway."
Maya made a face. "Whatever possesses you to commit such silliness?"
Vimbai considered the question she wanted to ask and then discarded it—there was simply no polite way of asking Maya about the way she spoke, about her carefully cultivated non-regional accent, without sounding offensive. She sighed and gave up on the idea—her mother was right: Vimbai, even though she was born and raised in New Jersey, was still a foreigner to most African-Americans, oblivious as she was to fine distinctions of speech patterns and code-switching. She was informed that she was not getting it when she was still in high school, and she was ashamed to admit that she had made little progress in the matter.
Instead, Vimbai poured herself another cup of coffee. "How do you like working in Atlantic City?" she asked.
Maya barked a short strained laugh. "What's not to like? Casinos surrounded by a ghetto. Land of contrasts, as it were. Plus, it's a good place to bartend, really—men are too preoccupied with gambling to hit on you. Which is, you know, a good thing. Like Martha Stewart."
"I've been at the casinos a few times," Vimbai said. "With my mom, mostly. She does some research there—her specialty is urban folklore, and there's a ton of it in Atlantic City."
"But not in the casinos."
"No. We went there for the buffets."
Maya laughed. "Oh my god. Those are such freak shows."
Vimbai's upbringing urged her to argue, to insist that all people deserved a claim to dignity and respect, and ought not to be called freaks. But she remembered these pale and lumbering shapes, their faces slack and remote, their eyes permanently dilated in the artificial semi-darkness. They seemed to live in the casinos—at least, Vimbai had never seen them anywhere else; they seemed shy underground dwellers, sliding softly through their habitual dusk with white porcelain plates heaped high with pasta salad and ribs, their only break from the life of sitting on a high stool and pulling a lever and putting shiny coins into a large Styrofoam cup, their lives augured by the fast-spinning cherries and lemons in tiny transparent windows.
"I know what you mean," she finally said. "Are the bars any better? I bet you have stories."
"You bet right," Maya said. "See, the casino bar is a great place—people come there when they are not gambling or eating, and that usually happens when they just lost a shitload of money, and cannot gamble anymore but are afraid to go back home. Some celebrate when they win, some are just there to hang out, you know? But it's always the losers who are interesting. This is why I remember them the most, I guess."
"Oh?" Vimbai smiled and refilled Maya's cup. This solicitousness felt natural to her, warm. "What's so interesting about them?"
Maya patted Vimbai's hand in gratitude, making her blush a bit. "I don't really know, but I guess this is when people are . . . honest, I guess. They know they've been beaten, and they are out of tricks for a while—they really know that they are fucked. And yet, there's this thing when they try to tell themselves that it's not that big of a deal. I don't know how to explain it, but it's like if they cannot lie to themselves about what happened, they start diminishing its importance. When they are honest, they almost have to be deluded, you know?"
Vimbai considered. "I'm kind of getting it, I guess." She wasn't really sure that she was getting anything, but she wanted Maya to like her so badly. Vimbai suspected that the spell of the house that lulled her so much tried to tie her not only to the house, but also to its inhabitants. Otherwise there didn't seem to be a reason for her to feel so invested in what Maya thought of her.
Vimbai attended classes, dutiful but disengaged, caught in the slow molasses of movement of time and the sucking embrace of gravity. The world came through muffled, and only the house and the dunes and the ocean remained real. Winter was coming, and there was a first dusting of miserly snowflakes scattered almost invisible on the frozen sand one morning in November.
That day, Vimbai stepped onto the hoary porch and saw that the very character of the dunes had been transformed—they lost their fluid, mutable aspect and even though they remained the same in appearance they now stood motionless, seized by the ice within, trapped into immobility.
Vimbai's bare toes curled instinctively, cringing away from contact with the cold boards of the porch (which, as her investigations had shown, harbored no nests of adorable foxes). She hugged her shoulders and stared at the leaden water, visible between the dunes, barely puckered by waves. Her fingertips grew numb, and the hairs inside her nose grew stiff with frost, singed with the smell of ozone. Still, Vimbai lingered in her robe, thinking of her mother—the first serious frost always put Vimbai in that frame of mind. As long as she could remember, it was the time when her mother grew pensive and quiet, and when pushed given to reminiscence. It was in November that Vimbai's parents left their home and came to the U.S.
Vimbai strained to see over the water—it just seemed impossible that the entire continent could be hidden by the curving razorblade cut of the horizon, bleeding now the first red streaks of dawn. Her breath formed tight white clumps in the air, like the memories of the still invisible clouds overhead.
Her mother had to regret something—and Vimbai suspected ever since she was little that her mother still, twenty years later, was not convinced that she had made the right decision. How could one know something like that, how could one not agonize over how life would've turned out if one had made different choices? Even Vimbai, with her sheltered existence and precious few choices with any consequences, wondered. Those were small things, insignificant perhaps, but she wished sometimes that she had chosen differently.
She breathed open-mouthed on her fingers, numb and discolored by cold, and thought about that kid, the little ten-year-old whose name she never learned. She was in high school then, old enough to largely ignore the kids playing in the elementary-school yard she passed on her way to classes. She walked alone, absorbed in her thoughts, and paid no mind to the persistent cries emanating from the schoolyard. The word that jettisoned her out of her preoccupation was 'lion'—not the sort of thing one heard often under such circumstances.
"Go hunt a lion," a largish and very pink boy shouted. "Go back to Africa."
Vimbai stopped and stared at the small black kid in ill-fitting white shirt and khaki shorts, backed up against a set of monkey bars. A few other children surrounded him in a tentative semicircle, not quite backing up the assailant but not dissuading him either. Non-committal, waiting to see how things shook out. Little vultures.
The small kid said nothing and just swallowed often and hard, as if trying to dislodge the words stuck in his throat.
The pink boy advanced half a step, and the semicircle drew up on itself tighter, the kids smelling blood now, just a moment away from taking part.
"Leave him alone," Vimbai said.
The pink kid turned to look at her; she still remembered the expression of contempt in his eyes. Without saying a word, he returned his attention to the cornered kid in the white shirt. "Go hunt a lion," he said again, with rather more force, as if challenging Vimbai to climb the fence and kick his plump behind.
Vimbai looked at her watch; she was already running late, and kids did this sort of thing all the time. "Stop it." She raised her voice to be heard over the rising hum of the other voices that had decided to join in.
Her stomach had ached when she turned and walked away.
In her darker moments, like that day watching the cold ocean over the frozen dunes, she wondered if she somehow upset her karmic balance that day, if everything that ever went wrong since then was the result of her failure. She had wished she would see this kid again, but no matter how many times she walked past the elementary school, he was not there.
Vimbai winced at the pain in her feet the moment she shifted her weight, and she hobbled inside, trying to remember how long it took for frostbite to develop. "Not clever," she mumbled, "not clever at all."
She decided to call her parents, just to tell them that she remembered what was important to them, and that she cared. Her roommates still slept, given to late hours and disorganized lifestyle; Vimbai would have disapproved if it didn't mean that in the morning she had the house all to herself. She walked in slow mincing steps, letting the sensation and accompanying pain revitalize her toes, to the phone—an almost extinct rotary affair, gleaming with slick black curves and the soft creamy ivory of the rotating disk. She picked up the receiver and listened for a while; she was puzzled by the static that inhabited the wires of the phone—it seemed haunted, like the rest of the house, alive with blurred disembodied whispers, and Vimbai thought that if only she listened carefully enough, she would be able to discern the words and the sobbing laments of the little ghost.
The static ceased just as she hovered on the brink of understanding, and the phone beeped and inquired whether she needed assistance from the operator. She sighed and dialed the number.
And once again it was as in a dream, with slow cloying molasses weighing her eyelids and her lips, as she whispered that she was sorry and that she loved them.
"Vimbai, are you all right?" her mother said. "You always sound so tired. Are you getting enough sleep? Are you staying up late?"
"No," Vimbai said, and then, "yes."
"Vimbai..."
"I am getting enough sleep. I'm not staying up late. I just miss you."
Her mother remained quiet for a while. "You can always come back home," she finally said.
"I can't. I have a lease."
"At least, you can visit. How's Saturday for you? I'm making stew."
These words coaxed a smile—Vimbai was unreasonably attached to the bland beef stew and rice, the food so generic it could be hardly counted as traditional. "Okay," she said. "I can make Saturday."
"Good," Vimbai's mother said. "It is decided then."
And then her voice faded, and the ghost in the wires spoke—clearly, for the first time.
Vimbai was not sure how much time had passed—she slumped on the floor, her frozen feet forgotten, the receiver pressed hard to her ear, listening to the stumbling, simpering words that poured out. She did not dare to ask any questions for fear of the ghost in the phone falling silent, spooked away by the fleshy human voice. So she let it talk, clutching the receiver with desperate force, afraid to loosen her grip and let go of the mystery inside it.
The ghost was not a ghost at all, or so it claimed—it claimed to be a psychic energy baby, birthed in some ethereal dimension, and pulled into the phone by the powerful magnetism of phone signals. It remembered with perfect clarity how it came to be—remembered coalescing from the reflecting membranous surface of the world, streaked with reflected light, humming with surface tension under the pressure of emptiness underneath. The Psychic Energy Baby found form among the emanations of people's minds and the susurrus of their voices, it found flesh in the shapes their lips and eyes made, the surprise of 'o's and the sibilations of 's's; its skin stretched taut like a soap bubble, forged from the wet sound of lips touching; its thoughts were the musky smells and the breath of fresh bread. Its fingers spread like ribcages, and its nerves twined around the transparent water balloons of the muscles like stems of toadflax, searching restlessly for every available crevice, stretching along cold rough surfaces. Its veins, tiny rivers, pumped heartbeats striking in unison, the dry dallying of billions of ventricular contractions. And it spoke, spoke endlessly, it spoke words that tasted of dark air and formic acid. It could speak long before it took its final shape.
And when it happened, when all the sounds and smells and words in the world, when all the thoughts had aligned so that it could become—then it found itself pulled into the wires, surrounded by taut copper and green and red and yellow insulation; twined and quartered among the cables, rent open by millions of voices that shouted and whispered and pleaded and threatened, interspersed with the rasping of breaths and tearing laughter. It traveled through the crisscrossing of the wires so fast that it felt itself being pulled into a needle, head spearing into the future while its feet infinitely receded into the past, until it came into a dark quiet pool of the black rotary phone, where it could reassemble itself and take stock.
When Maya woke up and came downstairs, she found Vimbai still sitting on the floor in her robe, the silent receiver in her hand, her face buried in her knees and her shoulders shaking with sobs—not grief-stricken but merely shaken and amazed beyond words.
To Vimbai's surprise and gratitude too deep for words, Maya was neither skeptical nor disbelieving when she heard the tale of the Psychic Energy Baby. "It happens," she said. "Don't you have classes to go to?" Maya's shift at the casino's bar did not start until eight p.m., and she left the house late.
Vimbai shrugged. "Who cares," she said. "There's that thing in our phone. I think it wants to get out."
"Of course it does," Maya said, her rich voice acquiring a soothing tone as if speaking to a cranky child. "Don't worry, we'll get it out. Just as soon as Felix wakes up. Come on, I'll make coffee."
Vimbai sat at the kitchen table as Maya went through the ritual of brewing coffee. They didn't bother with grinding whole beans, and Vimbai was getting used to the taste of coffee that came out of the can or a more fancy bagged variety—when it was Vimbai's turn to shop, she went for shade-grown and fair trade, more out of habit than any conscious choice. This is what her mother always bought. The clinking of the carafe and hissing of steam, the smell of coffee felt comforting, and with every passing minute Vimbai was more and more willing to believe that the Psychic Energy Baby was just a product of fatigue, cold, and bad reception.
The coffee bubbled and poured in a fragrant stream, and Maya sat down. "This ought to wake Felix up," she said. "He'll get that baby out of those wires. Poor thing."
"How?" Vimbai said. "What is Felix going to do?"
"What he always does," Maya answered. "You don't think he earns rent money by sitting around all day, do you?"
"I don't know what he does," Vimbai answered, and poured herself a cup. It warmed her hands and instilled a sense of serenity.
"Well, I'll tell you. He's a freelancer. Only what he does, no one else can. He separates things."
"Oh?"
Maya laughed and drank her coffee. "Things you can't see, like that baby in the phone. Felix says, they sometimes contaminate the things you can see, or the other way around."
"People pay him for it?"
Maya nodded. "Uh-huh."
"Like exorcisms?"
"Not those, the Catholics do them. Felix does more simple stuff. Like junkies with invisible insects under their skin, or amputees with phantom limbs."
"He amputates phantom limbs?"
"I suppose he does. In any case, we'll see what he can do, huh?"
Vimbai nodded. Somehow, the fact that Felix had an unusual occupation was easy to accept, and once accepted, any strange occupation seemed as reasonable as the next one. So if Felix made a living untangling the invisible babies out of the phone wires, what business it was of Vimbai's? Who was she to judge? She felt only intense curiosity, and the weakest pang of guilt for missing her classes.
# Chapter 3
Felix stumbled downstairs just before noon. His terrible eyes were mercifully closed, and his hair hung into his face in tangled clumps. Vimbai gasped—the long strands didn't just obscure parts of his forehead, but rather seemed to consume them entirely. His face seemed streaked by darkness, fractured like a tiger hidden in shadows. Her encounter with the Psychic Energy Baby had jolted her enough to realize that what she assumed was hair—had no other option, really, but to assume that—was a conglomeration of darkness, of absence of light; a black hole, emptiness of outer space, a jagged nothingness. It spilled over Felix's face, threatening to consume it and retreating when he tossed his head and smiled at Maya.
"Can I have some coffee?" he said.
"Of course," Maya said. "Help yourself."
Felix raked his insane hair out of his eyes, and his hands disappeared in blackness up to their wrists; he extricated them somewhat hastily, and his left eye rolled to look upward with a troubled expression.
Vimbai tried to think of a question to ask, but came short. She could only round her eyes and shrug at Maya.
"Felix," Maya said the moment Felix took his seat by the table. "Vimbai found a ghost in the phone wires, think you can get it out?"
"It depends," Felix said and winced at the too-hot coffee. "Does it want to come out?"
"I think so," Maya said. "Well?"
Felix blew into his mug. "What kind of ghost is it?"
Vimbai finally found her voice. "It's not really a ghost," she said. "It's a psychic energy baby."
"Did you Google it?" Felix asked. "Don't think I ever heard of one."
Vimbai brought her laptop downstairs, but the results were disappointing. Psychic energy baby turned out to be one of the very few things Google had no insight on.
"All right," Felix said. "It's in the phone now? I guess I'd better take a look."
It was the most words he had said since Vimbai moved in; in fact, he sounded remarkably coherent. It prompted Vimbai to blurt, "This is not really hair, is it?"
"No," Felix admitted. "I'll explain some other time."
Maya and Vimbai followed Felix to the hallway where the phone huddled, forlorn, on its dusty shelf. Felix picked up the receiver and listened for a while, his bloodshot eyes rotating quietly in their sockets in opposite directions; Vimbai found that he looked thoughtful.
"Just static," Felix said.
Vimbai sighed. "Just listen."
Felix did. He listened for a long while, slouched against the wall, and the thoughtfulness started giving way to boredom, but then there was crackling in the receiver, and he startled upright. "That's a Psychic Energy Baby all right," he said, covering the mouthpiece with his hand.
"Will it ever grow up?" Maya asked, and bit her fingernails in excitement. "Will it be a Psychic Energy Adult?"
"How should I know?" Felix glared a little, both of his eyes managing to simultaneously focus on Maya. "The thing is unGoogleable."
Vimbai cleared her throat. Her head swam, and she felt as if in a dream, able to do and say anything. "Seriously, what's with the hair?" she said.
Felix shrugged. "Not now." He grunted and picked up the phone, leaving the phone jack connected so as not to lose touch with the Psychic Energy Baby. Both Vimbai and Maya held their breath and each other's hands; Vimbai imagined that participants in a spiritualist séance would feel the same mix of disbelief, giddiness and fear lurking just under the surface as they did right now, watching Felix work his magic surrounded by peeling wallpaper and creaking floorboards, a black rotary phone the focus of his attention.
Felix thrust the phone into his hair; Vimbai whimpered a bit as the entire squat plastic box plus its dangling cord and the receiver were swallowed by the darkness. There was no way for it to fit—there was no way Felix could thrust his arm into his hair almost all the way to his shoulder, as narrow tongues of emptiness licked it, trying to pull it in. For a moment, Vimbai imagined Felix being sucked into the black hole of his hair and disappearing in a recursive black dot, but he managed to pull away, his hand still gripping the phone.
"That ought to do it," Felix said. "I hope."
Maya reached for the receiver and listened to the recorded incantation that suggested dialing 0 for the operator. "It's gone," Maya announced. "Where is it?"
"It's in my hair," Felix answered, as if referring to a moth or some other harmless but annoying insect. "Hold on."
Now both of his hands disappeared into his hair up to the elbows, and moved about energetically. Vimbai though that he looked like a man reaching for something slimy and nasty in the garbage disposal—his face acquired the same apprehensive expression as it did every time he had to touch his hair. Vimbai used to think that her own hair was unruly, but it didn't even come close to the existential horror of Felix's.
He finally grabbed a hold of something and pulled—judging from his wincing and the restless motion of his hands kneading some invisible dough, that something was either slippery or reluctant or both. A few times his arms were pulled back in and struggled out again, the resisting prize still hidden from view.
There was a shriek and a wail, and Felix grunted as he pulled out the wriggling shape.
"Yep," Maya said. "It's a Psychic Energy Baby all right."
Vimbai had not visited Felix's room before—in fact, she used to avoid thinking about Felix, because even after she had lived in the house for a while, she found that he didn't quite fit into her usual thinking patterns. He was the oddly shaped piece of a jigsaw puzzle that didn't seem to belong anywhere, and probably had tumbled here from some other, entirely different set, but there he was.
And there she was, following him and Maya up the stairs to the small bedroom at the end of the corridor. A not very mature "Keep Out—High Voltage" sign guarded the door. This is where they carried the Psychic Energy Baby; it wailed, distraught, and struggled and seemed to resent the unfamiliar surroundings and the lack of the binding (yet directing) phone wires. Vimbai half-regretted ever bringing it to Maya's attention.
Felix's bedroom was surprisingly clean and tidy—the bed neatly made, books on the shelves, the desk amazingly free of piles of paper and stray objects such as found their way onto Vimbai's. The only thing that was out of ordinary was the row of phantom limbs lined against the wall—there were hundreds of feet and legs and hands and arms, all cast in the same transparent substance as the Psychic Energy Baby, visible only by the curving of the reflected light stretched taut like a soap bubble.
"They are all . . . yours?" Vimbai said.
Felix nodded, his eyes rolling in rhythm with the bobbing of his head. "Well, they used to be somebody else's. But once you detach a phantom leg or arm, the owners don't want them. So I keep them—not like I can throw them away."
"How do you . . . " Vimbai posed, thinking of ways to better formulate the question. After some hemming, she gave up any hope of sophistication, and hoped only for coherence. "How do you do these things?" she pointed at the limbs and the baby that still lay transparent, cradled against Felix's narrow chest.
Felix seemed to have only a tentative hold of the ways in which his hair—or a small universe that orbited the dome of his skull, whatever one wanted to call it—worked; he understood it only enough to exploit it. The universe which he explored like a blind man would, by touch alone, contained primarily clean socks, a few household objects, and a desiccated head or two (he promised to explain the heads later as well). Also, it seemed to work as a prism of sorts—except that if a prism could split a beam of light into its component wavelengths, Felix's hair split any entangled objects into their components, be they material, spiritual, or both. Felix discovered it by accident, when he was quite young—a neighbor's kitten crawled into his 'do, and got separated from its voice—the disembodied meowing haunted the house until they moved.
"So you could separate a person from their soul," Vimbai said.
"If you believe in souls," Felix answered. "I suppose. But then the person would be dead, wouldn't it?"
"Oh for crying out loud," Maya interrupted. "Yes, Felix, your hair is a deadly weapon. Now, do something about this baby."
"Like what?" Felix said, and sat on his bed, the baby sniffling and waving its transparent limbs in his lap.
Vimbai reached for the apparition. To her surprise, the baby had some heft—not as heavy as a regular baby would be, but it felt as a being of substance. It cried some more.
"There there," Vimbai said. "You can talk, can't you? Tell us what you want now."
The Psychic Energy Baby (or Peb, as Vimbai mentally abbreviated it) stopped crying. "It was a dark and terrible place," it said in a blur of a voice, barely louder than a sigh.
"What, the phone or his hair?" Maya asked.
The baby pointed at Felix, and its lower lip, itself reminiscent of a bubble of spit, trembled. "Something held me there," it said. "It is not a good place."
"I bet," Vimbai murmured. "Now that you're here, what do you want to do?"
"I don't know yet," Peb said. "But I'm not going back into the dark, neither wires nor him."
"Fine," Vimbai said. "Can you move on your own?" It did look awfully tiny and insubstantial.
Peb could—it turned out, it could walk or float, and walls and floors did not baffle it or contain its movement. It started investigating Felix's room by sinking into the floor halfway, so just the transparent torso moved about, looking under the bed, until it finally crossed through the wall and disappeared from view. Vimbai and Maya sighed simultaneously.
"I don't suppose it will pay rent," Maya said. "It probably has no money."
"It doesn't have any pockets," Vimbai agreed. "We should tell it to keep away from the bathrooms when we're using them."
"Phantom limbs are so much easier," Felix said. "At least they stay put."
"And look creepy," Maya added.
Felix huffed. "And the Psychic Energy Baby that wanders through the walls at all hours is not creepy?"
"Not very," Maya said. "At least, it looks alive."
"It looks like a ghost," Vimbai said. "Only I don't believe in ghosts." It wasn't entirely untrue—Vimbai was never superstitious, and when she examined her belief system, she discovered that it was not sufficiently undermined to admit the possibility of ghosts. Or it was mere inertia, because if psychic energy babies indeed lived in phone wires and god knows what other hidden places, than there was no reason for ghosts not to exist either. And after all, weren't phantom limbs also ghosts of a sort? "Do you mind if I look at your . . . the phantom limbs?" she said out loud.
"Help yourself," Felix said. "Feel free to take any you like—I don't really need them; only it doesn't feel right to throw them away."
Vimbai studied the limbs—smooth like blown glass, with the same sleek appearance, they seemed mannequins, although no mannequin had ever exhibited that many purely human imperfections and malformations. There were deformed nails, ingrown hairs, bones too visible just under the soap skin. There were hammertoes and hitchhiker's thumbs, varicose veins, barely healed razor cuts and an occasional pimple or a scar. She touched one leg, cut off just below the knee, and almost jumped at the sensation of cool smooth and—most importantly—solid form under her fingertips, at the subtle humming of electricity just under the imaginary skin.
She didn't know what she wanted with a phantom limb, but she carefully picked up the half-leg and carried it to her room. It fit nicely by the window, next to the space heater. There was a cold stab of draft coming from the window where the frame didn't quite touch the wall, and Vimbai turned the heater on, letting its pink glow fall on the convex surface of the phantom calf. She sat at her desk and looked outside, where the leaden hem of the surf nipped at the frozen shore, and listened to the quiet rustling of the Psychic Energy Baby exploring the creaky old house.
Saturday came, and Vimbai drove reluctantly home. The street—so quiet on this cold day, so helplessly suburban—already felt alien. Like in a dream, the sidewalk familiar down to every crack and pockmark, the leafless peach trees in the front yard, the woven mat on the steps were just as she remembered them, seen clearly through the fisheye lens of separation. This is what coming home feels like, Vimbai thought, this is how her parents feel when they go to visit relatives in Harare—only even more so, their time and distance greater hundred-fold, thousand-fold than Vimbai's.
When she came in, she realized that her parents' house smelled of clean linen and a faint whiff of vanilla and nutmeg—something she never noticed when she lived here. She was separate from it now, separate enough to notice its smell. Separate enough to look at the kitchen table and admire the gleaming of white bowls in the slanted pale winter sunlight that poured into the kitchen through a large bay window. The things she had never noticed before, but now suddenly did.
At dinner, her parents talked the familiar talk—the department and the hospital, Africana studies and Zimbabwe politics. So Vimbai kept to her own thoughts and ate, rarely lifting her gaze off her plate. It was so easy to fall back into this pattern.
Vimbai's mother still complained about the new head of Africana Studies. "And he also said just the other day that Mugabe is the worst thing that ever happened to Zimbabwe. I told him that colonialism was really up there among the shitty things."
"But you hate Mugabe," Vimbai's father said mildly. "Why are you defending him?"
"I'm not," mother said. "I'm just sick and tired of hearing about African corruption. Sick and tired."
Vimbai made a small noise of sympathy. One of the things she had learned from her mother was that one did not disparage one's people or culture in front of outsiders. It's different for them, her mother said. They don't know what it's like, they have no sympathy, no kinship. They look and they criticize, they look for cracks, they look for proof of something they are already thinking in their hearts—that we are worse than them, that we should not be allowed to govern ourselves. So you argue and you don't show weakness. And you don't ever, ever agree with them if they speak poorly of your people. What if they are right, Vimbai had asked then. They are never right, her mother answered. They may appear to be right because of the words they use, but their hearts are wrong. To be right, you need to know, to understand, to have a kinship of spirit.
"I do hate what he did to the country though," father said. It wasn't news, and Vimbai nodded along, as one would to a familiar tune. This one was called 'The Land Reform'. Whatever they said, it always betrayed the Africa inside of them.
Vimbai ate her stew, the beef boiled flavorless and the rice—flavorless to begin with. She had nothing to contribute. Even though she knew the issues, she never felt them deep in her bones, resonating through the drum that was the internal Africa. She cringed at the sudden fear that one day soon her mother would be defending Mugabe and his cabinet from her, from Vimbai—and she thought that really, that was the price of growing up, cutting away the tenuous umbilicus that still attached her to her parents and, by extension, to the Africa within them. And soon she will have to find her own place in the world, somewhere in the dunes and the ocean, among the horseshoe crabs and phantom limbs and psychic energy babies.
Vimbai watched the scar on her mother's forehead—an almost invisible white line, so thin you wouldn't notice it unless you knew it was there. Vimbai knew. She remembered when her mother first showed it to her, along with the similar marks cut into her wrists and her ankles. Vimbai's father had more prominent scars, symmetrically bisecting his cheeks. Muti, Vimbai's mother said. When I was a little girl, my mother took me to a n'anga, a healer, and he put these marks on me for protection.
Vimbai used to have nightmares for months afterward, dreaming of a man with the razorblades that would cut her up (her own razorblades, much much later, an entirely different matter). That it was for her own good somehow made it worse, and she woke up crying, and her mother had to reassure her that they would never do anything like this to Vimbai. Still, the only time she visited Harare and they had to take her to a healer for her upset stomach, Vimbai had hyperventilated so badly that she almost passed out. Her mother's mother was still alive then.
Her father interrupted the stream of memory that threatened to sweep her along, take her into a different space. "What are you thinking about, muroora?"
"Grandma," Vimbai answered.
Her parents traded a look. "You remember her?" mother said.
Vimbai nodded. "Of course. I was what, thirteen?"
"Yes," mother said. "I really wish you'd get to know her better."
Vimbai wanted to say that she didn't, that anyway the old woman barely spoke English, and Vimbai's Shona could, if one was inclined to kindness, be described as lacking. Besides, grandma harbored an alarming number of strange beliefs, and tried to use Vimbai's short time in Harare to transfer the jumble of superstition and ignorance into her young mind. But she didn't say it out loud, of course—one did not speak ill of the dead, and even Vimbai accepted it as right. However, in her heart she had not forgiven the scars on her mother's face and limbs. "Did she really believe in ghosts?" she said, infusing her voice with proper respect.
"Spirits. Most people of her generation do," mother said. "Why?"
Vimbai smiled. "No reason. I was just thinking about ghosts. For that class I'm taking, about pre-Christian beliefs."
Her mother raised her eyebrows and started clearing the table. Vimbai helped, all the while thinking back to when she was little, and her mother embraced her freely and called her sahwira—girlfriend, and told her stories she had learned as a girl from her mother. Now, they moved past each other, stacks of dishes and empty bowls in their hands preoccupying their attention on the way to the sink. Vimbai shuddered as she imagined her grandmother, now a vadzimu, an ancestral spirit, summoned by a casual mention. Moving between them like a breath of cold air, pushing them away from each other, lacking even the tentative warmth of the Psychic Energy Baby who waited for Vimbai at home, and possibly cried.
# Chapter 4
When she drove home, the image of her grandmother solidified, until the tall wrinkled woman with white hair was sitting primly in the passenger seat of Vimbai's car. She had just left the Atlantic City Expressway and headed east, for the dunes. The vadzimu shivered a bit in this cold, and Vimbai studied her from the corner of her eye. The house in the dunes was close enough now, and in its sphere Vimbai could cope with ghosts and phantoms and ancestral spirits.
"Hello, grandmother," she said. "Did you come to give me protection?"
She hoped that vadzimu did not come because of some great danger—perhaps, it was not a vadzimu at all, since such ancestral spirits manifested in dreams. Maybe she was dreaming—the thought was reassuring, even though Vimbai hoped that she did not lose her ability to tell dreams from reality. Or maybe it was a chipoko, a simple ghost.
"No," the spirit whispered. " I was sent by the clan spirits, the mhondoro, to tell you a story. Listen, and learn well—ngano is how children learn."
The house loomed closer now, its windows yellow loving eyes, and under their steady staring Vimbai felt entranced as she parked the car. Her breath escaped in small careful puffs as she unbuckled the seat belt, but the cool and hard hand of the ghost lay on her wrist, transfixing her in her seat. A cold lump formed in her stomach, and Vimbai thought that really, it was shell shock, she simply did not have time to absorb everything that had been happening to her; as she thought that, her breath quickened and beading of sweat started forming on her forehead, until the vadzimu spoke.
There was a time once, a long time ago, when a hare decided to take the moon from the sky and put it in his home so that there would always be light. Hares are clever creatures, and our hare (whom we shall call Hare) realized that the sky and the moon were a high way up, and to get to the moon he would have to work at night, and he would have to come up with a clever plan to get there.
Hare waited for nightfall, and climbed the tallest tree in the forest. When he was halfway there, he came across a baboon who was slumbering in the branches. Everyone knows that baboons are dense and quarrelsome creatures, so Hare tried to avoid disturbing the Baboon and hopped over to another branch. He miscalculated his jump in the darkness, and almost fell. As Hare scrambled back onto the branch, he woke up Baboon.
"Hey," Baboon said. "What's all this racket?"
"It's just me, Old Uncle," Hare replied.
Baboon opened one bloodshot eye and gave Hare a mistrustful look. "And what would you be doing in the tree in the middle of the night, Old Grandfather?"
"I'm picking figs to feed my children," Hare lied. "I work in the field all day, and can only go fruit-picking at night."
Baboon went back to sleep, and Hare climbed up up up, all the way to the top of the tallest tree. By the time he got there, the moon rose, and Hare saw that it was just a thin crescent, hanging upside down. "That'll be good enough," Hare said to himself, and reached up. But the moon was still too far—it hung just inches away from Hare's paws, and smiled and laughed at his efforts to reach it (for that, it had to turn right side up.)
Hare shook his fist at the sky and threatened to give the moon such a beating, but the moon just laughed and remained wisely out of his reach. The noise woke Baboon who had been dozing off in the branches below. "Huh," Baboon said to himself. "Looks like Old Grandfather Hare is trying to get the moon, not the figs. I bet I could get it myself and then make him pay me a princely sum in figs."
But the crescent moon remained too far even for Baboon and his long arms, and the next night it did not get any closer. Only when the sickle grew thicker, it started to travel lower in the sky—as everyone knows, the bigger the moon, the heavier it is, and its weight pulls it closer to the ground. Because of that, the full moon is so low in the sky that its round belly can touch the tops of tall trees on a good night.
So on the day the moon was finally full and fat, both Hare and Baboon climbed to the top of the tallest tree. The moon was not laughing anymore, and only looking at them with its white fearful eyes. Baboon's arms were longer, and he grabbed the moon by its pudgy sides, and immediately yelped in pain.
"What's the matter, Old Uncle?" Hare asked and snickered.
Baboon sucked on his burned fingers. "It's hot," he said. "It burns like fire."
Hare, who was quite clever, picked a few leaves off the treetop—they were large and leathery like all fig leaves are, and perfect for carrying coals or other hot burning things. He grabbed the moon with its paws wrapped in leaves, but the moon slipped out like a silvery fish—the leaves were too smooth and slick.
"Let me do it," Baboon said. "You're doing it wrong, Old Grandfather."
"No," Hare argued. "It was my idea, and it is my moon, and my leaves."
Baboon reached for the moon again, burning his fingers the second time (I told you that baboons are none too bright), and Hare maneuvered the leaves this way and that, and he wove a basket in which to carry the moon. Only by the time the basket was finished, the moon had rolled across the sky, away from the treetop.
"I guess we'll never get the moon," Baboon said. "I thought I was quick and strong enough, but I was wrong."
"And I thought I was clever enough," Hare said.
They came down from the tree. In the clearing nearby, they saw a puddle of dark water and a tortoise who came to take a drink of water. Tortoise did not want the moon, he just wanted a drink; but as he drank, the moon reflected in the puddle, and its reflection, cool now, filled Tortoise's mouth and his belly with its milky light. Tortoise smiled and went home, shining like the moon among the trees.
Vimbai led the ghost into the house by the hand. Maya was at work and Felix, judging by the lights in his windows, remained cloistered in his room, doing whatever it was that he did—Vimbai imagined that he played with the phantom limbs as one would with dolls, or with whatever unpleasant things he pulled out of the black hole of his hair.
"There you go, grandma," Vimbai said. "You're welcome to stay here."
The ghost shuffled into the kitchen, looking disapprovingly at the empty coffee cups and saucers stained with syrup piled in the sink.
"It's Felix's turn to do the dishes," Vimbai said, apologetic. "Only he procrastinates."
The vadzimu heaved a tremulous sigh and glided up to the sink. Vimbai was about to argue but then realized it was silly to get into a tug of war about dirty dishes with her grandmother's ghost. The dishes clattered and the water poured, and the ghost stopped paying any mind to Vimbai. She hung around the kitchen for a while, unsure whether she should offer help. Then she decided to check on the Psychic Energy Baby, and snuck upstairs cringing at the creaking of the stairs under her socked feet.
Peb was in Felix's room, attaching a pair of phantom hands to itself.
"Should he be doing this?" Vimbai asked Felix.
He rolled his left eye up, and his right one leftward, giving Vimbai an impression of uncertainty. "Let it do what it wants—it stopped crying just now."
"I brought a ghost with me," Vimbai said. "It's my grandmother, so be nice to her."
"Okay," Felix said. "Ghosts sure do like you."
"Me? I thought it was the house."
"The house likes you too," Felix said. "But we sure never had so many ghosts before you moved in."
Vimbai perched on the windowsill, her back against the glass. "Do you mean I'm bringing them in?"
"You just told me you brought one with you." Felix pointed at Peb. "And you found this one in the phone."
Vimbai considered. "I don't know," she said. "I just moved here to be close to the ocean and to the horseshoe crabs."
Felix nodded. "I remember. Maya said you're a student. How's that working out for you?"
"Okay," Vimbai said. "I've been cutting classes a lot lately . . . I don't know what it is about this place, but I keep dreaming that I'm someone else, somewhere else, and nothing seems as important anymore. Is it weird that I'm saying that?"
"Not weird," Felix said. "I found it, you know. And when Maya showed up, things under the porch started shifting."
"What things under the porch?" Vimbai asked, alarmed. "I've been here for a month and never saw any things under the porch."
"Neither have I," Felix said. "But I hear them, and I know they're there and that they're Maya's."
"What are they?"
"Dunno. The point is, the house chooses."
"What for?" Vimbai asked. The house creaked and whispered in her ears, lulling her, convincing her that everything was as it should be, everything was perfectly normal. "Why does it choose and why us?"
"Dunno," Felix repeated, and shot her an irritated look. "Go play with the baby or something, okay? My head hurts."
Of course, Vimbai reasoned, it was easy to believe that they were special somehow, chosen, different, lost and adopted princes and princesses and their true parents would soon reclaim them and reveal their hidden destinies—isn't it what every book we read as children taught us to expect from life? Of course Felix decided that the house chose them for some unknown purpose, but in reality everything was much more banal. It appealed to them for whatever reasons, and they all came with baggage: Felix had his hair and Vimbai her ghosts, and Maya . . . Maya had whatever lived under the porch.
Peb had festooned itself with several hands and feet, and they remained attached to its transparent body through some otherworldly adhesion. Peb resembled an exotic fish decorated with grotesque appendages and outgrowths. Its skin stretched and shimmered with reflected light like a soap bubble, and Vimbai could not help but pick up the unsightly thing. "Come along," she said. "I'll introduce you to my grandma."
Peb babbled in response, talking about ethereal planes and dizzying stars. It seemed to miss other dimensions, too black or too fiery to describe.
"It's okay," Vimbai consoled. "You'll learn to like it here, and my grandmother knows so many stories—ngano, the folktales that tell children how to live in the world, and nyaya, the myths people make up to pass the time."
The vadzimu was done with the dishes and sat on the stool by the counter, her eyes hollow and her wrinkled hands folded in her lap. Such fragile birdlike hands, Vimbai thought, dry like twigs, wrapped in the cured leather of old skin that spent decades in the tropical sun. Vimbai barely remembered this woman, how she was in life—just her own passing embarrassment at the old woman's superstitions, and just as ephemeral a regret that they spoke different languages and thus were unlikely to connect.
Vimbai noticed with a measure of satisfaction that the ghost, at least, was more fluent in English. If it had also grown less superstitious remained to be seen.
"Grandmother," Vimbai said. "Look at this—it's a psychic energy baby."
The old woman looked and reached out, instinctively—as if there was really nothing else to do with babies but to pick them up and hold them, no matter how ethereal and burdened with unnecessary extremities; no matter how dead one was. And even after Vimbai went to bed that night, she heard quiet singing and cooing from the kitchen, along with the thin gurgling voice of the Psychic Energy Baby.
That night the tides had grown especially, inexcusably high—through her sleep Vimbai heard the lapping of the waves somewhere very close to the porch of the house, and through her sleep she thought that the sea was pulled so close by the gravity of the moon that sloshed happily in the darkness of Tortoise's belly. She dreamt of Tortoise, his smiling face smeared with moonlight, white and thick as milk, the oceans of the world following on his heels—oceans always followed wherever the moon went, tortoise or no.
Meanwhile, the waves whispered into the yard, their salty tongues singeing the roots of the few arbor vitae planted near the house; they poured under the porch spooking those who lived under it and chasing them up the steps, where they remained, wet and shivering, their backs pressed against the closed door and their fur growing slow icicles. They listened for Maya's sleeping breath in the depths of the house and whimpered softly.
The gentle fingers of the ocean pried the house from its foundation, carefully shaking loose every brick and every cinderblock, never upsetting the balance. The waves lifted the house on their backs arching like those of angry cats, and took it with them, away from the shore. In the darkness, the lighthouses shone like predatory eyes, and everyone in the house slept except for the vadzimu, who remained alert and awake, singing to the sleeping Peb, curled up in her lap like a cat, in a language no one but her understood.
The night continued much longer than usual—before the sun rose, the house had drifted far into the ocean, where water lay smooth as silk, wrinkling occasionally under the sleeping breath of the wind.
Under the several hundred yards of water, down on the bottom, horseshoe crabs burrowed in the sand, their movements sluggish in the cold water, the spikes of their tails pointing uniformly north. They had flat, almost round bodies that glistened pretty shades of dark green and light brown, and their blue blood flowed leisurely through their open circulatory systems. They were spent, depleted—bled almost dry and thrown back by human hands where they lingered in a disconcerting state between life and undeath. They had enough blood not to die—yet not quite enough to keep them living. So is it any wonder that the crabs—ancient, trilobitic—whispered stories of vampires that came in boats and then white medical trucks? Is it surprising that they told each other about people who stole blood from their veins and tossed them back, always back, so they could linger in the cold water never quite recovering?
Above them, the house floated, its inhabitants asleep inside. Vimbai was the first to wake up and come downstairs, where the ghost grandmother was entertaining the baby with some songs and hand-clapping. Peb clapped along, with all eleven of its hands, most of which were far too large for its tiny psychic body.
Vimbai glanced at the window and grabbed the kitchen counter for support—instead of the familiar landscape of dunes and sea, there was just a tapestry of green and pale blue and gray. The dunes had vanished, or so she thought until she looked out of the window and saw nothing but the ocean and the sky stretching as far as she could see, and felt a faint spongy rhythm of the floor below her feet.
"Where are we, grandmother?" she asked.
The ghost stopped singing. "We sail across the sea," she said.
"Where to? Why?"
"Perhaps it's a curse some witch, some muroyi, put on you," the ghost said. "Or perhaps it is you who started the journey to get where you need to be."
Vimbai groaned with frustration. Grandmother was just like Vimbai's mother (or the other way around)—both expected her to somehow comprehend her heritage, to become a Zimbabwean like her parents. They wanted her to have a clear purpose in life, even though Vimbai herself rarely thought past applying to graduate schools. And no matter how much they loved Vimbai, she could feel that they lamented the fact that she came out American, as if it were a sad accident, a birth defect of some sort. They wanted her to be like them, to care about the same things they cared about.
"I'm not going on any journeys," Vimbai said. "I have classes, and Maya has work. Where is she?"
"Sleeping," Peb said. "She is sleeping and dreaming of tall spires and the sad creatures on the porch."
"You mean, under the porch," Vimbai said.
"There're only horseshoe crabs under the porch," Peb corrected. "And even they are yards and yards below."
Vimbai faltered then, torn between the conflicting impulses to go check on her housemates, and to stare out of the window, and to see if Peb was lying about the creatures on the porch.
The latter won, and she tiptoed to the front entrance and peeked outside through the transparent window on top of the door. She could only see the edge of the steps, already crusted over with barnacles and wreathed in seaweed, and the tiny waves lapping at the porch. She opened the door and looked out through the screen.
There were three creatures, the size of smallish dogs or largish cats, covered in reddish-brown fur streaked through with yellow highlights. Pointy muzzles and pointy ears swiveled toward the creaking on the door, and the shiny black eyes stared at Vimbai with savage hope instantly supplanted by disappointment. They had narrow tails, bald save for the spiky tufts on their ends, and their needle teeth gleamed like icicles. They were like no animal Vimbai had ever seen, half-foxes, half-possums.
"What are they?" Vimbai whispered, looking at her grandmother's ghost out of the corner of her eye. Funny, at this moment of fear she looked to the ghost as her family, the only kin Vimbai had nearby. Blood always called to blood, no matter how distant.
"They are spirits," grandmother said. "Mashave, alien spirits that are following your friend."
"Why?"
"I don't know," the ghost answered, and picked up Peb to give herself something to do. "Everyone has one spirit or another following them, and who knows why?"
"What about Felix?" Vimbai wanted to know. "Is his hair—"
"Ngozi," the vadzimu interrupted. "It's the maw of an angry spirit that wants to devour him. He must've committed a truly abominable act!""
Vimbai decided that it was not the time to investigate this fascinating point. She had to wake up Maya, and together they would decide what to do. The house was working its subtle magic on Vimbai, and she did not consider the possibility of the house sinking—her concern was with finding her way back home, preferably before she missed any more classes.
She ascended the steps and stopped in confusion—the layout of the house had been changed dramatically. The hallway stretched farther than she ever remembered it being, farther even than her idea of the house's size would allow. Moreover, at the end of the hallway where she remembered her room being there was inside a solid wall of fragrant and green vegetation, twining along the walls and cascading from the ceiling like a curtain. Bright flowers bloomed and wilted, their petals falling on the floor as each flower transformed into green and yellow fruit; drops of dew condensed and slid along the midribs of large leathery leaves. Thankfully, the door to Maya's room was still visible.
Vimbai knocked.
Maya's hoarse voice mumbled something, and then rose. "Come in."
Vimbai did. She found Maya sitting up in bed, staring out of the window. "Did that old woman do this?" she asked Vimbai without ever turning.
Vimbai had not considered this possibility, but discounted it. "No," she said. "Of course not. That woman is my grandmother—well, her ghost, in any case. An ancestral spirit."
"Are they good?" Maya asked.
"Usually," Vimbai said and tried to remember what she knew of the relevant folklore. "They are the link between people and the creator, Mwari. Sometimes witches command them to do harm, but I don't think this is the case. She said she needed to tell me a story, and then she just stayed. Peb likes her."
"Great," Maya said. "A ghost babysitter for the Psychic Energy Baby."
"There are also animals on the porch," Vimbai said. "Everyone seems to think they are yours."
"Did you see them when you first came?" Maya asked. "I remember you looking under the porch."
"No," Vimbai said. "But I did see them today—they are on the porch now. The water chased them there, I think. They seem cold."
"What happened?" Maya said. "Do you know why we are floating?"
Vimbai shook her head. "The ocean carried us off."
"Or it's a flood," Maya said, grim. "Has it occurred to you? There's another flood and the only ones who survived are us and a few ghosts we have along."
"And your animals," Vimbai added. "What are they?"
Maya shrugged. "They do not like fish, that much I know."
Vimbai looked around the room, not because of any pressing curiosity but to distract herself from the sight of the water and the nagging fear that Maya might be right, that the world had simply disappeared overnight and there was no back to go to, no classes to catch, no parents to reassure. Vimbai rubbed her throat to chase away a large and cold stone that suddenly formed there. She looked at Maya's chairs and the shelf with knickknacks, at the stack of paperbacks, their covers worn into illegibility, and at the beanbag chair that sat in the middle of the bedroom like an imposing toad. It was a simple room, with precious few traces of personality—surprising for a dwelling inhabited by someone as distinct as Maya. In fact, Vimbai thought, the same could be said about Felix's room as well as Vimbai's own. In this house, there was no need for posters or furnishings or any other mass-manufactured claims to individuality, there was no need of proclaiming to the house that this was a room belonging to any specific person, with formed tastes and idiosyncrasies. The house took care of that—the very fact of them living here was enough to attest who they were.
"Get dressed," Vimbai said and headed for the door. "I'll check on Felix, and you take care of your creatures. Grandma is making breakfast, so we can eat and decide what we should do."
She avoided looking at the wall of greenery hiding the door to her room—instead she headed for Felix's room and knocked. Felix opened almost immediately, dressed and as alert-looking as his bloodshot eyes allowed. Vimbai thought that his hair did look a bit like the open maw of some spectral predator.
"Yes," Felix said. "I saw. And I don't know what's going on."
"Fine," Vimbai said. "Come and eat breakfast with us. And if you want to see the things from under the porch, you can."
In the kitchen, Maya had commandeered a few dishes, and fed the three shivering animals canned tuna. Vimbai was glad that they had just made a shopping trip, and at least there were plenty of cans in the cupboard. Despite being separated from the electrical supply, the refrigerator still hummed and sputtered, and the stove worked as well. Vimbai made a mental note to check the TV and the phone as she settled on her usual stool and poured herself a cup of thick, oily coffee her grandmother had made. She waited for Felix to come downstairs and take a seat, and for Maya to finish fiddling with the pack of half-foxes, half-possums. Even the chipoko, the ghost, ceased her shuffling and stood quietly by the stove, the Psychic Energy Baby and all its phantom limbs cradled in the strong crook of her arms.
Satisfied that all the house inhabitants—even the animal, even the immaterial—were present, Vimbai nodded to herself and took her first sip of coffee as a mariner.
# Chapter 5
Houses floating on strange and calm seas under frozen skies that only occasionally work up the energy to scare up a few clouds and sift a few snowflakes are bound to be guarded by different laws than ordinary houses. Dimensions, for example—as soon as the house in the dunes became unmoored from the very dunes that gave it its nickname, it grew larger on the inside, sprouting additional turrets and rooms and crawlspaces, often hidden behind the walls and impossible to get to—but existing nonetheless. And the proximity of the black hole of Felix's hair warped the spaces inside and pulled up additional layers and floors and realities in some phantasmagoric synergy.
At least, this is how it appeared to Vimbai. An act as simple as opening a bathroom door had to be performed with utmost care, because she could not be sure about what she would find on the other side—the best she could hope for was startling one of Maya's needle-toothed critters drinking out of the toilet bowl; they always turned, glowering, their bright eyes looking over their hunched and almost-human shoulders.
"I'm sorry," Vimbai said after the scattering footfalls of clawed and splayed paws. "I have to use the bathroom." Secretly, she was relieved that the bathroom remained as is, for now at least.
Peb lolled in the bathtub, half-filled with cold water. Vimbai regarded him and decided to pee despite his presence—the thing that now had absorbed all available phantom limbs, save for the one in Vimbai's room, always appeared in unexpected and inopportune moments, popping through the walls or the floor or the ceiling.
"And what are you up to?" she asked Peb as she sat thoughtfully on the toilet. "Grandma is probably looking for you."
Peb shrugged its shoulders and several legs. "She treats me as a child."
"You look like one," Vimbai parried. "You told me you were a baby."
"Not in any regular sense," Peb answered. "Do you know what it is like, in other planes?"
"Same as in Felix's hair?" Vimbai guessed and flushed the toilet. Miraculously, it acted as if it was still connected to a septic tank, but Vimbai felt guilty because she suspected that now it was connecting straight to the ocean.
"No," Peb said. "The planes are radiant and singing. Felix's hair is a dark and desolate place."
"But it is a place," Vimbai said. "An actual place, bigger than it appears to be."
"Oh yes." Peb sank underwater and spoke in small exhalations of bubbles. "I think it's a plane of some sort too, but not a very nice one."
"What's there?"
"Find out yourself," Peb said, suddenly petulant. "I am busy."
"You're awfully cranky for a ghost," Vimbai said.
No answer came and she exited the bathroom, ducking just in case there was a tree suddenly growing outside the door. With no classes to go and not much else to do but to explore the house, Vimbai headed for Felix's room.
He let her in. He had changed the least of them all, Vimbai thought, and it was probably a good thing—any more weirdness added to Felix, and he would be closer in nature to the ghosts and Maya's animals than to Vimbai and Maya. Maya, on the other hand . . . but that was something to consider later.
"Felix," Vimbai said, politely. "May I take a peek inside your hair?"
His eyes rolled wildly, like those of a spooked horse. "Why would you want to do such a thing?"
"Curiosity," Vimbai said. "And considering that we are in a floating house that sprouts new rooms every day, I think there may be some insight gained."
Felix slumped, and shuffled over to his bed, to sit on it in a pose of defeat and remorse. "You're blaming me," he said. "I tried to tell you."
"No one is blaming you," Vimbai said. "What was it that you tried to tell me?"
"That there are forces in the world," Felix answered. "Forces that run along invisible wires—like phone wires of the spirit, and sometimes you get trapped in them like Peb, and sometimes you stumble in the middle and get caught like a fly in a spider web..." He fell silent, shaking his head; the hair undulated along, with a barely noticeable delay—as if air provided too much resistance.
"You're telling me you know what's going on."
Felix shook his head again, with greater vehemence. "I only know that there are forces, and we are crossing their conduits. And we probably shouldn't. Like you shouldn't look in my hair—there's nothing there for you, nothing at all."
"I shouldn't or you won't let me?"
Felix sighed. "I'll let you but I do not think it's a good idea. But go ahead, look, see what I care."
Vimbai felt a cold wave of hesitation rise in her stomach. "I'm not going all the way in," she said to Felix as much as herself. "I'm just going to look, okay?"
"Whatever," Felix said and slumped some more.
Vimbai approached him in small childish steps. The black mass undulated closer to her face, and in it she saw quiet seething, like the surface of a cauldron full of boiling pitch—or at least what Vimbai imagined one would look like. It took an enormous effort for her to stretch her neck until her face—her eyelashes, her nose, her lips—touched its surface. It felt like sinking her face into a basin full of cold water—she was shocked at how cold it was, at how it singed her skin with frost.
She opened her eyes. It was dark at first, but as her pupils dilated and adjusted, she started to make out shapes at a distance—a mountain with a rounded top overgrown with what looked like trees bending in the wind and a faint white sickle (moon?) hanging above it.
Then the mountain shuddered, and two white round windows opened inside it. Vimbai jerked back as she realized that in the dusk she had misjudged the distance badly, and what she thought was a mountain in reality was a human head just inches away from her face, and the white circles were its dead eyes.
"Hello," the head said. "You new?"
"I'm . . . temporary," Vimbai said, and her heart—outside of here, distant—thumped like mad. "I'm just looking in."
"Like all the legs," the head said. "Funny, I see legs and hands and feet and only rarely—other heads."
Vimbai nodded. "I have a body too, only it's outside," she said. "I'm Vimbai."
"Balshazaar," the head said.
Vimbai studied the head—it was quite old and quite dead, and very desiccated; Felix was not lying about that. The sparse hair covering its parchment-yellow scalp did resemble trees—each hair stood alone and separate and rather straight up. Long and deep furrows covered the face, and Vimbai thought that she noticed traces of faint green luminescence hiding inside them. Balshazaar was a landscape in his own right, and Vimbai could not think of a single thing to say to him. "It's nice to meet you," she finally managed. All the other questions rising in her mind were cut off by their overwhelming mundanity—what did it matter who Balshazaar was or where he came from or if he ever owned a body? Now he was just a desiccated head living in the hair of a really weird teenage malcontent. The rest seemed trivial.
"Going already?" Balshazaar said politely.
"Yes," Vimbai answered. "I don't belong here—see, there's a whole other world outside, and—"
"I know," Balshazaar interrupted. "I've seen it."
"You used to live there?"
"No. Felix takes me out sometimes."
"You don't say." Vimbai was angry at Felix now, for not telling her more and certainly for not letting them know that he had dragged a disembodied head out of whatever unknown dimension. It was one thing to amputate phantom limbs, and quite another to show Balshazaar the world. It was just like grandmother said, one did not screw around with things one did not understand.
Grandmother. The woman who used to be so ridiculous was starting to make sense; or at least she lacked Vimbai's streak of rationality, which made her helpful in irrational circumstances. Grandmother lived—or used to, when she was truly alive—in the world where razor cuts protected from misfortune, and cunning muroyi, witches, could sic spirits on the living and make them ill. Grandmother would deal with a dead head like she dealt with all such problems—remember a remedy or go to a n'anga and have it fixed. Vimbai wished there were a healer nearby, someone who was versed in dealing with the supernatural rather than someone like her, who flailed and hyperventilated and tried to stay calm in the face of it—so far, that was all Vimbai could manage. Even her fear of the Harare healers had receded enough to think of them wistfully.
"I'll be going now," Vimbai said, and straightened. Balshazaar's face diminished as it hurtled away from her, and Vimbai looked into Felix's disturbing eye. She was not ready to tell him anything yet, and so she stalked away without saying a word. Felix was so disconnected from everything anyway that he probably didn't even think her rude.
Vimbai padded to her room along the hallway that had grown a covering of soft, slightly wet moss, and lay down on her bed. A mattress and box spring, really—not a proper bed. She resented her grandmother's arrival and the house's ill-fated journey. Why did it have to happen to her, a perfectly rational person? Were those the superstitions of her ancestors that dragged her along, people long dead but unwilling to let go? It just wasn't fair that someone she was related to by blood alone could do that, as if shared genetic background gave them some sort of power over Vimbai. She wondered if Maya too felt that same pull and resentment.
Maya. Maya who barely talked anymore and instead followed with her feral pack from one room to the next. They roamed like hunters, disappearing into the closets recently converted into thick suffocating forests, they swam in rivers that poured from the downstairs bathroom. Vimbai hated to admit that her worry about Maya was just a pretense designed to mask her envy and disappointment at not being invited. She too would enjoy a pack of furry familiars following her around, she too would like to be unconcerned about their future and the present circumstance.
Peb floated up through her pillow, its smooth skin and several feet and hands brushing cool against Vimbai's cheek. "Don't be sad," Peb said, its former petulance forgotten. "Why are you sad?"
"I miss Maya," Vimbai answered. "I wish I could go with her."
"There are creatures under the porch," Peb said cryptically. "The house found them."
"They were on the porch," Vimbai said. "And now they are gone."
"No," Peb argued. "Still under."
"There are only horseshoe crabs there." Vimbai sat up abruptly. "Is that what you're trying to tell me?"
Peb bobbed over her pillow, floating up then down, until it jetted higher up and disappeared through the ceiling.
Vimbai rubbed her face. "Horseshoe crabs," she murmured. Poor crabs, bled half to death by wazimamoto in medical trucks. Vimbai jumped off her bed, reenergized by the possibility of a new discovery. There would be time later for Maya and Felix and Balshazaar; now was the time for horseshoe crabs.
Vimbai had never anticipated that she would be sticking her face into strange and unfamiliar places so much, but there she was—on her hands and knees on the porch, by the very edge of the water. The house had mutated again, developing a coquettish hem of round pebbles and pieces of seaglass, polished and clear. The barnacles hung onto the edge, their quick ghostly feet kicking food in their mouths hidden somewhere inside their chalky shells. If it weren't for horseshoe crabs, Vimbai would've studied barnacles for the sheer weirdness of their anatomy and lifestyle.
Vimbai kneeled on the edge of the porch that currently fancied itself a littoral zone and studied the surface of water—smooth and clear, and so cold. Her breath fogged the air and touched the waves; Vimbai tried to cloud them with her breath like one would a pane of glass, but to no avail. She took a deep breath and thrust her face into the ocean.
The salt and the cold burned her skin, a million needles threading her cheeks. Her teeth ached. She opened her eyes underwater and they burned too, tears not helping the matters at all. The water around her seemed stationary, like a block of green ice. She couldn't see very far, and her breath tried to break out of her chest like a caged and panicked bird.
Vimbai came up for a breath of air, and gasped, still crying from the cold and the salt and the sadness of all this water, always separating her from something she wanted. How could one love something so cruel, something so terrible to her parents? As if answering, a withered ghostly hand lay on her shoulder.
"Sahwira," her grandmother said. "Girlfriend, my girlfriend. You look for thing no mortal eyes can see. Let me guide your vision."
Her grandmother's hands lay flat on Vimbai's temples and pushed her gently back toward the water's surface. For a moment, blind fear boiled in Vimbai—what if she were to hold her head down and never let her come up for air, no matter how much her heart thundered and her legs kicked and thrashed? What if she wanted Vimbai to be a ghost, like her, to finally touch the souls of her ancestors?
But it was foolish. The hands on her head were so gentle even without warmth, so kind, that Vimbai succumbed and let them guide her. She opened her eyes, and for a moment there was just familiar transparency without images, the endless wall of thick glass. And then her grandmother's eyes entered her own.
If she were asked to explain how it felt, Vimbai would've faltered for words, groping for images that best described what she was experiencing. Her grandmother's sight entered her own like a hand enters an empty glove. Vimbai had been hollow and now she had a center, a depth, a density—she felt three-dimensional and alive and aware. She focused her eyes and she could see every grain of sand in the bottom, every rock, every shrimp hiding in the crevices. She saw kelp forests and the silvering of a school of anchovies, the rapid quirk of a shad. On the bottom, hagfishes braided themselves into an incestuous, slithering nest of Gorgon's hair in the empty cavity of a dead shark's head, its gill arches protecting them like the barred windows of a jailhouse.
And beneath and beyond all that, under the sand sifting over the skeleton of a sunken ship, there were horseshoe crabs, pale and unwell. There were hundreds of them, or perhaps thousands, all of their tiny legs moving in unison, burrowing in the sand. As Vimbai looked at them, they stared back with their pinprick eyes. And as if one, they shifted, their legs working in reverse now, digging themselves out rather than in.
Among all the strange occurrences of the past weeks, the fact that the cold-blooded crabs were able to react with such speed and determination bothered Vimbai most of all. She wanted to pull away, to break the surface and to not have to see this, but her grandmother's quiet attention held her, their eyes—Vimbai's and vadzimu's—riveted to the creatures. They varied in size, from the tiny ones as small as a quarter to the adults as big as a dinner plate, and yet all of them moved together, the living carpet of them swarming onto the surface of the sand, their mouth parts open in plea or hunger. As Vimbai pulled away, the wave of crabs heaved after her; as she moved closer to them, they retreated but never far away.
"Why are they doing it?" Vimbai whispered.
"It's like that story I told you," the vadzimu answered. "Weren't you listening?" Her tone was impatient now, stern—just like she used to be in life, the kind of woman who would take her own daughter to be carved up by razorblades—but always for her own good.
Vimbai frowned. "What does the story have to do with horseshoe crabs?"
Grandmother heaved a sigh. "The tortoise," she explained slowly, patiently, as one would to a dim a child, "did not want the moon. But the oceans followed him nonetheless, as they always follow the moon."
"I dreamt about it," Vimbai interjected. "Seas following the tortoise."
"He did not want it, did not ask for it—and yet. He drank the moon, and the moon in his belly was bigger than he, and it commanded the ocean whether the tortoise liked it or not."
Vimbai considered she might have drunk that was so compelling to the crabs, and gave up. It wasn't her first beer in the house and it wasn't the tears she cried in secret, letting them soak into her pillow, her hair, her eyelashes. But there was something inside of her that made her find the house and bring her dead grandmother along, something that made her want to study horseshoe crabs—and now, apparently gave her a power of command over them. They were not as cute as Maya's half-foxes, but they were Vimbai's—at least, they seemed to think so.
When she opened her mouth, the salt water flooded it, numbing her tongue and pounding her teeth with its frozen hammer. Her face did not feel a thing, and she wondered idly how was she able to hold her breath for so long. Nonetheless, she managed to ask, "What do you want with me?"
The crabs answered in a quiet rustling of their legs and mouth parts, in the sad stares of their tiny eyes, We want you to take care of us, and let us take care of you. And this is how Vimbai found herself in possession of a horseshoe crab army.
Vimbai remembered the time when she was little, before the horseshoe crabs and her anxiety about Africana Studies. Back then, she considered New Jersey prosaic and hardly the place where one could hope to grow up while having important experiences. She listened to her parents as they spoke after dinner; when they talked to each other they did it in Shona. Back then, Vimbai did not concern herself with questions why it was so, but now she understood—even though both were taught English from an early age, Shona was a way to set themselves apart, to reaffirm that they were of the same cloth as each other, set against the rest of their surroundings. Later, Vimbai thought it an unnecessary affectation, and forgot most of what little language she knew back then. She did not realize the need to set herself apart—in fact, her childhood was dominated by the opposite impulse, to be one of many.
Her color did not help matters—even though they lived just a few miles away from Atlantic City, their particular town was white; they were the only black family on the block. And no matter how much one tried, there were things that simply could not be hidden.
Afterwards, as an adolescent burdened with an unfair amount of social conscience, Vimbai went through a brief but histrionic stage of embracing her heritage—she reasoned that if one could not blend in, it was better to exaggerate the difference. It brought about a brief resurgence in her interest in Shona and African lit, as well as the love of 'ethnic' fashions. Out of these, the latter lasted the shortest—all it took was one eye roll from Vimbai's mother to plant the seed of doubt. Nonetheless, it was during this time that Vimbai visited Harare and met her extended family on her maternal side.
Harare shocked her by its hasty urbanity—it felt like a city that was created too quickly, without giving a chance to people or the land to adjust to its presence. The tall skyscrapers that wouldn't be out of place in Philadelphia or New York City jutted out of red soil, as if plopped down by some magic tornado. The houses pushed upward among trees; her mother said that they were called jacaranda trees, and that when they all bloomed, Harare was the most beautiful city on earth. They went to African Unity Square, and Vimbai gawped at the flower market that shone with so many colors—several of them not found on the spectrum, Vimbai was pretty sure.
Much later, when Vimbai was eighteen, she watched her mother cry when she read in the newspapers that the flower market was destroyed at Mugabe's orders. She felt like crying too, but was too busy drawing a firm line between herself and Africana Studies and everything they entailed—even the flower markets in her mother's home city. Now she understood the deep hurt of that destruction, the most basic betrayal of one's childhood love. It was not just about the flowers; it was never just about anything. It was always about what one knew to be true about the world when one was a child, and the death of that knowledge.
# Chapter 6
"We have to get back to New Jersey," Maya said one morning, the ruddy fur carpet stretched by her feet across pale linoleum tiles. "We're almost out of coffee, and the milk is going bad."
Vimbai peered into Maya's cup, the murky coffee in it studded with pellets of milk coagulated in an unpleasant fashion. "Ew." The question of return had been on her mind, even though she tried not to think about her classes and her mother, insane with grief—the moment she did, her stomach felt sick.
"Exactly." Maya made a face and took a cautious sip.
"I suppose I could ask my horseshoe crabs to tow us back," Vimbai said.
Maya smiled. "So they are your horseshoe crabs now, huh? Do they have little harnesses?"
"No," Vimbai answered and drank her coffee black. "But I guess we'll need to give them something to grab on. They have to walk on the bottom—they are not great swimmers."
"Why couldn't you befriend dolphins?" Felix said.
Maya laughed, eyeing her half-foxes, half-possums tenderly. "Yeah. Mammals are smarter."
Vimbai shrugged. "I don't care. I like crabs. And they are the ones that can take us home, so be nice."
"Are you sure that they can?" Felix said.
Vimbai wasn't. "Pretty sure," she said out loud.
After breakfast of dry pancakes (they were low on syrup too), Vimbai went to talk to the crabs. Her grandmother came along, quiet and helpful as usual. She helped Vimbai see and helped her talk, and the words that bubbled out of Vimbai's mouth underwater were both of theirs. Moreover, Vimbai had noticed an increased frequency of dreams about Harare—especially the vegetable garden in her grandmother's backyard—to the point where she suspected that the ghost's memories were leaking in and coloring Vimbai's own. Or maybe the proximity of the ancestral spirit reminded her. Oh, jacaranda trees in bloom, Vimbai sighed underwater. Oh, horseshoe crabs. Will you take us home, to the sand bars and beaches of New Jersey, where you come every spring to spawn and dance through the tides on your little segmented legs?
It's not yet spring, they answered. It is cold and we will die if we leave the safety of our deep sleep.
Vimbai nodded, her hair floating in front of her face and crosshatching her vision like a mosquito net. Or a fisherman's one—she shuddered when she remembered the quartered corpses of horseshoe crabs sold as bait in every bait shop. They were good to put in eel traps, they said. No wonder they didn't want to go back without great necessity. "Do you know of anyone who can help us?" she said. "If we don't get home, we will die."
The crabs consulted among themselves, their whispers audible only to ghost ears. Finally, they said, Go back go back home. We will help you—just hang some ropes for us and don't look into the water until you get back home. And promise, promise to protect us from death if we come with you.
"I'll try," Vimbai said. "But how can I protect you?"
The vadzimu pulled her out of the water. "It is simple business, sahwira," she said. "Just don't let them die."
Vimbai had spent this morning braiding ropes that would be long enough to reach the bottom—she denuded the closet in her room of its vines, leathery and tough, and she twisted them together into long strands. She found supple branches in the young forest that had sprung where the attic door used to be, and she peeled off their bark. She teased apart the vascular bundles and twined them around the vines to give them enough strength to move the entire house.
She attached the ropes she made to the steel bolts in the porch, and hung them into the water.
It was difficult to avoid the temptation to look, but she resisted. The crabs asked her for a reason, and Vimbai knew enough fairy tales where a violation of explicit prohibition spelled an immediate and cruel disaster—all her sources agreed on that, European and African both. She just had to make do, and simply imagine the solemn crabs grabbing her ropes, clustering on them, their weak legs digging into the wavy sand studded with shells, and pulling, pulling with all their might. Or perhaps they could secretly swim, and no one ever knew about that—perhaps this is why they told her not to look. She imagined them, floating in the thickness of water, suspended like trilobites in amber, graceful as falling leaves.
She thought bitterly that those who featured in those cautionary fairytales had no graduate schools to apply to—if Vimbai could document such an interesting behavior as swimming, her application would be a snap. Then it occurred to her that the ability to talk was even stranger.
This collision of worldviews—one that allowed for talking horseshoe crabs and one that hinged on graduate school applications—made her breath catch in her throat, bowled her over, brought her to her knees, and she clutched her head in her hands. Ever since she had moved into the house in the dunes (which was now not quite the same house in the ocean), her mind, quite unbeknownst to her, had labored at keeping these two worldviews coexisting but never quite encountering each other. Now, accidentally, the two were brought together by the crabs, and Vimbai rocked back and forth on her knees, her head between her sweaty palms, and struggled to gather her thoughts. What am I doing here, she said to herself. Stupid Felix with his black hole coif and his pet desiccated head, stupid Peb, stupid half-foxes that weren't even all that cute. Stupid Vimbai for playing along with this nonsense rather than packing up and going back home; her mother would've been so happy. She wouldn't even bug Vimbai about staying out late.
Not that Vimbai had much of a social life, but she occasionally stayed late at her study group, and sometimes they went out for a pizza. No matter how sophisticated and urbane, Vimbai's mother had a real hang-up about Vimbai staying out after dark.
She had understood it better in Harare, where her grandmother explained to her that decent girls did not stay out late.
"Why not?" Vimbai had said, in a voice that made her mother frown dangerously. She did not approve of Vimbai mouthing off to her elders.
"Because you know what happens with young men and women after dark," grandmother said.
Vimbai laughed. "What? Sex? It happens during the day too, you know. People might stay out late not having sex, or have sex at 9 a.m."
"That's enough out of you," her mother interrupted, and dragged Vimbai out of grandmother's house by her arm, to go meet the family of her aunt twice removed.
Now she understood what clinging to habit, to tradition, because sometimes tradition was the only thing that kept one sane.
The house started moving again the very next night. They couldn't feel it at first, since for a change they all gathered together downstairs to watch the TV tuned to some foggy ghost channel—it showed nothing but snow-covered mountains and two women talking to each other on their cell phones; Vimbai found the women strange, since, despite the split screen, they appeared to be in the same room. Vimbai shifted in her chair and shot Maya a restless look. Maya smiled lazily back and whistled at the half-possums at her feet.
"Not hunting tonight?" Vimbai asked.
Maya shook her head. "Tired. And what's the matter with you? You hardly ever went . . . well, anywhere!"
"The house has mutated," Vimbai parried. "How do you know it's not dangerous?"
"I don't." Maya shrugged and stared at the TV screen again. "It probably is. But it is certainly more interesting than this."
"Yeah," Felix agreed from the couch, where he lay framed by the ghostly shapes of Vimbai's grandmother and the Psychic Energy Baby. "The TV here sucks. I wish the phone worked instead."
"It's the middle of nowhere," Vimbai said as mildly as she could. "But don't worry, the crabs said they will get us back to New Jersey."
"About time," Maya said. "Are we moving yet?"
Vimbai looked out of the window, and at first she thought that the house remained as it was—there was nothing but water and star-studded sky wherever she looked. But soon she noticed a small wave rising where the porch met the water. She rushed over to the window on the opposite side of the room, and had to clear away a thin, disconcerting layer of fresh meat that had grown over the windowpane just recently. She saw a luminescent wake, and cheered. "We are moving," she said to the questioning gazes of her roommates and ghosts.
And so they were. Vimbai tracked their progress by the rotation of the alien stars overhead, waiting for the familiar clangy shape of the Big Dipper to swim into her view. As the house traversed the waters, Vimbai found herself in little need of sleep, and she stayed up until morning, looking from one window or another, and feeling like a mariner and the discoverer of the world.
Her grandmother sat next to Vimbai, silent, but Vimbai knew that the vadzimu did not share her fascination with explorers and pioneers, discoverers and seafarers. They were trouble, they only brought bitterness with them, and they took away and even when they gave back it was not the same. They spoiled everything they touched.
Vimbai's grandmother was not nearly as politically aware as Vimbai's mother, or even Vimbai herself—she had seemed preoccupied with her vegetable garden and her family, and politics and history were dwarfed by these concerns, existing only as distant and vague hurts, a persistent feeling that things were worse than they could've been, and the blame was easy to find. Grandmother grew up in Rhodesia, and she never went to school because the black people were not allowed to. The occasionally heard claims that the English brought education to Zimbabwe sounded hollow to Vimbai because of her.
"Grandmother," Vimbai said, and tore her gaze from the darkened window. "How did you find us? Why did you come with me instead of staying with mom?"
The ghost's eyes clouded for a moment, by memory or regret. "Past speaks to the future," she said. "The present is already crumbling."
Vimbai's heart fluttered in her throat as she pictured her mother—her parents, both of them, still young and beautiful—getting older and smaller and more fragile, birdlike, and finally shrinking away to nothing, falling apart like a handful of ash. She shook her head, no, it cannot be like that.
"And you called me," grandmother said. "Your mother never did, but you called to me, through your anger, through your contempt."
"I never . . . " Vimbai started.
Peb floated up to the window and peered along with Vimbai for a short while; then it went to the vadzimu and nestled on her back, like an ugly festooned hump. When Vimbai looked at him and all his absorbed phantom limbs, she thought of the exotic fish that decorated themselves with fins and outgrowths until they resembled a piece of coral or an algal bed.
"Maybe I'll do better with you than I did with your mother," grandmother said.
"You didn't do badly with her," Vimbai said. Really, she didn't—it was not her fault, she did as she was taught, she meant as well as the parents all over the world do.
"She left home."
Vimbai smiled at that. She could not leave home, at least not now—the home was spacious but surrounded by a flat watery expanse that offered little in escape possibilities. Even with the horseshoe crabs in her command, Vimbai would not dare to dream of escaping. Then again, she did not really want to. "I won't leave. Your other daughters did not leave. Why aren't you home, with their children?"
"They have the whole clan. You have no one. No ancestor spirits to protect and guide you, to connect you to the creator."
"I appreciate that," Vimbai said. "And mom would too . . . if she knew, I mean."
Grandmother nodded, consoled or just playing along. "What are you thinking about, varoora?"
Vimbai looked around to make sure that neither Felix nor Maya was within earshot. When she was content that there was no chance of being overheard, she moved her head close to the ghost's. "Love," she said. "Being in love, I mean."
Boys—or, she supposed, in her age bracket they should be properly called young men or guys—were a minor puzzlement in Vimbai's life, and one more point of tension between her and her mother. Vimbai's mother was downright schizophrenic when it came to Vimbai's dating life—she warned her away from staying out late and spending too much time gawping at men, and yet she worried that Vimbai didn't.
Vimbai remembered going to the prom—just two years ago—and she remembered the dress she wore—a bright yellow silk sheath, golden even, the perfect color of the noon sun. She still kept it, in vain hopes for some occasion to wear it again. She did not remember the boy who took her to the prom. She remembered her parents being happy that she came home early but then whispering in the kitchen.
Vimbai did not know why she wasn't interested in them—like all her contemporaries, she went through the pre-assigned stages of development. When she was in middle school, she read encyclopedias on the sly, hunting for dirty words and hoping for illustrations. She pored in secret over art albums that her parents kept out in the open, but looking at the paintings with them present felt uncomfortable, like a too-tight scratchy woolen collar. And yet, the actual boys with their stained hands and hostile eyes did not appeal to her.
In high school, she wondered if perhaps she was a lesbian—she had a short but intense crush on a classmate named Elizabeth Rosenzweig, a tall British girl with long black hair who looked at everything as if it were too boring to even bother raising her eyelids all the way up for. They had a lot of classes together, and sat next to each during lunch, each thinking desperately of something to say to the other. Vimbai treasured one time they had stayed after school together because Elizabeth needed to copy a part of some assignment Vimbai and she were doing together, and her smooth cool hand touched Vimbai's, chapped and burning.
When Elizabeth went away for the summer, Vimbai missed her with the pointless urgency of first love, and she cut shallow marks into the insides of her arms and thighs—they had healed completely, but if one knew they were there, they could be seen as thin lines slightly paler than the rest of her skin, now just an annoying reminder of past foolishness.
When Vimbai started college, her classes preoccupied her too much to worry about not having a boyfriend or a girlfriend or some sort of significant other. It amused her to think that if she delayed dating long enough, her mother would be relieved if she brought home anyone—even a girl; even a British girl. Perhaps some day she would run into Elizabeth again, maybe at the mall or the coffee shop down the street, when Elizabeth visited home during the break—she went to college out of state. And maybe then they would reminisce and go to the movies, and at least then Vimbai would not have to worry about dating for a while. This passing thought grew into a justification with time—at least, Vimbai used it as an excuse to keep to herself and avoid any possibility of romantic involvement.
She considered it now, and wondered at the relief a part of her felt at being stranded at sea with some ghosts and two roommates—at least, she did not have to explain herself here all the time. And there was no possibility of Elizabeth showing up here, and making Vimbai feel awkward and inarticulate.
"Is there something wrong with me?" she asked the vadzimu. "I mean, shouldn't I want to love somebody?"
The ghost shook her head and patted Vimbai's hand. "There's nothing wrong with you or anyone here."
Vimbai smiled and moved closer, as if to cuddle up to the old woman—but then remembered the razorblades. She settled for touching the chipoko's hand instead. It felt (and, Vimbai supposed, was) immaterial, just a little warmer than air and light—yet solid enough to hold, just like Peb.
"Peb is falling asleep," Vimbai said. "Why don't you tell him a story?"
Grandmother smiled. "I suppose I could. Which one do you want?"
Vimbai shrugged. "Whatever you like."
"I do know quite a few," grandmother warned. "But I suppose there's no harm."
Her voice soothed Peb and made Vimbai sleepy. She dipped in and out of sleep, like a fisherman's bob on the surface of water, catching brief snatches of serpentine and seemingly endless stories about baboons and rabbits and other animals who all had active social lives and spoke on the phone a lot. Just as the sun rose in front of the window (they were heading east, it seemed), a word her grandmother said jolted her awake.
"Man-fish," grandmother said.
Vimbai sat up. "Is that from Marechera's book? I didn't know you read him."
"It's not just from the book," grandmother said. "It's a nyaya, a myth. Everyone knows it's not really true, but we tell it anyway, because it always contains truth, and not the boring part of it."
"Will you tell me?" Vimbai said.
The vadzimu nodded. "Sleep, granddaughter, and you will see everything you need."
Vimbai rested her head on the windowsill and dozed off, lulled by the quiet lapping of the waves against the house wall outside and the soft whining of the Psychic Energy Baby.
Her dreams, just like her eyes underwater, seemed an amalgam of her own and her grandmother's notions—a rather disconcerting situation, since the blend of the two had the quality of a comical nightmare about it. She dreamed of a broad river, Limpopo perhaps, or maybe Zambezi. There was rumbling of turbulent water off in a distance, but Vimbai stared at the smooth surface and large shallows by the bank. Her feet sank into the muddy soil, and a few grass stems brushed against her bare ankles. It looked like a good place for a swim, and she jumped into the warm muggy water cannonball-style, releasing a plume of spray and pungent, green smell of the river water.
She opened her eyes underwater, and squinted against swarms of silt particles that swirled before her, carried by the turbulence of her dive. She spotted a hippo with a calf, but remembered that she was just dreaming, and swam past them with nonchalance. A string of bubbles rose from her lips, but she breathed easily underwater, and she swam through the wide flats overgrown with grass, and into the deep channel of the river. There, she noticed that there was no more air escaping her lungs, and that she likely had no lungs left—her feet had fused into a wide lobed tail, and brown patterns covered her skin. A pair of whiskers hung from her lip and registered every disturbance in the flow of water. Vimbai did not need a mirror to figure out that she was now a catfish, and she sank all the way to the bottom and rested in the mud, feeling, listening.
A catfish can grow very large, and they do it by staying close to the bottom, eating anything organic, and growing fast enough to quickly become too large for most predators—crocodiles, perhaps, would still be a threat to a giant catfish, but catfish did not become large by being careless. The mud and the brown color of their skin protected them from view.
Something stirred in the water—catfish's whiskers sensed a wild thrashing, like a panicked impala trying to wrench free of a crocodile's jaws. The catfish headed in the direction of the commotion, staying close to the bottom, stealthy and cunning.
Catfish would eat pretty much anything. If there was a hunt, a death, they would pick up whatever remains fell to the bottom. And if the victim was a person—well, so much the better, so much more to pick up and savor.
It was a stupid boy who had decided to go swimming after a large dinner. He swam too far from the shore when a spasm in his belly twisted him into a knot, and his nose and mouth flooded with the taste of rich river mud. A stronger or a more composed swimmer would not have drowned—he would've calmed himself down and bobbed on the water, breathing, waiting for the spasm to pass. But not a scared boy who thrashed more as water flooded his mouth and his belly grew heavy with the river he swallowed. And everyone knows that a river is simply too large for a human child.
When the boy stopped writhing and sank quietly to the bottom, pulled along by the weight of river in his belly, catfish moved closer. He had no teeth to tear the flesh, and he waited patiently for those who would cut the boy open and let the catfish feed on morsels they dropped in their frenzy—bits of intestine, shreds of skin were welcome; being a catfish, he was not picky. But then he found something he did not expect.
There was fluttering in the water, a movement too small to notice for anyone but a fish with sensitive whiskers. It went past his face, and even though he could not see anything with his nearsighted beady eyes, the catfish opened his mouth and slammed it shut, and felt a small wriggling in his belly. It was the soul of the drowned boy, and the catfish became aware.
Vimbai—her consciousness still distinct within the tiny dull mind of the fish—wondered what the fish would do, and what would happen to the soul of the boy. It bloomed side by side with hers, filling the catfish with a new sense. His small eyes snapped open, and he saw the river for the first time, like he had never seen it before. Even though his eyes were weak, he discerned every undulation of the bottom, every silvery flash of the passing fish. He watched the water turn from green to red as the crocodiles arrived and started their meal; he watched with curiosity as his former body was torn apart by snapping jaws. He ate a few dropped morsels, but the hunger—the forever hunger that propels every catfish forward—had subsided, dulled by curiosity and flood of new sensations as the minds of the boy and the fish circled each other, sizing each other up. Vimbai remained an observer, lodged there in the catfish's mind like a foreign body, a dream splinter.
The catfish—or man-fish as he called himself—grew older and larger by the year, and the boy inside became a part of him. He remembered everything the boy remembered—the faces of his family and how many chickens they had, the name of the prettiest girl in his village, the address of some relatives in Harare. But the memories became mere decorations, baubles suspended in the vast and labyrinthine mind of the fish. He grew more cunning and more clever, but not more compassionate or introspective. He remained a catfish at heart, and he always hoped for another body and another soul.
# Chapter 7
The next morning they were still at sea; the waves were more restless than ever before—they reared up and flung their foam-topped crests against the walls of the house and expired in salty sprays. Vimbai ran from one window to the next, clearing away either meat or succulent green tendrils that always grew across the panes when she was not watching, anxious for any sign of motion. But the waves masked whatever trail the house had been leaving, and she feared that they had stalled or the ropes had torn or the crabs had died. The memory of her man-fish dream came back, and she imagined the scavenger fish crawling onto the nets, squeezing into the crab and eel traps to devour the gruesome bait left for them by the fishermen.
She imagined her horseshoe crabs now, dead on the cold pebbled bottom of the ocean, devoured by the wily fish—and, she thought, those fish would devour Vimbai's soul as well. If the crabs followed her because they had some connection to her, then, Vimbai reasoned, a fish could potentially get to Vimbai. More and more she relied on her grandmother's way of thinking, and with every passing minute the urge to check on the crabs grew stronger, almost physical, in her chest.
And yet, they had warned her. There were fairytales and forebodings, there was fear. Vimbai felt just like she did when she was little, when her mother left her by the supermarket's entrance and told her to wait. Vimbai waited until the thoughts swarmed: what if mother left without her? What if she forgot about Vimbai? What if she fell and needed help? She knew that she should wait, but anxiety would get the better of her every time and she would go looking, and then they would spend a good half-hour looking for each other along the endless rows of shelves that seemed to house everything except whatever one was looking for at the moment.
Before the memory finished flashing through her mind, Vimbai knew that she would have to check on the horseshoe crabs—whatever fairytale punishment was reserved for her would surely be better than the agony of not knowing and yet driving herself desperate with anxious imaginings, just like her mother's yelling at Vimbai for not staying put was far better than waiting in one spot.
She headed down the hallway but found that the steps leading downstairs had been overgrown by a particularly prickly variety of barberry bushes. She tried to struggle down the stairs, but the thorns left painful scratches on her arms and legs. Vimbai wished she had the machete they kept in the kitchen, behind the stove. She called for Maya or Felix, but no help or even answer came.
She tried to push through the prickly bushes but they pushed right back, gouging deep marks into her shoulders, tearing at her jeans like angry claws. She retreated and the bushes followed, pushing her into her room with unseemly glee. She backed away until her room was overgrown by barberry and a particularly nasty medicinal smell, and her back was pressed against the windowsill and thorny branches studded with bright red berries waved in her face. Only then she realized that the vegetation inside the house was rarely so aggressive, and felt the first prickling of fear and sting of her sweat in the new scratches on her forehead.
Vimbai had no other recourse but to open the window. It was close enough to the surface of water, Vimbai reasoned, and she could easily swim to the porch—despite the cold, she felt confident that the distance was short enough to cover with two or three long strokes. Plus, it would give her an excuse to sneak a look at the crabs, and then she would get the machete and deal with the insolent vegetation. She drew in a deep breath and pushed through the window, dangling ungracefully for a moment and then plunging, head first.
She did not expect the cold to be so cutting—the embrace of steel-cold water tightened around her chest, and Vimbai sucked in a breath and reached for the porch. It bobbed farther away than she expected, and with the resignation of someone in a bad dream she realized that the house was moving away. She tried to swim, but her lungs felt frozen and heavy, and her legs and arms weighed her down with useless bone and cramped muscle.
She called for help then, her voice too small to be heard in the house. Her legs kicked hard as she tried not to let panic set in, and she called for the crabs, for her grandmother, for anyone to come and help her. Her legs leaden, her arms useless, she felt herself slipping, sinking under the surface, and with no grandmother to keep her warm to guide her vision, there was only murky water; it poured into her mouth and filled her stomach, heavy like a brick.
And then, a hand—several hands, several arms, as many as an octopus, lifting her, pulling her head above the water. Several legs kicking by her, various in size, but all strong. Vimbai recognized Peb. She was too muddled and cold to feel real surprise, just extreme gratitude. So there was a reason why the silly thing was attaching every phantom limb it could find to itself.
And then, the porch swam into her field of vision, and she reached out her hand—clawed, unfeeling—to hook it on the edge. Peb helped her up, dragging her out of the water, and held her, protective and sympathetic, as she retched what felt like gallons of seawater. Her teeth would not stop chattering.
Maya and Vimbai's grandmother came out of the kitchen and hustled her inside, to sit close to the stove they turned on for just that purpose, and to be rubbed by large fluffy towels. Vimbai was too muddled to make sense of their exclamations, and only felt vague irritation when they persistently shook her by the shoulder and kept asking if she was okay and if she could feel her feet and fingers.
"Hypothermia," Maya kept repeating. "This ain't good."
The ghost brought blankets and warmed them by the open oven, and Peb hovered nearby. Vimbai closed her eyes—all the movement and noise distracted her from something nagging at the back of her mind, persistently enough to distract her from the fact that she had nearly drowned.
And like a photograph in a vat of developer, the image appeared in the black background of her eyelids. It was a palimpsest of the image she had seen underwater but was too frightened to absorb at the time. Now, it stood before her with a steady clarity.
She saw the ropes stretched taut and the horseshoe crabs festooned along them all the way from the bottom to the foundation. They did not pull but were carried—and Vimbai cried out and opened her eyes once she discerned the beasts that did all the pulling.
"She's in shock," Maya said to the ghosts, and patted Vimbai's hand. "Hang in there, sweetheart. You'll be fine, you just have to warm up a little."
Vimbai just shivered in response, thinking of the monsters—giant, ancient—pulling the house along. Monsters that left deep gouges in the sand, barnacles on their cracked carapaces, their eyes rotted out, their tails broken. They moved on clawed legs covered in cracked exoskeleton, exposing rotting bits of their flesh. Hagfish followed them, occasionally swimming up and ripping out chunks of putrid flesh, and still they moved—gigantic, undead horseshoe crabs, animated by some ancient and unknown will.
The vadzimu took Vimbai's hands into hers. "What have you seen, granddaughter?"
Vimbai shook her head and looked away, afraid that the terrible vision would leak from her eyes into her grandmother's. She did not want to share, not just yet—sometimes one had to be alone with knowledge to absorb the enormity of it. Sure, a burden shared was lighter, but sometimes one needed to appreciate the entire weight so that the future relief would seem all the more precious. So Vimbai swallowed and stared out of the window, feeling blood pulsing in her lips, warming them.
The kettle blew a sharp whistle, and Maya hurried to make her a cup of tea. Vimbai swallowed the scalding fluid, not caring that the skin in her mouth peeled, her stomach filling with warmth—filling with life, and the sensation was enough to chase away the terrible image crowding her mind.
She tried to make sense of it, as she always did—when she was little, she was taught that any problem had a solution, and if one just jiggled the pieces a little and squinted, looked at them sideways, then the general pattern would become apparent and everything would fit, suddenly, in a flash.
When she became older, she learned that some problems resisted such treatment—they were solved not by a flash of inspiration and sudden insight but by tedious, boring work—and too often, one did not truly solve them, just demonstrated enough of the ability to think to earn a passing grade, but the solution of the problem remained unknown.
But neither inspirational nor incremental approaches helped her to deal with the undead crabs. She was willing to accept that the house and the three housemates plus assorted ghosts fit together, that the horseshoe crabs were their allies and the fishes who devoured souls were enemies; she could live with her ability to control the crabs, just like she could forgive Maya her half-foxes and Felix his desiccated heads. But she could not move past the simple acceptance and start finding answers to why and how and who and for what purpose. She could only shiver in front of the stove and drink tea.
The two worldviews were at an impasse again, and there was not much Vimbai could do besides trying to incorporate them both; pinning them against each other so that either one would yield answers seemed far beyond her capabilities.
"Why did you jump into the water?" Maya asked, apparently judging Vimbai to have recovered enough.
"I couldn't take the stairs," Vimbai said. "I was attacked by the shrubs, and didn't have a knife. I called, but no one came. There were prickly shrubs chasing me all the way to my room, and they smelled like a hospital."
Maya arched her eyebrows. "I haven't seen them, but if you say so. It was still a stupid thing to do."
"I know," Vimbai said. "But it's not like there were other options."
"There are always other options," Maya said. "Come on, I'll take you to your room, and I'll show you a workaround for the stairs. And then you'll go to bed and sleep and feel better, okay?"
"Okay," Vimbai agreed and stood up, still shivering, clutching the warm blanket gathered at her neck.
The chipoko picked up Vimbai's t-shirt and skirt off the floor. "Don't worry, I'll dry your clothes and bring them to you. Go rest now."
Vimbai followed Maya, sulking a little—how come everyone but herself knew about this workaround? Then she remembered sitting by the window while Maya explored, but didn't feel better for it.
Maya led them into the living room and opened the doors of the cabinet that housed assorted plates, dishes and knickknacks they didn't quite have a place for. The knickknacks were gone now, subsumed by a path carefully marked in white sand, leading into a copse of tall and narrow trees—they lined the path like columns, and their branches twined overhead, creating a filigreed tunnel, black against the pale grey sky.
"Where are we?" Vimbai asked.
"Pantry," Maya said, and shot her a smoldering look. "You really need to get to know the house, you know. It's getting bigger every day."
"But why?" Vimbai whispered, overwhelmed with the weight of accumulated disbelief. "What is happening to us?"
"Who knows?" Maya shrugged. "Who cares? Enjoy it while you can, why don't you? There will be tons of boring shit in your life, okay? I promise."
"Okay," Vimbai sighed and followed along the path, next to a very clear and very fast brook that silvered between the trees. The path turned into a doorway Vimbai did not recognize, past a few skinned couches and a folded ladder dripping fresh white paint. Vimbai decided not to ask, since the questions were likely to yield only additional frustration instead of answers.
Maya grabbed her hand and squeezed hard. "Look, Vimbai," she said. "It doesn't matter why or how, don't you understand? Back home, girls like us, we're nothing. We work hard and make good, and sometimes someone might compliment you for it. But we don't run things; they are run by white guys and rich people. And here, now . . . we make the rules, see? It's ours. Maybe the house will grow bigger, and we'll get some milk from ShopRite and come back here. We can be queens here, queens of all we see, of crabs and ghosts and oceans. We can float like this forever, and no one will ever tell us what to do."
"What about Felix?" Vimbai said.
"What about him?"
"Nothing. Just haven't seen him in a while."
Maya shrugged and let go of Vimbai's hand with one last squeeze. "You can be the queen of Felix if you want, I don't mind."
Vimbai wished Maya would hold her hands just a bit longer. "I don't know," she said. "I mean, it sounds nice. But we're going back, you know? This feels like make-believe. A pretend world that doesn't really matter. Wouldn't you rather matter in the real world?"
"The queen of New Jersey," Maya drawled, and laughed. "Maybe."
They rounded the last outcropping of furniture and rock covered in what looked like fur, and Maya pointed at the mouth of a cave that yawned at them from between two striped signposts. "See? Can't miss it."
"Where does it lead to?" Vimbai asked.
"The hallway by Felix's room," Maya answered. "It's interesting, both Peb and I noticed it—no matter how much this house changes, the paths stay constant. So you can't get lost. Well, you could, but not really, not for long."
"You spoke to Peb," Vimbai said. Not really a question, just a statement of fact heavy with implication.
"Well, yeah. You were either staring out of the window or on the porch with your face in the water and butt in the air. What was I supposed to do?"
Vimbai shrugged, pleased that Maya was so willing to have this argument rather than dismissing Vimbai's words or denying her any demands on Maya's time and attention. "I'm just surprised. Where are your animals?"
"Roaming somewhere." Maya entered the cave, swallowed by the darkness, and only her voice reached Vimbai, strong and clear. "Funny thing how they always come when I call. I wonder now if everyone has a secret animal army."
"Then why doesn't everyone know about it?" Vimbai asked, and followed.
The cave was utterly dark, but just before Vimbai was ready to get scared and start flailing in search of a wall or anything solid, the darkness opened up before her, and she glimpsed the familiar walls with studs incompetently hidden under layers of paint, and the 'Keep Out' sign on Felix's door.
Maya waited for her in the hallway. "You'll be all right?"
Vimbai noticed that she was no longer shivering and nodded. "I'll be fine. I'm just going to go lie down for a bit."
"Okay. My dogs and me, we'll go roaming for a bit, but I'll check on you later."
"Dogs?"
Maya laughed. "I know, but I have to call them something. They act like dogs anyway."
Vimbai headed to her room, pushing aside the familiar curtain of dangling vines, bromeliads, and occasional orchids, their white roots twined like tortured fingers. It was pretty, she had to admit, and the shortcut from downstairs Maya showed her made her mind swell with possibilities. Maybe it was time to let the crabs do what they were doing.... And then she remembered the undead giants again and cringed.
Vimbai lay down on her bed, pulling her blanket over the one her grandmother wrapped her in. She squeezed her eyes shut, chasing away the gruesome images that crowded her retinas as if tattooed there. Funny, Vimbai thought; she was perfectly fine with the spirit of her dead grandmother making coffee in the kitchen and sometimes talking on the phone that was once again full of static and whispers, yet she refused to contemplate undead creatures. Spirits seemed cleaner to her, uncontaminated by rot and flesh. There was purity about the ghost, a creature of mere spirit, with its human flaws falling away, leaving the clean burning fire of the soul. The vadzimu was above and beyond her razorblades and her belief that politics was only relevant when it interfered with her vegetable garden.
Warmth came gradually, and Vimbai did not notice the exact moment when she no longer felt cold. She snuggled into the blankets as if they were a nest, and smiled. Tomorrow, she thought, tomorrow she would feel better and she would go roaming with Maya and her dogs, and would not think about the terror that dragged their house along, crawling across the sandy bottom on rotten broken legs. She would forget the aggressive shrubs and the hospital smell that still lingered in her room.
"Wake up, Vimbai, wake up, wake up." The voice droned as if from a great distance, and for a while it was possible for Vimbai to pretend that the voice was a part of her dream. Then a hand shook her shoulder unceremoniously, and she opened her eyes, annoyed.
Felix's hair had grown restless since she last saw him—it reared up and guttered like flames in the wind, reached out to lap at Vimbai's pillow. One especially long and hungry tongue stretched toward her face but Felix batted it away, his hand disappearing momentarily.
Vimbai pushed herself up on her elbows and yawned widely, too tired to care. "What?" she said. "Why'd you wake me?"
"I am troubled," Felix said. "We are out of beer."
Vimbai sat up, awake now. "Are those separate statements, or are you troubled because we are out of beer?"
"Separate." Felix sighed, miserable. "Although lack of beer doesn't help."
"What's the problem?"
"There are things happening . . . up there." Felix pointed upward to his hair, in a small gesture as if afraid that the hair would notice. "I don't know what to do. There were things . . . crawling out of there, and I didn't know anything could leave there."
"What sorts of things?" Vimbai asked.
Felix shuddered. "Dead things. With legs and long spiky tails, just last night. I woke up and almost died, I swear."
Vimbai swallowed and hugged her knees to her chest to ward off the chill thrumming along her spine. "They are the ones pulling the house now. But what were they doing in your hair?"
"I dunno," Felix sobbed.
"You said your hair separates things," Vimbai said slowly. The hypothesis was starting to form in her mind but lacked shape, and Vimbai hoped to coax it into proper expression by verbalizing it. "So there were crabs, undead ones, separated from their lives. In your hair. So I assume they crawled in there first, without you noticing."
"I was sleeping," Felix said and glared defensively, his eyes staring in opposite directions, giving him a simultaneously angry and confused look.
"They said it was too cold to go to New Jersey," Vimbai said. "Maybe they decided to become undead to get us there."
"A hell of a sacrifice," Felix said. "But . . . you were in there."
"Partially."
"And yet your soul did not leave you."
"I know," Vimbai said. A thought skimmed at the edge of her consciousness, too fast to grasp properly. "Say, do you know anything about man-fish? Njuzu?"
"No," Felix said. "What's that?"
"It's a Zimbabwean urban legend," Vimbai answered. "It's a fish who swallows the souls of the drowned, and then it itself becomes sort of human. Like it can talk and stuff."
"You thinking it might work with crabs?"
"I don't know," Vimbai said. "Only my grandmother was talking about man-fish, and then I dreamed that I was one. And then Maya's pets are afraid of fish."
"And then there's the house that attracts ghosts."
"And your hair."
They sat a while, puzzling, unable to tease any sense from the conglomeration of occurrences and half-baked ideas. That was the trouble with the supernatural, Vimbai thought—you didn't know what laws ruled it, and what was a coincidence and what was a sign and what was weird and what wasn't. It was like a whodunit, only the clues refused to be arranged into any sort of hierarchy or a straight narrative, and most of the time it wasn't even clear if they indeed were clues; a jigsaw puzzle where all the pieces were blank.
Felix's mind was apparently on the same track. "It's like life," he said. "I just don't know what matters and what doesn't and what I should pay attention to. But these crabs, they were just creepy."
"They are victims," Vimbai said, although she was not so sure about the undead variety. "They . . . they are killed and bled half to death, and thrown back in the ocean. And it's not just about them—so many birds feed on their eggs, and without feeding they cannot migrate. They would die without the crabs—these crabs carry so much on their backs."
Felix nodded. "Is this why you're studying them?"
"Uh-huh," Vimbai said. "And because they are so ancient . . . and we can kill something so ancient, so irreplaceable. It's just wrong, you know?"
"I didn't see it as a moral issue."
"All conservation is a moral issue," Vimbai said, and thought of her grandmother downstairs. "Be it animals or people or cultures. Some things are just . . . unique, and if you lose them, you can never get them back."
Felix touched his hair, cautiously. "I think this thing is unique."
"Probably," Vimbai agreed. "God, I hope so."
"You want to look inside again?"
Vimbai shrugged. "Maybe. Why don't you take Balshazaar out and ask him if he saw any crabs or if he knows anything?"
Felix slumped. "I knew he would tell you," he said. "I shouldn't have taken him out, only you have to understand. What, you think it's easy living with this thing?"
"I don't know," Vimbai said. "What is it like, and where did it come from?"
"It was a long time ago, sister," Felix said. "I don't even know what I remember and what I imagined. Does it matter?"
"I don't know. Listen, if you want to talk, maybe we should go for a walk or something. You can show me around."
"I haven't seen much myself," Felix answered. "But there is a place that's sort of nice. I'll wait for you outside."
Vimbai waited for the door to close behind Felix, and got dressed. She really needed to get some laundry done, and she quickly gave up on finding matching socks. One white and one striped, it didn't matter. Thankfully, t-shirts and shorts were abundant.
Felix led her down the hallway and into a closet that abruptly transitioned into a view of a desert, with a lake in the middle of it. The sand surrounding it was red and dry like Kalahari, apparently unaware of the abundance of clear, cold water in its midst. The sun was getting warm, and Felix offered Vimbai a handkerchief to cover her hair; she accepted with a muttered thanks.
A couple of lawn chairs reclined by the bank, surrounded by a sparse growth of dried up grasses and papier-mâché trees, tall enough to reach Vimbai's knee.
"Maya doesn't come here," Felix said. "There's fish in this lake."
"What kind?"
"Take a seat and you'll see."
Vimbai did, and Felix took the other chair. They sat in silence, watching the smooth surface of the lake, gray and reflective like mercury, until there was a loud splash and a fish came bounding out of the water and into the air. It somersaulted and entered the water. Vimbai did not need a second look to confirm what she knew from the moment Felix mentioned Maya—it was a catfish, and a large one at that.
As they watched, the catfish stuck its flat head out of the water and gave them a narrow-eyed, jaundiced look. "You have a tasty soul," it said to Vimbai, and leered.
# Chapter 8
Back in her room, Vimbai could not calm down her heart. She was too disturbed to be really embarrassed about running away from a fish, and only a small measure of her discomfort resulted from her recent show of cowardice. Not that Felix cared—he seemed to have fears of his own, fears of undead things that somehow crawled into his hair and separated their bodies from their mortality. That was enough to make anyone nervous.
They sat on Vimbai's bed, like scared children, and occasionally one or the other would steal a quick look at the door, as if the catfish would somehow follow them here. It was silly, of course, Vimbai told herself, just like it was silly of her to shower with her eyes open for weeks after seeing a horror flick. But some things were just not subject to rational reasoning, and recently that particular mode of relating to the world had been taking one hit after another.
"We have to talk to Balshazaar," Felix said. "I have to know what he'd seen. Only I don't think I should take him out again."
"Why not?" Vimbai breathed a nervous laugh. "He'll fit right in with the psychic energy baby and the undead crabs."
Felix winced. "Don't remind me," he said, and stood up.
Vimbai watched him pace from door to the window, until he noticed the phantom limb he'd given Vimbai the same day Peb joined them.
He smiled. "You still have this thing."
"What else would I do with it?"
"Give it to Peb."
"I like it," Vimbai said. " And Peb has plenty already. It seems so . . . delicate." The limb indeed resembled a work of art with its translucent veins and milky nerves twisting below the glassy skin like tree branches.
"Take it with you when you go talk to Balshazaar," Felix said. "If you convince him to come out, maybe he can use it to get around."
Vimbai raised her eyebrows. "Somehow you bypassed the point where I agreed to look back there. Besides, you just said that it wasn't a good idea to take him out."
"I don't know." Felix stopped pacing, his eyes simultaneously expressing great consternation in opposite directions. "Maybe you could look inside and see if there are still any crabs left there. I'm afraid . . . afraid to put my hand in there."
"So you want me to risk my face."
"You at least can see."
"What if it takes my soul?"
"It didn't before."
Vimbai thought that after jumping blind into a cold ocean she really ought to know better. Instead she sighed and carefully eased her head inside Felix's hair.
Balshazaar was there, floating vaguely as was his wont. "Hello again," he said.
"Balshazaar," Vimbai said. "Have you seen any horseshoe crabs around?"
"Sure did." Balshazaar bobbed, his chin pointing to his left.
It took Vimbai's eyes a moment to get used to the dusk in Felix's hair, and she saw several small translucent crabs that clung together in a tight cluster. The souls or lives or whatever it was they shed like old carapaces and left behind, just so they could take Vimbai back home. Acute pity made her catch her breath and whisper, "I'm sorry" to the crabs. They remained motionless, devoid of any spark that would indicate that they could hear and understand her.
"Balshazaar," Vimbai continued. "Would you help us? There are things happening we don't quite understand, and since you had a chance to observe the happenings here, perhaps you could explain them to us. Figure out what's going on."
"What's in it for me?"
"A leg," Vimbai said. "A phantom leg, but it is nonetheless functional."
"Interesting," Balshazaar drawled. "Why so nice?"
Vimbai considered telling him that she was usually nice, but instead settled for a reason he was more likely to believe. "We need you."
"I'll help you," Balshazaar said. "Only I'm not sure if I even want to leave here—it's nice. Secure. Bring the leg and then ask your questions. However, know that I promise nothing."
Vimbai extricated herself from the pocket universe, and reported on what she had seen. At the mention of the horseshoe crabs' souls, Felix made a small sound of terror.
"It's all right," Vimbai said. "They are not doing anything. And they are much smaller and a lot less scary than the ones that crawled out."
Felix shook his head and the long tongues of his hair stretched and contracted, reminding Vimbai of the way leeches moved—she had observed them in her invertebrate zoology class, and was endlessly fascinated by how they managed to grow long and thin one moment, and short and stout the next.
"I'll give him the leg," Vimbai decided, "and ask him about what else he saw. And what he knows about fish."
"Not yet," Felix answered. "Let me think about that. I'm not sure I really believe him."
And so Vimbai was left alone again, while Felix retired to his room to do his thinking. He puzzled her—his inability to make up his mind and his frank terror at the things living in his hair surprised and bothered Vimbai. He should've had enough time to come to terms with it, she thought, especially since he had been so nonchalant about extricating the Psychic Energy Baby from the phone wires. It took her a while to realize that he never got around to telling her about how he came to wear a personal-sized black hole around his head. Then again, men were good at avoiding questions.
She remembered how excited her mother had been when they traveled to Harare—especially excited to see her favorite nephew and Vimbai's cousin Roger. Roger seemed to be one of those kids who were so great one could never hope to compare to them—and Vimbai resented Roger before she even met him, even though he was not a kid anymore, but already a grown man, with a wife and intentions of starting his own business. Vimbai's mother talked to him on the phone for hours, making plans, and phone bills be damned.
When they had arrived, Roger was not home—he was not in Harare at all. The relatives said that he was on vacation, but by their sidelong glances and uncomfortable shuffling, Vimbai surmised that the vacation was a polite lie. Roger's wife had stayed home, and nobody seemed to know anything about his destination. Vimbai's mother did not believe the excuses either—she became thin-lipped and taciturn, and did not again mention Roger until they went back home.
It was two years later that Vimbai and Roger finally met. Roger had started his business—something to do with laptops or some other technology Vimbai had only pragmatic interest in, and he traveled to the US under some business pretense or other. In truth, they all knew that he wanted to see Vimbai's mother who was never good about hiding her disappointments—they came through even in long-distance phone calls.
Roger arrived on schedule, and quickly filled the house with his laugh that seemed to be coming directly from his diaphragm and his expansive gestures. He was smaller than Vimbai expected, and sadder—when he thought that no one was watching him. He did not have to apologize—he only hugged Vimbai's mother until she cried and hugged him back. Roger said, "I'm sorry, Auntie," and that was that.
But not as far as Vimbai was concerned. Roger was difficult not to like but she persevered, helped by the eternal teenage sullenness. She watched him across the table, her arms crossed in front of her with disapproval. For all his laughing and joking and telling stories and flashing pictures of his baby son, he noticed.
"What's the matter, Vimbai, cousin?" he asked her one day. Normally, Vimbai would've avoided a direct confrontation, but he caught up with her as she exited the bathroom, and there was simply no missing each other in the narrow hallway. "Did I do something to tick you off?"
"You blew off my mom when we went to Harare," Vimbai said.
He whistled. "That was a long time ago, muroora," he said. "You don't forget, do you? Take after your mom."
"Where were you then?" Vimbai said. "Just don't say vacation, or I'll have to slap you."
He laughed unexpectedly. "Why do you care so much?"
"You should've seen her face," Vimbai said. "She really missed you then, and you weren't there. She cried every night."
"That did not necessarily have anything to do with me," Roger said and frowned. It was strange to see him in their suburban wallpapered hallway, blue cornflowers on white background.
"Still." Vimbai leaned against the wall, her shoulder pushing against the familiar solidity of the wall. "Tell me."
"And you'll forgive me."
"Depends."
"No." He shook his finger with emphasis. "You forgive your cousin, okay? And then I'll show you."
She shrugged. "Okay. What did you want to show me?"
He turned his back to her, and Vimbai thought that he was about to head back to his room to bring some evidence—pictures or flowers or whatever to make it all right. Instead, Roger carefully eased the hem of his white shirt from his belt, partially obscured by his nascent love handles, and pulled it up.
Vimbai stared at the very white and very straight scar that slashed diagonally across the left half of his lower back. At first, she thought that it was a particularly vicious muti mark, or some other creepy magic her grandmother believed in and that required mutilation. "What is it?" she said.
"A scar." Roger lowered his shirt and turned to face her, blushing. "You're such a curious little cousin, and I just met you and you already asking me questions my wife wouldn't ask me."
"Maybe she should. What is it?"
He sighed. "I needed money to start my business. Twenty thousand dollars—where would I get that?"
"It's a lot of money," Vimbai said. Especially in Africa, she thought. That was a fortune enough to propel one forward in life, not just pay off student loans or credit card bills.
"Yes. So I sold a kidney."
She stared into his face looking for traces of jocularity, but he was serious, and the scar real. She felt herself blush. "I'm sorry I was a bitch to you, Roger," she said.
He waved his hand in the air. "Don't mention it, sister. And don't tell your mom. Believe me, some things only you want to know."
Vimbai had to agree as she remembered this conversation. She seemed to have a talent of getting hung up on questions everyone around her circumvented so smoothly—if people were leaves floating on the river surface, Vimbai would be the one that always got stuck against every obstacle, no matter how trivial and easy to bypass.
And now something else was nagging at her. She thought of the man-fish and how he manifested as soon as Vimbai dreamt him; then there was the vadzimu, who appeared when Vimbai imagined her as an entity that kept her and her mother so much apart. Now, the memory of Roger worried at her heart in the same way. What did it mean? she asked herself. Why did Felix's reluctance to speak remind her of her cousin?
The scar. That was it, the way Roger hid his scar and its origin. Vimbai jumped to her feet and rushed to Felix's room.
He was there, doing nothing, and only looked vaguely up when Vimbai came busting through the door.
"It's a scar, isn't it?" Vimbai said.
"Yes," he said, paling.
How does a man become a scar, or at any rate end up wearing one around his head? Only Felix knew, what it was like to cut an umbilicus that bound one to the universe that bore him, and to wear the spectral navel that still festered with the remnants of the enclosed space and its dark inhabitants. A dying tiny universe, and poor Felix dangled on the end of it, like a superfluous appendage.
And unlike the Psychic Energy Baby, he could never hope to disentangle himself from the wires that kept him suspended, the appearance of him standing on the ground a mere illusion. Still, he managed a small unconvincing smile. "I didn't know how to tell you or Maya. Or even what to tell you. And I still don't understand how the two of you play into it—you're dragging all those ghosts with you. And her, I don't even know."
"I'm dragging everything with me," Vimbai answered. "Even Africa—only it's not my parents' Africa, it's an imaginary one."
"What is Maya dragging?"
Vimbai shrugged. There were the half-foxes, of course, and there was the wild streak, the talk of being queens of some imaginary kingdom, be it New Jersey or somewhere else. For the first time, Vimbai thought that it might not be a bad idea—perhaps Maya was the expression of their purpose, the reason for them being here, at sea, floating somewhere . . . or perhaps standing still. Or perhaps the house stood still as the world moved under it, offering its watery and glistening curving back as they slid inexplicably toward some destiny, some mystical version of New Jersey.
The house . . . Vimbai gasped a little and sat down on Felix's unmade bed. "All of us," she said. "It's the three of us—your blind universe and my ghosts and Maya's dogs. We did it to the house."
"I assumed as much," Felix agreed. "So what?"
"So maybe we can control it," Vimbai said. "Maybe we can make it into something we want."
"Like what?" Felix asked, perking up.
"I don't know." Vimbai thought of the curdled milk in Maya's coffee and wrinkled her nose. "ShopRite, for starters. Or Farmers' Market, whatever. Something that sells food and milk. I'm really sick of canned ravioli."
"Me too," Felix agreed. He seemed quite eager to divert the conversation to a topic other than himself and his hair. Poor boy, dangled from some impossible hole like a piece of bait on a hook. "How do we make things happen?"
"I have no idea," Vimbai said. "Think of them really hard?"
"Okay," Felix said. "Here, or do you want to go somewhere else?"
"Somewhere else," Vimbai said. "Let's go to the porch."
There, they sat cross-legged, their backs hunched, bracing against the cutting wind that rose from the ice-cold water, slashing their faces like steel cables. Vimbai crossed her arms in front of her chest and stuffed her hands, numb already, into her armpits. She closed her eyes, and for a moment concentrated on feeling Felix next to her, his warm breathing present, so touchingly and surprisingly human.
There was a creaking of the steps and a soft jangle of the screen door.
"What are you doing?" Maya asked. Her foxes sniffed at Vimbai and circumvented Felix in a wide arc. "Can I help?"
"Sure," Felix said. "We're trying to make the house do what we tell it."
"What are you telling it?" Maya sat next to Vimbai, her warm elbow jostling against Vimbai's.
"To make us a ShopRite. We decided that we've changed the house, so might as well try to direct it."
Maya shrugged. "Makes sense."
The three of them sat in silence. Vimbai squeezed her eyes shut and felt her forehead furrow as she imagined the cool aisles of a supermarket, shelves upon shelves, a solid white front of gallon milk jugs and white gleaming egg cartoons. Boxes of butter and cream cheese, bagels stuffed neatly into plastic bags. Thick slabs of meat in their little Styrofoam coffins, yellow cheese, red apples. All of it.
And she pictured her mother, frowning at the row of canned beans. "These are all the same thing," she told Vimbai with irritation. "Same beans. All that's different is a picture on the can."
And Vimbai herself, scowling back, longing to go home. "Come on, mama. These are just brands—you know it."
Her mother rolled her eyes and tossed a few cans into the cart, not even looking at which ones she picked. "Today at the department someone asked me about culture shock, and if I was overwhelmed with choices when I first came here. Americans, they always expect us to be overwhelmed with food."
"I'm an American," Vimbai mumbled and followed the cart and her mother's receding back miserably.
"It's not what I meant. You know that it makes no difference how many different pictures you put on green beans—they are still the same green beans inside. It's an illusion of opulence they expect us to be impressed by and indulge in."
"It doesn't matter," Vimbai had said. "People are just curious, you know? It wouldn't hurt you to be nice once in a while."
"After fifteen years of answering the same questions, my niceness and my patience are almost gone," mother said.
"It's not a big deal," Vimbai said. She wanted to add, "Lighten up," but thought better of it. Nothing brought quicker and more thunderous retribution upon her head than suggestions that her mother should lighten up or relax.
She opened her eyes to the sight of the leaden ocean. It was beginning to snow, and heavy viscous waves swallowed up the snowflakes as soon as they touched water. Why did her mother always have to insinuate herself into Vimbai's daydreams? She did not want a replica of her—Solaris had scared her half to death when she first read it, and the thought of an intelligent needy fake with her mother's personality was too terrifying to contemplate for any length of time. She just wanted a gallon of milk and some fresh fruit. And yet, her mother hovered on the inside of her eyelids, insubstantial but persistent, her narrow face wearing its habitual expression of grim readiness to pounce every time a perceived slight occurred.
Vimbai's mother remained in her heart forever, her bitterness as familiar as the smell of coffee in the morning. She wasn't always like this, Vimbai reminded herself—there were times when she was happy and carefree, and laughed easily. There were times when her parents whispered and giggled like guilty children, and no matter how old Vimbai was, these times always made her feel like the rift between her and her parents simply disappeared, leaving no trace, no scar.
But Vimbai had to make an effort to remember the happy times—she often wondered if this was a defect in her, or if it was something common to all people, this reflexive dwelling on the anger and the distance, on all the times where her mother and she squared off and argued in circles, as Vimbai's gentle father sighed and tried to ask them be nice to each other; how desperately he tried to smooth the wrinkles that creased the surface of the life he would like to have, disfiguring it. Vimbai felt guilty for not thinking about him often enough, for focusing so much on her mother and the many ways in which she made Vimbai angry.
"Are you thinking about ShopRite?" Maya asked, jostling Vimbai back to the freezing porch and the cold waves, to the hidden horrors under the deep, deep water.
"Kind of," Vimbai answered, and dutifully imagined the beading of condensation on the sides of milk jugs and the doors of walk-in freezers fogged by breath, hiding stacks of frozen pizza boxes and foil packets of cauliflower and chopped spinach.
"Should we check on how it's going?" Felix said. "I'm cold."
Maya stood and stretched, her dogs following her lead as one. "I suppose, I only wish we knew where to check."
"Huh," Felix said, and stood too, shivering. "This house is very big."
"Can't your dogs sniff it out?" Vimbai said. "They have to be good for something."
Maya ignored the implied insult, and laughed. "A good idea, only they don't know what a supermarket smells like. I guess we'll just have to go look. Come along, Vimbai." She grabbed Vimbai's arm and pulled her to her feet. "You are just not content with hypothermia, are you? You want to add pneumonia to the list?"
"Or pleurisy," Vimbai mumbled, and followed Maya and Felix inside. "Maybe I like the cold."
They bade the chipoko and Peb to hold down the fort in the kitchen, and set out on a search for a supermarket, through the pantry and across a narrow jungle strip. Vimbai contemplated the mountains off in the distance, and did not bother to try and figure out how they fit inside the house.
# Chapter 9
Vimbai had to admit that there was certain fun in discovering a new world and getting to name everything. Thankfully, Felix was content with his stub of a universe, and did not presume to offer names. But Vimbai and Maya, oh how they argued. Martin Luther King Forest was not a problem, and Malcolm X Mountains had a ring to it; Vimbai insisted that the lake with the catfish (which they wisely circumvented, not yet ready to deal with the cunning adversary) had to be named after Marechera, and the thin gurgling brook that flowed from the lake and then roared to magnificence somewhere down at the basement was fit to maintain a literary kick—Achebe River it was, even though it was no Niger. They argued about whether a plain covered in nettles and rusted bed frames was impressive enough to name, and if so—whom would it belong to.
"You can just call it the Bedframe Valley," Felix suggested. "Or think about it later—now, I want to keep going."
"Fine with me," Maya said. "But the next thing will be named after Oprah."
Vimbai snorted. "No way. Wangari Maathai is next. Surely, a Nobel laureate is more important than Oprah?"
"And I want more literary tributes," Maya said. "How about Octavia Butler?"
"All right," Vimbai agreed. "And after that—Tutuola and Fay King Chung."
"Who's that?" Maya asked, and whistled for her dogs to get back as they chased something up a steep pebbled ridge. "I mean, the second one."
"Zimbabwe's former minister of education and culture," Vimbai said. "My mom says, she is Chinese, and in Rhodesia she could get an education and black people couldn't. And she wrote children's books."
"Good enough for me," Maya said.
Vimbai thought how happy would her mother be to visit the house—the only country in the world where not a single pebble was named after a white guy.
"Your mom misses Zimbabwe?" Maya asked.
"Yeah." Vimbai thought a bit about how to put it into words. "It's hard for her. She was a historian back home, she knew all there is to know about Zimbabwe folk traditions, and here she teaches Africana Studies."
"It's important," Maya said.
"And yet it's not the only thing she could do, but it is the only job they had for her. So it's hard, you know? There's tension between the faculty, who are all black, and the department chair who's white."
Maya rolled her eyes. "Figures."
"And Africans and African-Americans." Vimbai heaved a sigh. "The whole voluntary immigration thing."
Maya nodded that she understood and they walked in silence, pebbles and dry leaves crunching underfoot. The whole experience did not quite feel real—Vimbai noticed the especially artificial quality of the landscapes inside the house. Sure, there was a sun and a semblance of sky—at least, if she did not look too closely; if she did, the light fixtures and the whitewash of the ceiling became apparent, as if peeking through the illusion of the natural phenomenon. No, it was something more fundamental, and it took her a while to puzzle out that this quality was due to the absence of smell. She could smell neither water nor knee-high grass, only indeterminate stale and warm odor, like a pillow freshly slept on.
She was about to share her observation with Maya, when Maya pointed to their right.
"Look!" Maya said.
Something gleamed at a distance, just over the spiny ridge made of some unfamiliar rock layered like slate, baby cribs, and a tangle of steel cables, and they hurried up the slope. Vimbai breathed deeply, trying to taste something in the air, anything but the dull stale smell of the old house. The gleaming behind the ridge grew brighter and higher, as if there was a sun hiding behind the jumble of rock and the discarded trash. It spilled over the ridge, casting a hazy halo, and reflecting off the metal guardrails of the broken cribs. Vimbai would've thought that they were in a landfill at sunset, if it weren't for the cursed absence of smell.
Maya hurried ahead and stopped as she crested the ridge. She was cast in silhouette against the golden light, and Vimbai felt her breath catch—there was such beauty in the outline of her roommate, such elegant simplicity in the cast of her shoulders, the set of her chin. Such strength and confidence in her legs and feet planted slightly apart; she was an explorer surveying the new land opening in the water gap, a discoverer of unknown lands and landmarks, the namer of things. Her dogs crowded around her, their black shapes filled with a quiet dignity their usual selves woefully lacked. Vimbai, enchanted, wanted neither to move nor look away.
Maya turned, tossing her hair over her shoulder. "Coming, Vimbai?"
"Yeah," Vimbai said, and reluctantly moved up the slope, into the bright light. "What is it?"
"See for yourself," Maya said.
Vimbai hurried ahead, now that Felix also reached the crest and stood quietly staring down; Vimbai could not see whether he was impressed or awed, or merely waited for Vimbai to catch up to them and share in the view.
She stepped onto the crest, wobbly under her feet, shifting with all the inclusions of broken handles and rolled up spools of cable. She looked down and cried out in surprise and wonder.
The light they'd seen came from the second sun hovering over the rooftops, as if it were about to set, but it never quite dipped below the line of buildings. But it wasn't the houses that drew Vimbai's attention—it was a line of trees covered in blue and purple blooms, blue fire flickering around the branches but never consuming them, the pure ferocity of jacaranda trees in bloom. Even though Vimbai had not seen them for herself, the memory she shared with her grandmother and her mother's stories left no doubt in her mind.
Yet, as she looked at the buildings, she decided that it was not Harare—at least, not the one she remembered. Town homes from the richer parts of the city mingled with traditional round huts one could still find in the provinces, and suburban New Jerseyan Cape Cods and bungalows. It was Harare of Vimbai's dreams which jumbled things she did not quite remember with those she knew well. These were the streets she sometimes drove in her very first car, a Geo her parents got her, she suspected, as a joke; she drove along them in her dreams, frustrated that she was unable to find home and that all the streets led in random directions, never intersecting in any satisfactory way. The city where she and her mother never fought, and friends and relatives from New Jersey and Zimbabwe dropped by without any rhyme or reason, and dead grandparents were alive and spoke English and told Vimbai they loved her . . . just like the vadzimu did.
Vimbai rubbed her face. Oh, my jacaranda trees, she thought. Oh how I missed you and yet I cannot smell your sweet blooms, I cannot feel your breath on my face. The trees and the flowers and the buildings shifted and multiplied, and rotated and blurred, then swam into focus again like beautiful images in a kaleidoscope. She realized then it was tears that twisted and purified her vision.
Maya touched her shoulder—such a habitual gesture by now, the curve of Vimbai's shoulder felt like it was shaped by Maya's hand to fit into it, just like by her mother's hand before. "What's wrong?"
Vimbai looked up, into Maya's worried face and Felix's eye rotating away and then toward her, like a possessed bloodied apple. "It's nothing," she said. "This city . . . this is Harare, but not really. This is my dream Harare . . . in the Africa of the spirit."
Maya smiled then. "This is it," she said. "This house doesn't become what we ask of it. It's what we dream about—it's our dreams that shape it, not us."
Without saying a further word, Maya started the perilous descent down the crumbling precipitous slope. Felix and Vimbai followed, slipping and trotting awkwardly at times, sliding among the small avalanches of pebbles and refuse. Their feet left deep troughs as they descended, and already Vimbai was worrying about how they would get back up this steep slope.
She forgot all about it when she stood in the street, her heart sinking. From the distance, the place had seemed alive and real enough, but once inside she could not help but feel that she had wandered into a movie set—there were no people, and the houses seemed mere cardboard facades, and a single push would bring the entire street tumbling down. But the trees seemed real enough, and she reached up and touched a knotted branch, leaves like green spearheads, with bright stars of flowers clustered among them. With the slightest of pulls, the branch came off, and Vimbai cringed expecting this violation to dispel the mirage under the forever setting sun.
"It's so pretty," Maya said, and picked a branch too. "To bad I can't smell anything."
"It's not you; it's this place," Vimbai said. "Listen, if this is where our dreams go . . . what's with the dogs?"
"A Freudian nightmare," Felix volunteered.
"Hush, silly boy," Maya said. "I do dream of all sorts of creatures, you know. About being a queen of animals. I always wanted to work at the Philadelphia Zoo." She said nothing else, but Vimbai could feel the sting in Maya's words all the same—the acute hurt of someone who wanted to work in a zoo and instead served drinks at a casino. Life really had to work on having fewer discrepancies like this, Vimbai thought. And here they all were, surrounded by the ghosts of the dreams they gave up. Maya's foxes-possums howled a bit and wagged their tails, and their beats resonated on the dry ground.
They did not find the supermarket or anything that would provide any variety in their menu. Instead, they collected great armfuls of blue flowers—Vimbai thought that the vadzimu would enjoy them, since she seemed as fond of these trees as Vimbai's mother. And Vimbai herself felt deep gratitude that she was finally able to see them for herself, however distorted they were by her dream-memory, blue-purple, ice-cold. However devoid of scent.
The chipoko was pleased with the flowers, and as Vimbai told her about the dream Harare they had found, the ghost nodded along, her hooded eyes lowered to the opulence of flowers in her arms—so thin, so wrinkled. Vimbai piled the branches higher and the ghost held them like one would a child.
Peb hovered nearby, whispering of supernovas, but seemed drawn to the flowers. The ghost of Vimbai's grandmother noticed too, and gave Peb a branch, which he immediately absorbed. His transparent hide grew suffused with the gentle purplish-blue color, and the twisted twigs of the branch poked out of his back like a grotesque fin. Vimbai did not question his need to incorporate everything that appealed to him, like she never questioned her grandmother's attachment to jacaranda trees and her ability to possess Vimbai's body.
"Maya thinks we're dreaming this house," Vimbai informed the vadzimu as soon as the old woman was able to tear her gaze away from the flowers. "Do you think we are dreaming it?"
"Some dreams you leave behind," the vadzimu answered, her voice especially old and desiccated today. "Some dreams you discard along your way, like your baby clothes. They litter your past, like small corpses, like shed skins."
Vimbai nodded, thinking, listening to the bubbling of the kettle on the stove—the vadzimu did not approve of the whistle, and wrenched it free from the kettle's nozzle, like a pacifier from the lips of a recalcitrant infant. The dead air and the strange apparition, the taste of longing and dust settling over everything, testified to the veracity of the ghost's words, and Vimbai felt like crying as she thought of the expanse the three of them had created and populated with sad little remnants of themselves. And it was so hard to decipher sometimes—was the man-fish Vimbai's or Maya's? Had the cribs comprising the ridge they named after Fay King Chung sifted out of Felix's dead universe, or were they Maya's forgotten memories? It was impossible to tell sometimes.
"Have something warm to drink," grandmother said. "It's getting chilly."
Vimbai poured herself a cup of boiling water, sweetened it with a spoonful of sugar (there was still plenty), added a drop of lemon from a bright yellow squeeze bottle (getting low), and headed to the porch. She had no desire to see the undead horseshoe crabs or their underwater secrets, she just wanted to be away for a while, separate from the crushed hopes that sprawled everywhere and filled the house to near bursting.
She stared into the horizon, gray sky welded to gray ocean, with barely a shadow to separate the two. Vimbai imagined what it would be like, to see a passing ship in the distance; to notice a darkening of the horizon that would then grow into a humped shape, and to yell, "Land ahoy!" To see a bird—an albatross, perhaps, or a seagull—circling above. But the ocean remained as quiet and lifeless as the house, and Vimbai suspected (without verbalizing it, because it would be too painful) that it was not the real ocean but a product of the house, its sick effluvium. And yet, and yet . . . she smelled the salt in the air and the sharp sting of iodine, of crushed seaweed, and she hoped. She hoped that the horizon would split open and finally admit a welcome sight—a sandy beach with humped dunes in the background, boardwalks bleached by wind and salt into a gray weightlessness of driftwood, and a tall figure, her neck craning, her head tilted back to see better. Come home, baby, come back home. We miss you and we forgive you and we promise that everything will be all right, we promise.
I'm coming mama, I'm coming back, Vimbai thought but did not dare to whisper. In her mind, the small figure on the beach grew more distant, retreating, until there was nothing but the sky and the heavy sluicing of cold waves.
Felix finally decided to let Balshazaar out, to let him roam around the house—it didn't seem fair, to keep him all alone among the silent souls of the horseshoe crabs; besides, the universe that had been growing and mutating around them did not really seem all that different from the one Balshazaar currently inhabited.
"I think," Felix told Vimbai and Maya at dinner, "it's like letting him from one dream into another. If it were a real world, that would be a different story."
"What is the real world?" Maya said, and gave a cock-eyed look to her fork and the anemic piece of ravioli impaled on it, drizzles of tomato sauce like blood. "But whatever; I don't suppose it would alter things in any way."
"Famous last words," Vimbai mumbled, but did not argue further.
"It's settled then." Felix beamed. "I sort of felt bad about keeping him there after I showed him the world . . . at least if he did not know what was outside, it wouldn't matter."
"Can't miss what you don't know about," the vadzimu said.
Vimbai shivered—her mother spoke in these words, her stern intonations bleeding through. "You can't show people the western lifestyle and expect them not to want," she would say. "It's cruel, to show and to lie like this—in a hundred years, people in the rest of the world won't be able to live like we do, but they will want it even more. Greed and jealousy, that's the problem with cultural imperialism." Another speech Vimbai knew by heart, another one of her daily conundrums where disagreeing would be monstrous but agreeing unbearable.
"Peb," Vimbai said out loud. "Would you mind fetching the phantom leg from my room? Just don't take it—it's for someone else."
Peb rose like fog from the vadzimu's back where it was clinging, blue and smoky like a Picasso painting. For some reason, Vimbai wanted to show him Guernica, and see what he thought of it, if he liked the blind eyes of the little girl who seemed oblivious to the limp hand cradling her. Peb nodded and floated away, like mist, like smoke, like an elusive fish skittering and disappearing in the thick of water.
When Peb returned with the leg, he and the vadzimu watched with a mix of curiosity and, Vimbai suspected, a trace of jealousy. Peb had relinquished the phantom leg with a quiet sigh, and now the leg stood on the kitchen counter, perfect and smooth like blown glass. Maya sat back, her arms crossed, her plate bearing an arabesque of tomato sauce forgotten in front of her. She frowned slightly, and her front teeth bit her lower lip.
Felix dug through his hair two-handed. A few times his face twitched into a grimace, and Vimbai guessed that he was touching the horseshoe crabs' little immobile souls. Finally, he gave a small cry of triumph and pulled out the desiccated head.
Balshazaar looked around him, and smiled when his eyes met Vimbai's. "Good seeing you again," he said.
"Hi," Maya interjected, still frowning. "I'm Maya."
Balshazaar was introduced in turn to the vadzimu and Peb, and Vimbai thought that he seemed quite unperturbed by the new environment. Perhaps Felix had shown him more than he told Vimbai, or perhaps he could see from the inside of his hair somehow. She chased the thought away as silly—she had seen the inside of Felix's hair, isolated from the rest of the world by inky blackness.
The phantom leg took to Balshazaar, despite the yipping and growling of Maya's dogs—they cowered away from the pruned face perched atop of the transparent leg, which was growing clouded, as if diseased by the contact with alien and dead flesh.
Balshazaar wobbled and made an awkward hop on the kitchen counter, knocking over an empty ravioli can.
"How does it feel?" Vimbai asked him.
"Fine, fine," Balshazaar answered, his thin scarred lips shaping a slow smile. "Will take a bit of getting used to, but I'll manage."
They watched him hop and bounce along the countertop, then jump down to the floor. He traversed the kitchen from the counter to the screen door, and from the screen door to the pantry. He then disappeared inside—presumably, to investigate the rest of the house.
"He'll be back," Felix said, and gave Vimbai a hopeful look from his right eye. "Won't he?"
"I'm sure he will," Maya said. She sounded as though unsure if that was a good thing. "You realize that now real people are in a minority, right?"
"Depends on what you mean by 'minority,' " Vimbai answered, and shot an apologetic look to the vadzimu. "She is my grandmother."
"She's a ghost," Maya corrected.
"Ghosts can be vengeful," Vimbai said.
Maya shrugged. "Should we go look for a supermarket again? Or if you want, there's a new forest by the attic. We could go name it, and see if anything cool lives there."
"In the morning," Vimbai said. "I want to check on the crabs."
"I'll come with you," Maya said.
They sat on the porch for a while. Vimbai looked underwater, her grandmother's sight letting her see the creatures as if they were close by. When she came up for air, she shook the water out of her hair. "I'm not supposed to see them," she said. "And yet I'm supposed to keep them alive somehow."
"While they are undead," Maya said.
"It's temporary, I think," Vimbai said. "They left their souls in Felix's hair."
Maya laughed, the sound resonating far over the ocean. "I can't believe this sentence makes sense to us. That there would be a world in which it's normal shit to say, you know?"
"I know," Vimbai said. "In any case, I suppose they are safe. They will get us home, I have no doubt of that." She did.
"Yeah," Maya said. She sidled up to Vimbai and dangled her feet in the ocean despite the cutting cold and darkness. "Provided we want to go home."
"Sure we do," Vimbai said. "We'll still have the house, you know? If our dreams are changing it, then there's no reason for it to change back once we're in New Jersey."
"Perhaps we cannot dream as well in New Jersey." Maya pulled her feet up and rubbed them with her feet. "The water's freezing. I better go get a pair of socks."
"I'm going to bed," Vimbai said. "I'm tired. And it is hard to look under water, even with my grandmother helping me." She could not quite describe the heartbreak, the dull sickness in her stomach when she saw the creatures covered now with a thick mat of barnacles, hagfishes sliming through the cracks in their carapaces. But their legs kept moving, always moving, like the long restless fingers of a sickly pianist.
"Okay," Maya said and squeezed her arm. "Dream us something nice for tomorrow, will ya?"
# Chapter 10
Obedient, Vimbai dreamt. Her dreams were vivid—more vivid, it seemed, than the waking landscapes inside the house. She dreamt of smells and sounds, of saturated solid planes of color. She dreamt of Africa as she had half-remembered it from her trip, half-imagined from the coloring books her mother bought her, and then got upset when Vimbai colored children on the pages pink instead of brown. These books had lions and vast open plains Vimbai colored rust orange and brick-red, blue oceans populated by smiling whales (green polka dot) and their fountains (yellow, like the champagne her parents drank on special occasions).
Now Vimbai dreamed of a rust-colored savannah, with green umbrellas of acacias scattered at a distance. Two plush giraffes grazed among the leaves, their long and unrealistically pink tongues twining and snaking between black thorns shining like volcanic glass. A stuffed lion slumped in the shade, inanimate at the moment, and it did not even stir when Vimbai passed right by it.
There was a lake on the horizon, a smooth blue mirror, but Vimbai was weary of fresh water rife with catfish. Instead she headed for a group of gigantic stones—she guessed them for the Great Zimbabwe, the ruins that gave her country their name, even though they seemed grievously misplaced in the dream. Gray stones towered over Vimbai, their fissures greening with moss and slender grasses, and she thought that if the Great Zimbabwe was to ever fight Stonehenge, the latter would have its ass handed to it.
She passed through the arches and between walls, the remnants of a giants' house, and came to the other side where round grass huts—arranged in a semi-circle, like one would see in a Discovery Channel documentary—teemed with people and dogs.
"Run away," people shouted to Vimbai, and dogs barked. "They are coming, they are coming."
Only then did she notice that they were packing bundles of their belongings and carried children, fleeing from some dream disaster.
At that point, Vimbai was quite aware that she was dreaming, and so she decided to stay behind and see what all the commotion was about. She waited until the huts emptied and the people all climbed into aerial boats moored nearby—long speedy hollowed-out tree trunks, fashioned with bright golden wings where oars would've been, topped by great scarlet sails. The sails filled with sunlight—like a gust of giant breath—and the boats took off through the air, fast as arrows, the wings on their sides beating in unison, the speedboat engines mounted in the rear of the boats strangely helpless and superfluous.
Vimbai watched their departure and disappearance, how they grew into tiny dashes on the horizon and dissolved in the expanse of the molten sky. She smelled dry grass and a whiff of motor oil, and she breathed hastily, lustily, in order to retain and remember them when she woke up.
And then she heard the sound of motors rumbling. It did not come from the boats but from between the stones of the Great Zimbabwe, and she surmised that it signaled the approach of whatever caused the mad flight of the aerial boats.
She heard a siren, and her feet moved against her will—a fear too visceral to overcome, the nightmare given to her by the Kenyan babysitter when she was just a baby, the sound of the medical trucks.
They came from among the stones, emerging from and between them, coming up from the sifting, puckering soil that spat them out like something distasteful. They came like ants fleeing a forest fire, like impala fleeing the drought . . . They swarmed like locusts.
The trucks looked just like Vimbai imagined them—old-fashioned things, reminiscent of army trucks from the twenties, with wheels of solid metal that thumped softly on the dry ground. Brass rails ran along the open cabs and beds of the trucks, one on top and one on the bottom, and several men in blue surgical scrubs stood on the lower rail, hanging onto the top one, giving Vimbai an impression of children peeking over a split-rail fence. Large red crosses were painted on the cab doors.
Vimbai could not see what was in the open beds of the trucks, but she could hear a quiet and terrible slurping that filled her with quiet dread. It's just a dream, she reminded herself. They cannot hurt you. And yet the trucks slurped and sluiced and thumped and moved closer, surrounding her in a ring.
The men in surgical scrubs, their faces hidden behind gauze that only left their tired, kind eyes and sweating white foreheads visible, jumped from the truck closest to her, and Vimbai saw a large flat cistern filled with pale blue blood. Several hoses snaked around its base, and one of the medical men grabbed a hose and motioned for his mates to hold Vimbai. Too late, she thrashed in their arms; too late she tried to will herself to wake up. But the hose got closer to her face, and now she could see a large needle glinting on the end of it. She struggled but the man's gloved hands cradled her face, and the needle jabbed her neck. She felt her life draining away from her, her soul hanging by a thread, as the cistern got fuller. She did not struggle anymore, and the medical men let her arms dangle by her sides, her legs segmenting and treading the dry sand, her gills dry and desperate for the cool embrace of water.
"That's a huge horseshoe crab," one of the medical men said. "Toss it back."
Vimbai wanted to scream and protest, she wanted to ask them to take her to the sea, to the life-giving salt water. But instead, they tossed her into the lake, where a hungry catfish waited for her, wise, smiling with its hard toothless mouth.
In the morning, Maya and Vimbai went for a walk in an aspen grove, which Maya recognized as her own. Her dogs tagged along, their fluffy-tipped tails swaying gracefully and their pointed possum faces grinning, bristling with white conical teeth. Their eyes gleamed brightly.
Vimbai kicked up the leaves littering the path, and they rustled and rose, and then fell back. Maya seemed pensive.
"I had a strange dream," Vimbai informed, and told Maya about the men in medical trucks.
"That is a really messed up dream," Maya said. "Jesus. Bad dreams are a hazard here, aren't they?"
"I don't know," Vimbai said. "I thought the house had our old dreams . . . the ones we have discarded and forgotten about."
"Maybe," Maya answered. "I hope so." She whistled to her dogs and they perked up: their tails wagged, and their tongues hung out. They crowded closer to Maya and stared at her expectantly, as if waiting for her to do something amazing or entertaining.
"They love you." Vimbai sighed. "It's really cool how they follow you."
"Somebody has to," Maya mumbled and bent down to scratch a few dog heads. When she straightened, she shot Vimbai a quick smile. "Don't mind that, I'm just being silly."
"It didn't sound silly," Vimbai said. "It sounded serious, actually."
Maya shook her head. "So I whine a little every now and again. I'm allowed to."
"I'm not saying you're not," Vimbai said. "Only you sounded so sad . . . is there anything I can do?"
"No," Maya said. "There's nothing, really. It's just sometimes I think that all I have is these dogs and Felix and you."
"No family?" Vimbai felt guilty that she had never thought of asking Maya such simple questions. Her own family occupied so much of her internal space that she assumed it was the same for Maya, unable to recognize a sucking emptiness in another's soul.
Maya shook her head. "My grandmother died two years ago, and I never had anyone but her. But I'd rather not talk about it now. Maybe later."
"Okay," Vimbai said. "Fine. You want to go to the lake?"
"No." Maya nodded at her dogs. "They are afraid of lakes—I think it's the fish. The same fish you dreamed about."
"This is what I wanted to see," Vimbai said. "To make sure. Maybe it's not even there anymore, or the lake is gone."
"You're not afraid?"
Vimbai considered her latest shameful flight from the catfish. "A little," she admitted. "But if I don't go in, I can't drown, and it can't take my soul. It can't hurt real live people, can it?"
Maya shrugged. "I'm not about to find out. I don't think you should either."
Vimbai hesitated. It was so tempting, so sensible. But her promise to the horseshoe crabs beat in her heart, like ashes of Klaas. She promised not to let them die, and she had to make sure that the sinister catfish was not hatching any evil plans, like her dream seemed to suggest. "I must," she told Maya.
Maya sighed. "At least, take someone with you. Felix or Peb or your grandmother." There was a hint of struggle as she pronounced the last word, and Vimbai thought that Maya had to feel a little sore, that it was not her grandmother who showed up to look after them and to make them coffee. That the vadzimu was Vimbai's, even though Vimbai had a full set of parents and did not really need a ghost. And Maya . . . Vimbai could not be sure, but she suspected that Maya would trade all of her dogs for a glimpse of her dead grandmother.
"The vadzimu? She doesn't leave the kitchen—only to go to the porch. I don't think she wants to be anywhere else."
"Why?"
Vimbai shrugged. "No idea. You know how ghosts are always restricted to one room. Or a hallway or whatever."
Maya nodded. "I guess. But you did drive her here."
Vimbai stopped, awkward. It was time for them to part ways—Maya seemed intent to continue down the path covered in yellow leaves, to the bluish grove of firs on top of a hill they named after Oprah. And Vimbai had to head back, past the rich deposits of old mattresses and into the desert, the yellow sand with two chairs on the shore of a silent, smooth lake, where the unspeakable cannibalistic horror of a catfish lurked beneath.
It was silly, Vimbai told herself. Perhaps it was just a fish with no intelligence and no accompanying malice; perhaps she just imagined its words. Perhaps the house was just a canvas onto which she projected her silly fears and believed them to come to life in the shifting, uneven light—much like her own carelessly tossed shirt transformed a peaceful chair in her childhood bedroom into a monster. Perhaps all she needed was a closer look, a light switch, that would let her see her own foolishness.
"I'll see you later," Maya said. "Just be careful."
Vimbai headed for the lake. At first she thought of stopping by the kitchen and talking Peb into coming with her, but it was a bit of a detour. Besides, Peb being the creature of mostly spirit (and soap skin) could be vulnerable to the man-fish—provided the latter was real. It was better to go alone, she decided.
Vimbai waited on the lakeshore, looking for telltale circles in the water. She waited for the flip of a tail, the silvering of a side and the splash of a large slithering fish. She stood among the green cattails and succulent patches of sedges, their green inflorescences tilted like bayonets.
There was no sign of the catfish, and she considered retreating back to the lawn chairs, perhaps sitting down, kicking up her legs and enjoying a nap—for all the weird absence of smell here, inside the house it was much warmer than outside, subtropical even. Vimbai decided not to contemplate the heating bill—and after all, who said anything about a bill? For all she knew, they would never have to encounter anyone from the gas or electric company again. She finally understood what Maya was so jubilant about, imagining a life with no bills and no responsibilities, free to roam the endless plains of this dream Africa, with its forever flowering Harare and plush lions who cuddled and never bit, its mountains and ridges named and explored by them, by Maya and Vimbai, and not some dead unknown people. Their own world, their endless circus that had the good sense to run away with them.
Her feet sank into the soft soil of the bank, and she wiggled her toes, enjoying the sensation of thick ribbons of warm mud squeezing between them. She longed for the rich smell of river, of the green and decay and silt warmed by the sun, and she sighed. Their dream refuge had a serious flaw, no doubt about that.
She lifted one foot, and the mud made a sucking noise, an obscene slow kiss, as it released her. She turned around and froze at the sight of Balshazaar hopping along on his phantom leg, away from the lake. She was not sure if he had seen her, but crouched low just in case he turned around. There was no particular reason for her to hide, it just seemed like a good idea. Balshazaar roamed freely now, and Vimbai saw no harm in keeping a secretive eye on him, even if it meant crouching on the bank and getting mud on the knees of her relatively clean jean overalls.
Balshazaar never turned, and Vimbai watched the back of his shriveled head, parchment skin with a few long wisps of gray hair, disappear behind the straight line of the horizon—she knew that a grove of palms and couches waited just behind it, embraced by a clear gurgling brook with a pebbled bottom, where mayfly larvae built their strange delicate houses from straw and tiny shells cemented with silt.
"What do you want?" a voice came from behind. An unpleasant voice, with a strange suffocated quality to it—it sounded like a person talking without breath, a mouthed voice with no lungs behind it to give it strength.
The man-fish peeked out of water, his fins propping him up not far where Vimbai had previously stood. He was a large fish, beautiful in his way—brown and green patterns covered its wetly glistening sides, like a snakeskin boot. His eyes, small and golden, bulged a little out of his flat head, staring at Vimbai with an empty feline expression.
Vimbai studied him a while. There was no hurry for her to speak, and she hoped that her silence came across as unnerving rather than timid.
The catfish smirked a little, his whiskers hitching up to expose a wide lipless mouth. "What's the matter? Cat got your tongue?"
She spoke only when she felt certain that her voice would come out without trembling. "Who are you?" she said. "What are you doing here?"
"I live here," the fish replied in the same breathless voice. "As to who I am?" He paused, swallowing air in large gulps, his gill covers falling and rising like bellows. "I think you know that. Or have you lost touch with the stories you learned like you lost touch with your past?"
Vimbai smiled. A few months back these words would've stung. Today, she knew well enough that they were not the truth—at least, not the real truth. Sure, her Shona was lacking and her knowledge of her parents' culture was patchy, to say the least. Yet, Vimbai refused to feel guilty for being the way she was. "I remember," she said. "You're the man-fish."
"That's right," the catfish answered. "And what are you doing in my dream?"
It had not occurred to Vimbai that the house might not be their creation entirely; yet, she dismissed the thought that the catfish was the architect of this place. It seemed too influenced by human things—furniture everywhere, and very little water. Besides, dreams of dreams sounded awfully recursive to her. "It's not your dream," she said. "You're lying."
"Maybe not yet," the man-fish answered. With a single beat of a strong blunt tail, he sent a spray of murky water splashing into Vimbai's face. When she rubbed her eyes dry, he was gone—not even a trace on the surface of the lake, not even a tattoo of concentric circles as if after a fallen stone.
Back in the kitchen, Vimbai's pensive mood was dispelled by Peb's frantic cries. He hovered over the stove and wailed and whimpered, inconsolable, despite the vadzimu's and Felix's efforts. Peb cried and cried and fluttered frantically about, like a moth trapped under a lampshade.
"What's the matter with him?" Vimbai asked. She had to raise her voice to be heard over the racket.
"Aaaaaaa!" Peb wailed in response, opening his mouth wide. Only then did Vimbai notice that his tongue was missing—his mouth was an empty cave bordered by two rows of transparent teeth, smooth and devoid of any features, like the inside of a teacup.
Vimbai turned to Felix, who was following Peb around the kitchen, his hands flapping helplessly. "Who did that to him?"
"No idea," Felix said. "He just showed up like this."
"Can you help him?"
Felix shook his head and stopped pacing. "No, of course not. How can I? There are forces, and I don't understand them, and no one ever had a phantom tongue—at least the one I know of."
Cat got your tongue? Vimbai remembered the hissing voice of the man-fish. More like catfish got your tongue, she thought. Perhaps that was the original expression—cat getting someone's tongue just didn't make sense; then again, neither did catfish.
Vimbai's throat constricted, and when a sob squeezed out of it, it startled her, as if it came from an extraneous source. She had surprised herself; she did not expect to feel such acute grief for the poor Peb and his stolen tongue. The Psychic Energy Baby, birthed in some ethereal realm, had grown to be a part of Vimbai with his festoons of feet and hands, with his relentless desire to absorb colors and parts of people. He became a part of the household, and without Vimbai's ever noticing, they all had learned to love him—even Felix, as unhinged and disconnected as he was most of the time.
Vimbai could think of nothing better to do than to pick Peb up—he struggled in her arms at first and then quieted and felt silent save for an occasional sob. She held him awkwardly, having little experience with babies, psychic or otherwise. As Peb relaxed in her arms, Vimbai thought about what it would be like, to have a baby sibling.
The vadzimu touched her elbow, startling Vimbai from her thoughts. "Don't cry, granddaughter."
"But . . . but they took his tongue," Vimbai said, and swiped her open palm over her watering eyes. "What will he do now?"
"You can always find what has been misplaced," the ghost said. "You just need to know where to look. Now, who could've taken it?"
"The catfish," Vimbai said. "The man-fish, I mean. Only I don't know how—he was in his lake and I spoke to him."
The vadzimu gasped. "Why would you do such a thing?"
"I don't know," Vimbai said. "I wanted to see what he was up to, I guess. And just to make sure he wasn't planning anything . . . I was worried about the crabs because of the dream I had."
"You should be careful with the man-fish," the ghost said. "He is cunning—more cunning than you can imagine."
Felix stopped his pacing. "Maybe," he said, "maybe he took Peb's tongue so that Peb couldn't tell us something."
"Like what?" Vimbai asked, still sniffling. So small, so ethereal. So helpless. Impossible, and yet alive, and yet mutilated. The conglomeration of wrongness was so great that Vimbai felt like crying again.
"I don't know," Felix said. "But Peb, he floats everywhere. He babbles . . . babbled about all sorts of abstract stuff, but he notices things. Right, Peb?"
Peb nodded, his forehead brushing against Vimbai's shoulder, light like sleeping breath of a real infant—or at least, that was how Vimbai imagined it.
She hugged Peb closer to her, and he felt like an air-filled balloon in her arms, smooth and light and real. "What have you seen?" she asked. "Who did this to you?"
Still sobbing, Peb pressed his face into her shoulder.
"Are you afraid to tell us?"
Another brush against her shoulder signified another nod.
"Don't worry," Vimbai said. "We'll protect you. You tell us when you're ready."
Peb wailed a little.
"He can't talk," Felix said.
"I know. He can still point whoever did this out, or answer yes or no questions."
The vadzimu patted Vimbai's shoulder reassuringly. "When he's ready," she said. "When we all are ready."
Vimbai fled to the small tropical grove that currently separated her and Felix's rooms. She left Peb, still distraught but quiet, with her grandmother's ghost, and sought solitude and time to think. She felt exposed and betrayed, as if a dream she was enjoying had taken a sudden and unwarranted turn toward nightmare. She wished Maya was here so that Vimbai could ask her what she thought now, now that Peb's tongue had been stolen, about having their own domain and being wild queens of the dream realm she suspected was Africa of the spirit. What she thought now, when the man-fish was stalking them from its lake and paying no attention to all the great names for rivers and mountain ridges and furniture deposits they had come up with.
She wandered among the thick trunks, flared at the bases like trombones, covered in green ribbons of moss and twisting ropes of vines. She craned her neck to see interweaving branches hundreds of feet above her, right under the painted fiberboard sky. Orchids and bromeliads cascaded from the branches, and Vimbai squinted at the bright red and yellow flowers.
She grabbed onto an especially sturdy vine and yanked it a few times. The vine held, and Vimbai pulled herself up, her toes finding footholds in deep fissures in the bark. She had been a good tree climber when she was younger, and now the skills still remained. The whole thing about tree climbing was not being afraid to fall, and having faith that the next foothold or a branch would be there when one needed it; and Vimbai had this faith. As she climbed, the bark opened in accommodating cracks and the branches offered themselves to her reaching fingers, until she settled in the intersection of several sturdy branches that offered a perch and a canopy. Her back resting against the trunk and her gaze settling on the idyll of a basket fern housing a white and pink orchid among its feathery leaves, she felt alone and at peace; and most of all, she felt secure from the catfish, so far away in his lake.
Vimbai wished she could stay up in this tree forever, without ever having to go down and to deal with the man-fish or Peb's missing tongue. She wished she could stay here until they safely touched ground in New Jersey, and there she would go home. Her mother must be worrying herself sick about her by now, and her father was probably quiet and reassuring at home, but at work he would spend his breaks phoning morgues and hospitals; he would pull favors with both Camden cops and Camden drug dealers, both of which he had been patching up for years now. He would look for information and come home late after stopping by every morgue and looking at every dead black girl between sixteen and thirty, each body a simultaneous stab in the heart and a sigh of guilty relief.
Vimbai regretted that she had been so focused on her mother and she on Vimbai that they both pushed her father to the sidelines, his relationship with them uncomplicated, reduced to the function of arbitrator and peacemaker. Her father who only lost his cool when either Mugabe or Rhodesia was mentioned. Her father with a secret political past that barred him from ever visiting home, and his vague and undefined fears Vimbai wished she asked about.
It struck her as profound, that she could see her parents' grief with such clarity. It was not an eternal childish they'll-be-sorry-when-I'm-dead mantra but rather her intimate knowledge of how they were, how they functioned in the world, their responses as predetermined and predictable to her as her own. Perhaps even more so. She wished she could go home now, to reassure them and to stay with them so they would never have to worry again.
She wondered about Maya then, about how she managed to survive in the world and to function without such supporting love, invisible and strong, even if it was far away and only imagined. No wonder she clung to her dogs.
As if answering her thoughts, short thin barks reached her from below, and she peered down between the branches. Ruddy backs and fluffy tail tips appeared in the greenery and disappeared again, hidden by the lush vegetation. And there was Maya, her unruly black hair and yellow t-shirt as unmistakable as her half-foxes half-possums.
"Maya," Vimbai called.
Maya stopped and looked around, puzzled.
"Up here." Vimbai waved with both arms when Maya looked up.
"What are you doing in this tree? Have you heard about Peb?"
"Thinking," Vimbai answered. "Yes, I saw him. Terrible, isn't it?"
"Yes," Maya yelled, craning her neck. "Felix thinks it's the catfish in the lake. Did you see him?"
"Come up here, and I'll tell you."
Maya shook her head. "My dogs can't climb, and they'll go nuts if I leave them down here by themselves. You come down."
Vimbai sighed but obeyed. Climbing down was always harder for her. On her way up, she could just keep her gaze on the sky above; coming down, she had to look at the ground, aware how far away it still was. When she finally stood next to Maya, panting, she smiled. "You have to climb with me one day. It's really gorgeous up there. And the flowers!"
"Maybe," Maya said. "What happened with the fish?"
Vimbai recounted her adventure and her conversation with the catfish. When she mentioned Balshazaar, Maya frowned. "You don't think he could be mixed up in it, do you?"
"Why would he be?" Vimbai said.
Maya shrugged and looked around. "I don't know. He just creeps me out, that's all."
"All he knows is Felix."
"Precisely. Maybe he wanted to make some new friends, friends of his own."
Vimbai looked around too, watching for the glistening of parchment skin in the underbrush. A shrunken dome hopping around on its single non-existing leg like it owned the place. "You're a bit quick to jump to the conclusions."
"Who else then?" Maya said. "Not Felix and not the ghost. Not you, not me. Who else is here?"
"The wazimamoto," Vimbai said. "The men in medical trucks. In my dream, they were with the man-fish—in cahoots with him, I mean."
"I haven't seen any of them around," Maya said.
"This place is huge," Vimbai said, and felt a chill. "There could be anything hiding in here somewhere and we wouldn't even know it."
Maya seemed worried for a second. "If they came to Peb, they know where we all are."
"Of course. There's always someone in the kitchen. Or the living room, at least."
"We need to find another place to hide out, in case of emergency," Maya said. "I think I know one."
"From your dreams?" Vimbai guessed.
Maya nodded. "Come along. I'll show you my secret, and I'll explain on the way."
Vimbai followed Maya, and the dogs barked and bounded ahead, as if they knew the way very well.
# Chapter 11
Vimbai was pleased to have earned enough of Maya's trust for her to talk to Vimbai so openly; yet the story Maya told her left her worried and upset. It just didn't seem either normal or fair, to seek refuge in one's own nightmares. And Maya was a nightmare factory.
When Maya was younger, she used to live in Northern Jersey. Not in the projects as such, she said, but pretty damn close. In Newark there just aren't too many what one would consider 'good' neighborhoods, and she learned early on what gunshots sounded like, and what 'alley apple' meant.
Still, it wasn't all bad, she told Vimbai. As a teenager, Maya carried a switchblade, as a kind of bravado rather than against any real danger. The neighborhood she lived in, while not exactly wealthy, was not unsafe—there were flowerboxes on the windowsills, and geraniums bloomed in them. Late night in August, people sat on their porches, having long and slow conversations, waiting for the heat of the day to let up enough to allow sleep.
Up in her room, Maya used to listen to the drawl of voices outside, as she lay on top of her blue and yellow quilt, her forehead beading with sweat. She listened to the TV in the apartment downstairs, to the sounds of traffic on East Kinney Street. Through the window by her bed, the streetlamps looked like ghostly white globes through the haze rising off the heated pavement, only snatching glimpses of the peeling siding and the wide brick porch of the house across the street. She could not see the porch of her own house, but she knew that her grandmother was there, sitting on the steps with her feet in black shoes planted firmly on the sidewalk, her knitting in the wide hammock of her dress stretched taut between the knobby old knees.
It was too hot to sleep, and Maya tossed from side to side, willing the white curtains to flutter, to bring a breath of fresh wind, any movement of the sticky humid air. Only in retrospect did she realize how happy she had been back then. How sad it was when you left your happiest days behind when you were fourteen.
She was eager to reassure Vimbai that really, it wasn't so bad—so many had it much worse, and Maya was lucky in many ways. She would not call it a hard life; it was just that shit happened with alarming frequency—to everyone, so why not her? She was not too special not to catch some shit every now and again.
Statistically speaking, being raised by one's grandparents put one at a certain disadvantage—life expectancy was a bitch, as Maya's grandmother was fond of saying. Well, maybe not in those very words, but the meaning was the same. This is why she made sure that Maya studied hard and did not slack off at school; this is why when she found the switchblade she chased her down the street, swinging a long leather belt she must've acquired for that very purpose—at least, she never wore trousers, and the belt clearly had no fashion-related application. Even if Maya wanted to grow up ghetto (and she had toyed with the idea), she never had a chance—not while her grandmother was alive, and not after she passed away. Even in death, she threatened Maya's conscience with the leather belt, forever branded into her imagination.
Maya's grandmother threatened to die so frequently that when she actually did, in December of Maya's sixteenth year, Maya felt betrayed and puzzled—she hadn't done anything wrong, and therefore did not warrant this most severe of all punishments, the kind that was supposed to exist only as a threat but never to be carried out. Especially not when Maya was doing so well in school, taking AP lit and biology, and acing her practice SAT tests.
So it was not surprising that Maya's nightmares centered around the simple pine coffin, with a small old lady (and she was small, despite Maya's early memories when her grandmother towered over her, gigantic like God) dressed in black shoes and black dress, a pillbox hat, and a pair of white gloves clutching her knitting needles—she was quite clear on requesting her outfit and her knitting in her last will she dictated to Maya just two weeks before her failing heart had finally given out.
Maya remembered the shining organ pipes in the church and the coffin, but not much else. Maybe this is why her dreams placed the coffin into the building she could never forget, no matter how much she tried.
It was a multistoried monstrosity just a few blocks to the north from Maya's house, and the epicenter of the gunshots that reached their relatively peaceful enclave of the working poor. The building itself seemed the very cause of the violence and other improprieties—at least, this is what Maya's grandmother said. "They stuff people into these egg cartons," she would say, frowning, "and it's so big that no one knows who their neighbors are, and they never have to look them in the face. So they break and steal and put their graffiti on the walls, because only God can see them, and he ain't saying anything. And here, on this street, we know our neighbors, and this is why everyone behaves—shame keeps people decent."
The city authorities were apparently of the same mind, and in their continuing quest for gentrification, they decided to demolish the projects and build a center with shops and coffee bars instead, where people would come to spend money rather than kill each other and sell drugs. Maya's neighbors weren't fond of the projects, but they were even less fond of shopping malls that were designed to drive housing prices even higher, and people who lived there away. Maya and a few of her friends visited the gutted building just before it was razed, to pay their ambivalent respects.
This was the building Maya dreamed about, twisted and distorted by time and sleeping mind, and enough time had passed that she would not be able to say how accurate the dream version was, or even if it bore any semblance to reality. She dreamed of the central staircase ensconced in concrete slabs, spiraling through the center of the hollow, empty building—just the outer walls remained, with all the internal constructions, floors, partitions and doors completely gone. Just a giant echoey brick of space, with a staircase boring through its empty heart, and when Maya looked up it seemed to go on forever.
There were no banisters remaining, and Maya and two other kids, Phil and Janet, climbed the staircase, scared that they would get dizzy on its endless turns and plummet all the way down to the naked foundation, with nothing there to break their fall. And in her dreams, Maya still climbed this staircase, all the way to the top. It ended in a simple wooden platform that held a coffin with a small old lady inside, her gloves and hat and knitting needles just like Maya remembered them—the dead heart of a gutted monster building, useless and unbearably sad.
The monstrous high-rise stood on a sharp cliff that jutted out of a seemingly endless sea of old shoes and handbags. Vimbai decided not to contemplate the origin of them, but rather concentrate on the cliff itself. It appeared to be made of the same material as the climbing wall in the college gym, and Vimbai felt an acute pang of nostalgia for her classes and the campus and a sky that was not just painted on the ceiling.
There was a path leading to the top, steep but passable, and the dogs bounded ahead—they knew the way.
"Weird," Vimbai said. "You have this place, and I have a Harare. It's like each of us gets our own little fiefdom."
"Or a queendom," Maya said.
"The point is, who decides? Who gives us those things?"
"The house," Maya said. "Our dreams. I don't know; does it matter?"
"We keep saying that it doesn't," Vimbai said. "But we're just saying that because we cannot find out, and it is terrible, living in a place you don't understand. It has some laws we don't know, and there's someone . . . some thing that makes everything happen. Doesn't it bother you?"
"A little," Maya admitted. "Maybe it's like one of those stories they tell children, like a morality tale. About kids who ask too many questions, or look when they're not supposed to, and lose everything."
Vimbai thought back to her spying on the horseshoe crabs—involuntary, drowning, and yet she broke an explicit agreement and felt guilty. "I know what you mean," she said. "How much farther?"
The path had been turning and twisting, and Vimbai could not see the house on top of the cliff, only the rough rock face ahead, with the path growing precipitous enough for them to start using their hands. The rock offered convenient handholds, like the ones on the rock wall at the gym, made of metal and plastic. Like everything here, it seemed to hide artifice under the surface appearance of natural things, as if the house tried to disguise itself as a forest or a rock, and still its studs and dry walls and paint showed through the camouflage. She wondered if it was more successful pretending to be a different house.
The apartment building appeared before them as soon as they rounded the side of the hill, as if it decided to meet them halfway—Vimbai was quite certain that they were not yet at the top, and this was not where the building was first visible.
Maya shrugged as Vimbai's puzzled look met hers. "It does that. I never know where it's going to pop up."
Inside was just like Maya had described—an empty shell of a building, so hollow that it was a miracle it did not collapse on itself, supported by nothing but four walls. The staircase, the concrete and iron rods and the steps winding round and round and up, was the only structure inside, and it made the building look even more vulnerable, as if they caught it in a second before the whole thing imploded; the second stretched, liable to end at any time, giving the place an air of simultaneous stillness and the impending catastrophic movement, inevitable tumbling down in a cloud of dust and grime and cement slabs.
Maya motioned for Vimbai to follow her, and the two of them ascended the staircase. Vimbai lost track of the floors signified only by turns of the staircase, and she lost all sense of direction, winding and winding around. The empty windows offered no other sights but the blind brick wall on all sides, as if the building was enclosed in another, larger one; Vimbai supposed that it was technically true, but it did not lessen the fear that was rising in her stomach. Round and round they went, as if trapped on some awful merry-go-round. "I don't like it here," Vimbai whispered, addressing herself more than anyone else.
Maya continued her ascent just ahead of Vimbai, her buttocks moving energetically under the jean fabric of her cutoffs. "Neither do I."
"Then maybe we should go back."
"Not yet," Maya said. "Soon. I have to show you something first."
Vimbai's words flooded her mouth yet refused to leave it—but how? She wanted to ask. Is she still here, your dead grandmother, not even a proper ghost but an apparition of her dead body, lifeless? What cruelty was this, when even our dreams and wish fulfillments offered not comfort but relived heartbreak? It seemed shockingly unfair.
They arrived at the top, to the small wooden platform mounted on top of the staircase like a crow's nest. And there was a coffin and garlands of flowers, wreaths and condolences written on ribbons; there was a small dead woman in a coffin, her small face pruned, her black shoes polished to a mirror shine. But worse, so much worse were the traces of life around her—there was a tent built of blankets and couch cushions, a pillow fort children build when they are trapped indoors for too long, some mysterious squiggle in their genes commanding them to convert every blanket and pillow into a den, regressing to the early days of the species' existence. The dogs were there too, stretched comfortably as if they were home—they were home, Vimbai realized with a trickle of cold sweat between her shoulder blades. There were soda cans and candy wrappers, a small pile of clothes, a book, a flashlight. Maya had moved here from her room, this is where she spent every night and most of her days, climbing here away from Vimbai and Felix, to be next to a small woman in a small coffin, to sleep under the funereal wreaths.
"Oh Maya," Vimbai whispered with dry lips. Oh, to be so alone—Vimbai could hardly imagine such a thing, such a separation between self and the world that a pack of mutant foxes and a dead body would be desirable company. And it hurt a little, too—she had to admit, to herself if not out loud—that Maya would prefer this to her bedroom, to the kitchen and to Vimbai and Felix and the poor tongueless Peb.
"It's not that bad," Maya said, answering not so much Vimbai's thoughts, which remained unspoken, but her expression. "It's cozy, even."
"But . . . " Vimbai fell silent, unsure how to say what nagged her. The fact that Maya's grandmother was dead, that she couldn't dream her alive, would sound too much like an accusation. "Do you think it's healthy for you?"
"Why not?" Maya shrugged and sat down, her back defiantly propped against the coffin wall. "And even if it's not, so what? I don't owe it to anyone to do only what's healthy for me. Not even to myself."
Vimbai could not argue with that, and she sat down next to Maya. "I'm sorry. I don't mean to tell you what to do. Do you really think we can hide out here?" She almost kicked herself—her words sounded condescending even to her.
But if Maya noticed, she did not let it show. "Sure," she said. "As good place as any. And no fish or medical truck could get up here—see, safe."
Vimbai nodded. "What about Peb? What do we do about his tongue?"
"I don't know," Maya said with a hint of irritation. "You can go back to that catfish and ask him. Or you can go chasing after the trucks, anyway if they exist at all. Or go talk to your crabs. I'm staying here. You can stay, or you can go for Felix and the rest. Do what you want, but I'm not leaving."
Vimbai sat by Maya for a while, until the silence between them acquired the taut quality of stretched fabric, ready to tear any second. Then she stood up. "Thank you for showing me. I hope you'll be home for dinner."
Maya made a noncommittal sound and jerked her shoulder.
"In any case, I'll see you later."
Maya remained silent, and Vimbai started her lonely descent down the endless stairs and then the rocky hillside, down and away, farther and farther from the dead woman and her coffin and her granddaughter, carrying on the vigil through all the intervening years.
Vimbai's face grew numb from the cold, and the smell of salt and seaweed assaulted her, making her eyes water—she had been spending so much time indoors that the natural smell of the ocean she used to love felt astringent and too strong. She wrinkled her nose and rubbed her eyes with the back of her hand. "Come on, little horseshoe crabs," she muttered. "Let's see how you're doing."
The chipoko stood beside her, ready to help and guide and breathe as if one with Vimbai. Her quiet posture and hands, roped with veins, folded in front of her, filled Vimbai's heart with heavy, regretful blood. There she was, her grandmother who moved and talked, and yet she was as dead as Maya's. Just a ghost, a dream of a vague memory from many years ago.
"I'm ready, grandma," Vimbai said, and felt the ghost's hands fill hers, and her grandmother's eyes look through Vimbai's with the wisdom and sadness of too many years. She wanted so bad to be kind to the ghost, even though it was most likely just a product of her imagination; she so wanted to show her kindness—kindness she had been too young and too arrogant to show her in life, her heart hidden away from the old woman by the thin crisscrossing of scars on Vimbai's mother's wrists and ankles.
She submerged her face in the stinging, harsh water, sharp little bites of salt pinching her cheeks. She opened her eyes to look at the crabs. Her mouth opened of its own volition, salt flooding her mouth and nostrils, her eyes disbelieving.
The crabs—undead, terrible—stopped their movement and turned as one, looking back at her. She could not see their eyes, hidden deep in the fissures of their shells, but she could feel the age-old fatigue and stone-cold fear, the disappointment and sadness that seared like a knife across an open palm.
You promised us, they whispered. You promised.
"I know," Vimbai said. "I'm sorry—it was an accident. I tried not to peek, honest."
And now you see our disgrace and degradation, our soulless shells, our bodies thrown into death so that they could crawl, crawl forever across the sandy ocean bottom, crawl without fatigue or fear or hunger or thirst or lust. And you promised to protect our souls from diminishment.
"They are safe," Vimbai said. "I saw them—they are safe."
The crabs seemed to heave a sigh, although Vimbai was not quite sure if such a feat was possible without either lungs or air. Are you sure? We feel uneasy.
"I'm sure," Vimbai said, the creeping sickness of doubt settling in her stomach. "I'll check again as soon as I have a chance. But meanwhile, I have a question for you—do you know who could've stolen a ghost's tongue?"
Those who don't want anyone to speak, those who keep everyone mute. Those who hate life while they vow to protect it.
Vimbai's lungs felt ready to explode, and she came to the surface with a wracking gasp. Water dripped off her chin, froze in thin icicles in her hair. The vision of the medical trucks and the mute men dressed in surgical scrubs passed before her inner eye. The man-fish splashed in his lake, grinning, his yellow cat eye sly and laughing, cold. The vampires and the stealer of souls, somewhere close. Inside the house, Vimbai almost cried out, inside the house! So close, so ridiculous—like one of those urban legends, she thought, when the victim realizes that the phone calls are coming from inside the house. Ridiculous lies, like the one about a man waking up in a tubful of ice.
She remembered the scar on cousin Roger's back, and cringed. There was no tub of ice, like there was no phone. And yet, the man-fish, the urban legend of a distant place, laughed and frolicked in his lake, and his gravelly voice rubbed the insides of Vimbai's ears raw.
"Where's Felix?" she asked the chipoko as soon as the two of them separated and the ghost stood next to Vimbai once again.
"He went for a walk," the ghost said. "Come home with me—the baby needs comfort."
Vimbai felt guilty about forgetting the tongueless Peb's troubles. She only thought of his misfortune as a mystery to solve, to get to those who would harm the rest of them, and did not consider how he felt, alone and mutilated. "What can I do?"
"Tell him a story," the ghost said. "Stories always help."
# Chapter 12
Vimbai took over the Peb-consoling duties as soon as she entered the house and found Peb curled up in the oven. Peb whimpered, and the ghost nudged Vimbai—she said she had ran out of stories; not entirely, she was quick to mention, just for the time being. Surely, she would be able to think of something later. Meanwhile, she said, would Vimbai think of a story to tell poor Peb?
Vimbai thought of all the fairytales her African babysitters told her—Ghanian and Kenyan tales mixed with each other in her memory, and she felt ashamed that she had become one of the people who so intensely aggravated her mother—people who could not tell one culture apart from another. But Peb cried, and she sighed. All the fairytales, all the Tutuola she had read would have to do, and her mother was not here to criticize the mishmash. Her mind crowded with images of women turned into beasts and the ghosts calling each others on the phone. Vimbai drew a breath and said, "All right, don't cry and listen. This is a story about a boy named Munashe. His mother turned into a lion one day—or at least, this is what he thought."
Oh, how she wailed. The sky shuddered and storm clouds split open at her hoarse, inhuman cries. Munashe cringed at his mother's unarticulated, bare suffering, at her voice rising higher and higher, lunging for heaven. He looked at blood that came out of her mouth and curdled on the earthen floors and rank pallet, black and granular like coffee grounds. He listened to the sound of her fingernails biting into the floor, dragging across it with the jerky movement of the dying.
He sat by her, trying not to be annoyed at her eyes, white with fear, swiveling in her hollow-cheeked face. He made nice, and brushed her long hair out of her face, stroked her cheek with filial attention.
"Let me go," she pleaded in staccato gasps.
He tried to make his voice soothing, reassuring, as if talking to a child. "Where would you go, mother? You're too weak to walk, and no village would take you."
"Munashe."
"I can't, mother. You should be grateful that I am staying here with you."
"Please."
He sighed. "You should've thought about that before you went and turned into a lioness."
She gasped and cried some more, and he could not help but laugh. The woman was deluded enough to think that she was still human. She tried to convince him, thrusting her dark, withered arms into his face. "Look at me. I am not a lion, I am your mother." As if he couldn't see the hungry beast looking out of her eyes, the red glow of its pupils burning hotter than the embers of the cooking fire. He heard from old men that women went wild, turned into beasts, and there was only one way of turning them back into humans.
He took a charred piece of impala meat from the coals, and offered it to his mother. "Will you eat now?"
She cried. "It is too hot, too black. I can't eat this."
He nodded to himself. She wanted raw meat, of course, like any lion would. He tried to do good by her, taming her with cooked meat, but so far she hadn't taken any. And her time was running short. AIDS was killing her, and if she went as a lion, her afterlife would be bleak—if she would even have an afterlife.
He ate alone, in the retreating light of the fire. The darkness reached for him, spreading its hungry fingers like a wrathful spirit, its bottomless mouth opened wide to swallow him whole. His mother made no other sound but her labored breath, and the faint scratching of her fingernails on the floor. Like a beast, she wanted to crawl away, to find a secluded place in the savannah grass, where she would expire alone, lamented by wind, buried by ants, kissed by red dust. Fortunately, she was too weak to do so. He waited for the scratching to stop before he went to sleep, curled on the earthen floor of the grass hut. Far away, hyenas gloated. They knew that a lion would be dead soon.
When Munashe woke up, his mother was dead, her eyes opened wide but blind, her pallet stained with sweat and blood. Munashe grunted his discontent, and hurried toward the doorway of the hut. There, he stopped and clamped his hands over his mouth to hold back a wail of terror that swelled in his chest. Instead of the yellow, undulating expanse of the savannah, punctuated by lopsided umbrellas of acacias, a solid green wall of forest surrounded him. There were no lions or hyenas, but only colobus monkeys chattering up in the trees.
The monkeys saw him, and wrinkled their faces, baring tiny, needle-sharp teeth that curved inward. "Munashe," they sang in nasty childish voices, "Munashe, mother-killer."
Their taunt, as direct as it was cruel, brought him out of the daze. "No," he yelled back. "It was not my fault. AIDS killed her, not I."
One of the bigger monkeys swung on the bough and leapt from branch to branch, until its face was level with Munashe's. The monkey's breath smelled stale, and its inward-curving teeth glistened like small yellow fishhooks. "Really?" it hissed. "Did you take her to the doctor, did you make sure that she ate well? Did you care for her in her comfortable home, or did you drag her away from people, from help?"
"I was trying to help. She turned into a lion—she wouldn't eat anything but raw meat."
The monkey's eyes gleamed; its terrible mouth opened wide, and the monkey cackled, the sound of its laughter like scratching of dead leaves. The monkey leapt and landed on Munashe's shoulders. Before he could toss off the unwelcome rider, the monkey's hind legs and long tail wrapped around his neck, and the sharp claws of its hands dug into tender cartilage of Munashe's ears. "Run now, donkey boy, mother-killer!"
Munashe twisted and struggled to get out of the monkey's hurtful grip, but it only laughed and tightened the chokehold of its tail, and wrenched his ears until they bled. Exhausted and terrified, Munashe ran, as the monkey steered him by the ears, deeper into the forest.
It was dark and stuffy under the canopy of the tall trees, and thorny lianas snagged the sleeves of his shirt and his trouser legs, ripping them, digging into his skin until he bled. His lungs expanded and fell, but sucking in the humid air was like trying to breathe underwater. His vision darkened and he took a faltering half-step, stumbling on the ropy roots, falling, anticipating the touch of soft ferns that lined the forest floor. A sharp tug on his ear made him cry out and right himself, picking up his step.
"You don't get to rest, mother-killer," the monkey screeched in his ear.
He ran until the air turned purple and then black, and strange noises filled the air. Something hooted, something chuckled, something else whined in a plaintive, undulating voice. Before the darkness swallowed him, he saw a single bright light beckoning him from behind the trees. The monkey made no objections as he directed his torn feet toward the light.
He came across a grass hut nestled between two strangler figs. The light he saw came from a small lantern perched atop the flat roof.
The monkey gave him a quick, vicious smack on the back of his head, and Munashe bent low, and hurried through the blanket-covered doorway.
"I brought him as you asked," the monkey said, and leapt off his shoulders, to take place next to a military-style woodstove that filled the hut with unbearable heat.
In the glow of the embers, he saw a low cot, and an old, fat woman that reclined upon it. Her bare breasts glistened, framing her swollen abdomen, from which a belly button protruded like an upturned thumb. Her bright eyes held Munashe's for a moment. "Well, well," she said. "Looks like Tendai did a good job." She gave the monkey a fond glance, and it hopped and chittered.
"Who are you, lady?" Munashe's cracked and swollen lips moved painfully.
"I am Tapiwa," she said. "You will serve me until your debt is paid."
Munashe was about to protest, to say that it wasn't his fault, but only sighed. The salt of his sweat burned like fire on his cracked lips. He felt certain that no matter what he said, he was already judged and found responsible for his mother's demise. His only hope of returning home was to listen and to obey; perhaps then they would let him go. "How may I serve you?" His gaze wandered involuntarily to her elephantine thighs circled by rims of fat, and to the dark, curly vegetation of her pubic hair.
Tapiwa noticed the direction of his glance, and shook with a booming laugh. "Ah, not that way, boy. I have bad bedsores, and I need someone to take care of them. Tendai and Robert are not strong enough."
"I'll do whatever you need me to, lady. But can I have a drink of water?"
Tapiwa nodded. "You may drink and you may rest. Tomorrow morning, you start."
The morning brought feeble light and the smell of dead embers and sweat, as Munashe started on his task. It took him a few tries to roll Tapiwa's bulk to her side. Waves traveled under her skin with every move, and his fingers slipped on her smooth, damp skin. Two monkeys—Tendai and his brother Robert—watched from the perch atop the woodstove.
Munashe puffed, but finally Tapiwa was stable on her left side, her left breast flopping to the floor. Munashe looked at her back and gagged—where her skin should have been, there was nothing but an open sore, running from her shoulders to her backside. A white mass shimmered and moved inside the wound, filling it, spilling to the pallet with every breath Tapiwa took. Maggots.
"What are you waiting for, boy?" Tapiwa said. "Clean them up."
Munashe extended his shaking hand to the living carpet of vermin, and a few maggots popped under his touch. Still, he gathered a handful, looking for a place to throw them.
"On the floor, on the floor," Tapiwa said, impatient.
He obeyed.
Tendai and Robert left their roost, and gathered the maggots with their long fingers, stuffing them in their mouths.
"You want to help me?" Munashe said.
The monkeys chattered and laughed, and shook their heads, their jaws moving energetically.
And so it went—Munashe scooped out the maggots by the handful, and the monkeys ate them, showing no signs of getting sated. Munashe kept his eyes half-closed, and breathed through his mouth; his mind wandered far away, back to his home village, to the fields worked by women and children, to the smells of manure and upturned soil, to the proud cassava mounds, surrounded by yam and cowpeas.
Munashe missed home every day of his joyless labor. While Tapiwa was not unkind, her wounds grew re-infested every day, and Munashe was starting to suspect that his labor would never be over. And he gave Tapiwa the care he did not give his mother, care he could not give to all the people in his village—hollow-cheeked men that came home from the city one last time, to their patient wives, thin and hard and strong like strips of leather. Tapiwa, the fat spirit—for he was sure that he was in the spirit forest—was all the sick, all those destroyed by the new way of life that he could not heal. Her sores wept for all.
At night, when the woodstove blazed, burning the already hot air of the hut, Munashe crept outside, under the sultry starless canopy of the forest, and prayed to the ancestral spirits to free him. He cried until his eyes ran dry, and rested in a crouch, listening to the night-sounds; there was chittering and chirping, sighing and moaning, wailing and weeping. And grumbling. His muscles tensed as he listened to the approaching roar—could that be a leopard? Twin lights shone through the treetops, and moved closer, like falling stars. Munashe's mouth opened in awe as he realized that the sound and the light issued from a very old, very large Cadillac, painted bubble-gum pink. The Cadillac descended, leaping from branch to branch like a most agile monkey.
The Cadillac gripped a low horizontal branch with its front wheels playfully, swung, and somersaulted, landing in front of Munashe with a flourish.
"Hello, Mr. Cadillac," Munashe said, shaken, but present enough to remember his manners.
"Hoo! What a dim boy!" the voice came from behind the tinted window. The window rolled down, and a smiling skull with red eyes blazing from under an old khaki baseball hat stared at Munashe. "Why would you think that the car was alive, hm?"
"I . . . I don't know, sir."
"It's a spirit car." The car door swung open, letting out a tall skeleton dressed in a tattered tuxedo, with sleeves and trousers that were too short. The skeletal remains of his neck were wrapped in a dirty red tie. "Now tell me what you need. You didn't call me here for nothing, did you?"
Munashe told the skeleton his story, all the while marveling at the ease of spirit summoning in the spirit forest.
The skeleton listened with an inscrutable expression. "So, you want me to rescue you from your servitude?" he said once Munashe had finished.
Munashe nodded. "Please."
"Maybe. But first, tell me—what did you learn from all this?"
Munashe stumbled for words. "I don't know, sir. Maybe that everyone needs to be taken care of?"
The spirit skeleton nodded. "I suppose they do. What will you trade me for my help?"
"I don't have anything," Munashe said.
The skeleton's eyes flashed. "You have flesh, boy. How much flesh will you give me for my help?"
Munashe closed his eyes, and thought about his mother. How emaciated she was. And still she lingered, grasping onto life with her stick hands. "Take as much as my mother had lost," he offered.
The skeleton's grinning mouth moved close to Munashe's face, breathing out the smell of liquor and stale meat. It drew a great breath, and Munashe felt millions of tiny teeth gnawing on him, moving under his skin, shaving off his flesh pound by pound, yet never spilling any blood or damaging his skin.
When he opened his eyes, the skeleton seemed bigger and fatter—as much as a skeleton can be fat. He nodded to Munashe and got into his car. "Tomorrow night wait here for my uncle. He'll help you."
"Wait!" Munashe waved his arms after the Cadillac as it started its graceful ascent. "What's your name?"
"Fungai," the skeleton answered, and he and the Cadillac were gone, swallowed by the weakly glowing branches.
The next day, Munashe felt weak but almost cheerful as went about his task. Tendai and Robert, the monkey brothers, noticed, and each gave him a vicious smack and an ear-boxing. Even that could not dispel Munashe's good mood, and he grinned through the tears.
"Ah, you're learning," Tapiwa said. The living shroud of maggots that simmered on her back did not seem to inconvenience her in the least.
Munashe looked up. "Learning what?"
She shrugged, sending the maggots spilling over the pallet and the floor, where Tendai and Robert made quick work of them. "That there is a point in every pointless task," Tapiwa said.
Munashe was not sure if he agreed. A pang of guilt coursed through his body—taking care of his mother was a pointless task; she would have died anyway. So instead he chose a task he thought he could accomplish—taming the lion back into human form.
When the night fell, he snuck outside and waited by the giant strangler fig. He wondered if Fungai's uncle also drove a Caddy.
Something tugged on the shreds of his trouser leg, and he looked down. He almost cried out at the sight of a small baby next to him that stood on all fours, its tiny, long-fingered hand clutching the fabric of Munashe's trousers. Worst of all, the baby's face was projected on a large TV screen; instead of a head, the TV perched atop the baby's shoulders, dwarfing his small, withered body.
Munashe swallowed hard a few times. "Are you Fungai's uncle?"
"Yes," flashed the letters on the TV screen. Then, they were supplanted by a large red question mark that took up the entire screen.
"How can I heal Tapiwa's wound?" Munashe said. "How can I go home?"
"One or the other," the screen said.
"Both, please. I can't leave until she's better."
The baby's face reappeared, smiling. "You could. I could help you leave right now," it said. Apparently, the TV had sound too.
Munashe bated his breath. This was better than he dared to hope. Still, he resented abandoning his hopeless task, no matter how pointless. "Help her first, and then help me leave."
"One or the other," the screen said.
"Then help her. I know I can leave after she's better."
A question mark again.
Munashe sighed. "I don't know for sure, but I think this is how it works."
Fungai's uncle shrugged his tiny baby shoulders, and showed his face again for a moment, before displaying a chart. "Find the kobo tree -> Find the Lady-Who-Lives-Inside -> Ask for a wishing thread -> Ask for her price."
"What's a kobo tree?" Munashe asked, but Fungai's uncle was already crawling away, the mahogany casing of his television head striking tree trunks that stood too close to his path.
The next night, Munashe set out looking for a kobo tree. He wasn't exactly sure what he was supposed to be looking for, but reasoned that it would be easy enough to recognize. His expectations were fulfilled once he saw a majestic blood-red trunk, crowned with blue foliage and peppered with small yellow flowers.
"Lady?" he called. "Lady-Who-Lives-Inside?"
The Lady-Who-Lives-Inside stood before him as soon as he uttered her name. She was a tall young woman, the most human-looking creature he had encountered so far. Munashe thought that she was just like any woman in his village, until he noticed her stomach—or rather, that she did not have one. There was a large round hole in her midsection, where her belly should've been, framed by the arches of her ribs and pelvis, and festooned with red fragments of gore that fringed the empty space, as if her organs had been ripped out of her.
"I need a wishing thread," Munashe said. "What is your price?"
The spirit reached inside of the hole and pulled out a thin string of sinew, red and blue and yellow. "For this," she said in a high nasal voice, "I want the same from you."
Munashe nodded and clenched his teeth as the spirit's clawed fingers—ten on each hand—pried apart his skin and muscle, sinew and bone, until a tiny piece of Munashe dangled, dizzying, in front of his face.
"There," the Lady-Who-Lives-Inside said, and gave him her thread. "Touch it to whatever wound you wish to heal, and it will be done."
Vimbai's story ended abruptly when Felix walked into the kitchen. Vimbai momentarily pitied him for his pale translucent skin—she could see a blue vein pumping away on his temple, just where the darkness of his hair started. He looked so vulnerable, so distraught with his eyes pointing in opposite directions. He rummaged through the refrigerator—the habit all of them had developed recently, even though they knew there was little of value there; still, the foolish hope that they had somehow overlooked a soda bottle in their endless searching refused to leave.
"There's nothing there," Vimbai said.
"I know," Felix answered, and commenced rummaging. "Have you seen Balshazaar?"
"Not recently," Vimbai said. "Why?"
"I was looking for him, and he's nowhere to be found."
"This place is big." Vimbai adjusted Peb who was starting to doze off on her lap. "Say, can I take a look inside your hair?"
"Good idea," Felix agreed. "Maybe he went back in there."
"You would've seen him, wouldn't you?"
"Unless I was sleeping." Felix sighed and slammed the fridge door closed. He stood in front of Vimbai and bent down dutifully, letting her inspect his hair.
She pressed her face forward, cautious of what could be waiting for her inside. It could be a trap within a trap within a trap—she was not convinced that the house was safe, let alone Felix's hair. It took her a while to adjust to the darkness inside, and the sleeping movement and inarticulate mumbling of Peb in her lap were disorienting.
She squinted, looking at the familiar dusty-gray landscape. Balshazaar was not there, and she felt relieved for a moment—until she realized another, more troubling emptiness. The empty spaces were gray like the rest of the contents of Felix's head, and it took her a while to realize what it was she was not seeing, what it was that reminded her of her promises with its nagging absence.
The horseshoe crabs' souls were gone—not a carapace, not an errant leg or a tail spike remained. They had disappeared, and for a moment Vimbai's eyes looked back and forth, searching for what could not be found. The souls her crabs had entrusted her with were gone. She had failed them, and—she suspected—she had failed her own hope of ever returning home, her parents, so sick with worry, and Maya; if they never got home, there would be nothing left for Maya but to play with her dogs and to sleep by the coffin, until she grew weaker and weaker, until the terrible men in the medical trucks came for her—came for all of them, to drain their blood and to toss their weak, not quite alive bodies into the lake, where the man-fish would make short work of their souls, consuming them like he had undoubtedly consumed those of the horseshoe crabs.
Vimbai freed her face from Felix's hair. Her eyes met his, and she frowned. "Oh Felix," she said. "We're in so much trouble right now."
Felix swallowed hard. "I see. What do we do?"
Vimbai drew a breath and petted Peb absent-mindedly, like one would a sleeping cat. "We have to go and find the man-fish and the men in the medical trucks. And we have to get Peb's tongue and horseshoe crabs' souls back."
# Chapter 13
"Enough is enough," Vimbai said. She had sent Peb to retrieve Maya, and as soon as she and her dogs showed up, she swung into action. As little as it appealed to her, Vimbai decided that now was the time to take serious action. It was her failure that the horseshoe crabs' souls were stolen. It was her job to set it right.
She made everyone assemble in the kitchen, which she thought of as her command post. She also suspected that here they were protected by the benign magic of the stove and the refrigerator, guarded from the eavesdropping of the man-fish and other entities she was not yet sure about. She had decided that Balshazaar was an enemy—after all, who but him knew where the horseshoe crabs kept their souls?—as well as the men in the medical trucks. And she especially did not want the horseshoe crabs to overhear her and to learn about her failure.
She looked at the chipoko and Peb in her arms, at the intense, open-mouthed Maya's face, who looked at Vimbai as if she had just met her, and was expecting something profound or interesting. Vimbai noticed Felix standing by the window overgrown with flat hairy leaves, his shoulders hunched over and his hands buried in the black hole surrounding his head with an equal measure of despair and concern—he seemed to be constantly checking for things going in or out, if anything was being stolen away.
"So this is what we're going to do," Vimbai said. "We'll go to the man-fish, and we tell him to give the crabs their souls back. And I bet he would know how to get us home."
"Or how we got here in the first place," Maya interjected.
"Maybe that." Vimbai considered banging her hand on the kitchen counter, but decided against it. "But now we need to take care of business."
"How do we do that?" Felix asked.
"We'll talk to him," Vimbai said. "He's a fish. Maybe we can threaten him or something."
"How do you threaten an eater of souls?" the vadzimu asked.
"Surely there's something he needs," Maya said. "Or is afraid of. I'm with you, Vimbai—let's go."
"Felix and Peb should come too," Vimbai said. "We need Peb—he can point out whoever hurt him."
They set out to the lake. Vimbai gritted her teeth and felt altogether grim: she felt her forehead furrowing with long horizontal lines, and her jaws and fists clenching, as if in a movie. She thought that it was the first time in her life she felt such resolve, such simple realization that she had to do something, and there was nothing that could stop her from doing it.
She missed her mother then—her mother who went to work every day with the same clenched fists and jaws, the same stern faith spilling out of her eyes. It had been easy for Vimbai partially because she had a mother like that, a mother who could march into the office of a department chair or school principal, and put forth her demands. She would not be swayed by the appearance of reason, by the soothing voices and sober explanations of why her demands could not be met. She would cross her arms and wait in silence, until they either caved or asked her to leave, thus granting her a moral victory at the very least. Vimbai wished she could be like this.
Then again, her mother had the dubious advantage of having to fight for everything, and most of these fights Vimbai was not privy to. She only caught tail ends of arguments and meaningful exchanges of glances between her parents, or occasional phone conversations with other faculty members in Africana Studies. Of talks over tea, of complaints about white people setting the Africana agenda, and how unfairly colonial it was.
Vimbai felt embarrassed of her ignorant indifference toward these battles, of her dismissal of things that had anything at all to do with Africana Studies or African politics or Africa anything. She was an American, she used to tell herself, and it had nothing to do with her, the only person in her family who spoke English without an accent. It was her parents that carried Africa within them, who could not let it go and kept obsessing over it years and years after it became irrelevant to them—and after they became irrelevant to it, immigrants, deserters, people who left their country and were in turn left behind, as it moved on without them.
"Everything had changed so much," Vimbai's mother kept repeating with quiet wonder as they walked through the streets of Harare, and she insisted that she knew these streets like the back of her hand but kept taking wrong turns and getting lost anyway. At night, she cried about it when she thought Vimbai could not hear her.
But if it was not Vimbai's, this burden, this memory, why did she have an ancestral spirit following her and telling her stories, filling Vimbai's eyes with her sad visions—jacaranda trees in bloom—despite everything? Why did she have her own Harare here, in this dune house from South Jersey? Why did the man-fish and the fairy tales, the wazimamoto of her Kenyan babysitter, follow her and refuse to let go? She could not shake them like she could not shake her parents and their sins and memories. Tied to them by the tenuous bond of blood, and through them, tied to the continent she neither knew nor particularly liked. She wondered if Maya felt this ancestral bond too, through the intervening generations and the accumulated twin heartbreaks of colonialism and slavery.
They approached the lake that stretched, deceptively peaceful and smooth, before them. The surface remained undisturbed, like a pane of green glass, and Vimbai decided that it meant that the man-fish was at the very least cautious, and possibly, she hoped, concerned. "You should be concerned, you bastard," she muttered through her teeth. "You better fucking worry."
Maya, who stopped at the lakeshore just ahead of Vimbai, looked over her shoulder. "Whom are you talking to?" she asked Vimbai. "And why are you swearing?"
"The man-fish," Vimbai answered. "And sorry about the swearing."
"I don't care." Maya laughed and turned back to stare at the lake. "In fact, you don't swear nearly enough."
Normally, Vimbai would've felt resentful: she hated it when people told her how she should talk or what she should act like, especially if they accused her of acting white—oh, how it turned her stomach. She suspected that Maya never said things like that because she had had the same words thrown in her face too. "I just never picked it up, I guess."
Felix nudged Vimbai's side. "What if it . . . the catfish. What if it doesn't come out?"
"He always does," Vimbai said. "Let's just wait a little."
The water remained still, and Vimbai picked up Peb who hovered by her elbow, as if having accepted her authority and the hope of help. She cradled him, his grotesque hands and feet brushing against her cheek like soft strands of seaweed. "It'll be all right," she whispered. "Don't be afraid, little Peb. We'll get your tongue back from the bad fish."
Peb moaned and shook his head.
"What? It wasn't him?"
Peb nodded, wailing for emphasis. It tugged Vimbai's hair and pointed with seven or eight of its limbs, at something behind Vimbai.
She whipped around, only to see the quick movement of something disappearing in the low brush behind the stacked couches, just a few dozen yards away from the lake. She was not sure what it was, but it was low to the ground and moved in swift but jerking motion, sending the branches that concealed it into spasmodic trembling. "Balshazaar," Vimbai said.
Felix turned. "Where?"
Vimbai and Peb pointed at the bushes, and Felix took off toward them, with a speed Vimbai had not suspected in him.
Maya looked after him. "Poor Felix," she said. "She chases this stupid thing like it means something."
"Maybe it does mean something to him," Vimbai answered. "I won't pretend that I understand anything about Felix."
Maya nodded her agreement. "He's a strange one, that's for sure. I wonder how it is, to have the remnant of a universe hovering around you?"
"Or rather hanging down from the remnant of a universe," Vimbai said. "Still, do you know what happened to him? Where he was before, and how he came to be here? Can we even comprehend that?"
Maya shook her head. "No way. I don't even think about that—once you start, you can't stop, because then you start asking how come he speaks English and if everyone there does, and how was he able to get a New Jersey driver's license, or even if he did get it—maybe he always had it or found it in his hair, and what is he even doing, existing like that, you know?"
"Yeah," Vimbai said, and cradled Peb closer. "I'm just creeped out by Balshazaar, and Peb seems to imply that it was he who had taken his tongue."
"Could be." Maya walked up to the water's edge and tried it with her toes. "Warm. Anyway, maybe Balshazaar is pissed at Felix and at the rest of us because we're Felix's friends. Maybe he likes the fish for whatever reason."
"And the men in the medical trucks," Vimbai added. At this, Peb stiffened in her arms but did not utter a sound. Vimbai decided to let him be for now.
"You keep saying that there are these guys in trucks." Maya crouched down and splashed water with her hands. "But I haven't seen them, and no one else did either. How do you know they are even here?"
"Oh, I know," Vimbai said. "Sometimes you just do."
Sometimes, you just did. Vimbai did not believe in ESP—rather, she trusted that human instincts, having evolved over hundreds of thousands of years, were better at picking up signals indicating danger than her rational mind would ever be. Sometimes, one had to trust the gut feeling, whether it came from quick but persistent observations that had not reached her conscious mind yet or from internalized knowledge, too old and too deep for words. She did not need to hear her mother's voice or see her face to know that she was angry—the anger colored the air in the house, pumped it full of tension that Vimbai could feel as soon as she entered the house.
And just like that, she felt the electric charge in the air, she felt the unseen and unspoken menace—the men and their trucks, the sense of wazimamoto crouching nearby. She had developed this fear as a kid, and now it came in handy—or in any case, it felt more constructive than the blind childhood panic that made her dart to the bathroom at night, running so fast that her feet seemed to barely touch the cold hardwood floors. The same panic that forced her to take showers with her eyes open, fearful that the moment she closed them, she would feel the cold hand of wazimamoto on the inside of her elbow and feel a long needle go in, so deep, scraping against the bone.
Now, there was still fear and the long sucking sensation in her stomach, and the rising hairs on her arms at the thought of violent needles. She bit her lip and tossed her head back.
Vimbai considered Felix compared to Maya—he did not seem like the same kind of roommate. With Maya, they could bond and argue; with Felix, any illusion of understanding was aborted before it even had a chance to take hold, with just one look at his eyes and his hair—inhuman, inhuman. There was no chance of casual chat, of friendly bickering—as much as she had tried, all she could do now was to try to accept his presence and help him as much as she could; not out of friendship as she would do with Maya, but rather some generalized compassion, the ethical obligation one felt to help other creatures or at least to be reasonably nice to them in order to consider oneself a good person. Even Peb seemed more human: no matter how many phantom limbs it had attached to itself and no matter how many flowering branches it had absorbed, Vimbai could understand its suffering and its pain. She could relate to it. There was nothing to relate to in Felix. So she let him go, chasing after Balshazaar through the low scrub, and let him disappear from her mental landscape as soon as she looked back to the lake. It was not indifference, she decided, just the mind's inability to hold onto something so incomprehensible and smooth like an egg, missing any angles her attention could snag in. Instead, she stood next to Maya, Peb in her arms, and waited for the man-fish.
The man-fish finally decided to show himself, when Vimbai was about to give up and suggest that maybe they should come back tomorrow, although that would certainly kill the momentum of her accumulated decisiveness and rage. He popped up among the reeds, his transparent fanned fins propping him up. He looked bigger now—so huge, big enough to swallow Peb whole with his thick-lipped fish mouth. He smiled a little bit, and Vimbai held her breath, as if afraid that the fish would suck her soul out with the next exhalation. It also gave her time to look over the fish.
The lips and the whiskers, she thought, were just like Vimbai remembered them—undoubtedly catfish, and yet suffused with very human sarcasm as the fish thrust out his lower lip and eyed Vimbai. The eyes, golden and cat-like, seemed to smirk and wink, a difficult feat without any eyebrows or eyelids. His flat head, mottled gray and brown like a stone, seemed too heavy for his weak fins—it wobbled, and the massive long body had to follow suit, tilting slightly from side to side, compensating for the head's appearance of feebleness.
"Did you take Peb's tongue?" Vimbai asked as sternly as she could.
Maya did not say anything, but her right fist gave a short, resonant punch to the open palm of her left hand. A simple but highly suggestive gesture, Vimbai thought, and smiled.
"No," the man-fish said, studying Maya with some curiosity. "Don't have any tongues, I really don't. But I do wonder why are you threatening me—I've done nothing to either you or your despicable half-breed rats."
"I'm not threatening," Maya said. "But since you've mentioned doing things . . . you wouldn't happen to have a few dozen horseshoe crab souls, would you?"
"Don't be silly," the man-fish said. "Crabs don't have souls—even fish don't, unless we swallow some drowned ones."
"Is it true?" Maya whispered into Vimbai's ear.
"Don't know," Vimbai whispered back. "But makes sense, sort of. Only those things I've seen in Felix's head—what were they?"
"Apparitions," said the man-fish, whose hearing turned out far superior to what one would expect from two holes on the sides of his head. "Accretions. Come closer, and I will show you."
"Vimbai, don't." Maya's hand wrapped around Vimbai's forearm, the strong protective warmth of her fingers encircling like a sigil guarding from evil.
Vimbai gently freed her arm and handed Peb to Maya. "I'm just going to listen. Nothing will happen to me while you are watching over, right?"
"I'll see what I can do," Maya mumbled, and showed her fist to the man-fish. "Don't make me cave your skull in, fishstick. And don't you think this lake will protect you—my dogs will drink it dry if need be."
The man-fish rolled his eyes. "I do not mean you harm—not at this very moment, at least. But perhaps once you understand what it is you're defending you would be more inclined to leave me alone."
"I won't leave you alone until Peb has his tongue back, and the horseshoe crabs are whole again," Vimbai said, and regretted it immediately—perhaps, this was not a good time for threats she could not really fulfill, especially not so close to the man-fish's hypnotic gaze—he floated in the shallows now and she stood knee-deep in warm water, fat mud oozing between her toes, and wondered how she got here. Before she could verbalize her question, the man-fish bobbed up and down on the waves, and swam closer. "Do you even know what horseshoe crabs are?"
Arthropod and other assorted invertebrate classifications turned out to be irrelevant, and Vimbai almost regretted memorizing their mouthparts, tiny, numerous, and confusingly named. Mouthparts did not make the horseshoe crab—or at least this is what the man-fish said.
When something is as ancient as these crabs, when it lives on the bottom, scavenging, for so long, it is only a matter of time before spiritual accumulation becomes as significant as the chitinous growth of the shell. Tail spikes and fragile little legs, eyes hidden behind the spiked bumps of their armor—all this was just surface. But there were other shells, other eyes, built from things less tangible than chitin.
The ocean is awash in the souls of drowned sailors, or rather in their remnants—time passes, years wear on, and the souls are rent apart by the constant action of the waves, their endless back and forth over the seesaws of coral reefs and jagged cliffs. The souls become small fragments of memories and preferences, of vague longings and dreams one could not forget no matter how hard one tried. These soul fragments, small as grains of sand and just as inconspicuous, permeated everything on the ocean floor. And they became accumulated and accreted in the shells of the horseshow crabs—forming a similar but spiritual structure that gave them not only consciousness but also fortitude and memory, persistence in the face of being quartered and stuffed into eel traps and bled half to death for profit.
And listen, listen, here's the best part: even if you end up in the sea a few minutes after your death, your soul would still be liable to become a part of it. And the Atlantic coast—there are so many bodies there, there are so many people thrown overboard because they were dead or dying. Yes, yes, little girl—I mean the slave ships.
The wazimamoto are real, and they are not fools. Even if your soul is in shreds and a thousand miles away, a part of a horseshoe crab with a spiked shell and a long tail, even then they will find you, even then they will steal your blood. It is their nature, see, and it doesn't matter to them if you're a man or a crab—as long as you are helpless and alone and vulnerable; as long as there is blood (red or blue, doesn't matter) for them to steal.
"So you see," his gravelly voice tinted with hidden laughter said. "Your precious crabs are just people in different guises, and what you thought were their souls are just simple carapaces, same as their regular shells. They have no meaning or importance, they are not at all like human souls."
"And yet, you took them." Vimbai opened her eyes—she did not remember closing them—and stared into the man-fish's, yellow and bright, so close to her face. She swam in the golden ocean, weightless, bathed and suspended in pure sunlight, just slightly blurred by the film of tears. "What did you do with the souls . . . shells you took?"
"I didn't take anything," the man-fish replied. "If you don't believe me, come to my lair with me, I'll show you."
"Vimbai, don't!" Maya's voice reached her from the shore distorted by water as she sank, slowly and obediently, following the whipping tailfin, the serpentine twists of the mottled green and brown scaleless body. The water turned muddy around her, and soon she could see nothing but the clay-colored murk and the undulating tailfin before her.
It had occurred to her then in a lazy, sleepy way reminiscent of a sluggish dream, that she was not being wise, following the catfish like that. Perhaps she should turn back or swim for the surface, where she could breathe air instead of water slowly filling her lungs . . . that was not good and she started, as if jolted awake.
The vadzimu was not here to guide her senses and vision, she was not here to help her breathe—and the water was not the cold, singeing salt of the ocean, but tepid bland mud. How could she be so stupid, forgetting not to inhale? There was no time for it now, and Vimbai swam for the surface, already struggling against a pressing cough rising in her chest. If she coughed now she would swallow even more water, and that she could not allow—already her lungs strained under the weight of water as well as the overwhelming sense of suffocation, of absence of oxygen.
She kicked her feet and propelled herself upward, and she saw the sun through a thick layer of dung-colored water and thought herself saved, the motes of silt playing in the amber light, the surface so near now. But then there was a shadow darting over head, a large shadow—as long as Vimbai, or perhaps even longer. A shadow with two fanned fins and a blunt, flat head.
Vimbai kicked faster, almost reaching the surface, but the shadow returned now, and its flat face, momentarily close and clear, blotted out the sun and Vimbai felt a strong nudge as the fish butted its head against Vimbai's, forcing her underwater. She tried again, pushing the fish away with her hands and feet, kicking it away, reaching for the surface, but the fish was too strong and too slippery, too old and too large. The impact of its massive head felt like a hit by a basketball, rubbery and yet heavy, disorienting.
Soon enough, Vimbai was not sure which way was up, and the silt particles in the water swarmed like myriads of tiny flies, blotting out the light and sense of direction, even the sight of the man-fish. Vimbai only wished she could breathe, and covered her face with her crossed arms.
She felt him approach again and waited for him to get close, within striking distance. As soon as his face touched hers, her hands shot out and grabbed at the sensitive whiskers, the only part of him she could hope to grasp and to hurt. She pulled and punched, aiming for the eyes, and the catfish thrashed, one of its whiskers held firmly in Vimbai's hand, wound for security across and around her palm.
Vimbai's fingers clawed blindly until they felt a glassy slippery fish eye underneath; then they tore. The catfish thrashed more, the paddle of its tail whipping Vimbai across her chest and face. Every second stretched and went on forever, and even their fractions dragged like funeral hearses—at least, this is how it felt to Vimbai's flooded and exhausted lungs.
Another slap of the tail, and Vimbai closed her eyes; but even through closed eyelids the flood of sunlight was unmistakable and welcome. She gasped, sputtered, and spat out half a gallon of tepid water just as her lungs expanded, drinking in tasteless but welcome air. She thought at first that she had managed to struggle to the surface, but then she realized that her feet were planted firmly on the bottom of the lake.
Her gaze cast about, to see the man-fish, his whisker still in her left fist and his face under her right, flapping in the shallow water—the level of the lake was no higher than two feet now, and falling. Vimbai saw the Psychic Energy Baby kneeling on the lakeshore, his face in the water, his chest and back rising and falling with great measured gulps, and only then did she realize that he had saved her once again—he had drunk the water of the lake, saving Vimbai and trapping the man-fish. As if sensing her looking at him, Peb raised his face and gave Vimbai his new and terrible tongueless smile. Then he resumed his drinking.
# Chapter 14
Maya helped Vimbai out of the mud in the lakebed, and sat with her on the grass by the lawn chairs, rubbing her shoulders with her large, warm palms. Her touch was comforting to Vimbai, and she struggled with an overwhelming desire to rest her head on Maya's shoulder, to let her hold Vimbai and make her feel at peace and at home.
The Psychic Energy Baby swelled up with all the water he had swallowed—an entire lake's worth!—and sat back on the bank, great quantities of lake water sloshing inside him, as if he were a giant distended wine-bag, half-sunken into the soft mud by the shore. The cattails and sedges nodded in the breeze, sleek and green in the subdued glow from the sky-ceiling. It seemed so peaceful here, so calm—if one were to ignore a gigantic catfish flailing and thrashing in the wet mud where barely two inches of water offered it some comfort.
"I'll die like this!" the man-fish rasped. "I swear to you, I won't harm you again."
"And you'll give back his tongue and the horseshoe crabs' spirit shells," Vimbai said. "Right?"
"I promise!" the man-fish pleaded. "I'll do what I can, but I don't have those spirit things . . . they are of no use to me."
"The wazimamoto," Vimbai said. "Where are they? Do they have what we want?"
"Yes," the man-fish said. "I'll help you, I swear. And I'll tell you now, that bald head on one leg is also helping them. See? I am on your side."
Vimbai nodded to Peb and he leaned forth, a thin stream of water dribbling through his slightly parted lips. He let out just enough water to let the man-fish lie on his side, gills submerged, but the other side still exposed to harsh drying air.
"Tell us more," Vimbai said. "Tell us about the wazimamoto, and where they are." She did not ask about their purpose—after all, it was their nature.
The man-fish remained silent for a while, greedily pumping the tepid muddy water through his gills. "All right," he said. "I'll tell you how to find them."
What he told them did not surprise Vimbai—as the man-fish spoke, Vimbai realized that she had known it already, but was too embarrassed to admit that it was her part of the communal creation within the house that gave the vampires shelter. It was too painful to think that the blooming jacaranda trees, their branches heavy with purple and blue flowers, sheltered and blessed those who sought to harm Vimbai and Maya, and had already harmed Peb and the crabs.
The man-fish hemmed and hawed, but finally told them that the vampires did have a truck and all sorts of medical equipment. They asked the man-fish about things, and they promised him favors—he hinted at it obliquely but Vimbai felt a cold hand constrict her throat when she realized that the man-fish was looking forward to swallowing her and Maya's souls, after their blood had been drained away.
He only shrugged at her terror and disgust. "We all do what we have to. This lake here, there aren't many drownings, as you can imagine. Very little to feed on. And if you find someone who can help you—hey, why not?"
"You understand why we would be unsympathetic," Maya said. Her arm wrapped around Vimbai's shoulders in a protective gesture, and Vimbai felt gratitude flood her eyes, making them suddenly warm. Being held like that . . . it felt like being home from school, back when she was still a kid and getting out of school was precious because it was rare, and it was made even better by her mother's cool hand smoothing Vimbai's burning forehead. She was also reminded of the touch of Elizabeth Rosenzweig's smooth hand, and thought that she rather liked Maya holding her—almost as much as she would if it were Elizabeth.
"My dogs wouldn't even come near you," Maya said. "Although right now I do have half a mind to call them and let them have their way with you."
The man-fish thrashed, and Peb let out a bit more water—just enough to let the man-fish flip onto his belly and remain submerged save for his dorsal fin and its sharp spikes. Vimbai rubbed her forearm, which bore four long protective gouges, and winced. "So they are in the city. Is there any special weapon and tool we could use to defeat them?"
"Always looking for shortcuts," the man-fish admonished. "Always wanting the easy way. Think about it—if there was a vulnerability, would they tell me? Would you?"
"No," Vimbai said. "I see your point."
"But I can explain their nature to you," the man-fish continued. "I don't know if it would help, but please accept it as a show of good faith—I do expose myself as much as the others when I talk about such things."
"Of course," Maya said. "Go ahead, talk."
That's the thing about injustice, the man-fish said. Those who are affected by it naturally wish for vengeance, for a manifestation of their rage and pain; and manifestation comes, although rarely in the form it is expected. When Lilith was banished from Eden, they say that she was the mother of giants, but really, the giants were just a sign of the injustice done to her. They roamed and rumbled and shook the earth.
Monsters followed Cain to the land of Nod, and monsters bred and lived in the shadows, on the underside of history—like thin fabric grown transparent in the sunlight, it showed them briefly and in shadowed outline inhabiting humanity's dreams. They bared their teeth and claws, and their eyes watched people from every fold of darkness, waiting for them just beyond the edge of sleep.
So were the wazimamoto, the vampires, born out of injustice, as its manifestation and burden. They took residence in Harare built by Vimbai's imperfect recollection, the closest they could get to the Africa of dark dreams and cruelties not talked about, and did what they were imagined to do, embodying the terror and the despair of those who had birthed them.
"I think I get it," Vimbai said. "I just don't understand why you . . . and them, I guess—why all of you are here? Are you just my nightmares?"
"It's never that simple," the man-fish said. "Now, give me my lake back and go—I mean, if you care at all about your friend."
Maya and Vimbai stared at each other.
"Shit," Maya said. "Where's Felix?"
"I hope to God you're not lying," Vimbai said to the man-fish, and turned to Peb. "Come on, sweetie. Spit out the water so I can carry you."
Peb obeyed, and Vimbai marveled at the stream of water spewing endlessly from his mouth, as Peb himself deflated gradually. The man-fish bounded and swam to the bottom.
"Don't you worry," Maya told Vimbai, "we can always get to him if we need to. Do you think they really got Felix?"
"I haven't seen him after he went chasing after that freaky dried up head," Vimbai said. "Then again, I wasn't paying attention with all the drowning."
Maya laughed and patted her shoulder. "You really have to cut it out," she said. "It's the second time this has happened, and the second time Peb saved you."
Vimbai nodded, and Maya pulled her to her feet. "Come on, let's go. It's over the Malcolm X ridge, right?"
Vimbai smiled. "We've named everything there. I wish I'd written it all down."
"I remember," Maya said.
Vimbai nodded. "I do too."
The two of them almost ran now, through the kitchen where Maya's half-foxes joined them, and into the closet. They crossed the plain of discarded sisal rugs and mattress boxes, past the mound of gumboots and handkerchiefs. They passed through the valley of Five Percenters (named on Maya's insistence, since Vimbai's understanding of the doctrine consisted of the vaguely remembered class on African-American History, where it shared a lecture or two with hoodoo and other not-quite-religions for which the lecturer seemed almost apologetic. Even back then, Vimbai could not understand why the professor thought that these religions were less legitimate than the big three, or even the African religions and voodoo and muti magic.)
They reached the Harare of Vimbai's dreams late at night, when the sun was already setting. They looked from the ridge at the long shadows falling over the city, starting at the no man's land surrounding it and reaching deeper into the streets, serpentine, both familiar and strange—as, Vimbai supposed, a dream city ought to be.
"Perhaps it is not wise to go there in the dark," Maya said. "For all we know, they can see in the dark."
"They can," Vimbai said. "And it is a real problem for Felix right now."
"That's right." Maya frowned. "Felix. How do we find him here?"
"We let them find us." Vimbai sighed. "I just don't see any other way."
"Unless my dogs can sniff them out." Maya turned to her animals, smiling. "Go search," she told her pack. "Search for Felix."
"Wait," Vimbai said, and dug through her pockets. "This is the handkerchief he gave me—maybe it still smells like him."
"And don't forget Peb," Maya added. "He must retain some smell of Felix—after all, he and all his limbs came from his hair."
They made sure that the dogs got a good and thorough sniff of the handkerchief and Peb both, and Maya sent them into the streets below. Maya and Vimbai followed the silent pack as they sniffed the air, no doubt stumped by its lifeless quality.
"It's win-win," Vimbai told Maya and Peb. "Either we find them, or they find us. In any case, I hope we get to Felix in time."
"Wait," Maya said. "Should we take Peb with us?"
"Good point." Vimbai propped Peb in the branches of the nearest jacaranda tree, blue and languid like the night itself. "Stay here, little Peb, and if something bad happens, go get my grandma, okay?"
Peb nodded that he understood, and smiled a little. In the dusky gloom, he seemed transparent, but happier than he had been ever since his tongue was gone. He was either aware that his tongue was nearby, Vimbai thought, or the ability to help them in their search had distracted him from his troubles. Vimbai kept turning to look at him, glowing like a ghost of the moon in the low blue branches.
As they wandered through the streets, following the meandering track of Maya's dogs, Vimbai looked for landmarks, for any signs that signaled that this city came from her dreams. She recognized the painted stone, the stone friezes, stiff and intricate like frozen lace. She looked into the windows, dark on the inside, and saw stone carvings everywhere—birthed from her memory of the small coop stores that sold such carvings by the artisans. Stone green and black, simple flowing lines hinting at the outline of a face with a single sweeping turn. There were flowers inside, heaps upon heaps of them, as if every house Vimbai peered into was a stall at the flower market. There were people—or rather, signifiers of them, little more than dark faces in the dark corners, hovering above moth-white crucifixes of t-shirts. She remembered how much of a shock it was to her, walking down the street with her mother, and the two of them not being the only black people around—in fact, almost everyone was black in Harare. She expected that, of course, but her heart could not be prepared for the exhilaration she felt then; the sheer intensity, the reality of it could not be anticipated.
Then there were houses that seemed to belong more to South Jersey suburbs than Harare, but Vimbai's careless dreams plucked them from her memory anyway and dropped them among the trees and houses they did not belong with. In those, vinyl siding reflected the moonlight in fuzzy, opaque pools, and the floor lamps inside lighted the endless repetition of Vimbai's parents' dining room—the sturdy formal cherry table and the straight-backed chairs that surrounded it, haughtily expecting guests whose bottoms they would soon cradle. The tables were covered in the same white cloths with red trim—unusual, some sort of a heirloom, Vimbai suspected, but never cared enough to actually ask. The TVs glowered from the corner, with blue artificial static of their fisheye screens.
There were houses with tricycles on the lawns and plastic toys, large and bright and terrible in their garish innocence, strewn across driveways. There was asphalt and red dirt, and the signs for streets one would find in Zimbabwe mixed together with the ones from New Jersey. There were underpasses too steely and desperately industrial to be properly connected to a place, steel and concrete and humming of wires—the same in Zimbabwe as they were in northern Jersey and everywhere else in the world. Vimbai thought that humanity always managed to dream these not-quite-places everywhere—structures and interiors that remained the same from one continent to the next, airports and highways and hospitals, the dining rooms of franchise restaurants, prison cells. Even if the small details differed (and they rarely did), the overall sense of alienation remained the same, marking them as similar to each other and separate from the rest of the world, from the vibrant life that flowed and smelled differently in different cities, that made them all unique and recognizable—even in dreams.
Vimbai stopped as soon as she saw the sign. It winked at her from afar, its sideway neon grin fractured by the dark outlines of tree branches. "Hospital" the sign read. Of course, Vimbai thought, and pointed out the sign to Maya. "This is where they are."
"You're taking the medical truck literally, huh," Maya observed, but moved closer to Vimbai, ever so casually.
Vimbai smiled—she did not begrudge others their fear. "Yeah. Plus, if they collect blood, they would have to keep it somewhere, right?"
"If it's a dream, they can keep it in an old hat," Maya said. "But I do see your point."
The two of them walked toward the hospital, the slash of its sign disfiguring the night like a scar. The hospital seemed familiar, and with a squeezing of her heart Vimbai recognized it as Cooper, the University hospital where her father worked—the same tower of glass and steel and painted concrete, the looping driveway and the parking lot, and the parking garage—a towering structure alongside with the hospital proper, the path inside it winding endlessly, corkscrewing into the sky.
Vimbai motioned for Maya to be careful, and the two of them bent low, holding hands, moving in short dashes between the wrought iron gates and a small copse of trees and shrubs surrounding a couple of bird baths and benches—a handkerchief-sized piece of nature, wedged mercilessly among all the death and artifice of the towering stone.
Maya's dogs waited for them by the bird baths, and only signaled their joy at Maya's appearance by drumming their tails on the ground.
"Good dogs," Maya whispered.
"I think we better go through the garage," Vimbai said. "I've been at this hospital before."
"You know where the blood bank is?"
Vimbai shook her head. "No, but I know how we can get to the offices and the patients' rooms and the nurses' stations without disturbing anyone."
"Okay," Maya whispered. "Lead the way."
Sneaking by the turnstile that dispensed tickets and let the cars through was not a problem, and they tiptoed under the white dead light of halogen lamps that lit rows upon rows of rusted cars on cinderblocks, cars that would never drive anywhere—not even in this dream made substance. The pavement between the rows of cars had cracked, letting through thin, anemic stems of grass. Large chunks of asphalt had been cleaved off, as if by the stomping feet of giants, but Vimbai knew that it was grass and the young saplings that pushed upward among the cars that did it. Young trees, jacaranda and cherries, apple trees and maples reaching eagerly toward the fluorescent lights. They did not know any better, and mistook their artifice for the real sun.
They walked to the floor marked 'D', and Vimbai judged that they were sufficiently high above the ground. Maya's dogs stayed subdued and pensive, and clustered around Maya's ankles like a rust-colored, clumped and very scared rug. Vimbai calculated that they were somewhere on the fourth floor, and it suited her—she figured that if the wazimamoto expected them, they would watch the ground floor and the security checkpoints with vigilance. A quiet entry through the service corridor linking the parking garage to the hospital was a stroke of brilliance, Vimbai thought.
The emergency exit linking the parking garage with the main building was closed, and Maya heaved a sigh. "I suppose we have to go back down now, and risk the main entrance."
"Not yet," Vimbai said. "Let me try something." She patted her pockets and smiled when she found her wallet—habit was stronger than reason in her, and even though she had not anticipated a need for an ID when she left the house this morning, she still stuck her wallet into the back pocket of her jeans.
"You'll set off the alarm."
Vimbai shook her head. "My dad works here. I mean, in the real Cooper. He showed me how to do this."
"You think it will work here?"
"It should. It is my Cooper, I think." Vimbai pressed the edge of the credit card between the doorjamb and the dented edge of the door, where the underlying blue-gray aluminum showed under the chipping yellow paint. She wriggled the card until it clattered and caught something—a sense of solid metal transmitted to Vimbai's fingers as she felt the tapered edge of the lock and pressed, pushing the door smoothly open.
"Wow," Maya said. "You're good." She and her dogs followed Vimbai inside, into a short and blind corridor ending in a set of swinging double doors. Vimbai remembered those doors—they led through storage closets and sometimes surgery recovery rooms, the utility spaces filled with rolled up cables and wire to the actual corridors, wide and well-lit, which would take them to the patients' rooms, and various doctor offices and the nurses' stations.
Oh how little Vimbai loved them, those small islands of order and clean-smelling paper, tables and desks where the ragged doctors and interns could sit down to catch their breath or eat a meal or catch up on paperwork. So clean, so sane—and among this order in chaos, these islands in the stormy sea, was her father, like a king of his atolls and the captain of the ship, always calm and composed even when people hemorrhaged on the gurney while he fitted the IV bag, even when there were so few free beds they had to park the gurneys by the nurses' station. He moved among them, elegant and dignified, like royalty in charge of morphine pumps and gauze packs, the lord of disposable syringes and enameled bedpans. With the same smooth motion, he slid a needle into a collapsed, pale vein and handed Vimbai a cup of hospital Jell-O, the taste of which was still one of her favorite things in the world. She was proud of him, and unlike her mother he never felt compelled to say more than was necessary, and thus largely avoided being embarrassing to her.
Now, she poked her head through the double doors, to survey empty corridors—not even the memories of patients' shadows graced them, and even the nurses' station—this forever source of light and comfort—remained silent and dimly lit.
"Where to now?" Maya whispered.
It was a good question, Vimbai thought, and the one she had no answer to, except peering into every room on this floor and then going to the next. How many floors? Ten? Fourteen? How long would it take them? "Can your dogs sniff him out?" she said.
Maya crouched down next to her dogs. "Come on," she told them. "Go search. Search, okay?"
The dogs pummeled their tails on the ground and smiled, their open mouths and bright tongues colored scarlet-red. Finally, they stood as one, and walked tentatively toward the stairs on the other end of the hallway.
The dogs yelped a little until Maya hushed them, and started up the stairs. One floor, two, three—Vimbai was starting to lose count, and followed mechanically, barely noticing the turns of the stairs, reminded of the hollowed out building that had become Maya's grandmother's shrine—just as cold the stone, just as endless the stairs. Vimbai shivered and wished the morning would come.
The dogs led them to the top floor, and then into the hallway. Vimbai saw a sign for some medical department—a Bone Clinic? She did not remember one being there. She followed the dogs and Maya, her legs tense as if ready to take flight at the slightest provocation, into the reception area of the Bone Clinic and then into the office.
At first, she thought that she was looking at a row of chairs, and half a second later she realized that these were backs of the medical men, clad in green scrubs, all the same height and size as they crowded together, side by side, around a narrow surgical table. There were tubes conducting some black and foul liquid, and there was a pale body, translucent even—and Felix's disjointed eyes looked at them (one at Maya, one at Vimbai) with raw suffering.
The wazimamoto turned to follow his gaze, and as they parted, Vimbai wanted to scream—Felix's hair, his little cursed universe was gone, taken apart and slurping down the tubes. Then she heard Maya gasp and clutch her hand, and as she looked at the wazimamoto, she felt like gasping too. Their faces were concealed by gauze surgical masks and caps pushed low over their white brows beaded with sweat. But even these contrivances could not disguise the fact that the wazimamoto had neither noses nor eyebrows, neither lips nor chins; even their eyes were the barest hints, slight depressions in faces otherwise smooth as eggs. Vimbai only made a sound when she realized that, despite these limitations, the wazimamoto managed to smile at her somehow, with the invisible predatory smiles of nightmares.
# Chapter 15
Vimbai's fear blinded her to everything but her faceless opponents—her field of vision narrowed into a tiny spotlight over the bloodless, featureless faces that managed to leer at her. On the edges of her vision, a black vortex swirled, blotting out Maya's pleading mouth and the incredible paleness of Felix's face, the rusty-colored dogs. She only saw the green cloth masks moving slowly in and out, like an air sac on a frog's neck, with a terrible mockery of breath.
The words she had carefully prepared and rehearsed in her mind were nowhere to be found, and Vimbai wished that her mouth wouldn't be so dry and so sour, and the wazimamoto didn't advance on her so slowly and menacingly. She took a step back, and felt the smooth surface of the door with her back.
Her vision slowly returned, and she could take in the buckets filled with sloshing fluid, viscous and black like tar—the sad remains of Felix's universe, she guessed. The buckets were so many, the tar in them so unlike the air and the vibrant movement of Felix's coif . . . it made her want to cry.
"Why?" she whispered, addressing no one on particular. "Why did you do that to him?"
The answer came in a crowding of words and images thrust forcibly into her mind, without any gentle mediation if words—this felt like an assault, like any true telepathy would, thought Vimbai. This flood of images, this relentless and redundant droning that penetrated even into the secret places behind closed eyelids. The words insisted that Felix's universe had to be destroyed—must be destroyed, they said, it must be destroyed because with too many conduits there were too many drafts blowing the ethereal dimensions through and through. It had to be destroyed because Balshazaar made it a condition—he did not want to go back in, or even risk having to go, and he promised to deliver the delicate soul shells of the crabs in exchange for their promise to get rid of the stupid remnant, an appendix of a universe. They did so, they kept their promise, and that was a good thing, wasn't it?—they got rid of it in the same way they went about accomplishing anything: draining. They drained Felix, and now his face was as white as the sheets underneath him, and his skull, fragile like an egg, traced with a web of veins like cracks, shone in the dusk of the Bone Clinic, unprotected and pitiful.
And then there were the crabs—the soul-shells, the crab-ghosts—scattered about as at a market. Vimbai though back to the time where she was driving home from college, along one of the many quaint little roads linking the behemoths of the Atlantic City Expressway and Black Horse Pike, and she saw a small shop by the road, with a hand-painted "Fresh Crabs!" sign. She pulled over, figuring that a quick dinner of local crabs would be both delicious and socially responsible, and walked into the store. The crabs were indeed there—stuffed by dozens into buckets, they struggled and churned, a seething mass of captive bodies, too dumb to understand that the ocean was too far away to escape to, and Vimbai ran from the store, gripped by sudden disgust and despair. It seemed too cruel, too indifferent somehow—and now she wished that instead of running she should've bought as many as she could and driven them to the shore and released them into the ocean.
That would've been a noble thing to do, she thought as she watched hundreds of ghost crabs strewn about the ward. Some were cracked open, with long needles stuck in their gills and carapaces, the needles that pumped the blood (life force, the intrusive voices corrected) out of these soul shells and into the plastic bags, like the ones hospitals used for IVs. They drained everything, Vimbai thought, and remembered the words of the man-fish—it was their nature, to drain. They wanted to get to the horseshoe crab bodies, trudging restlessly along the bottom, getting them closer and closer to home, but meanwhile they were not going to pass up the opportunity to drain their life essence instead of blue, material blood. It mattered not a whit—like all colonial creatures, the wazimamoto were vampires, concerned only with taking and not so much with putting anything back in, or even giving any thought to the results of their actions. Even now, they told Vimbai about what they did without a trace of deceit or embarrassment—they could be ashamed about stealing blood no more than a bee could be ashamed about collecting nectar, or a beaver could be embarrassed about building a dam.
This, Vimbai thought, this was the trouble with evil—it was rarely malicious, usually born out of single-mindedness and narrow views. She wanted to share this insight with Maya, and Vimbai forced her eyes to find Maya, and to absorb the sight of the wazimamoto fitting a long rubber cord around Maya's well-muscled upper arm, encircling her narrow but heavy biceps, and waiting for the vein in her arm to swell to the surface like a deep purple river upwelling with rain, to puff up under the skin like a tense wire.
Maya kicked at her captors, and her dogs growled and tore at their legs—but there was nothing for them to either kick or grab with toothed narrow jaws, nothing but the billowing green scrubs with an outline of shadow underneath. It was something neither of them had considered, and the man-fish of course did not warn them—the wazimamoto had no flesh and could not be hurt, they had no conscience and could not be deterred.
They grabbed the dogs and tied them together, lashed their paws and jaws with rubber hose and ropes, and tossed them in the corners, like they had done with the crab souls.
"Leave her alone," Vimbai pleaded. This was neither a game nor an adventure anymore. "Please, let her be."
The wazimamoto did not answer, absorbed as they were in their gruesome business. Their movements, spare and terrifying in their calm efficiency, seemed matched together, as if they had been working side by side for an eternity—and Vimbai guessed that they had been.
Before she could move, they surrounded her, moving swift and silent and smooth like water, and their hands found her arms and her neck, her eyes, her face—she looked and looked, in unrelenting terror, once she realized that each one of their hands bore ten fingers, long, sinuous and multi-jointed. They wriggled in a complex, spiderlike manner, as if following an internal rhythm.
They held Vimbai fast, just as they held Maya and her dogs—just like they held Felix on his narrow stainless steel table. Not a surgical one, Vimbai realized—at least, it was not intended for human surgery, it was too short and too narrow, a stainless steel table more suited for Maya's half-foxes than full-sized humans—a vet's table, just like the one Vimbai's childhood cat was put to sleep on, meowing and distraught under the harsh lights and foreign hands that held it down.
She felt the stainless steel under her own back—quick and cold like water—as well as the tightening of rubber cords around her wrists and upper arms, the wrenching sense of bones being pulled against the coiling of muscles as she tried to bend her elbows, to keep her arms close to her sides.
The fingers on her skin felt cold and slippery, slightly trembling as if in fear, as they lashed her wrists to the restraints built in the side of the table. She waited for the inevitable kiss of the rubber hose around her arm, for the sting of a needle and the slow, lightheaded descent into unconsciousness. She half-welcomed it, as one welcomes relief from fear—so exhausting that one had to smile a little at the prospect of finally surrendering and not having to be afraid anymore, not having to tiptoe under the glare of white fluorescent lights awaiting a scare or betrayal at every step. Giving in and letting go was easier, and a moment of pain would be worth it.
She closed her eyes, but the prick of the needle did not come—instead, there were voices. The wazimamoto were speaking to each other, and although Vimbai did not understand the words, she recognized the intonations, wobbly with doubt and abrupt with panic.
She opened her eyes, almost regretful, to see that Maya had been left alone. Her hand swelled and turned a disconcerting shade of dusky purple, but her blood was not being drained, and her dogs, restrained but unharmed, did not dare to bark at the wazimamoto.
Vimbai was about to ask their captors what was going on, when the scars on the insides of her arms started to itch. She wished her hands were not tied to the table and that she could scratch the maddening burning. Oh, Elizabeth Rosenzweig, she thought, why did you have to be so insidious, why did I have to be stupid enough to think that cutting these sigils into my skin would make you love me, or at least protect me from heartbreak?
She forced her head to her shoulder (it felt heavy now, disobedient and dumb with fatigue), and her eyes snapped wide open and her breath caught in her chest at the sight of the scars. They had changed from the barely visible, slightly raised traces of connective tissue into bright red, burning rivers shooting small flames and exhaling pungent sulfurous smoke. They twisted into fiery dragons and straightened into moats spewing fire, they coiled and flowed into complex patterns, and otherwise behaved in a manner no scars had any reason to.
She had reached a state of fatigue and surprise that made everything appear as a dream, and she accepted the dragons and the flames, as she accepted the thought that the wazimamoto were deterred and terrified by her scars, as if they were magic somehow, a charm against them.
"Hey," she called to the assembled faceless surgeons. "Untie me, or else." She did not know what to threaten them with and was afraid to bluff and make a bad mistake that would make it obvious to everyone that she had no understanding or control over her sudden power.
The medical men consulted among themselves, their surgical masks rising and falling, rising and falling, and one of them approached her cautiously, to untie the restraints on her right hand, and then quickly jumped back to join the others.
Vimbai used her free hand to free the other one, and sat up, rubbing her wrists. The scars on her arms flared, glorious tattoos of fire that burned but did not consume, and Vimbai jumped off the table, protected by their halo. She untied Maya, and smiled at her. "Are you okay?"
"Yeah." Maya looked at her with a new expression, of deep respect and surprise. "Boy, did I ever underestimate you."
"Get the dogs," Vimbai said. "I'll tend to Felix. And the crabs." She turned to briefly glare at the wazimamoto and to show them her fist, just in case they forgot that they were afraid of her. No one had ever been afraid of her, and Vimbai found the new experience not altogether unpleasant—she suspected she would've enjoyed it more if she were not so shell-shocked by the experience, if indifference did not seem like the best coping mechanism available to her.
Felix remained on his slab, his head a defenseless egg, so unfamiliar and strange that Vimbai felt like weeping every time she looked at it. Her eyes met Felix's tormented gaze—for once, his eyes seemed to be pointing on the same direction. "Felix," she whispered. "Can this be fixed?"
He shook his head and cringed as the newly exposed skin touched the cold steel. "No," he whispered back. "I think this is it—I hope there wasn't anything valuable in there." The loss of his private tiny universe had not seemed to reach him yet.
Vimbai unfastened the archaic leather belts that affixed his wrists and ankles to the table. "Come on," she said. "Let's get going."
Felix made no attempt to move, listless and disoriented like a cat without whiskers.
Vimbai grabbed his hands and pulled him off the table. He let her, inert, and stood, swaying slightly, making no attempt to walk or even flinch away from the medical men who clumped tightly together, watching them with the blind eyeless depressions on their featureless faces.
"I think we'll have to lead him," Vimbai said to Maya, who had finished freeing her foxes. "Help me to pick up the crabs."
Maya nodded and motioned to her dogs—they seemed the least affected, and growled at the apparitions in green scrubs as they scuttled about the Bone Clinic, picking up the ghosts of the horseshoe crabs in their red mouths.
Vimbai and Maya stuffed the remaining crabs in their pockets and down their t-shirts, and Vimbai linked her arm with Felix's. The glowing sigil on hers crossed over to his skin, and he shuddered a little, as if waking up. "Come on," Vimbai urged. "Come quick."
She felt uneasy now—the medical men seemed to have come to some decision, and even though they made no attempt to stop Vimbai and her roommates from leaving, there was a new sense of purposefulness about them, as if they just waited for them to leave the room to spring into action. Vimbai was also uneasy about the man-fish and Balshazaar, free and roaming the depths of the house somewhere.
Vimbai took a deep breath and pushed open the door, Felix hanging limply on the crook of her arm, and took a step into the silent and bright corridor of Cooper Hospital, inexplicably thrust into the center of her dream Harare.
They found Peb where they had left him—he bobbed up and down in the tree branches, seemingly content. Vimbai wondered if he experienced time in the same way they did, if he ever worried or became bored or counted to sixty to gauge how long did a minute take.
"Come with us," Vimbai told him. "I can't carry you now."
Peb sulked but bobbed along, silent and obedient. Vimbai felt her stomach churn—she had forgotten all about his tongue. "I don't have it," she said out loud. "But we know where the wazimamoto are, and I can probably make them give your tongue back to me, but I need to talk to my grandmother first."
It wasn't exactly a lie—Vimbai did hope that the vadzimu would be able to shed some light on the mystery of the sudden burning signs appearing on her skin. Even though she probably didn't know anything about the wazimamoto, she probably knew more about magic than Vimbai. Vimbai wished her mother was here too, because she was the one who wrote papers comparing voodoo with hoodoo, and correcting the many misconceptions she believed white Americans to have—it always puzzled Vimbai that those articles always seemed to be written for white people; possibly the ones who read those articles and became qualified to run Africana Studies departments. However, they still managed to focus primarily on the aspects of muti that used human organs cut out of living people, which annoyed Vimbai's mother to no end. "It's as if they only want to see the folk magic practitioners as savage mutilators," she would say. Vimbai would then think of cousin Roger and nod in agreement; economics simply didn't make for as compelling a monster as a dark-skinned medicine man in traditional garb. And mutilation certainly held a kind of grim fascination that always made the headlines.
They made it home when the sun was rising among the slanted ceiling beams of the main hallway, and they made it to the kitchen by midmorning. Vimbai's dead grandmother had a coffee pot waiting for them, and apologized for the lack of sugar, even though it was certainly not her fault—the ghosts did not eat or drink a thing, and only cleaned excessively.
When the vadzimu saw the marks on Vimbai's arms, she gasped and looked closer—even though the flames had died down somewhat, the patterns still glowed the angry red of molten iron. "These are muti marks," she proclaimed. "And they protected you from danger—did your mother have them done?"
There was an intensity of hope on the grandmother's voice that made Vimbai cringe. The ghost still waited for forgiveness, and seeing her daughter follow in her magical footsteps would be a certain sign that Vimbai's mother was not angry with hers anymore. "No, grandma," Vimbai said. "I made them myself—only I didn't know then what they were." (Protect me from a broken heart.)
The vadzimu shook her head. "One needs to be a n'anga to make those; one needs to practice Un'anga, the folk medicine. Or voodoo, like witches do."
Vimbai wished she paid closer attention to her mother's articles—she only remembered that Un'anga used both medicinal herbs and spiritual cures, and that most people frowned upon it nowadays. People had no use for spirits anymore, the ghost grandmother said, shaking her head from side to side. They even called the creator by a different name, just like they called their country by a wrong name. It was always the British missionaries, renaming things and demanding respect for their god, the same respect they were so unwilling to offer anyone else.
According to the vadzimu, the protection marks that glowed so brightly on the inside of her arms could not be made willy-nilly, by just anyone. One had to go to a n'anga for things like that, or to muroyi, the witches shrouded in mystery of such a malignant and disreputable nature that only the truly wicked and desperate dared to inquire into it. Of course, Vimbai had spent enough time with her mother to interpret it to mean that the witches were mostly unpopular with the white Christian missionaries and thoroughly vilified by same. Just another form of control, but right now it seemed of little use to her. She was more curious about the roots of the magic, the source of muti power.
On the other hand, she wondered at her ability to perform such magic—was it that the vadzimu was wrong and muti marks could be drawn by anyone with enough conviction? And really, how often did they get tested anyway, in the outside world so devoid of magic, be it in Harare or Atlantic City?
Or—and this is what gave Vimbai such a headache—could it be that she was special somehow, that in her genes there were little coils of African nucleotides that knew somehow about the muti and the scars, about protective and injurious magic alike? "This is such a stereotype," she heard her mother's voice in her mind, and smiled at the ridiculousness of it all. "Like being of African ancestry means that you automatically know voodoo—it's such offensive nonsense." Still, the thought lingered, even though she knew full well that revealing herself to be a conjure woman would be a political disaster in her mother's eyes.
Maya and Vimbai had put Felix to bed, to let him recover from the awful draining he had just undergone and quite unsure of what else they could do for him. They convened in the kitchen by the coffee pot to survey their progress and plot further plans of action.
"It doesn't look great," Maya said and made a face at her black and bitter coffee. "Not terrible, but not great. Pluses: we got the crabs back, and Felix too. We know where they are. We know that your marks repel the medical men but not the fish."
"And we still need to get Peb's tongue back," Vimbai said and sighed. "And I feel so bad about Felix—we should've protected him."
"I'm not the queen of Felix," Maya said, scoffing. "He took off all by himself, and we had other things to deal with, remember? Like you jumping straight into the catfish's mouth."
"I did not," Vimbai objected weakly and without much conviction. "In any case, we did not protect him. But the crab souls are back, and maybe we should hide them somewhere where neither the wazimamoto nor the man-fish would find them."
"Good idea," Maya said. "Where?"
Before the word left her lips, they both knew the answer. The safest place there was—a tall hollow tower, glass and concrete, the lone platform on top where there was a coffin with an old woman, and blankets and empty candy wrappers betrayed Maya's secret nest. "Will you take them there?" Vimbai asked. "The dogs can carry them, and no one will get to them there."
Maya nodded. "It's okay, I suppose, as long as it's temporary." She breathed a short laugh. "That's a silly thing to say. I guess everything is temporary, especially here, right?"
"Right," Vimbai said. "Just make sure you keep an eye out for Balshazaar."
"I don't think he would ever bother us again," Maya said. "I mean, he got what he wanted, right? Felix's universe is destroyed and he would never have to be locked up in there."
"Maybe." Vimbai thought about Balshazaar's parchment skin and sunken eyes, the grotesque phantom limb fused to the withered remnant of his neck, and sighed. "I just don't think we can trust him."
"Of course we can't." Maya smiled. "We just have enough shit to worry about without him, so I'm saying don't worry about him unless he pops up."
"Sounds good." Vimbai momentarily envied Maya this clarity, this ability to separate the essential from the secondary. Vimbai lacked that skill, doomed to forever swim in the soup of relative values and conjectures, where everything was conditional and everything seemed to have equal importance, always competing for her attention. It was good to have Maya around.
"Okay then," Maya said. "I'll go take care of the crabs. What about you?"
"I'll try to figure out what the deal with my scars is," Vimbai said. "I'll check on Felix, and then I'll figure out how to get Peb's tongue back."
# Chapter 16
Vimbai sat by Felix's bed, with Peb bobbing nearby like an obedient and grotesque fishing float. Felix slept, or possibly descended into a deeper and more disturbed state—his eyes flickered back and forth under the closed eyelids, like quick little mice in the grass.
Vimbai put her hand onto his forehead—smooth and cool, not a sign of fever —and considered whether she should keep it there for a while longer, to offer comfort, until he moaned and thrashed, twisting from under her hand as if it were too heavy or burned his skin.
Vimbai sighed and withdrew, under Peb's silent and, she imagined, accusing stare. To avoid it, she studied the bare walls of Felix's room, just slightly covered with lacy lichen and peppered moths camouflaged between the lichen patches, only their black eyes and long, twitchy antennae betraying that they were still alive. She leaned closer to one of the moths, to take a closer look at its small furry body and the delicately powdered white and gray wings. The moth fluttered, and Vimbai could hear the high-pitched squeal of the scales rubbing together and the soft whispering of the body hairs brushing against each other.
The chipoko stood on the threshold—Vimbai only noticed her when she looked up from the trembling velvety moth, and her gaze stumbled over her grandmother's. She seemed as troubled and as silently accusing as Peb. I didn't do anything wrong, Vimbai wanted to say, but the burning on the insides of her arms belied her innocence. Somehow, she had managed to do it to herself, Elizabeth Rosenzweig or no.
Oh, Vimbai remembered her face so well, the curve of the soft cheek fuzzed with tiny hairs only visible under direct sunlight, light and dear like the crosshatching of a peach. Same color, and, Vimbai imagined, same taste—would it be that a girl with such a sweet blush, such soft creamy cheeks was not so different from a piece of fruit, not animal but plantlike in her innocence and sugary sweetness? Bees should be following her around, attracted by the invisible dripping of soul nectar; birds should be building nests in the dark thickets of these eyelashes, long and tangled like the branches of sagebrush.
The memory ached, and the ache resonated in the curves of her inner arms, doubled and tripled and twined around her elbows as the scars puckered and reopened, rivers of gray ash shot through with some residual sparks, playing and skittering across the surface. Now it seemed that just a memory of Elizabeth was enough to bring these formerly dormant charms to life; so Vimbai decided to remember.
She was not particularly good at love—never had been, too awkward and easily discouraged, too self-conscious and ungainly. It had always been easier to back away and cry quietly after dark, so that her parents would not hear, so that her hot tears soaked into her hair and the cool cotton of the pillowcase, so that they burned her eyes like coals. It was easier to treat love as something imaginary, as something one indulged in in one's head, guiltily yet zestily, like daydreaming. Loving Elizabeth Rosenzweig from a distance was a snap—it was even easier when they went to different colleges and never even talked anymore, since Vimbai was too preoccupied with love to have bothered to develop a friendship or even a casual bond.
The scars, she realized now, were just like those daydreams—not action but a symbol, a substitute for doing. Neither cutting nor daydreaming accomplished anything but they offered a refuge, an escape from otherwise painful thoughts—painful enough, she realized, to have possibly pushed her into action as long as she didn't let herself become distracted. And yet, her longing was potent enough—important enough, she told herself—to imbue her scars with some protective magic. She made them to protect herself from having to go out there and declare her affection, and probably being rejected—to protect her from a broken heart. Who could have imagined that they actually worked?
"Enough staring, grandma," Vimbai said. "I'm not a witch, and you know that. You should be grateful—you should be happy I have this magic, or I would've been dead otherwise."
"I realize, granddaughter," the vadzimu answered. "Would you like to see the crabs?"
"Yes, please." Vimbai smiled—she missed the sight of her silent underwater army working so hard—their legs so brittle and segmented!—to get them home, despite the cold and the season and the cruelty of the man-fish. They were entirely too good, Vimbai thought, and she promised to herself to dedicate her life to making sure that horseshoe crabs were no longer chopped up for eel bait or bled into near-oblivion by the faceless monstrosities that holed up in the Cooper Hospital of the Harare of her dreams.
The weather had grown milder—the wind outside died down, and the smell of the ocean did not seem as sharp. It had grown almost spicy, heated by the tremulous and pale sun that reminded Vimbai of spring rather than fall. Could it be? No, she chased the thought away as ridiculous. No, just a slightly warmer-than-usual day, common enough at any time of the year. She waited for the vadzimu to enter her, to occupy the same Vimbai-shaped amount of space as she herself occupied, and pressed her face underwater.
"You are safe," Vimbai reassured them. "Your souls are safe, waiting for you in the tallest tower where neither fish nor truck can get to them."
The horseshoe crabs mumbled and whispered their thanks, reassured, and their legs worked faster—Vimbai watched the shadow of the house, a square small outline that did not at all match the bounty of space and landscape within, crawl and flicker over the long narrow sandbars, glide like a manta ray over the deeper trenches where small fish played in silver schools. The horseshoe crabs picked up the pace—they almost flew now, their cracking undead legs working so fast that Vimbai feared that they would suffer a final break and fall apart, splinter and disintegrate like termite-infested wood.
Vimbai was no longer terrified of the crabs' unnatural undead state but rather felt profound pity and anxiety, now that the crabs' souls were back and protected by the death magic of Maya's grandmother. Vimbai did not know whether the wazimamoto had any ways of finding out information except for what Balshazaar and the man-fish told them. What if they had some hidden sense, the way villains always know everything in horror movies? What if even now they and their medical truck were on their way to intercept Maya and her dogs, to steal the crab souls back, to be drained and dissected, and quartered afterwards to be stuffed into eel traps, fish bait, useless in death as they were in life?
She chased these thoughts away as she watched the bubbles of her breath rising to the surface. They stretched and danced, their surface radiant, as they multiplied and shimmered and burst as soon as they reached the surface. There were none coming out of the crabs, which wasn't surprising, but Vimbai wished she did not know that the gills—delicate feathers she had studied under the microscope so many times—remained unmoving and useless inside their chitinous shells.
"It's okay, little crabs," she said. "We're going to make it home soon, and it'll be warm and nice, and you will all come ashore and lay your eggs—there will be plenty for the birds and still there will be thousands of new crabs hatching and playing in the waves. And I'll keep your souls safe for you."
The crabs chittered back, excited. Soon, they agreed. We know that we are close—we recognize the signs, the sandbars. We feel the scent of the familiar water, we recognize the manky stench of river silt flowing into the ocean. We know these salt marshes, we know the screeching of terns and gulls overhead. We are close, so close.
Vimbai smiled and moved to straighten—her face had grown numb under water, and the vadzimu's eyes felt indistinguishable from hers, a sure sign that she had spent too much time with her grandmother's ghost inside her, and that the two of them were at risk of confusing self with other. But before Vimbai's face had breached the surface, she felt a blinding pain in her side and stomach, as if from a kick strong enough to knock the wind out of her, and her arms and knees buckled under her, sending her toppling sideways into the cold, cold water, where nothing but the undead crabs waited for her.
The vadzimu's presence saved Vimbai—because of the ghost, she had grown more impervious to cold, and her breathing underwater, while labored, was still reasonably comfortable. She grasped the rope that linked the house above to the crabs beneath, and hovered in the thickness of water, next to the clusters of dormant crabs. Her chest and stomach still hurt, and she could feel an ugly bruise spreading under her shirt, tracking its progress with a sensation of intense heat. That would be not even a bruise but a hematoma, Vimbai thought. Of course, there was also a question of who it was that kicked her under.
Balshazaar, her own and her grandmother's voices whispered in unison. Who else but that conniving, shriveled old man? What other appendage was capable of such a swift and decisive kick if not a phantom limb they had voluntarily given him?
She considered getting out of the water, but decided against it—surely, Balshazaar was waiting for her on the porch, waiting for her to surface and to betray that she was still alive, still presented a danger or, at the very least, an obstacle. In the water, she would be vulnerable to him—apparently, her scars did not protect from any desiccated heads (or, as far as she could tell, from anything but the wazimamoto, which made some twisted sense—African magic deterred African phenomena.) He could drown her or hurt her while she trod water, helpless. It would be better to wait him out, to let him think that she was gone and drowned, so that he could tell the man-fish and they both could regret that her soul fell to the horseshoe crabs instead of the wily catfish. Then, it would be safe for her to come out.
Even though she had not seen her assailant, she felt sure that it was not the wazimamoto—they could not come into the house. Her Kenyan babysitter was quite clear on that—she insisted that they left people in their houses alone, and only drained blood of those who were destitute enough to sleep in the streets. Poor people, migrant workers, prostitutes, homeless children—they were the preferred wazimamoto targets.
And then there were her scars, her protection. She wondered if the wazimamoto were so scared of them because they did not expect muti but if they would find a way to work around them. She found thinking easier in this thick green water, bobbing halfway between the bottom and the surface, a perfectly balanced float, her hand holding the rope that dragged the house home. Maybe she could stay here for all time, she thought—it wasn't bad, and she would be perfectly positioned to study her favorite crabs, with her mind and her grandmother's special vision and the ghostly ability to breathe underwater.
Then she worried that she would spend too much time with the ancestral spirit inside her, and that their souls would get entangled somehow, would become one. Vimbai certainly did not want to become her grandmother: even though she liked her better now than before, she still did not want the old woman's superstition or the conviction that muti, the mutilation magic, was somehow good for her children. She did not want her laments of the old days and the insistence that things used to be better when the British were in charge, just like she did not want her death, her endless stories that went nowhere, her narrow-minded ways.
A memory niggled at the edges of her mind, a half-forgotten fact from a botany lecture. She remembered the delicate, steady crosshatching of her drawings, the smiley faces of monocot vascular bundles and the perforated plates of the sieve tubes. The branching of the leaf's veins, and the delicate internal structures of the anthers and pistils. And then there was something, something else—she remembered tracing the thin fibers snaking in-between the tissues of a vine's stem, almost invisible filaments that penetrated the plant's food and moisture supply, coiled into every cell and narrow space between vessels.
The parasitic plant, Vimbai remembered, the thing that hid inside another plant and only became apparent when it bloomed with its horrible febrile flowers—gigantic, three feet across, red and warty white. A gruesome flower that looked like slabs of meat and stank of rotting flesh; Rafflesia it was called, she remembered. Still, it took her a while of silent bobbing and being dragged through the numbing cold water to realize what the flower reminded her of—she recognized in its quiet creeping the same deceiving calm and even tenderness that she had felt as the vadzimu's memories blended with hers, seamlessly twining between the threads of Vimbai's life. She recognized the imperceptible shifts and subtle rearrangements of what made Vimbai the girl that she was, she felt the memories of her first love (Elizabeth, Elizabeth, her memories and dreams sang in unison) being pushed to the side to give just a hair's breadth more space to the memories of red soil and dry summer months, of the red dust that hung relentless in the vegetable garden, the squash and the yams ailing in the heat. Her mother's face shifted and flowed in her memory—from a stern woman with sharp cheekbones to a soft-faced girl and back again, the marks on her face changing from fresh cuts to almost invisible scars.
Vimbai resented herself for thinking of her dead grandmother as a parasite—everything else aside, the vadzimu was the reason Vimbai was still alive in the freezing ocean, breathing underwater. Her memories offered Vimbai a glimpse of a life so different from her own that really, she should be grateful for the opportunity. Her mother always told Vimbai that she was too sheltered, too ignorant of how the rest of the world lived—and yet, Vimbai thought, she was the one doing all the sheltering. Just think about what it took for Vimbai to move out of her protective fierce embrace—it took a house that was filled with landscapes and contained the entirety of Vimbai's idea of Africa, and two very strange roommates.
Vimbai thought of the jacaranda trees and the horseshoe crabs, their fine delicate claws combing the white sand of the bottom with the speedy mechanical motion of a small windmill; her memories of her children intertwined with her memories of her parents, and for a few dizzying moments she could not tell which was which. Her eyes filled with tears of either childhood helplessness or sadness of old age, useless underwater and superfluous in the ocean already filled with salt, and unknown hard words filled her mouth—Shona words she had neither used nor remembered since she was little, too little for kindergarten; it was kindergarten where she stopped talking Shona, she remembered now. Before, Shona and English were inseparable and the same, one become the other in her mouth as easily as in the mouths of her parents—they did not discriminate, and the languages switched in a joyful leapfrog of words not bound by rules. When Vimbai was little, she found the words that best filled the void, be they Shona or English; she had lost this ability on the first day of kindergarten, when she answered her teacher's question in Shona and everyone in the class laughed. Her parents still spoke a mix of languages at home, and how she envied them! She wished she could forget the laughter of the kids and just speak the way she had been doing before.
And now Shona forced its way back into her throat, just like the memories of Harare forced themselves back into her mind, and Vimbai closed her eyes, her warm tears flowing into the cold ocean and disappearing there without a trace, leaving no imprint. She had to go back now, she thought, back to the surface, into the warm embrace of the house and its smells of dank domesticity and over-boiling coffee, where she could disentangle herself from her dead grandmother and be herself; she just wanted to catch her breath and examine what was and wasn't her anymore.
But the feat of returning proved more difficult than she had hoped; it always was, after all, the impossibility of the act implied in so many language clichés and morality tales. She kicked her way to the surface only to discover that the surface as such had disappeared—instead, there was a thin layer of oily and impenetrable darkness, as if some mythological version of Exxon Valdez had suffered an accident and spilled whatever mysterious substance it carried and that could poison an imaginary ocean.
Not Exxon Valdez, Vimbai realized when her face touched the murky substance and entered it as one enters a summer night, clammy and humid and warm, from the crisp chill of an air-conditioned house—the too-warm, too-humid air wrapped around her skin and beaded it with sweat. There was darkness and nothing at all to see, and there was neither sky nor the house anywhere in view.
Shocked, Vimbai sank again, back into the comforting and familiar chill of the ocean. Her thoughts raced and her heart thumped harder against the delicate cage of her ribs—she could feel every contraction, every pump resonate through every bone in her body with a hollow echo. She looked around her, at the newly vivid green of the water and the white of sand, studded with black and blue bivalves, twined in the yellow and brown and green of seaweed. She could not comprehend what had happened to the surface, but there was no doubt in her mind that it was the work of Balshazaar and the wazimamoto.
And then it came to her, the memory—thick, viscous fluid, the extraneous membranes of space in Felix's universe. It was the remnants of space from his denuded head, she realized, it was what had been drained from him—a layer of foreign dimensions floating atop of the ocean like oil, cutting her off from the house as effectively as any barrier. There was no crossing this pocket universe, even as it spread in an infinitely thin layer—she could only stick her face inside and squint at the impenetrable darkness and the faint taste of rain-beaten dust in the thick, immobile air.
The urgency of Vimbai's situation caught up with her occasionally and then she would bound for the surface, crazed and weightless like an air bubble (and those had ceased leaving her lips a long time ago), only to find again and again that the barrier persisted, and she still did not know how to breach it. Then she sank back, to the crabs, and floated by them as they dragged the house and Vimbai clinging to the rope along with them. There, she thought that maybe it would be okay, maybe she could stay there until someone—either Maya or Felix or Peb—found her. Or perhaps she would be okay all the way to New Jersey—didn't the crabs say that they were close?
But she knew that these were idle fancies, and that she did not have time to be found or rescued. With every passing second, the wrinkles on her grandmother's face grew more and more familiar, with the same inevitability as one's face is recognized in the mirror. Soon, the vadzimu and Vimbai would not be able to tell where one ended and the other began.
"I'm so sorry, grandmother," Vimbai whispered, and immediately answered herself, "I'm so sorry, granddaughter."
It had occurred to Vimbai that lately she had been spending quite a bit of time drowning or otherwise under water; she wondered if there was some significance to it.
"What say you, crabs?" she said out loud.
The crabs chittered and whispered among themselves, such disconcertingly high-pitched and birdlike sounds. "What pierces the darkness?" they finally asked.
Vimbai sighed. Stupid riddles, she wanted to say, sthe ame riddles that surfaced in fairy tales—just feeble-minded guises for simplistic morality lessons, not at all challenging or enlightening. "Light," she said out loud, struggling not to let her irritation show. "Light pierces the darkness. Thank you, this is very helpful. Only I have no sources of light here, and neither do you."
This is not true, the vadzimu in her mind answered. Remember the story I told you, remember the story you told Peb. Both are ngano—and ngano is how children learn. Your task may be hopeless or you might not even know that you have a task in the first place, but there are things within you that you can reach.
Vimbai only sighed in response. It seemed silly, the same psychobabbly message of hope she'd been hearing from school counselors and the books that were supposed to instill 'values' (no one ever told her what those values were supposed to be) into her. It didn't change, she thought. There was always someone offering a simple solution, there was always this belief that only if you try hard enough, want something bad enough, there would be a wellspring of miracles and you would always get whatever it was you wanted. One could always triumph—but she knew, she had learned through a long and disappointing string of letdowns that sometimes there were circumstances beyond one's control. Sometimes one was too short for basketball or too stocky and thick-boned to seriously consider gymnastics. Sometimes one did not have the complexion to play Snow White, no matter how much enamored one was of this role at the age of five. Sometimes one had to throw away the dreams, no matter how dear or powerful, after first experiencing a bitter sting of reality. But then again, this is what this house was for, wasn't it? The old dreams that everyone had forgotten about, so she really had no right to get angry at them.
She thought of the tortoise in her grandmother's story, and hated the smug beast who got everything everyone else wanted without even trying. Humility indeed. And yet, and yet . . . there were dreams in this house, she thought, and what were dreams if not irrational wish fulfillment? What was the point of ever dreaming if one could not be a ballerina anyway? And didn't the sea follow her? It followed her in her dreams as if it was her, Vimbai who was the moon, round and heavy like an old silver coin—a coin tossed by a careless hand, heads or tails, and now stuck in the middle of the sky. The coin that attracted all the seas in the world, heavy and smooth, grave and yet pouring out bucket after bucket of pure light, reflected though it might be—it didn't matter in the slightest.
Vimbai closed her eyes and imagined pure white light, white as milk, as the tortoise's beak slurped it up as if it was candy. She pictured all this cold, pure whiteness sloshing inside her belly, heavy and round like the moon, with enough gravitational pull to attract all the oceans in the world, and then she thought of her slender fingers reaching inside, into all this light, asking for a wishing thread—and receiving a white burst of light instead, the kind of light that burned bright as a carbide lamp, and before which no darkness could resist.
# Chapter 17
There was one memory Vimbai rarely thought about—not because she had forgotten and not because the memory was in any way unpleasant. Rather, Vimbai felt that some things were too precious to tarnish with frequent reminiscences, and thought around it, obliquely, while always retaining the warm feeling the memory gave her.
But now she felt it was a good time to remember it—she chose this memory from among all the others for its golden light and the overwhelming sense of joy that radiated from it. It all happened when both Vimbai and Elizabeth Rosenzweig were in eighth grade, when Vimbai was still too clueless to realize what was happening to her.
She remembered that day with such clarity—it was May, and their class was mercifully sent on a fieldtrip to one of the dinky little museums that peppered the shore towns like lighthouses and souvenir shops. Vimbai did not remember which town it was, and she did not remember much of what she had seen at the museum—there were vague memories of old fishing nets and handmade fishing floats, rusted antique anchors, and the musty smell, the same as every tiny and ill-conceived maritime museum she had visited over the course of her life in South Jersey. There were stuffed blue marlins mounted on the walls and insipid paintings with white-sailed ships frozen on the brink of white-capped waves, and things rescued from shipwrecks of dubious authenticity mixed in with preserved specimens of octopi and other strange-looking invertebrates; if Vimbai was so inclined, she could've traced her fascination with marine biology to these dusty jars with discolored eyes and tentacles in them.
What made this museum different, though, was Elizabeth's presence—she had just transferred in from whatever glamorous life she had previously lived, and Vimbai tried really hard not to follow the new girl too much but found such restraint difficult, due to Elizabeth's interesting way of speaking. At the museum, Vimbai spent little time looking at the exhibits and a lot trying to maneuver herself next to the new girl so that it looked like an accident, in case anyone actually paid attention to Vimbai.
She had finally managed to stand next to Elizabeth, who yawned and looked at an old, sepia-toned photograph of one fishing vessel of the bygone days or another.
"Hi," Vimbai said, staring at the photograph with a greater intensity than it warranted and keeping her tone casual.
"Hi," Elizabeth answered and smiled at the photograph. "Vimbai, right?"
Vimbai felt a happy little flutter in her stomach that the glorious new girl remembered her name, and even pronounced it correctly. "Yes," she said. "And you're Elizabeth?"
The girl nodded and finally tore her gaze away from the photograph and gave Vimbai a slow, half-lidded look, which gave Vimbai goosebumps on the back of her neck and head. "I don't like any of the diminutives of my name," she said. "And I'm bored. Is there anything else to do around here?"
"Well, sure," Vimbai stuttered and looked for the teacher who was just ahead of them, pointing something out on some stupid diorama. "There's the beach, and the shore towns always have a boardwalk. But we're supposed to be here . . . I think."
Elizabeth shrugged one shoulder, took Vimbai's hand—so confidently and thoughtlessly, as if it was her right to grab Vimbai's digits like they belonged to her doll or stuffed bear—and dragged her along, to the diorama. "Excuse me, Ms. Burns," she said to the teacher. "My allergies are acting up because of the dust, and I don't have my medication with me. Vimbai will take me outside, and we will meet the group by the bus later."
Maybe it was that snooty accent, Vimbai thought. Maybe it was the way Elizabeth carried herself—she didn't even ask, she told the teacher what she was going to do, and Mrs. Burns just nodded and told them to stay out of the sun. Or maybe, Vimbai realized much much later, maybe their overworked teacher, who was looking back to the summer recess more than her students did, had a small moment of mercy and just decided to let them go and enjoy themselves on the boardwalk. Such a small kindness that seemed such an enormous stroke of luck back then—such incredible escape from the dark and dusty and boring sepia-colored museum and its stench of formalin, into the blinding sun and the smell of salt in the air.
The boardwalk was not crowded, since it was only May and a work day, too early for the tourists to swarm in earnest. But there was salt taffy and small shops that sold cheesy t-shirts and painted shells and hermit crabs that allegedly made excellent pets.
Elizabeth dragged Vimbai along, laughing, stopping at every store that caught her attention. She bought Indian lapis-lazuli jewelry and canvas bags that said nonsense like "Visit Ocean City." Vimbai passed on the jewelry, but made up for it in cotton candy and funnel cake—the latter invention Elizabeth had been woefully unfamiliar with, and Vimbai did her best to remedy the situation. They quickly got covered with powdered sugar and Elizabeth complained that her hands dripped with oil and she couldn't possibly clean it off with napkins. All in all, it was the best day of Vimbai's life, even though she was too inexperienced to realize that it was due to the fact that she held hands with Elizabeth, rather than the boardwalk and the funnel cake.
The memory of that golden, sun-drenched day was her most precious one, and as she recalled the sun and the red and white stripes of the shops' awnings, the smell of salt from the ocean and the fried dough on the boardwalk, the overwhelming sense of freedom and happiness at being allowed to break out of the museum—just the two of them, marked as special enough for such privilege by the teacher—and to roam the town instead of doing dull and educational things. Oh, how she missed Elizabeth now, how she missed her magic.
The magic was never far from Vimbai's skin's surface—the scars, the sigils glowed again. Not with a red hot protective fire, not with the hidden lava of painful love magic Vimbai had not realized she knew so well—but instead they burst open, split like the seams of a pea pod and released not some prosaic seeds but sunbeams, the light Vimbai had drunk in years ago, like the tortoise who did not know what he was doing—he thought that he was just slaking his thirst, just like Vimbai used to think that she was just cutting school and eating funnel cake. The acts of great personal replenishment went unnoticed and unrecognized, and their significance could only become apparent in retrospect.
Vimbai's skin split and narrow sunbeams shot out. The horseshoe crabs, the undead ones running below and the little ones encrusting the ropes as if they were Christmas ornaments stood out in bright relief, in their true color—emerald green concentrated into dark khaki by the water. The beams crossed the thickness of water and sliced into the oily darkness floating on the surface, and Vimbai let go of the rope and swam after them, trying to keep inside the narrow road of light that seemed to lead—somewhere.
To her surprise, she did not reach the house or the surface. To make the matters worse, the vadzimu's mind inside her grew stronger and louder, jamming her memories, chasing away the perfect recollection of a perfect day in adolescent love. The light pouring out of her opened cuts grew dimmer and the skin, held open and taut by the sheer intensity of the light stream, sagged and wilted, closing the open scars and diminishing the light further. Oh, this was not good, and Vimbai scrambled for more.
The only trouble was, there was not enough in her love for Elizabeth to sustain a long examination or contemplation—surely, there was plenty of material for navel-gazing if she was so inclined (and she had been in the past). But the simple truth of the matter was that Vimbai's first love was an exercise in cowardice, where she never dared to say anything first, and expressing her feelings remained entirely out of the question. She had failed by clinging to this one infatuation well into her college career, as the means of letting herself escape any other kinds of entanglements—she was even too afraid to find out whether she only liked girls, or if boys were an option as well. I lied to you, grandmother, she whispered, mournful and dimming, almost lost in the darkness that approached from all sides, engulfing her once more. I'm sorry for lying—I don't really worry about boys, I worry that I would never be able to love anyone for real, and this is what I'm afraid of.
Don't worry, granddaughter, the vadzimu replied. We all have fears, and none of us knows a perfect way of dealing with them. But maybe you just need to take a look at the person you love now, and learn bravery there.
Vimbai felt neither outrage nor shock, just weary acceptance, and she had no strength to deny. There was no point—from the first day she sat in the (now unrecognizable) living room of the house, from the moment she watched Maya sling her long legs over the armrest of the worn chair, she knew that she wanted to stay. Yes, the house pulled her in—but so did Maya's voice and face, so did the prospect of having a roommate such as her, seeing someone as breathtaking every day. And she thought of Maya's lonely tower, where she slept like a fairytale princess, among the crumpled candy wrappers and empty soda cans, her sleep guarded by an unmoving and dead grandmother. She had nothing left but to coax this reluctant love (those who had lost, she remembered now, those who were honest were the most delusional) into its real form, and she tried to coax herself into admitting what it was and why it mattered.
Vimbai's grandmother did her best to help as well—she pushed on Vimbai's eyes from inside, forcing forth the visions of her daughters, of them growing up. She lamented the deaths of her friends with the same quiet clarity as she lamented the passing of the country she once knew—she did not miss the British, but she found a total collapse frightening. She pushed forth the memory of independence and the jubilation in the streets when the Land Reform was first announced. She grieved about the failure of the 'willing seller, willing buyer' paradigm. Then she thought about all the strength and all the love she had seen in her life, and it filled Vimbai's heart with hope, and her scars with light. The light pulsed and pushed them open, forcing Vimbai's own confessions out of them.
Maya poured out of Vimbai's scars and her eyes, and there was darkness parting before her. The little eddies and layers of darkness floated and separated, and tore like stormy clouds in the November sky. Through the holes she could see snatches of the real sky above and the shingles of the house's roof. With one final thrust and a kick of her tired legs Vimbai pushed herself through one of the openings and came to the surface, just a few feet away from the porch.
Balshazaar was nowhere in sight, and Vimbai swam up to the porch in quick strokes. Her vision doubled, and she feared that the vadzimu would become too entrenched to ever be separated from Vimbai. Her own hands already looked strange to her—pruned from being in the water for so long but young, too young, with pinkish full moons of fingernails and the skin that was lighter than what it was supposed to be—and it took her a while to remember that she was neither eighty nor dead. She clambered onto the porch, simultaneously panicking at her grandmother's insidious presence and addressing herself as sahwira, trying to talk herself down and thinking that young people spooked entirely too easily nowadays. Despite her confusion, she felt the cold in the air, and the sticking of heavy wet clothes, and she crouched on her hands and knees, shivering violently and vomiting gallons of salt water—now that there was air to breathe, the water in her lungs become heavy and unwelcome, and Vimbai remembered that it was unnatural for human beings to breathe underwater like that, even though spirits could.
Somewhere along with all this salt water and an occasional tiny fish, Vimbai managed to expel the spirit too—or perhaps the vadzimu had extricated herself without Vimbai's help, and now she stood by her, patting her back solicitously, as if burping a baby. Vimbai spat out a couple more mouthfuls and stood up, her legs trembling under her, and queasiness filling her stomach.
"It's all right, sahwira," the ghost said, seemingly unperturbed. "Go change your clothes, I'll make you some tea. And then, then you better go and set things right—poor baby still doesn't have his tongue."
Maya had returned from her expedition, and reported on the successful stashing of the horseshoe crabs' souls. Vimbai felt almost relieved that Balshazaar had chosen to turn his questionable attention to Vimbai and away from Maya—the fact that the crabs were safe and undiscovered by him made it almost worth the blind, panicked flailing underwater, with nowhere to go but the oily dying space off Felix's head.
"Oh, poor Felix," Vimbai said out loud. "Should we check on him before we go looking for Peb's tongue?"
"This is an awful way to pose a question," Maya said, her voice teasingly scolding. "I can't say either 'yes' or 'no' without agreeing to go look for Peb's tongue. And I have to say, I don't like these weird quests. What's with the body parts, anyway? Can't we go looking for some Book of the Dead or Amulet of Awesome Power?"
"No," Vimbai said. "Looks like for us it's all about Felix's Hair and Phantom Limbs and Peb's Tongue. Speaking of which . . . "
Maya heaved a sigh. "I know. Don't nag, please. We'll go as soon as I get a chance to take a nap, okay? I've been climbing stairs all day long. And you've been drowning, so maybe you should do the same. Then we check on Felix and go looking for the tongue."
Vimbai nodded. "I'll check on him first. See you in an hour or two."
She tiptoed past the forest of what looked like coat hangers and fishing poles and hat racks, and across a brand new meadow, sprinkled with white and pink flowers, that hadn't been there this morning, on her way to Felix's room. Thankfully, he remained unmolested, and asleep. Vimbai thought of waking him but decided against it—there seemed no point in exposing him again to the shock of his transformation, to the realization that his parent universe was gone forever, his umbilicus to his home world, however tentative, severed with unnecessary brutality and machine-like efficiency, and that the effluvia of the dead universe were dumped onto the ocean surface. Vimbai wondered if it would affect marine life at all, or if they would be able to surface through the ghastly space remnants with no problems. Maybe waves would disperse it, she thought; maybe it was the sort of thing only people noticed, like time. No, it was better to let him sleep—perhaps his dreams would help him when he woke up.
Vimbai reached her room and curled under the blankets. She thought idly that she probably should hang up her wet clothes from earlier today so that they could dry properly, but dismissed the thought as something best left for later. Now, she had to concentrate on the strategic dreaming—she had to dream of something that would help them to retrieve Peb's tongue. Vimbai closed her eyes and, before uneasy and fitful sleep claimed her, pictured the grotesque body of Peb, with many arms and hands and feet bristling from it in every direction, and the empty hole of its black crying mouth.
Vimbai's dream felt strangely sedate, even ordinary —she dreamt of being a petulant twelve-year-old, shopping for shoes with her tight-lipped mother. It was important for some reason to get new shoes right before the school started again, and Vimbai's mother was determined to make this experience as stressful as possible—even worse than the rest of obligatory back-to-school nonsense.
First, there was the issue of the overall effect malls had on Vimbai's mother—there was something about the sheer volume of the superfluous consumption that put her in a foul mood as soon as she parked her car. Wherever they went afterwards, there were more and more irritating things, and the stream of muttered commentary never ceased; it eventually grew in volume, causing the shoppers nearby to look at them. Vimbai felt embarrassed and hissed at her mother, and she snapped back. And it went downhill from there.
Second, there were the shoes themselves. Vimbai, being twelve, liked them square-nosed and funky, with chunky heels and bright colors; her mother tended toward more demure and practical styles, preferably of the Mary-Janes variety; lime green and three inch wedges or platforms were out of the question.
Third, there was the political side—it took them forever to find shoes that were not made in a sweatshop, and made by those who were either the US unionized workers or at the very least fair wage workers in China or elsewhere in the world. That took forever, and it drove Vimbai insane—nothing she liked could possibly meet her mother's approval, and if by some miracle it did, there was almost no chance that it would pass the fair labor test. Vimbai thought that it wasn't fair that her shoes had to be a political statement by her mother, but there didn't seem to be a way around it.
In the dream, Vimbai saw herself as if looking on from the outside, hovering disembodied and invisible, and looking with her adult eyes at the sulky and young version of herself—was she really that chubby as a kid? Young Vimbai scowled at the brown shoe that enclosed her foot like an ugly polyp. Her mother kneeled before her, tying the laces with uncalled-for vigor, as if she were trying to strangle Vimbai's foot.
"It's ugly," Vimbai said. "I hate it."
Her mother looked up—one of the very few moments in Vimbai's life when her mother was looking up at her. "Vimbai, sahwira, please. These are the only ones in your size."
"Mom, this is ridiculous. There are tons of shoes here. And some of them are not even ugly."
"You know why we can't get these," her mother said, exasperated, her pupils narrowed into needle points, her voice so taut it was ready to tear into a scream at any second. Dangerous, dangerous, not the woman to toy with or to piss off just now.
Vimbai rolled her eyes. "Mom, buying a pair of shoes is not a political decision. It's just shoes. It's not fair to put it on me, you know? There are countries and governments and all these people in the world who could make sure that there are no sweatshops or child labor, so I can just get a pair of fucking shoes without drama and without you telling me how everything is my fault."
To her surprise, her mother's lips relaxed and her shoulders sagged, as if the tension wire had just been pulled out of her, leaving her without the ringing terrible support she relied on all these years. "It's not your fault," she said. "I never said anything was your fault—why would you even think that?"
Vimbai shrugged and nodded at the ugly hoof on her foot.
Her mother laughed, unexpectedly. "I suppose it feels like punishment, doesn't it?"
"Yes." Vimbai kicked off the shoe, now that she realized that argument and screaming had been miraculously averted.
Her mom sighed and stood. She took Vimbai's hand and pulled her along, away from the imitation leather benches and the low mirrors on the floor, away from the shelves crawling with mismatched shoes, away from the smooth hardwood floors and the restless children and annoyed mothers. "We'll find you something," Vimbai's mother said. "Just understand one thing for me, all right? It's not your fault, but sometimes we have to do what we can to correct wrongs done by other people. Sometimes those who committed them are dead or they don't care or they don't see it as a wrong. But this is what makes us human, this—the fact that we are able to fix other people's mess. Even when it's not fair."
Vimbai nodded that she understood. "You can just get me canvas sneakers," she said. "Now let's go get a pretzel."
Her mother smiled, nodding, and her firm warm hand squeezed Vimbai's in unsaid gratitude. When Vimbai woke up, it was dark outside, and she felt like crying.
Maya woke up before Vimbai. She sat in the kitchen, darker than a storm cloud. "We're out of coffee," she announced as soon as Vimbai came downstairs.
"Bummer," Vimbai mumbled, unwilling to meet Maya's eyes. "Any tea?"
"Just loose green tea that's been sitting in the cupboard since the previous tenants," Maya said. "Whoever they were."
Vimbai sniffed at the yellow paper package with green lettering, and laughed. "That's not tea, that's mate."
"What's the difference?"
"It has a better kick than coffee," Vimbai said. "Only you have to drink it through special straws, otherwise the leaf debris would get into your mouth."
"I don't care," Maya said with a suddenly renewed enthusiasm for life. "How do you make this thing?"
Vimbai put the kettle on and poured boiling water over what she judged to be sufficient quantities of a substance that resembled dried grass in appearance and smell.
Maya drank greedily. "Yuck," she said. "Then again, it does have a kick." She looked at Vimbai and stopped smiling.
"What's the matter?" she said. "You look bummed out."
Vimbai heaved a tremulous sigh and sniffed, all the while aware that it wasn't really fair to Maya who did not have any family left. "I miss my parents," she said. "Especially my mom."
"I thought you fought a lot."
"We did. We do. But it doesn't matter; I still miss her."
Maya nodded. "I suppose families are like that. Anything you want to talk about?"
Vimbai considered the offer—it was tempting, to tell Maya about her mother's obsessive social consciousness and the liability it brought to her teenage daughter, amplifying the usual embarrassment every offspring had suffered while interacting with their peers in their parents' presence. Vimbai suspected that her social status suffered doubly—for her mother's insistence on responsible consumption and her accent. Even though she was a college professor, her accent and her color marked her as an immigrant, a first-generation, and Vimbai preferred to downplay her mother whenever possible—which was not often. Yet, all these complaints seemed petty now, especially in Maya's presence. Maya, who did not have any parents, did not deserve to listen to Vimbai's unsubstantiated bitching. Instead, she said, "I just regret that I never invited my mom over to this house. I think she would like it, and she would love you."
Maya laughed, took a hasty sip of her mate, and coughed, her face turning dark purple.
Vimbai patted Maya's back, trying to dislodge whatever renegade maté leaves had lodged in her throat. "No, really, she would. She would try and adopt you, of course. And then she would drive me nuts telling me how I should be more like you."
"Why would she say that?"
Vimbai shrugged and sucked in a mouthful of maté through her teeth, trying to filter out the debris in the manner of whales. "She wants me to have more positive role models. See, I'm not African enough, and then I'm not American enough, and I'm not really anything proper. And my mom . . . she means well and she tries hard, but I know that she secretly wishes that I had grown up in Zimbabwe so that she wouldn't have to deal with a spoiled American kid. Or she wishes that I would know more about the Diaspora, at the very least. She wants me to understand why it matters to everyone but me that my parents came over here voluntarily."
Maya nodded. "We all have our problems, I guess."
"I guess. And I know that mine are not important; they are just the ones I know."
Maya finished her mate. "I understand. Well, I have a job and a roof over my head, so I have no reason to complain either; still it doesn't matter if I do. Meanwhile, let's take care of those who can't complain even when they want to."
Vimbai finished her drink and stood. "Oh, grand. We'll go find the wazimamoto and ask them for Peb's tongue."
"That's right," Maya said. "What are you afraid of? You seem to have power over them."
Vimbai sighed. "I hope it still works."
# Chapter 18
Vimbai and Maya decided to look for the truck—it was day, and the wazimamoto were more likely to be roaming around. Maya's dogs trotted ahead, sniffing the ground, barking in short bursts and occasionally peeing on the ground, excited.
Vimbai thought guiltily that she wouldn't really mind walking like this, through the plains overgrown by skeletal umbrellas and yellowing sedges, with rare clusters of what looked like forks piled high with calamari salad off in the distance—just walking and talking to Maya, about anything they wanted to talk about.
"Your mom sounds really cool," Maya said. "And smart, too. I think it is awesome that your parents came from overseas and managed to make a good life here."
"I guess it is good," Vimbai said. "Only my mom complains so much, you never would guess that she is happy."
"Maybe she complains because she sees how things could be better."
Vimbai nodded, all the while imagining bringing Maya over to visit her parents. They would hit it off, Vimbai thought, her mother and Maya; they would really like each other. They would probably understand each other better too—they would sift through their collected experience, looking for similarities in stories of privation, shutting out Vimbai who really never missed anything. Maybe this is why her mother got so angry—maybe it was because they were too good as parents, they provided too well, spoiled her too much. They made it too easy for her, and thus failed to raise a child they could relate to. Vimbai could not decide whether it was truly sad, or if it was a ridiculous thing to feel bad about.
She was distracted from her thought by the appearance of something tall, stone, and domineering on the horizon—even if she hadn't seen it in her dreams, she would've recognized it anyway. The Great Zimbabwe, this version made of concrete slabs and wrought iron. When they traveled closer, Vimbai saw that there were occasional Legos and plastic building blocks sprinkled in the great seams where one slab joined the next.
Maya's dogs dispersed over the grassy area between the giant structures—houses of giants, Vimbai thought, temples of dragons. The sort of thing that made one want to believe in ancestral spirits and their ability to bring messages from the creator. Vimbai smiled and looked around her, a vague pride filling her heart with joy.
She wanted to look for people's houses, for the round houses she remembered from her dreams as well as her travels to the outskirts of Harare, so perfect and almost fairytale-like, with their smooth walls and grassy roofs. She wanted to find people from her dreams and their winged boats, the delicate contrivances that allowed flight from the terrible draining of the wazimamoto. She wanted to hear the powerful whooshing of these wings, displacing the air with great beats, and the shouts of people in the boats, not looking back but intensely staring ahead of them, already forgetting what they had escaped, intent only on finding out what waited for them in whatever new place their boats carried them to.
But there were neither houses nor boats, and Vimbai sighed with disappointment. Maya wandered between the great stone contraptions, her mouth alternatively hanging open and shaping a delighted smile. "This is yours, isn't it?" she asked Vimbai as if it was something she had made herself. "You have such wonderful dreams."
"Thank you," Vimbai said, and felt a bit silly at being complimented on the quality of her subconscious. "This is something I've really seen—it's The Great Zimbabwe." She explained to Maya what it was, all the while keeping her gaze on the openings between the stones, where the green canopy of the surrounding forest, punctuated here and there by tall gray spires of unknown origin, met the grass of the clearing. Vimbai could not see any roads, and yet it offered no comfort.
She was not surprised when she heard the sound of engines, getting closer and closer. She thought then that the wazimamoto were like European ghosts, unable to do anything but revisit the places that had mattered to them when they were still alive. Like clockwork, their truck went in circles, regardless whether there were victims to be had.
"Quiet," Vimbai whispered and took Maya's hand. A normal protective gesture, she told herself, no reason for Maya to think anything was up and to reject Vimbai on the spot and outright. She pulled her along, to hide in the tall grass between the jutting cliffs and leaning slabs of the construction, parts of it resembling not so much the Great Zimbabwe but radioactive spill sites—those were always covered with concrete slabs, in indifference or foolish optimism, it was so difficult to decide. All Vimbai knew that every single one she had ever seen had cracked concrete with thin tree saplings pushing through the cracks, nothing contained, and thoughts about where the radioactive spill went were best left unthought and unanswered.
Maya followed her, and the two of them lay on their stomachs, behind a piece of concrete that jutted partway out of the ground, forming an inclined smooth surface that was so easy to hide behind. The noise of the engine came closer, and Maya barely had enough time to whistle to her dogs, who came to her call and lay behind the slab too, a rusty river of fur and pricked up ears, of bright black eyes and long pink tongues separating Vimbai from Maya like a legendary sword.
Vimbai waited for the sound of the car engine to get closer—so close, it seemed to be shuddering in her heart now, the ashes of Klaas, the thunderous choking beats that made her want to jump up, her hands over her ears, screaming, enough, enough, please stop!
Instead, she clung closer to ground, trying to disappear in the narrow space between the concrete slab and grass, her eyes squeezed shut. Maya's elbow pressed against hers, and only this warm touch offered a measure of comfort. The scars on her inner arms glowed with a pale yellow light, as if they felt the approach of this specific danger. Or perhaps something different—just as the engine fell silent, Vimbai felt a gentle tap on the shoulder, and turned around to come face to face with the man-fish.
He did not seem much inconvenienced by being out of the water, and perched among the small saplings sprouting through the cracks in the stone. The man-fish managed to maintain a semi-upright position; his fins must've gotten stronger since the last time, Vimbai thought. Or maybe he found more souls to swallow, and this is what sustained him.
"Hello," the man-fish said in his gravelly voice. Vimbai thought that if he only twirled his whiskers, he could've passed for an operetta villain. "What are you doing here?"
"What are you doing out of your lake?" Vimbai countered. "Don't you need water to breathe?"
"Eventually, O girl who would not drown," he said. "But now I am here to help those who help me—they cannot deal with you, apparently."
Maya moved closer to Vimbai, crouching by her side, her knee touching the side of Vimbai's thigh. "Neither could you."
The man-fish ignored her, and turned his slightly glassy eye to the dogs, who whimpered but stayed close to Maya—out of loyalty, or possibly out of fear of something else hidden within this dream replica of a great monument. His mouth gulped air in quick, convulsive breaths, and his gill covers rose and fell like miniature beating wings. "What have we here?" he said. "Little fox-creatures, little girl's imaginary friends—all little pieces of her soul, all tasty morsels."
"What is he talking about?" Vimbai whispered to Maya.
Maya only paled in response and gathered her pets in a protective embrace.
"That's right," the man-fish said, leering. "You know I can suck them all in as if they were candy, slimy gummi worms. You know that these misshapen mutts are just little freewheeling bits of you, and if I swallow them, what will become of you, hm?"
Vimbai drew herself up, straightening between the man-fish and Maya and her whimpering creatures. "You won't be swallowing anyone today," she said. "You better tell us where Peb's tongue is, and then we'll be on our way."
The man-fish seemed taken aback—he deflated somewhat, shrank away from Vimbai and looked smaller than he ever had. "And you think you can command me: why?"
Vimbai thrust her carved-up forearms that glowed brighter and spilled their pale yellow light in narrow beams, like the weak spring sun, into the man-fish's face.
He backed off a bit. "Where did you get this magic?" he asked, with curiosity rather than fear.
"I made it myself," Vimbai answered, deciding that going into great detail would be counterproductive.
The man-fish nodded with respect. "Very nice," he said. "With magic like this . . . it's very impressive, really. If one had such magic, one wouldn't need to beg for soul scraps from others."
"You mean—" Vimbai started.
The man-fish nodded again. "I mean that with such magic, I wouldn't have to go to wazimamoto or even help them drain your blood and your soul—and I could, I'll have you know, I totally could. Child's play. I'll even strike a bargain with you—you carve me a spell like this, you muroyi, you. You little witch. You carve me a spell and I tell you how you can get the psychic tongue back."
"So you lied to us the first time," Vimbai said. "It wasn't in the hospital."
"Oh, it was. Only not where you'd think. The wazimamoto, see, they are just nightmares, blind and dumb. They are nothing—they need psychic energy to even exist, let alone talk. The tongue you're looking for was there with them the whole time."
Maya gasped. "So Peb's tongue is what's keeping them talking."
"It's what keeping them existing," the man-fish said. "Which is a good thing for me, because this place is not exactly rich in life, and therefore in souls. They had drained what they could off your friend, and then off that funny head on a single leg. They give me what they don't use. But if you offer me something better . . . "
"I'm not going to let you steal more souls," Vimbai said. "Or help you to do so."
"You don't have to," the man-fish said. "If you make me a spell that would let me live without souls, that would let me collect the energy I need from the air and the water around me, then I would be content."
Maya nudged Vimbai. "You sure he's not lying?"
Vimbai shook her head. "Of course not. I mean, he probably is. About some of these things, at least."
"Can you put a spell on him?"
"Yes." Vimbai's fingertips stroked the scars on the insides of her arms, left hand to the right arm and the other way around, crossed, entwined. "I'm not sure I understand how it works or why I even can do that, but I think I could. But only after we get back Peb's tongue."
"That's rather inconvenient," the man-fish said. "If you banish the creatures that sustain me and then your spell fails, what will happen to me then?"
"We'll release you into the wild," Vimbai promised. "There are plenty of lakes in New Jersey, and there are dead people's souls you can swallow to your heart's content—if the spell fails, that is. As soon as the crabs get us there."
The man-fish appeared to scowl, even though Vimbai was not quite sure how he managed that without any eyebrows. "And I should trust you: why?"
"Because we cannot trust you," Vimbai said. "You tricked us twice already—it would be stupid to believe you again, you have to agree."
The man-fish muttered but conceded the point.
"So you see, you'll have to trust us, or we'll be at an impasse," Maya said. And added, in a flash of brilliance, "Besides, how long do you think before they decide to drain your blood?"
"That's a very good point," Vimbai said. "Can you really trust any creatures who do nothing but rob everything alive of its blood?"
The man-fish considered, his small eyes slowly moving from one girl to the other. "You won't trick me?" he finally asked.
Vimbai rounded her eyes at him. "How could we? You are quite smart, we wouldn't dare to."
"Yeah," Maya said. "And we give you our word—it actually is worth something."
The man-fish sighed, his gill covers fluttering. "All right," he said. "Now, get closer."
Vimbai and Maya approached the man-fish on their hands and knees, cautiously, as Maya's dogs hung back, whimpering with their fluffy tails lodged between their hind legs.
"Now," the man-fish said. "The tongue you're looking for is shared between all of them, split into many fine energy strands, psychic energy fibers, if you will. And to draw it out of them, you will need something inert, something that would accept this energy and hold it. It's like osmosis, see? Spirits would move into a greater spiritual vacuum—so you just need to find something that is a greater spirit vacuum than the wazimamoto."
"Is there such a thing?" Vimbai asked. "Is there anything more devoid of soul than colonial vampires?"
"Undead crabs?" Maya suggested.
"Their souls are too close," the man-fish argued. "But something close, something dead . . . "
"Oh no," Maya interrupted. "You're not touching my grandma."
The man-fish chuckled softly. "Even if it's just a memory of her death? Even though it would let you fix your little psychic energy friend?" His flat head and beady eyes thrust forth, his slimy skin almost touching Maya's face, his wet cold lips almost on hers. "Even though you could make her alive again, even for just a little while?"
The wazimamoto's truck had fallen silent and the man-fish crawled away, muttering dark obscenities and vague promises; he begged them to come and visit him at the lake as soon as Maya made up her mind. Her dogs had grown bored and dispersed, hunting crickets and whatever other small and timid life crawled between the great concrete imitations of real boulders—real somewhere in the outside world, the world beyond these walls, the world that seemed a dream sometimes.
Vimbai had run out of comforting words, and could only sit next to Maya, her legs folded under her cold and numb, with only occasional prickling of phantom pins and needles suggesting that they were still alive; Vimbai's arms, goosebumped and heavy with fatigue, wrapped around Maya's unresponsive shoulders. How long had they been sitting like this?
Forever, Vimbai thought. Her head grew heavy with thwarted sleep, leaning against her will on Maya's indifferent shoulder, merciless gravity pulling her eyelids close. Galaxies were born and fell to dust, constellations swirled and traversed the skies millions of times, changing their position slowly, imperceptibly—and still, the two girls sat in the ruins of a great civilization, quiet and uncertain about the fate of a dead grandmother.
"Come on," Vimbai coaxed gently. "Let's go home—you don't have to decide anything here."
Maya shook her head, and Vimbai was unsure what she was objecting to. Finally, Maya shook her head one last time and stretched, breaking open the protective ring of Vimbai's arms. "I don't suppose I have a choice now, do I? Let it be, then. How do we get her to the wazimamoto?"
"We can wait here," Vimbai said. "We can wait for them to show up. I will ask the horseshoe crabs to bring her to us—on their little backs, on their slender legs. They have no souls that would leak into her."
Maya nodded. "Ask your crabs to bring her then, I don't mind." She heaved a shuddering sigh. "I wish my grandmother was more like yours."
Vimbai understood what she meant—a bare spirit was better than a lifeless body. The separation of flesh from soul was a terrible thing—all death was terrible. But the ghosts, the vadzimu and other spirits, were pure and comforting, offering protection and advice, telling stories and doing dishes, really alive rather than dead. After separation of flesh and spirit, it were the spirits that remained alive. Vimbai shuddered at the memory of Maya's grandmother—nothing but a flesh suit, grotesque and unwieldy in its white gloves and floral hat, especially meaningless because the corpse did not need protection from the cold or embarrassment. It was inert and still, its trappings betraying the anxieties of the living.
And this is how she was when the horseshoe crabs, summoned by Vimbai's insistent call, aided by the vadzimu (who Vimbai could not see but easily imagined as she stood on the porch, peering into the water and gathering the undead arthropods for their mission), marched to the concrete tower somewhere on the edges of the house, as far as Maya's consciousness could reach, and picked her up on their backs. They moved like a lumpy river of olive-slick carapaces and gray infested meat and pale broken legs, dragging through the dust and the linoleum of the house, a small dead woman on their backs—her face peaceful, as if she were lying in state with her hands crossed on her chest and her starched white slip barely peeking from under the edge of her black skirt. Dead—certainly dead enough to suck away whatever psychic energy the wazimamoto had squirreled away, inside of their faceless, soulless bodies—if the man-fish was to be believed, at least.
# Chapter 19
When the procession of crabs appeared, so solemn, so grotesque, carrying the small dead woman, Maya suppressed a quick sob and covered her face with her hands. Vimbai could not help but hug her again, a mute comfort of companionship and implied understanding the only offering she had to give. And after the crabs had settled in a patient unmoving (undead) circle, surrounding the dead woman and her hat.
"You better leave," Vimbai told the crabs. "Please. If you stay, Peb's tongue . . . the psychic energy we're after might jump into you. I mean, you're not wearing your souls now, are you?"
No, the crabs whispered, mournful. Our brethren have their souls, but ours were drained and stolen, damaged forever—why didn't you tell us that you couldn't keep them safe?
"I'm so sorry," Vimbai answered, a deep blush blooming forth under her skin, ready to reach the surface as soon as she finished speaking. "I thought I found them in time." And then, the blush; the memories of the horseshoe crab souls came forth, unbidden—their spirit shells impaled on long needles, the ghastly wazimamoto contraptions penetrating their gills and their eyes, their shells, with the casual brutality of those who wanted nothing but blood blood blood, medicinal blood for their antibodies and serums, blood they could drain from those who did not consent to it and then toss them back, used up, half-dead. Or quarter them and cut them up, stuff them into traps that would soon be crawling with thick slimy eels.
She had seen one of these traps being pulled out of the water once—the mesh bag reinforced with steel hoops that kept the trap open and barrel shaped, with a half-decomposed mass within it, unbearable to look at—it dripped and stank, and the fisherman who held the rope on which the trap was suspended seemed oblivious to the stench.
Vimbai had felt like gagging and looked at her classmates (it was a fieldtrip for her very first marine bio class, and they were doing a unit on fisheries, which meant fieldtrips and talking to the fishermen a lot, and going on their boats to check their nets). Those trips always made Vimbai so nauseous.
"And this is how you catch eels," the bearded old man said.
Vimbai had been peering into the trap, puzzled—there didn't seem to be anything in there, except the fishy organic rot and a few broken segmented legs tipped with pale pincer claws. And then the mass started to move—seethe, churn, roil, like a pot of stew left in the sun for a few days roiled with maggots—and Vimbai had to look away just as the writhing black eels started falling through the mesh and slithering across the wooden pier.
"I'm never eating eel sushi again," one of Vimbai's classmates muttered in her ear.
"These are usually not for sushi," the professor explained cheerfully, his stereotypically gray goatee shaking with glee. "These are used as bait for the large-mouth bass."
It was then that Vimbai had decided that she would work on horseshoe crabs, on saving them from the awful destiny of being bait for bait, so recursively demeaned.
And now, standing in the ruins of the Great Zimbabwe and faced with their accusing stalked eyes, she blushed and looked away. "I am sorry I could not protect you. I swear to you that I tried. I did my best."
She feared their accusations, but they remained silent, looking at her with an indecipherable expression in their beady eyes. Vimbai couldn't figure out if they had forgiven her, these crabs that were now doomed to forever remain undead, or if she should apologize further.
Maya broke the awkwardness. "Can you hear that?"
Vimbai listened to the distant sound—lapping of waves, she had thought initially. Rustling of leaves, pounding of surf, beating of wings. For a moment, her hope of seeing the people in winged boats once again flared up, but she soon recognized the thudding of a car engine.
"They are coming back," Vimbai told the crabs. "You better leave if you don't want to be fed to the catfish."
The crabs finally listened and skittered away, one wide glistening river of olive and gray and brown, and Maya's dogs followed them at Maya's command—there was no use for them, and no point in them trying to protect Maya from the rapidly approaching danger.
Vimbai and Maya were left alone again, with only the dead body as their protection from the wazimamoto. And Vimbai's scars—she almost regretted not having offered Maya to carve some protection into her skin, and simultaneously found this way of thinking horrifying and repellent, just like she found her grandmother's beliefs in mutilating her own daughters wrong.
The medical truck pulled up to them, squealing to a slow laborious stop by the largest of the slabs. The old-fashioned cab of the truck was filled with the faceless surgeons, and they also clustered along the railings, their hands mere pale latex gloves that looked as if they contained nothing but air. Their faces remained hidden behind the gauze masks that could not disguise the absence of human features beneath the draping of their folds.
They disembarked from their vehicle as it groaned under their shifting weight, its shocks raising it higher above the ground as its passengers stepped down from it. They gave the body of Maya's grandmother only a cursory glance, immediately pegging it for something that offered no drainage possibilities, and moved past it, toward Maya—they must've remembered that Vimbai was not accessible to them.
As they moved by the dead body, Vimbai saw thin wisps of rainbow-radiant energy start peeling from them, swirling into a complex geometric shape in the air right above the dead woman's forehead. The wazimamoto faltered in their tracks and grasped with their gloved hands at the dancing, laughing apparition—the soap bubble, the shining glassy skin stretched over the countless phantom legs, a teasing smile of ethereal dimensions.
The shimmering shape remained floating and suspended, and time itself seemed to slow down and hover, twining around it like a strange dimensional pretzel. The wazimamoto slowed their motions, and their gloved hands grasped at the apparition as it leaked out of them—rainbow and brimstone!—as if trying to force it back into their pale hollow chests.
And yet, Vimbai thought, it was not their fault—it was not their fault that they kept draining life out of everything they saw, like it wasn't their fault that now they were struggling, like ragdolls, against falling apart as the bright tongues of light exited them and filled the prostrate body of Maya's grandmother with their ghostly light. Was it just Peb's tongue? Vimbai thought. Or were there are other psychic energies, other dimensional body parts that the wazimamoto had collected somewhere along their dream travels—and now they were all unraveling.
Vimbai thought—or some uninvolved part of her did, as the rest of her mind alternately recoiled away from the spectacle and drew closer to it, not to miss a single spark, a single wisp—of how pretty it was, how reminiscent of Peb himself, of his shimmering misshapen glory. And she was reminded of the aurora borealis, which she had seen only once, on her class trip to Alaska; the shimmery stretchy wisps lifted the dead woman off the ground and filled her—they seemed to be wearing her like a suit, as if the dead flesh was just a mask to hide the terrible and glorious lights inside.
Vimbai pressed Maya's head against her shoulder, wishing she could hide her from the traumatic scene unfolding before them, protect her with embrace. The sigils on her arms glowed with a molten color of burning tissue and embers, and she kept hugging Maya closer. "Don't look, don't look," she muttered. "It'll be okay, I promise you, I promise." Maya's tears burned through the thin cotton of Vimbai's t-shirt.
But Vimbai herself looked—she looked at the conflagration of plasma and earthly fire, at the sputtering sparks from her own unintentioned charms and the spirit lights, at the flames that were springing up to consume the surgical scrubs and the latex gloves—so obviously empty now. The facemasks and the gauze, the hats and the rubber hoses were all going up in a giant bonfire that sprang where the wazimamoto had previously stood.
The fire roared up, up, and it spread sideways, bathing Vimbai's face in a blast of hot air, like the lick of a tremendous tongue. Vimbai retreated behind one of the slabs, Maya still held securely in her arms, Maya's face turned carefully away from the fire and the shambling puppet of her dead grandmother who lumbered slowly away from the spreading flames, her white eyes wide open and pouring out the same tormented fire that spread on the ground.
The grass turned to ash and the saplings bent and sputtered sap, crackling and groaning, their green branches bending and twisting like thelimbs of contortionists. The fire circle spread until it reached the truck, and Vimbai ducked behind her slab, expecting an explosion, just like anyone who had ever seen an action movie would.
To her surprise and secret disappointment, the fire did little more than melt the metal tires—apparently, the vehicles of colonial vampires did not use gasoline; the paint on the sides burned and crackled, swelling up in blisters and bursting. The rails and the cab heated to bright red and then white, and then they buckled and melted, turning into a soft clay and then viscous liquid. As it flowed to the ground, covering the burned grass and whatever ash was left of the wazimamoto's former shapes, Vimbai realized that the vampires and their truck were gone now, and whatever terrible essence had animated them was not trapped in Maya's grandmother's body, which stood silently in the clearing, unaffected by either fire or molten metal. The old woman looked quiet, pensive almost, and if it wasn't for the white light streaming from her wide-open eyes the color and appearance of boiled eggs, she wouldn't have warranted a second look from a passerby; just an ordinary old lady, wearing a hat and gloves as if heading to church on a Sunday morning.
Maya sobbed behind her. "Grandmother?" she whimpered, sounding disturbingly like a small girl. "Granma, is this really you?"
The old woman turned with a clockwork-like motion, and opened her arms to Maya. "How you have grown," she said by way of greeting, and Vimbai did not know whether she should've held Maya back as she rushed into her undead grandmother's embrace.
The vadzimu had made a pot of maté, and even thoughtfully filtered it through a cheesecloth (Vimbai did not even know they had any cheesecloth, let alone what to use one for). Vimbai made a small cry of relieved gratitude and poured herself a cup, for a moment abandoning the unpleasant thoughts that had been swirling in her mind all the way home. Was Maya's grandmother really herself, just animated by energies better not to be contemplated, or was it just a sham, the wazimamoto disguised to assume a new form? And, most importantly and most impossibly, if they were to return Peb his tongue (another matter the feasibility of which Vimbai could not possibly assess), would it mean the destruction of Maya's grandmother, be she real or illusory?
The vadzimu took the presence of another grandmother well. The two of them shook hands and engaged in some small talk about the best way of cleaning off residue from the inside of a coffee machine carafe, and Vimbai and Maya sat by the table, momentarily reduced to the age of twelve or thereabout, and drank their maté and listened.
Peb floated into the room, and Vimbai tensed as soon as he zeroed in on Maya's grandmother. He hovered up to her and started crying—a terrible wordless yowling, like that of a cat.
"He wants what's his," said Vimbai's grandmother, and reached for Maya's, reassuring. "Don't worry, dear. He can wait a bit longer."
As Peb wailed and whined, demanding, Maya turned to Vimbai. "We can't just give his tongue back to him, can we?"
"I don't think we have a choice," Vimbai said. "Look at him—he's so little."
Maya heaved a sigh. "This is my grandma you're talking about."
Vimbai patted her friend's hand; it looked so alone and weak splayed on the Formica surface of the kitchen table that Vimbai felt like crying. "I know. But Peb . . . he's been with us since the very beginning, remember? Sure, he looks weird and all, but he's our friend, like Felix."
Maya stroked some shallow cuts on the table surface, running her fingertips along their ragged ridges. Vimbai thought about the kitchen table back home—her parents' house, which she still considered home, no matter how much the house in the dunes grew on her. That table bore no cuts or irregularities of any sort, its surface smooth and polished daily by a soft cloth—it was so soft, in fact, that little Vimbai used to sleep with this cloth after she managed to liberate it from the kitchen cupboards.
"Do you think it will hurt her?" Maya said. "If Peb gets his tongue back, will my grandma go back to being dead?"
"I don't know," Vimbai said. "Ask her."
Maya gave her a tormented look. "I can't. What if she says yes?"
"Then we get the man-fish and beat the fuck out of that slimy bastard," Vimbai said and scowled, feeling rough and dangerous for once. "Then he would have to help us to sort things so that Peb gets his tongue back, and your grandmother can stay. Or at least—" Vimbai saw Maya's face, and didn't finish her sentence. There was too much hope mixed with fear in her dark eyes.
Maya drained her cup. The grandmothers had gotten acquainted by then, and chatted amiably, with Peb lolling and crying nearby, refusing to be ignored by the grandmothers.
"Grandma," Maya called. "Will you survive if you give Peb his tongue back?"
"Which one is his tongue, child?" the grandma answered.
"Oh damn it," Maya said. "We might need Felix again."
Vimbai clapped her hands over her mouth. "Oh, poor Felix! We left him all alone since last night!"
"Or longer," Maya confirmed. "Plus, that universe of his was drained."
"There's still some floating on the surface outside," Vimbai answered. "Maybe. Balshazaar poured it out. Let's go check."
"I would say that you're talking crazy," Maya said and stood, "if I wasn't used to all of us talking crazy."
On the porch, they stood a while, both surprised that the sun was so bright and large and real outside—and there were smells, familiar smells of the ocean and a new, coppery odor Vimbai could not immediately place. It didn't matter though—she thought that they were spending so much time indoors, in the constantly growing, mutating house, in its musty smell and its fake sky painted over the ceiling; with its sheetrock ridges and furniture mountains, carpet lawns and meat windows.
"It's nice to be outside," Maya said.
Vimbai nodded. She stared at the water, choppy with small stubborn waves, solid and angry. The waves butted against the porch, and there was no trace of Felix's remaining universe as far as the eye could see. "Damn it," Vimbai said.
Maya kneeled on the porch, peering between the boards—it wasn't too long ago, Vimbai remembered, that she imagined a nest of foxes under it, thought she observed a quick liquid movement of a long-tailed creature. "Look at this," Maya said.
Vimbai kneeled next to her. The water was dark under the porch, and until Vimbai's eyes adjusted to the shifts of light and shadow, to the narrow stripes of sunlight and sudden collapses of darkness, she wasn't sure whether she was seeing just water, or something else. Soon enough her eyes grew sensitive enough to discern the nuances, and she sighed with relief as she recognized the dark oily substance trapped under the porch. "How do we get it out?"
"I think I know," Maya said, and jumped to her feet. "You stay put, I'll go get Felix."
Felix was revived somewhat by the mention of his errant black hole, and he rushed outside, his lips, white as sheets, trembling with weakness and relief, a savage hope battling the familiar fears. He stuck his hands between the floorboards and wept as the thick oily fluid flowed up his pale arms and slopped over his neck and face and head, in an orgy of recognition and achieving completeness.
There were times, Vimbai thought, when things just came together—the constellations aligned and the world turned in such a way that the Coriolis forces of the world pushed all the disparate things and influences so that they came together in a beautiful swirl. Perhaps the house was too big to truly see that, but the kitchen was not—and Vimbai held her breath, wishing for this moment to stay with her as long as it could. There were Vimbai and Maya, standing by the sides of the screen door, their backs propping up the walls. They held hands in mutual support and anticipation, the jointed lock of their fingers hanging by the doorknob, as if it too was capable of admitting them somewhere else.
The grandmothers sat by the table, opposite each other, their eyes locked—one ghostly and one undead, but grandmothers nonetheless, and one could not help but love them, love them in ways one could not love one's parents out of pride and embarrassment and too much baggage and adolescent arguments. Perhaps those resentments too would burn away in a clean spiritual fire, Vimbai thought; for now, grandmothers sufficed.
The horseshoe crabs Vimbai thought of as hers and Maya's dogs were not in attendance, with the crabs being under the ocean and industriously pulling the house along, and the dogs temporarily exiled to the porch, where they squinted at the sun and panted with their tongues lolling; neither seemed to mind much at not being included in the ceremony.
And then there was Peb, floating grandly over the kitchen table, and Felix standing nearby, somewhat less pale, somewhat more animated ever since he managed to collect the remnants of the oily universe from under the porch and reattach them to his skull. It wasn't anything like his old do—there was no magnificence left there, it was barely enough to cover his head with a thin film, but, as they all had observed in turn, it was better than nothing at all.
Felix swallowed a few times as he looked from Peb to Maya's grandma and back. His Adam's apple, suddenly large and fragile under the transparent skin like a porcelain egg, bobbed in rhythm with the swallowing. He licked his lips a few times. "Here goes," he said, and dipped both hands into his hair. Both came away covered in what looked like tar but Vimbai guessed at the gooey space of his former universe, and squeezed Maya's hand tighter.
Maya squeezed back. It was a bad time, Vimbai thought, but then again, is there ever a good time for anything? And just as Felix reached his stained fingers into Maya's grandmother's mouth, Vimbai whispered under her breath, I love you, looking straight ahead and addressing no one in particular.
She had been getting used to the otherworldly light shows; still, when Felix pulled out a writhing, rainbow colored fish, Vimbai gave a little gasp of surprise. The small thing flapped and strained against his fingers, ethereal, and for a moment Vimbai thought that Felix had extracted something he wasn't supposed to. But Peb reached out with seven or eight of his limbs and grabbed the brightly colored appendage and stuffed it in his mouth. He smiled then, and babbled happily about the ethereal dimensions and the deepest chasm filled with molten sulfur and black iron.
Vimbai's eyes turned to the grandma—the old lady gave Felix a disapproving look of her white eyes and coughed, delicately covering her wrinkled mouth with one white-gloved hand. She coughed for a while, as if clearing her throat from dust and grime accumulated over the years (and Vimbai suspected that was the case); grandma hacked and caught her breath and hacked again, with great inhales and sharp coughs reminiscent of the tearing of butcher's paper. When she was finally able to stop, she extracted a small white handkerchief demurely tucked away into her sleeve, and mopped the corners of her eyes and her mouth. "Good Lord," she said. "Maya, are you wearing cutoffs made from man's jeans? Have you lost your mind?"
Maya let go of Vimbai's hand then, and she rushed across the kitchen floor elbowing Felix out of the way, and she gave her grandmother a great hug, crying and laughing at the same time.
And what was Vimbai going to say about that? Nothing, that's what—she kept her lips sealed, because she too had her grandmother who had no reason or way of being here, because sometimes hows and whys did not matter as much as the greatest gift in the world, the biggest privilege imaginable—the ability to look at someone whom one had lost, and to tell them all the things you always wanted to say but did not have a chance. The second chance, the greatest gift—and who was Vimbai to deny it to anyone, least of all Maya?
# Chapter 20
Vimbai watched the sun rising over the ocean—still a thin silvery stripe, with the clouds just barely turning pink and golden.
Maya came out onto the porch and stood next to her. "Nice sunrise. I don't remember the last time I've seen one—properly, I mean. Driving home after the night shift doesn't quite count."
"Sunrise is a sunrise," Vimbai said. "You okay?"
Maya nodded. "Yeah. Just trying to get used to the idea that my grandmother is, you know, a zombie."
"Mine is a ghost," Vimbai pointed out.
Maya heaved a sigh. "Do you think ghosts are cooler than zombies?"
"No way." Vimbai smiled. "Zombies are way cool. Although the ghosts are too."
"And undead horseshoe crabs?"
"They are in their own category," Vimbai said. She pointed at the dark stripe on the horizon, something she had been trying to discern the shape of since it was light enough to see anything. "What does this look like to you?"
Maya looked, her hand shielding her squinting eyes, and grinned, wider than Vimbai had ever seen her smile. "This is land, Vimbai. This is land! We'll get us some milk soon."
"And afterwards?"
"I still vote for staying in the house and being queens of all we see," Maya said. "I don't think we'd get to keep our grandmothers in the outside world."
"I'm sure they'll be fine in the house," Vimbai said. "But we . . . you and I, we need to go outside now and again. I want to see my parents, and I would like you to meet them, and I hope they are not too mad at me—I mean, they would be mad, but they'll forgive me, I hope. And I want to go back to school, and I hope to get away with academic probation for such a long absence, and I worry that I won't be able to."
Maya smiled. "I know. Still, it's tempting to imagine what it would be like, to leave everything behind like that, and just explore and name things, wouldn't it?"
"Sure," Vimbai said. "Maybe we'll be able to—maybe the house will keep growing on the inside, and we can take weekend trips to its distant reaches."
"And we can go and visit the man-fish."
"Oh, damn it!" Vimbai clasped her hands to her chest. "I completely forgot that I promised him a spell—I better take care of it before he gets pissed off and starts walking around swallowing souls."
"I'll come with you," Maya said. "Do I need to bring the dogs with me?"
Vimbai shook her head. "Nah, let them be. I will need a knife, though."
Maya followed her in the kitchen and stayed close, as Vimbai rummaged through every drawer looking for the sharpest knife. "You think you know what you're doing?"
"Yeah," Vimbai lied, and shot a reassuring smile in the direction of both grandmothers. She headed for the door before the grandmothers got suspicious and started asking what they wanted with sharp knives. Maya followed, her expression alternating between giddy and doubtful.
Vimbai and Maya hurried to the man-fish's lake. The way was so familiar now, so ordinary that Vimbai barely paid any attention to the usual overnight terrain alterations—there was a small hill built of rolled up, twisted laundry, and on the other side of the path a freshly sprung puddle of Jell-O and rich mud. Weak and pale stems of rye fringed the path and brushed against Vimbai's bare calves, like stiff cat whiskers.
"Do you think that land we saw was . . . is really New Jersey?" Maya said. "I mean, could it be some never-never land or something?"
Vimbai shrugged. "I doubt it. The crabs know where they are going, and they know New Jersey. They would tell us if they were lost . . . wouldn't they?"
"Of course," Maya said. "Just wondering, you know? These past few weeks have been a bit—"
"Weird?" Vimbai interrupted.
Maya smiled and nodded. "Yeah. If you're aiming for the understatement of the century."
"You've been coping well." Vimbai sucked on her lower lip, considering her words. "I was getting an impression that you were rather . . . reveling in it."
"I never thought it would make sense to freak out or whine about shit," Maya said. "Roll with the punches, dontcha know. But this stuff, this . . . this house and the ghosts and my dogs—all this has been great. I like it, and sometimes you just have to stop worrying about what's possible and what isn't, and how it's all going to play out, and what will happen to you and if you're losing your mind."
"You thought that you might?" Vimbai said.
"And you didn't?"
Vimbai shook her head. "I would've, if it was just me. But with you around, being so cool about all this . . . I never really doubted."
The cattails and reeds fringing the man-fish's lake greeted them with sage nodding to the nudgings of a light gentle wind, and the sun reflected in a thousand facets on the lake's surface.
There was a movement, a splash by the small island of wild rice not too far off the bank. The man-fish waited for them, his wide mouth twisted in a grimace of acute displeasure that eerily reminded Vimbai of her mother. "Finally," he said. "Took you long enough to show up."
"Sorry," Vimbai said. "We've been battling blood-draining monsters."
"Successfully, I assume." The man-fish gave Vimbai a long measuring look. "Of course. You wouldn't be here otherwise."
"And you would." Maya shook her head. "You really know how to hedge your bets—you would be here no matter what."
"You don't seem to understand the man-fish," the man-fish said.
Vimbai nodded. It seemed impossible to understand such a creature, like it was not possible to understand the vampires. There was that solitary drive, the terrible single-minded obsession that Vimbai lacked and yet did not envy. "I suppose we don't," she said. "Then again, what are you going to do?"
"You don't understand the survival." The man-fish crawled closer to the bank, his wet browned skin glistening in the sun as his humped back and the stiff dorsal fin breached the water surface. "You don't know what it is like, constantly thinking of not dying, and finding enough to eat so you can live another day."
"Maybe not," Maya said. "But you need Vimbai now, so you better be nice to her."
The man-fish grumbled and crawled closer still. "All right, all right. So what say you, witch-girl?"
"I'm definitely not a witch," Vimbai said, even as she doubted her own words. Her thoughts ran in so many directions at once now—was she really a witch, or was her magic born merely of love and desperation? Would her mother be mad at her if she knew? Of course not, Vimbai thought. She would be mad at Vimbai's long disappearance first of all, and if she was ever kicked out of school or even put on academic probation, she would be even madder. Magic was quite far down on the list of things Vimbai's mother would get mad about. "But I can cut you good. It's a pity I didn't bring a fish knife."
Maya snickered, and the man-fish rolled his beady eyes. "Very funny," he said. "Go ahead—less talking and more cutting, and I only pray that you're more proficient in the latter than in the former."
Vimbai eyed the wide expanse of the man-fish's flank and back, brown and green like silt, like river mud. He swelled immense, and his eye still glimmered with malice he never bothered to conceal. And what sort of magic could she pour into such a creature? She could only rely on her vague understanding of how these things worked, and she reasoned that if her love, her unrequited desire fueled the protective spell she had somehow carved into her forearms, then to quench the insatiable thirst for souls she had to offer something the opposite of it—satisfaction, satiety, contentment. Vimbai smiled, since this was something she knew quite a bit about.
She turned to Maya. "I'm going to tell a story—it's a traditional ngano, and I'm a sarungano, a storyteller. You'll be the audience and you'll have to ask me questions when I stop, all right? And answer mine when I ask you."
"I'll do my best," Maya said, and looked puzzled. "Is it like a spell?"
"It's like a story," Vimbai answered. "Ngano is how children learn." She cleared her throat and started, the knife in her hand rising and falling in rhythm with her story.
"Who is the wisest animal in the forest?" Vimbai said.
Maya opened her mouth and laughed. Then said, "I don't know. Who?"
"Is it a jaguar?" Her knife fell, leaving a long thin mark on the catfish's smooth skin.
"No."
The mark swelled with blood.
"Is it a baboon?" Another cut, crisscrossing the first at a sharp angle.
"No, it is not." Even Maya fell under the spell of her rhythm and swayed along, and gave her answers in a singsong voice.
"Is it a hare?" The new cut fell, and the overall patter of crosshatching grew apparent to Vimbai.
Maya hesitated, and Vimbai shrugged at her, indicating the correct answer. "Maybe," Maya said.
"Is it a tortoise?" The skin of the catfish was now developing a pattern of blood-stained, elongated rhombi.
"Yes?" Maya offered.
Vimbai nodded and smiled. "Go on, ask."
"Why is it wise?" Maya asked.
"Because it does not chase after things." Cut.
"Because it is satisfied with what it has." Cut.
"Because it carries his house on his back and does not covet a new one." Cut.
"He is never aggressive and yet he gets his way." Nod to Maya.
"How?" Maya said.
The knife in Vimbai's hand trembled and paused, raised over the devastation it had wrought—the skin of the man-fish was a pattern of bloodied diamonds, a horrible jester's suit. "Because he knows that he already has everything he needs, and if he ever needs more, the creator will give him more. It is up to the Mwari, the creator, and the mhondoro, the tribal spirits, to give everyone what they need. Otherwise, the eyes grow greedy, the hands feel empty, and there's never any satisfaction and no one is ever sated and happy with what they have."
"Except the tortoise," Maya offered, sounding more confident.
"Except the tortoise," Vimbai agreed. "May the moon forever slosh in his belly."
At her last words, the man-fish's mouth snapped open—a dark tunnel of unquenched hunger—and he lunged, his jaws snapping shut just a hairbreadth away from Vimbai's nose. She screamed out and jerked away, the hand holding the knife lashing out in a reflexively protective gesture.
Maya gasped nearby and out of a corner of her eye Vimbai caught a blur of motion as Maya struggled to her feet, as the man-fish slithered and snapped, trying to get to Vimbai's soul, his evil reaching out in a final desperate gesture. Vimbai's knife caught him across the throat, and the last cut, ragged and cruel, traced the pale skin below his jaw, carved away a good chunk of his snout.
The fish fell back, exhausted, the bloodied chunk of his face in Vimbai's lap. She pushed him away, kicked his limp slippery body away from her, and struggled to catch her breath.
She dropped her hand with the knife down into her lap and looked at her handiwork. For a moment she worried that the man-fish would bleed to death, expire because of her incompetent magicking—and even he had tried to drown her not too long ago, even if he tried to steal her soul it would seem wrong to her, having killed someone who voluntarily went under her knife, preferring it to the needles of the wazimamoto and the eternal hunger of the cursed.
Then, the man-fish stirred, and the flow of blood stopped. The diamonds of his savaged skin glowed and silvered, and Vimbai and Maya could not quite believe their eyes and had to touch them with their fingers, to make sure that those were indeed scales—something no catfish ever had.
"Whoa," Maya whispered. "How'd you learn to do this?"
"I didn't," Vimbai said.
"And that story?"
"I just made it up." She tossed the knife to the ground and sat back, her weight pushing her heels deeper into the muddy soil. "I don't really know anything. I just make shit up, you know?"
"Seems to work just fine." Maya crouched low next to Vimbai, close enough for their knees to touch, and watched the man-fish's continued transformation. His skin was now covered in perfect silver scales with small shimmering white and green spots, and his face was changing too—the whiskers had disappeared and his upper jaw curved into a haughty beak, extending his face forward, covering up the disfigured lower one. His head and body did not look flat anymore, but acquired the graceful proportion of a fast fish that did not feed on the bottom but propelled itself with strong strokes of its lobed tailfin.
"That's a lake trout," Vimbai said. "I think."
"Is it good?" Maya dared to pat the fish's head, and it flared its gill covers in response.
"Hey, Mr. Fish?" Vimbai said. "Can you still talk?"
The fish opened its mouth as if in silent laughter, splashed its tail in the shallow water like an oar, and—one, two, three—it was gone, disappeared under water. In just a few moments, the surface of the lake grew smooth like silk again, and did not betray the presence of a large fish underneath anywhere.
That night, Vimbai could not sleep. The thoughts of the previous day kept churning in her mind, and her imaginings of the day to come charged the air with great anticipation. She wasn't the only one—the previous evening, even though no one had said anything about it, had been taut with barely concealed excitement.
The two grandmothers in the kitchen argued about what needed to be done food-wise, seeing as how they only had some preserves and canned soups and ramen and a bag of flour left. They compromised on pancakes but barely spoke to each other afterwards. Felix retreated to his room, but seemed to be in high spirits—the universe around his head, drained and ravaged and discarded, had been growing again, and Vimbai supposed that soon enough it would resume its normal undulation—although without Balshazaar, whose demise in the hands of wazimamoto passed unlamented by anyone; only Felix was kind enough to acknowledge that he had ever existed.
Peb would not stop babbling now that he had his tongue back, and he traveled all over the house, his many limbs bristling like the fins of a lionfish, yelling cheerful nonsense about brimstone rivers and blue electrical storms, of the worlds made of ball lightning and fire and of black unfathomable chasms populated by creatures capable of swallowing entire galaxies.
Maya's dogs and Vimbai's horseshoe crabs remained outside—the former curled up on the boards of the porch, their bushy tails covering their glistening wet noses from the cold, their eyes looking up wetly at whoever ventured onto the steps. The crabs stayed hidden, but Vimbai could imagine the restless churning of their legs, the clusters of their soul shells waiting for them on the ropes, waiting for the day that it was warm enough for the creatures to become whole again.
Maya and Vimbai had left the kitchen with its squabbling old women and the overexcited Peb, and sat on the porch, under the stars, the hunk of land black against black sky, its outline only hinted at by the absence of stars. In the darkness of her room, Vimbai smiled at the memory, at the contentment she felt whenever she and Maya could be away from everyone else and sit side by side, listening to the quiet sluicing of the waves and talking in low voices, as if sharing secrets even though they discussed quite mundane topics.
"What will you do when we get back to New Jersey?" Maya had asked. "I mean, besides going back to school and freaking out that your mom would yell at you."
Vimbai smiled at the barb, at the fond familiarity of it. "I will start looking into horseshoe crab conservation. I mean, there are initiatives now—like they don't allow fisheries to use them as bait anymore, but I'm sure there are more things I could do. And no one thinks that the medical research is damaging them, but I know it does—you can't just drain away most of someone's blood and think that you're not harming them."
"I'll say." Maya's face was hidden by the night, but her voice was smiling.
"Anyway," Vimbai said. "Shouldn't you be spending more time with your zombie grandmother?"
"Not when you put it this way." Maya laughed softly. "No, I will. I'm just . . . it takes getting used to, you know? And then there are all these crazy notions that she would be disappointed in me for not finishing college, for not making more of myself."
"You still can."
"I know." Maya sighed. "Still."
"I'm sure she won't be disappointed." Vimbai continued.
"Well, maybe not. But it's strange for me too, having her back and yet not quite knowing if it's really her, you know? How did you cope with your grandma?"
"I barely knew her when she was alive." Vimbai stroked the wooden plank by her side. "I don't know if it's really her, but I can't know—I have very little idea of what she is supposed to be like. But you'll figure it out."
"I guess so."
"But again, does it even matter?" Vimbai said. "Isn't it better than having no grandmother at all?"
"You're right." Maya shifted in the darkness, petting the dogs, and stood. "We better turn in—we'll be there tomorrow. Need to get some sleep."
"Yeah," Vimbai said and rose too. "Good night, Maya."
And now she lay in her room, her mind racing. Occasionally, she drifted into brief snatches of sleep, and dreamt of the crabs coming ashore where Vimbai's mother waited for her, her hand shielding her eyes from the sun, forever vigilant, forever waiting. She dreamt of the sun rising and touching the silvery ocean surface behind her back, lighting the land outline in front of her. And as she dreamt, the house touched the beach softly, its porch sliding over the sand compacted by the surf, over the tops of the dunes, until it found its old foundation, left free of sand. The house sighed and creaked and stretched its roof corners and its wainscots as it settled into the familiar grooves—but carefully, as if afraid of disturbing the delicate contents that filled it to brimming.
The half-foxes, half-possums crawled under the porch, sighing contentedly, as they curled up in the familiar dark cave, the sand underneath still wearing the rounded troughs left by their bodies. They wondered if they would be allowed back inside, and if they would go hunting tomorrow, fording rivers and running across the great golden plains of straw and couch cushions.
The horseshoe crabs remained underwater, sleeping, soulless for now, under the freezing waves, and dreaming of the days when the sun would rise high and warm the chilly waters, when the tides would rage high on the beach and they would put on their soul shells and perhaps fix them with the remnants of the souls still sluicing in these waters, become themselves again and come dancing through the surf, raising their legs high like chitinous ballerinas. They dreamed of the bygone days when wave after wave of spawning crabs flooded the beaches and crashed upon them in a frenzy of whipping tail spikes and burrowing legs, where the eggs of the crabs outnumbered the grains of sand.
The ghosts in the house slept too—unusual for the ghosts, but they welcomed the relief. Peb curled in the chipoko's lap as she nodded off in a living-room chair, and both dreamed of the branches of jacaranda trees. Maya's zombie grandma closed her terrible white eyes for the first time since she walked again, and she conjured up visions of downtown Newark and church service on Sundays, of the gospel choir whose singing reached through the honking, screeching traffic, all the way down the street.
And the human inhabitants . . . their dreams were more vague, more difficult to pin down—but they were the ones that filled the house with the forlorn memories of the past and the regrets of the present, they were the ones that gave the walls and the valleys and the ridges their shape. They were the namers and the creators, the wills that shaped the house so that it could remain itself, even now, when it was moored securely on solid land, in the forever shifting dunes.
# About the Author
Ekaterina Sedia resides in the Pinelands of New Jersey. Her critically-acclaimed novels, The Secret History of Moscow and The Alchemy of Stone, were published by Prime Books. Her short stories have sold to Analog, Baen's Universe, Dark Wisdom and Clarkesworld, as well as the Japanese Dreams and Magic in the Mirrorstone anthologies. She won a World Fantasy Award in 2009, for Paper Cities: An Anthology of Urban Fantasy.
BOOKS BY EKATERINA SEDIA
The Alchemy of Stone
Heart of Iron (forthcoming)
Paper Cities: An Anthology of Urban Fantasy (edited)
Running with the Pack (edited)
The Secret History of Moscow
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Faculty / Unit Faculty of Architecture Faculty of Arts Faculty of Business and Economics Faculty of Dentistry Faculty of Education Faculty of Engineering Faculty of Law Li Ka Shing Faculty of Medicine Faculty of Science Faculty of Social Sciences Asia Global Institute Centre of Development and Resources for Students Communications and Public Affairs Office Development and Alumni Affairs Office Graduate School Hong Kong Institute for the Humanities and Social Sciences Hong Kong University Press Institute of Human Performance Jao Tsung-I Petite Ecole Knowledge Exchange Office President's Office Sustainability Office Technology Transfer Office University Libraries University Museum and Art Gallery Women's Studies Research Centre
HKU's artificial mussel technology helps fight marine pollution globally
HKU Director of School of Biological Sciences Professor Rudolf Wu Shiu-sun has developed a technology called artificial mussel to test water quality. Professor Wu has been invited by Partnerships in Environmental Management for the Seas of East Asia (PEMSEA) and International Atomic Energy Agency (IAEA) to provide a two-month training scheme for scientists from Tunisia, Tanzania, Ghana, Thailand and the Philippines to learn how to use the technology to test for heavy metals that affect ecosystems and food chains.
Knowledge Exchange Awards 2013
The Faculty Knowledge Exchange (KE) Awards were introduced in 2011 in order to recognize each Faculty's outstanding KE accomplishment that has made demonstrable economic, social or cultural impacts to benefit the community, business/industry, or partner organizations. Results of the 2013 Faculty KE Awards are now available.
HKU academic urges less consumption of groupers which face extinction
A study conducted by HKU Biological Sciences Professor Yvonne Sadovy has found that at least 20 grouper species are facing extinction. Professor Sadovy said threatened species such as HK grouper and longtooth grouper, which were found in HK waters, should be protected by law. She has urged the government to protect them through legislation and tighter monitoring of grouper trade, and to set a good example by serving sustainably caught seafood in official functions.
Mineral treasures of China on display at HKU Stephen Hui Geological Museum
A special exhibition of mineral treasures of China will be held at the HKU Stephen Hui Geological Museum, in association with the Mineralogy Society of HK, until August 30, 2013. Over 50 world-class mineral specimens from private collection of Dr Guanghua Liu will be on display.
HKU academic discovers selective use of natural condiments may reduce cancer risk
HKU Biological Sciences Associate Professor Dr Wang Ming-fu discovered that selective use of natural condiments such as grapefruit, tomato sauce and green tea as meat seasoning may significantly reduce cancer risk. By incorporating appropriate cooking methods, such condiments can also lead to the formation of some novel food components that helps preventing cancer. Dr Wang and three other HKU researchers will be presented with University Outstanding Young Research Award 2012 in the Award Presentation Ceremony for Excellence in Teaching, Research and Knowledge Research on March 27.
The largest ever database on night sky brightness developed by HKU reveals very serious light pollution problems in Hong Kong
An HKU research team, led by Dr Jason Pun Chun-shing of the Physics Department, developed the HK Night Sky Brightness Monitoring Network (NSN) in May 2010, which is the largest ever database on night sky brightness. Over 5 million night sky brightness measurements have since been collected. The Hong Kong urban night sky was found to be 100 to 1,000 times brighter than the international dark sky standard between 8:30-11:00pm, making it one of the most light-polluted cities in the world. In European cities like Madrid and Florence, the readings were normally below 100 times the standard. The NSN was funded by the Environment and Conservation Fund of the Environmental Protection Department, and the HKU Knowledge Exchange Fund. Results of the project will provide the scientific basis for the Government in possible developments of regulations on outdoor lighting usages.
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AltairsJailBird- Welcoming Predator the master of hunting, and welcoming Ezio the Master Assassin!!
AltairsJailBird- Hello Predator do you think your going to win this fight and have a nice skull that you can be proud of and stare at remembering your hunting skills?
AltairsJailBird-Ezio, Do you think you could you could bloody your hidden blade florescent green?
Ezio-I can never know only suspect, but I suspect that this will be the case!
AltairsJailBird- Well then since your both enthusiastic that you'll win we'll be getting started ready your blades for the hunt is on in the Centro district in Rome!
Please watch with the size/bolding of your words. Such measures can give people eye sores.
Also, the Yautja probably wins, but i need way more information to vote.
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Damien Chazelle's tense biopic detailing the life of Neil Armstrong and his iconic mission to the Moon arrives on blu-ray this week. Does the Oscar contender stick the landing on its home release? Check out our full review to see what we thought!
Like many movies of the last year, I missed out on First Man when it hit theaters in the Fall despite feeling a great deal of eagerness to check it out. Biopics aren't normally high on my list of "must-watch" films, but as a history buff I was really excited to see how this one pulled it all together.
Thankfully, with the film's blu-ray release, I finally got my chance to check it out and now I'm even more sad that I missed it on the big screen. The film chronicles the portion of Neil Armstrong's life leading up to his historic walk on the surface of the Moon. It begins shortly before he applies, and is accepted, to the astronaut program and his eventual landing. While that may not seem like a lot of ground to cover, there's a significant amount of time included that details his work at NASA while providing insights to his personal life.
Obviously we know how the story ultimately ends (it's part of history), but seeing the journey in this film is still impressive. One of the best things the film does is immerse you deeply into the experience, making it feel as though you're watching it unfold in real time before your eyes. It manages to make the idea of space flight terrifying and unsure. There was literally a point in the film where I asked myself if they were going to make it...Before I remembered I already knew the answer!
That's how tense First Man is, and just how powerfully it manages to engross you in the story. The acting all around is superb, with Claire Foy being the stand out in my mind. She stole every scene she was in, upstaging everyone and doing it effortlessly. Couple that with a strong script and there were several points throughout the film where my emotions got the better of me.
By the time the credits rolled I almost immediately wanted to watch it again and I kept thinking back to the film and its performances for days until I had the chance to watch it again. I'm happy to say, the film remains just as impactful the second time around as the first. It's such a well put together film and hits all the right notes. If you saw it in theaters and was worried about it not holding up for multiple viewings, have no fear.
This is a film you don't want to sleep on. Even if you're not big on biopics or historical dramas, the film has a lot to offer. From the stunning portrayals, excellent character work, and touching story there's a lot to love.
Between Whiplash and La La Land, Damien Chazelle showed he has a knack for delivering stunning and unique visual style, while staying within 'realistic' settings. First Man is no different and despite adhering to the 60s aesthetic, manages some gorgeous shots scene-after-scene, with a variety on the color pallete you wouldn't quite expect.
All of this is on grand display in the HD transfer to the blu-ray. The image is sharp, allowing for crisp edges on the cast while keeping all the smaller details in focus. The blacks are sufficiently deep without crushing, which makes the colors and characters pop on the screen (especially when we get that sweet, sweet Moon footage).
The sound design, which plays a crucial part in the accuracy of the presentation, has made the transition equally well. The multi-channel surround sound makes it feel as though you're strapped in along with the astronauts and immersed in the sounds of those older space-faring vessels. It adds another dimension to the historical factor without detracting from the dialog priority. All in all, from a technical aspect, the First Man blu-ray does a great job of showcasing the beauty of the film.
All of these offer some nice behind the scenes material, though none of them are terribly long. The longest featurette comes in around seven minutes (Putting You In the Seat), which means there isn't a lot of time to delve as deeply into how this biopic was created as you'd like. Still, they're pretty fun, though I'm not entirely sure you'll come back to them that often. The highlight of this disc is definitely the film.
If you missed out on First Man when it arrived in theaters last year (as many seemed to), there's no reason I can think of to pass on it's home entertainment release. The film itself is impressively well told, with excellent performances, and tension that keeps you glued to the screen even on multiple viewings. It looks gorgeous on blu-ray, with the stellar filmmaking on display. Don't miss out on this release and add it to your shelf as soon as you can!
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Below is a video of my class presentation.
IN-CLASS PRESENTATION Below is an informal presentation shared with a pod of nine individuals also participating in this assignment and course. EXPO DEMONSTRATION Below is a short demo of the dynamic component of my design recorded at the 2017 Mechanical Engineering Exposition.
The video of my presentation can be seen above.
Watch final presentation here: https://www.youtube.com/watch?v=2ZJpg1roPRw Watch prototype presentation here : https://www.youtube.com/watch?v=y6-sinySjLE Pictures of the final product at the ITLL Design Expo on April 29th, 2017. Curtesy of Professor Jean Hertzberg.
placeholder for video Cannot, for the life of me, get my video to upload!!!!
This was the video of my final presentation for Aesthetics of Design. For the project, I built a bamboo fly rod by hand, in an effort to convey a "naturalist" aesthetic as a tribute to the craftsmen and early pioneers of the sport.
This post serves as a place to store comments about the Rick Totem presentation, which can be found in the video below.
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\section{Introduction}
Let $f_k: \hat \C \to \hat \C$ be a sequence of endomorphisms of the Riemann sphere of degree $d\geq 2$ that diverges in the space of all endomorphisms. Concretely, this means that at least one zero and pole of $f_k$ are colliding in the limit. Our main goal is to understand the degeneration of the dynamical features of $f_k$ and, ultimately, to extract useful information from a ``limit dynamical system." In this article, we concentrate on the measure of maximal entropy.
The existence and uniqueness of a measure of maximal entropy $\mu_f$ for a rational function $f$ of degree $\geq 2$ were shown in 1983 \cite{Lyubich_Measure, Freire-Lopes-Mane_Uniqueness_1983, Mane_Uniqueness_1983}. Shortly after, Ma\~n\'e observed that the measure $\mu_f$ moves continuously in families \cite{Mane_Weakly_Continuous}, with the weak-$*$ topology of measures and the uniform topology on the space of rational functions. By contrast, the Julia set $J(f) = \supp \mu_f$ fails to move continuously (in the Hausdorff topology) in the presence of bifurcations \cite{Mane:Sad:Sullivan}.
The space $\Rat_d$ of complex rational functions of degree $d \geq 2$ can be identified with the complement of a hypersurface in $\Ratbar_d = \PP^{2d+1}$. In \cite{DeMarco_Boundary_Maps_2005}, the first author showed that for ``most" degenerating sequences $f_k \to \del \Rat_d$, a limit of the maximal measures $\mu_{f_k}$ can be expressed as a countably-infinite sum of atoms. (The measures $\mu_{f_k}$ themselves are atomless.) There it was also shown that Ma\~n\'e's continuity property for maximal measures \textit{does not extend} to all of $\Ratbar_d$. Although weak limits of maximal measures for degenerating sequences may not be unique, our first main result shows that every weak limit is purely atomic.
\begin{thmA} \label{sequences}
Let $f_k$ be a sequence that diverges in the space $\Rat_d$ of complex rational functions of degree $d\geq 2$, and assume that the measures of maximal entropy $\mu_k$ converge to a probability measure $\mu$ on $\Chat$. Then $\mu$ is equal to a countable sum of atoms.
\end{thmA}
Our second main result shows that Ma\~n\'e's continuity property \textit{does extend} to degenerating 1-parameter families. Moreover, we are able to give a refined description of the limit measure using an associated dynamical system on the Berkovich projective line.
\begin{thmB} \label{main theorem}
Let $\{f_t: \; t\in \D\}$ be a meromorphic family of rational functions of degree $d \geq 2$ that is degenerate at $t=0$. The measures of maximal entropy $\mu_t$ converge weakly on the Riemann sphere to a limiting probability measure $\mu_0$ as $t\to 0$. The measure $\mu_0$ is equal to the residual equilibrium measure for the induced rational map $f: \Berk_{\mathbb{L}} \to \Berk_\mathbb{L}$ on the Berkovich projective line, where $\mathbb{L}$ is the completion of the field of formal Puiseux series in~$t$.
\end{thmB}
\begin{remark}
The continuity of maximal measures on $\Chat$ can fail for degenerating families over a parameter space of dimension 2; see \cite[\S5]{DeMarco_Boundary_Maps_2005}.
\end{remark}
\begin{remark}
While we prefer to work with the more ``geometric'' field $\mathbb{L}$, one can replace it with the field of formal Laurent series $\mathbb{C}(\!(t)\!)$ in the statement of the theorem.
\end{remark}
One should view the Berkovich dynamical system $(f, \Berk_{\mathbb{L}})$ as the limit of dynamical systems $(f_t, \hat \C)$ as $t\to 0$. This fruitful perspective was introduced by Jan Kiwi in his work on cubic polynomials and quadratic rational maps; see \cite{Kiwi_Puiseux_Dynamics_2006, Kiwi_Rescaling_Limits_2012} and \cite{Bonifant-Kiwi-Milnor}. A closely related construction, viewing degenerations of polynomial maps as actions on trees, can be seen in \cite{DeMarco-McMullen_Trees}. Charles Favre has recently constructed a compactification of the space of rational maps, where the boundary points are rational maps on a Berkovich $\Berk$ \cite{Favre:personalcomm}. Our work is very much inspired by these results. The Berkovich space viewpoint allows us to recover the results in \cite{DeMarco_Boundary_Maps_2005}, and it provides a conceptual explanation for the form of the limiting measures. In a sequel to this article, we will describe a countable--state Markov process that allows one to compute the residual measure explicitly.
As with non-degenerating families, the Julia sets of $f_t$ may fail to converge to a limit as $t\to 0$. Consider the example of $f_t(z) = t(z + z^{-1})$ in $\Rat_2$. As $t\to 0$ along the real axis, the Julia set of $f_t$ is equal to the imaginary axis, while there is a sequence $t_n\to 0$ (tangent to the imaginary axis) for which $J(f_{t_n}) = \Chat$. Ma\~n\'e used the continuity of $f\mapsto \mu_f$ to deduce that the Hausdorff dimension of $\mu_f$ is a continuous function of $f$, but this property does not extend to degenerating families; for example, the measures for a flexible Latt\`es family have dimension 2 while the limit measures always have dimension~0.
The measure of maximal entropy $\mu_f$ for a rational function $f$ of degree $d\geq 2$ is characterized by the conditions that (a) it does not charge exceptional points, and (b) it satisfies the pullback relation
$$\frac{1}{d} \, f^*\mu_f = \mu_f.$$
To prove Theorem~A, we show that any weak limit of measures of maximal entropy on $\Chat$ must satisfy an appropriately-defined pullback formula (Theorem~\ref{paired pullback formula}); we then show that any measure satisfying this formula (for all iterates) is atomic. The pullback formula is phrased in terms of ``paired measures,'' which is an ad hoc object we introduce to keep track of weak limits of measures in two sets of coordinates simultaneously. This is all accomplished in Section~\ref{Sec: Complex Surface}.
The proof of Theorem~B (which inspired our proof of Theorem~A) is more conceptual and can be divided into three parts, each with its own collection of results that are of independent interest. We sketch these results here.
\medskip\noindent
{\bf Step 1. Dynamics on a complex surface.}
In Section~\ref{surface def}, we view the holomorphic family $f_t: \P^1 \to \P^1$ as one (meromorphic) dynamical system
$$F: X \dashrightarrow X$$
on the complex surface $X = \D\times\P^1$, given by $(t,z) \mapsto (t, f_t(z))$ for $t\not=0$. By hypothesis, $F$ will have points of indeterminacy in the central fiber $X_0 = \{0\}\times\P^1$. If $F$ collapses $X_0$ to a point, we let $\pi: Y \to X$ be the (minimal) blow-up of the target surface so that $F: X \dashrightarrow Y$ is nonconstant at $t=0$; otherwise, set $Y = X$ and $\pi = \mathrm{Id}$. By counting multiplicities at the indeterminacy points of $F$, we define a notion of pullback $F^*$ from measures on the central fiber of $Y$ to measures on $X_0$. We prove (Theorem~\ref{complex pullback}) that any weak limit $\nu$ of the measures $\mu_t$ on the central fiber of $Y$ satisfies a pullback relation:
\begin{equation} \label{eq 1}
\frac{1}{d} F^* \nu = \pi_* \nu.
\end{equation}
The proof relies on the Argument Principle to handle the measure at the points of indeterminacy for $F$.
\medskip\noindent
{\bf Step 2. Dynamics and $\Gamma$-measures on the Berkovich projective line.}
Let $k$ be an algebraically closed field of characteristic zero that is complete with respect to a nontrivial non-Archimedean absolute value. The Berkovich analytification of the projective line $\PP^1_k$ will be denoted $\Berk$; it is a compact, Hausdorff, and uniquely arcwise connected topological space. A rational function $f: \PP^1_k \to \PP^1_k$ extends functorially to $\Berk$. If $d = \deg(f) \geq 2$, then the equilibrium measure $\mu_f$ may be characterized similarly to the complex case by the conditions that (a) it does not charge exceptional points of $\PP^1(k)$, and (b) it satisfies the pullback relation $\frac{1}{d} f^*\mu_f = \mu_f$ \cite{Favre_Rivera-Letelier_Ergodic_2010}. See \cite{Baker-Rumely_BerkBook_2010} for a reference specific to dynamics on $\Berk$, or see \cite{Berkovich_Spectral_Theory_1990} for the more general theory of non-Archimedean analytic spaces.
The goal of Section~\ref{Sec: Measures} is to define a notion of pullback $f^*$ on a new space of quantized measures relative to a finite set $\Gamma$ of vertices in $\Berk$. Every Borel probability measure $\nu$ on $\Berk$ gives rise to one of these ``$\Gamma$-measures'' $\nu_\Gamma$. And if $\nu$ is a solution to the standard pullback formula $\frac{1}{d}f^* \nu = \nu$, then $\nu_\Gamma$ will satisfy a quantized version:
\begin{equation}
\label{Eq: New Berkovich pullback}
\frac{1}{d}f^* \nu_\Gamma = \pi_* \nu_\Gamma.
\end{equation}
(One must push $\nu_\Gamma$ forward by a certain map $\pi$ in order to have a meaningful equation since $f^*\nu_\Gamma$ lies in the space of $\Gamma'$-measures for a potentially different vertex set $\Gamma'$.) A solution to the pullback formula (\ref{Eq: New Berkovich pullback}) is typically far from unique. However, we will show (Theorem~\ref{Thm: One-vertex Unique}) that uniqueness is restored if one considers simultaneous solutions to pullback equations for all iterates of $f$, after ruling out measures supported on classical exceptional cycles.
\medskip\noindent
{\bf Step 3. A transfer principle.}
Now, let $k=\mathbb{L}$ be the completion of the field of formal Puiseux series in~$t$, equipped with the non-archimedean absolute value that measures the order of vanishing at $t=0$. (See \cite[\S3]{Kiwi_Rescaling_Limits_2012}.) By viewing the parameter $t$ as an element of $\mathbb{L}$, the family $f_t$ defines a single rational function $f$ with coefficients in $\mathbb{L}$. We define a vertex set $\Gamma\subset\Berk$ consisting of one vertex only, the Gauss point. In \S\ref{residual measures}, we define a correspondence between measures on the central fiber of our surface $X$ with $\Gamma$-measures on $\Berk$. From Step 1, any weak limit $\nu$ of the measures $\mu_t$ will satisfy the pullback relation \eqref{eq 1}. The corresponding $\Gamma$-measure $\nu_\Gamma$ must satisfy the non-Archimedean pullback relation \eqref{Eq: New Berkovich pullback} on $\Berk$, by Proposition \ref{transfer}.
We repeat the argument for all iterates $f_t^n$. From Step 2, we deduce that $\nu_\Gamma$ is the equilibrium $\Gamma$-measure, and consequently, the limit measure $\nu$ is the ``residual" equilibrium measure. See Section~\ref{Sec: Transfer}.
\medskip
\bigskip
\noindent
{\bf Acknowledgements.} We are grateful for the opportunities that allowed these ideas to germinate: the 2010 Bellairs Workshop in Number Theory funded by the CRM in Montreal, and the Spring 2012 semester on Complex and Arithmetic Dynamics at ICERM. We would like to thank Charles Favre, Mattias Jonsson, Jan Kiwi, and Juan Rivera-Letelier for helpful discussions, and we further thank Jonsson for inviting us to speak about this work at the December 2012 RTG workshop at the University of Michigan.
\bigskip
\section{The space of rational maps: complex-analytic arguments}
\label{Sec: Complex Surface}
In this section we prove Theorem~A along with a number of preliminary results that will be used in the first step of the proof of Theorem~B.
\subsection{The space of rational maps} \label{space}
We will let $\Rat_d$ denote the set of all complex rational functions of degree $d$. It can be viewed as an open subset of the complex projective space $\P^{2d+1}$, by identifying a function
$$f(z) = \frac{a_0z^d + a_1z^{d-1} + \cdots + a_d}{b_0z^d + b_1z^{d-1} + \cdots + b_d}$$
with its coefficients in homogeneous coordinates
$$(a_0:a_1:\cdots:a_d:b_0:b_1:\cdots:b_d) \in \P^{2d+1}.$$
In fact, any point $\Phi \in \P^{2d+1}$ determines a pair $(P, Q)$ of homogeneous polynomials in two variables, and $\Rat_d = \P^{2d+1} \setminus \{\mathrm{Res}(P, Q) = 0\}$. We set $\Ratbar_d = \P^{2d+1}$ so that $\del \Rat_d = \{\mathrm{Res}=0\}$. For each $\Phi = (P,Q) \in \del\Rat_d$, we let $H = \gcd(P,Q)$, and let $\phi$ be the induced rational function of degree $< d$ defined by the ratio $P/Q$. To match the algebraic language of the later sections, we refer to the map $\phi$ as the \textbf{reduction} of $\Phi$.
A 1-parameter \textbf{holomorphic family} $\{f_t: \, t\in U\}$ is a holomorphic map from a domain $U\subset \C$ to $\Rat_d$. A \textbf{meromorphic family} is a holomorphic map from $U$ to $\Ratbar_d$ with image not contained in $\del \Rat_d$. A meromorphic family is \textbf{degenerate} at $u\in U$ if the image of $u$ lies in $\del \Rat_d$.
\begin{lem} \label{nontrivial limit}
Let $f_k$ be a sequence in $\Rat_d$ converging to a point $\Phi \in \del \Rat_d$. After passing to a subsequence if necessary, there is a sequence of M\"obius transformations $A_k \in \Rat_1$ so that $A_k \circ f_k$ converges in $\overline{\Rat}_d$ to a point with nonconstant reduction. If $B_k$ is any other such sequence in $\Rat_1$, then $M_k = A_k \circ B_k^{-1}$ converges in $\Rat_1$ as $k\to\infty$ (along the subsequence determined by $A_k$). If the $f_k$ lie in a meromorphic family $\{f_t: t \in \D\}$, then the sequence $A_k$ may be chosen to lie in a meromorphic family $\{A_t: t\in \D\}$.
\end{lem}
\proof
Existence is carried out, algorithmically, in \cite[Prop.~2.4]{Rivera-Letelier_Asterisque_2003} and appears also in \cite[Lemma 3.7]{Kiwi_Rescaling_Limits_2012} when the sequence lies in a holomorphic family; the strategy is as follows.
At each step of this argument, we may pass to a subsequence. Write
$$f_k(z,w) = (P_k(z,w): Q_k(z,w)),$$
normalized so that $(P_k, Q_k) \to (P,Q)$ in $\Ratbar_d$. Note that at least one of $P$ and $Q$ is nonzero. By replacing $f_k$ with $S_k \circ f_k$, where $S_k(z) = \alpha_k z$ with $\alpha_k>0$, it can be arranged that the limiting $P$ and $Q$ are both nonzero. If $P$ is not a scalar multiple of $Q$, we are done.
Suppose $P = c_0 Q$ for some constant $c_0 \in \C^*$. If $m = \deg_z P = \deg_z Q$, write
$$f_k(z,w) = (a_k z^mw^{d-m} + \hat{P}_k(z,w): b_k z^mw^{d-m} + \hat{Q}_k(z,w))$$
where $\hat{P}_k$ and $\hat{Q}_k$ have no term involving $z^mw^{d-m}$. Now, postcompose $f_k$ with a translation by $a_k/b_k = c_0 + o(1)$, replacing $f_k$ with
$$f_k(z,w) = (P_k(z,w) - a_kb_k^{-1}Q_k(z,w): Q_k(z,w)).$$
If $P$ and $Q$ are not monomials, then we are done; the new limit has nonconstant reduction. If $P$ and $Q$ were monomials, the resulting limit in $\Ratbar_d$ will have constant reduction ($= 0$); we rescale and repeat the initial argument. It follows that the new $P$ cannot be a scalar multiple of $Q$ because it has no term involving $z^mw^{d-m}$. This completes the proof of existence of $\{A_k\}$.
If the given $f_k$ lies in a meromorphic family $f_t = (P_t, Q_t)$, then the scaling and translation maps can be chosen meromorphic in $t$, since they are built from the coefficients of $f_t$.
Now suppose $A_k\circ f_k \to \Phi_A$ and $B_k \circ f_k \to \Phi_B$ in $\overline{\Rat}_d$, with nonconstant reductions $\phi_A$ and $\phi_B$. Set $M_k = A_k \circ B_k^{-1}$. Again passing to a subsequence, $M_k$ converges to $M_0 \in \Ratbar_1$. Away from finitely many points in $\PP^1$, we have
\[
\phi_A(p) = \lim_{k \to \infty} A_k \circ f_k(p) = \lim_{k\to\infty} M_k \circ B_k \circ f_k (p) = M_0 \circ \phi_B(p).
\]
As $\phi_A$ is nonconstant, so is $M_0$, and therefore $M_0 \in \Rat_1$. This also shows that $M_0$ is uniquely determined, so the full sequence $M_k$ converges.
\qed
\subsection{Counting pre-images}
\label{Sec: Counting pre-images}
Fix a sequence $f_k$ in $\Rat_d$, and assume that $f_k$ converges to a degenerate point $\Phi\in \del\Rat_d$ with gcd $H$ and {\em nonconstant} reduction $\phi$. For each point $x\in \P^1$, we define multiplicities
\begin{equation} \label{complex m and s}
m(x) = \deg_x \phi \qquad \text{and} \qquad s(x) = \operatorname{ord}_x H.
\end{equation}
The quantity $m(x)$ is the local degree of $\phi$, and the quantity $s(x)$ will be called the {\bf surplus multiplicity} at $x$.
Let $\eta$ be a small loop around $\phi(x)$ bounding a disk $D$, and let $\gamma_x$ be the small loop around $x$ sent with degree $m(x)$ onto $\eta$ by $\phi$. Choose $\gamma_x$ small enough so that it does not contain any roots of $H$, except possibly $x$ itself. Because $f_k$ converges locally uniformly to $\phi$ on $\P^1\setminus \{H=0\}$, for each $k \gg 0$ there is a small loop $\gamma_k$ around $x$ that is mapped by $f_k$ with degree $m(x)$ onto $\eta$. Let $U_k$ be the domain bounded by $\gamma_k$.
\begin{prop} \label{counting}
Assume that $f_k$ converges to $\Phi\in \del\Rat_d$ with nonconstant reduction. Fix $x\in \PP^1$. For all $k$ sufficiently large,
$$ \# ( f_k^{-1} (z_0) \cap \overline{U}_k) = m(x) + s(x)$$
and
$$ \# ( f_k^{-1} (p_0) \cap \overline{U}_k) = s(x).$$
for all points $z_0$ in $\overline{D}$ and all points $p_0$ in $\P^1 \setminus \overline{D}$.
\end{prop}
\proof
The proof is an application of the Argument Principle from complex analysis. Assume first that $z_0=0\in D$ and $p_0 =\infty \not\in\overline{D}$. Then
$$\# ( f_k^{-1} (z_0) \cap U_k) = \# ( f_k^{-1} (z_0) \cap \overline{U}_k) = \#\; \mathrm{Zeroes}(f_k) \mbox{ inside } U_k,$$
and
$$ \# ( f_k^{-1} (p_0) \cap U_k) = \# ( f_k^{-1} (p_0) \cap \overline{U}_k) = \# \; \mathrm{Poles}(f_k) \mbox{ inside } U_k.$$
By the Argument Principle, for all large $k$ we have
$$\#( f_k^{-1} (z_0) \cap U_k) - \# ( f_k^{-1} (p_0) \cap U_k) \; = \; \int_{\gamma_k} \frac{f_k'}{f_k} \; = \; m(x).$$
On the other hand, we may compute directly that
\begin{equation*} \label{poles}
s(x) = \# \; \mathrm{Poles}(f_k) \mbox{ inside } U_k
\end{equation*}
for all sufficiently large $k$, since $f_k \to \Phi$. Indeed, $H(x) = 0$ with multiplicity $s(x)$ (and $\phi(x) \not=\infty$), so there are exactly $s(x)$ poles converging to $x$ as $k \to \infty$. (Compare \cite[Lem.~14]{DeMarco_Boundary_Maps_2005}.) It remains to handle the case where $z_0\in \eta = \del D$. By construction, the boundary $\gamma_k$ of $U_k$ is mapped with degree $m(x)$ over $\eta$; and by viewing $z_0$ as the point $\infty$, we see that there must be $s(x)$ preimages of $z_0$ converging to $x$ as $k\to \infty$.
\qed
\subsection{Paired measures}
\label{Sec: Paired measures}
Let $C,E$ be two copies of $\PP^1$. A \textbf{paired measure} $(\mu_C, \mu_E)$ is a pair of Borel probability measures $\mu_C$ on $C$ and $\mu_E$ on $E$. Let $\{A_k\}$ be a sequence of M\"obius transformations in $\Rat_1$. We say that a sequence of Borel probability measures $\{\mu_k\}$ on $\PP^1$ \textbf{converges $\{A_k\}$-weakly} to the paired measure $(\mu_C, \mu_E)$ if
$$\mu_k \to \mu_C \qquad \mbox{ and } \qquad A_{k*} \mu_k \to \mu_E$$
weakly.
Let $\Phi$ be an element of $\Ratbar_d$ with reduction $\phi$. We define a measure $\Phi^*(\mu_C,\mu_E)$ on $\PP^1$ by
the formula
\begin{equation*}
\Phi^*(\mu_C,\mu_E) := \phi^*\mu_E + \sum_{x\in \PP^1} s(x)\delta_x.
\end{equation*}
Recall that $s(x)$ is defined in (\ref{complex m and s}).
\begin{lem} \label{prob}
For any paired measure $(\mu_C,\mu_E)$, the measure $\Phi^*(\mu_C,\mu_E)$ has total mass $d$.
\end{lem}
\begin{proof}
The proof is a simple degree count:
\[
\Phi^*(\mu_C,\mu_E)(\PP^1) = \deg(\phi) + \sum_{x \in \PP^1} s(x) = \deg(\phi) + \deg(H) = d. \qedhere
\]
\end{proof}
\subsection{Weak limits satisfy the pullback relation}
Fix a sequence $f_k$ in $\Rat_d$ that converges to $f_0 \in \del\Rat_d$. We also fix the sequence $A_k$ of M\"obius transformations guaranteed by Lemma~\ref{nontrivial limit}, such that $A_k \circ f_k$ converges to a point $\Phi \in \Ratbar_d$ with gcd $H$ and reduction $\phi$ of degree $>0$. If the reduction of $f_0$ is nonconstant, we let $A_k$ be the identity for all $k$, so that $\Phi = f_0$. (Note that if the reduction of $f_0$ is constant, it is possible that $\deg \phi = d$.)
Let $C,E$ denote two copies of $\P^1$ as in \S\ref{Sec: Paired measures}. If $f_0$ has nonconstant reduction, then $A_k(z) = z$ for all $k$ implies that $\mu_k \to (\mu_C, \mu_E)$ $\{A_k\}$-weakly if and only if $\mu_C = \mu_E$ and $\mu_k \to \mu_C$ weakly.
\begin{thm} \label{paired pullback formula}
Any $\{A_k\}$-weak limit $(\mu_C, \mu_E)$ of the maximal measures $\mu_{f_k}$ will satisfy the pullback formula
$$\frac{1}{d} \, \Phi^*(\mu_C, \mu_E) = \mu_C$$
as measures on $C = \PP^1$.
\end{thm}
\begin{proof}
Without loss, we may replace $f_k$ with a subsequence in order to assume that $\mu_{f_k}$ converges $\{A_k\}$-weakly to $(\mu_C, \mu_E)$. By the definition of $\{A_k\}$-weak convergence, and because $d^{-1} f_{k}^*\mu_k = \mu_k$ for all $k$, we know that
\begin{equation} \label{convergence to pi*}
d^{-1} f_{k}^*\mu_k \to \mu_C \quad \text{as} \quad k \to \infty.
\end{equation}
We need to show that the weak limit of $f_{k}^*\mu_k$ can also be expressed as $\Phi^*(\mu_C,\mu_E)$.
Let $I(\Phi)$ denote the union of the roots of $H$. Let $U$ be a small neighborhood of $I(\Phi)$ in $\PP^1$. Choose a partition of unity
$$b_r + b_s \equiv 1,$$
subordinate to the open cover $\{\PP^1\smallsetminus I(\Phi), U\}$
so that $b_r \equiv 1$ on $\P^1 \smallsetminus U$ and $b_s \equiv 1$ on a small neighborhood of $I(\Phi)$ inside $U$; as usual, $b_r$ and $b_s$ are non-negative continuous functions.
Fix a non-negative continuous function $\psi$ on~$\PP^1$. Recall that the pushforward of $\psi$ by $f\in\Rat_d$ can be defined by
$$f_*\psi (y) = \sum_{f(x) = y} \psi(x),$$
where pre-images are counted with multiplicity. Because $b_r$ vanishes near $I(\Phi)$, and because $A_k \circ f_k$ converges uniformly to $\phi$ on compact sets outside $I(\Phi)$, we have uniform convergence of functions
$$(A_k \circ f_{k})_* (b_r\psi) \to \phi_* (b_r\psi),$$
and therefore
\begin{equation} \label{regular}
\begin{aligned}
\int b_r \psi \ (f_k^* \mu_k) &= \int b_r \psi \ \left( (A_k \circ f_k)^* A_{k*} \mu_k \right) \\
&= \int \left(A_k \circ f_k \right)_* (b_r \psi) \ A_{k*} \mu_k \rightarrow \int \phi_*(b_r \psi) \ \mu_E
= \int b_r \psi \ \Phi^*(\mu_C, \mu_E),
\end{aligned}
\end{equation}
by the weak convergence of $A_{k*} \mu_k$ to $\mu_E$. Upon shrinking the neighborhood $U$, \eqref{convergence to pi*} and \eqref{regular} together will show that
\begin{equation} \label{notIndeterminate}
\int_{\PP^1 \smallsetminus I(\Phi)} \psi \, \mu_C = \frac{1}{d} \int_{\PP^1 \smallsetminus I(\Phi)} \psi \, \Phi^*(\mu_C, \mu_E)
\end{equation}
for any test function $\psi$.
Fix $x\in I(\Phi)$. As in \S\ref{Sec: Counting pre-images}, let $\eta$ be a small loop around $\phi(x)$ that bounds an open disk $D$, and let $\gamma_x$ be the small loop around $x$ sent with degree $m(x)$ onto $\eta$ by $\phi$. Choose $\gamma_x$ small enough so that it does not contain any point in $I(\Phi)$ other than $x$ itself; we shall further assume that it is contained in the neighborhood where $b_s \equiv 1$. Because $A_k\circ f_k$ converges locally uniformly to $\phi$ on $\P^1\smallsetminus I(\Phi)$, for each $k\gg0$ there is a small loop $\gamma_k$ around $x$ that is mapped by $f_k$ with degree $m(x)$ onto $\eta$; for large $k$, this $\gamma_k$ is also contained in the region where $b_s \equiv 1$. Let $U_{x,k}$ be the domain bounded by $\gamma_k$.
We now apply Proposition \ref{counting} to the sequence $A_k\circ f_k$. For $x \in I(\Phi)$, let $\psi_{\inf}(x)$ denote the infimum of $\psi$ on the component of $U$ containing $x$. For all $k$ sufficiently large,
\begin{align*}
\int_{\P^1} b_s\psi \ (f_{k}^*\mu_k)
&\geq \sum_{x\in I(\Phi)} \psi_{\inf}(x) \int_{\overline{U}_{x,k}} \ (A_k \circ f_k)^* A_{k*} \mu_k \\
&= \sum_{x\in I(\Phi)} \psi_{\inf}(x) \int_{\P^1} \# \left( (A_k \circ f_k)^{-1}(y) \cap \overline{U}_{x,k} \right) \, A_{k*}\mu_k(y) \\
&= \sum_{x\in I(\Phi)} \psi_{\inf}(x) \Big[ s(x) A_{k*}\mu_k(\P^1\setminus \overline{D}) + (m(x)+s(x)) A_{k*}\mu_k(\overline{D}) \Big] \\
&= \sum_{x\in I(\Phi)} \psi_{\inf}(x) \left[ s(x) + m(x) A_{k*}\mu_k(\overline{D})\right].
\end{align*}
Letting $k\to\infty$, the $\{A_k\}$-weak convergence of measures gives
$$\liminf_{k\to\infty} A_{k*}\mu_k(\overline{D}) \geq \mu_E(\{\phi(x)\}).$$
Because $d^{-1} f_{k}^*\mu_k$ converges weakly to $\mu_C$, we deduce that
$$\int b_s\psi \, \mu_C \geq \frac{1}{d} \sum_{x\in I(\Phi)} \left[ s(x) + m(x) \mu_E(\{\phi(x)\}) \right] \psi_{\min}(x).$$
Shrinking the neighborhood $U$ of $I(\Phi)$, we obtain
\begin{equation} \label{singular}
\int_{I(\Phi)} \psi \, \mu_C \geq \frac{1}{d} \sum_{x\in I(\Phi)} \left[ s(x)
+ m(x) \, \mu_E(\{\phi(x)\}) \right] \psi(x) = \frac{1}{d} \int_{I(\Phi)} \psi \, \Phi^*(\mu_C, \mu_E).
\end{equation}
As $\psi$ was arbitrary, adding \eqref{notIndeterminate} to \eqref{singular} yields the inequality of positive measures
$$\mu_C \geq \frac{1}{d} \, \Phi^*(\mu_C, \mu_E).$$
But both are probability measures (by Lemma~\ref{prob}), so we must have equality.
\end{proof}
\subsection{Proof of Theorem~A}
Let $f_k$ be a sequence in $\Rat_d$ converging to $f_0 \in \del \Rat_d$ and with maximal measures $\mu_k$ converging to a measure $\mu$. From Lemma~\ref{nontrivial limit}, there is a sequence $A_k\in\Rat_1$ so that $A_k\circ f_k$ converges to $\Phi\in\Ratbar_d$ with reduction $\phi$ of positive degree. Passing to subsequences for each iterate $n$ and applying a diagonalization argument, we choose sequences $\{A_{n,k}: k\in\mathbb{N}\}$ in $\Rat_1$ so that
$$A_{n,k} \circ f_k^n \to \Phi_n \quad \mbox { as } k\to\infty$$
in $\Ratbar_{d^n}$ with reduction $\phi_n$ so that $\deg \phi_n >0$ for every iterate $n$.
By sequential compactness of the space of probability measures on $\PP^1$ (and another diagonalization argument, if necessary), we may assume that $\mu_k$ converges $\{A_{n,k}\}$-weakly to a paired measure $(\mu, \mu_{E_n})$ for each $n \geq 1$.
Since the measures $\mu_k$ are also the measures of maximal entropy for iterates $f_k^n$, Theorem~\ref{paired pullback formula} implies that
$$\mu(\{p\}) = \frac{1}{d^n} \Phi^*_n(\mu, \mu_{E_n}) (\{p\}) \geq \frac{s_{\Phi_n}(p)}{d^n}$$
for any iterate $n$ and any point $p\in\P^1$; recall that the integers $s_{\Phi_n}(p)$ are defined in (\ref{complex m and s}). Degree counting shows that $\sum_{p \in \PP^1} s_{\Phi_n}(p) = d^n - \deg \phi_n$, which yields
$$
1 \geq \sum_{p \in \PP^1} \mu(\{p\}) \geq 1 - \frac{\deg \phi_n}{d^n}.
$$
If $ \deg\phi_n = o(d^n)$ as $n\to \infty$, then we see immediately that $\mu$ is a countable sum of atoms. It remains to treat the case where $\deg \phi_n \not= o(d^n)$.
The next lemma shows that the reduction maps $\phi_n$ are not unrelated.
\medskip
\begin{lem} \label{composition}
The reduction maps $\phi_n$ form a composition sequence. That is, there exist rational functions $\phi_{n+1, n}$ of positive degrees $\leq d$ so that
$$\phi_{n+1} = \phi_{n+1, n}\circ \phi_n$$
for each $n\geq 1$. Moreover, $A_{n+1,k} \circ f_k \circ A_{n,k}^{-1}$ converges to $\phi_{n+1,n}$ away from finitely many points in $\P^1$.
\end{lem}
\begin{proof}
This lemma follows from uniqueness in Lemma \ref{nontrivial limit}. Write $\Phi = H\phi$ for any $\Phi\in\Ratbar_d$, where $H$ is the gcd of the two polynomials defining $\Phi$ and $\phi$ is the reduction. As $k\to\infty$, we have $A_{n,k} \circ f_k^n \to H_n\phi_n$ and $A_{n+1,k}\circ f_k^{n+1} \to H_{n+1} \phi_{n+1}$. Consider the sequence $f_k\circ A_{n,k}^{-1}$ in $\Rat_d$. Passing to a subsequence, there exists a sequence $C_k$ of M\"obius transformations so that $C_k\circ f_k\circ A_{n,k}^{-1} \to H\phi$ with $\deg \phi>0$. But then, by the continuity of degenerate composition (exactly as in \cite[Lemma 2.6]{DeMarco_JAMS_2007}), we have
$$(C_k\circ f_k \circ A_{n,k}^{-1} ) \circ (A_{n,k} \circ f_k^n) = C_k \circ f_k^{n+1} \to (H_n^d \cdot (H\circ \phi_n)) \; \phi\circ\phi_n.$$
But uniqueness in Lemma \ref{nontrivial limit} then implies that there exists a M\"obius transformation $B = \lim_{k\to\infty} A_{n+1, k}\circ C_k^{-1} $ so that $\phi_{n+1} = B\circ \phi\circ\phi_n$. We set $\phi_{n+1,n} = B\circ \phi$.
\end{proof}
\bigskip
Lemma \ref{composition} implies that the degree of $\phi_n$ may be computed by
\[
\deg \phi_n = \deg \phi_1 \cdot \prod_{j = 1}^{n-1} \deg \phi_{j+1, j}
\]
In particular, $\deg \phi_n \neq o(d^n)$ implies there exists $n_0>0$ so that $\deg \phi_{n+1, n} = d$ for all $n \geq n_0$. For the remainder of the proof, we will operate under this assumption.
Suppose for the moment that there exist nonnegative integers $m > n \geq n_0$ such that
\begin{equation} \label{preperiodic orbit}
A_{n, k} \circ A_{m,k}^{-1} \to L \in \Rat_1 \quad \mbox{ as } k\to\infty
\end{equation}
(after passing to a subsequence, if necessary). From Lemma \ref{composition} and the continuity of composition,
$$A_{n,k} \circ f_k^{m-n} \circ A_{n,k}^{-1} = A_{n,k}\circ A_{m,k}^{-1} \circ A_{m,k} \circ f_k^{m-n} \circ A_{n,k}^{-1} \longrightarrow
L \circ \phi_{m,m-1} \circ \cdots \circ \phi_{n+1,n},$$
and the limiting function has degree $d^{m-n}$. In other words, the sequence of conjugates $A_{n,k} \circ f_k^{m-n} \circ A_{n,k}^{-1}$ will converge in $\Rat_{d^{m-n}}$. But properness of the iteration map $\Rat_d \to \Rat_{d^{m-n}}$ \cite[Corollary 0.3]{DeMarco_Boundary_Maps_2005} implies that the sequence $A_{n,k} \circ f_k \circ A_{n,k}^{-1}$ must also converge uniformly to some rational function $g \in \Rat_d$. The continuity of measures within $\Rat_d$ then implies that $\mu = \lim_{k\to\infty} (A_{n,k}^{-1})_* \mu_g$. The sequence $\{A_{n,k}\}$ must diverge in $\Rat_1$ (because the sequence $\{f_k\}$ diverges in $\Rat_d$), so the limiting measure $\mu$ will be concentrated at a single point.
It remains to treat the case where
$$A_{m, k} \circ A_{n,k}^{-1}$$
diverges in $\Rat_1$ for all $m > n \geq n_0$. A diagonalization argument allows us to assume that the limit exists in $\Ratbar_1$, and we set
$$a_{m,n} := \lim_{k\to\infty} A_{m,k} \circ A_{n,k}^{-1}(p)$$
for all but one point $p$ in $\P^1$, say $p = h_{m,n}$. Recall that we continue to assume that $\deg \phi_n = o(d^n)$ as $n\to\infty$, so there is a constant $0 < \kappa < 1$ such that $\deg \phi_n = \kappa d^n$ for all $n \geq n_0$. We wish to show that $\mu = \lim \mu_k$ is purely atomic. For the sake of a contradiction, we suppose otherwise and write
$$\mu = \nu + \tilde \nu,$$
where $\tilde \nu$ is a countable sum of atoms and $\nu = \mu - \tilde \nu$ is a nonzero positive measure with no atoms. Similarly, write $\mu_{E_n} = \nu^n + \tilde \nu^n$, where $\nu^n$ and $\tilde \nu^n$ are the ``diffuse part'' and the ``atomic part'' of $\mu_{E_n}$, respectively. Applying Theorem~\ref{paired pullback formula} to the $n$th iterates $f_k^n$ and comparing diffuse parts, we find that
\[
\nu = \frac{1}{d^n}\phi_n^*\nu^n \quad \Rightarrow \quad 0 < \nu(\PP^1) = \frac{\deg \phi_n}{d^n} \nu^n(\PP^1) = \kappa \cdot \nu^n(\PP^1)
\]
for all $n \geq n_0$. Hence, there exists $N$ so that
$$\sum_{n = n_0}^N \nu^n(\P^1) \geq 2,$$
Fix a small $\eps > 0$. For each pair $n_0 \leq m,n \leq N$ with $m \neq n$, choose small pairwise disjoint closed disks $D_{m,n}$ and $D'_{m,n}$ around $a_{m,n}$ and $h_{m,n}$, respectively. Let $U$ be the complement of all of these disks in $\PP^1$. Since $\nu^n$ is atomless, by shrinking $D_{m,n}$ and $D'_{m,n}$ as needed we may assume that
\[
\nu^n(U) > \nu^n(\PP^1) - \frac{\eps}{2^n} \qquad (n_0 \leq n \leq N).
\]
Weak convergence of measures $(A_{n,k})_* \mu_k \to \mu_{E_n} = \nu^n + \tilde{\nu}^n$ implies that
$$(A_{n,k})_*\mu_k (U) > \nu^n(\P^1) - \frac{\eps}{2^n}$$
for all sufficiently large $k$ and all $n_0 \leq n \leq N$. (Restricting to finitely many $n$ allows us to do this uniformly.)
For distinct indices $n_0 \leq m, n \leq N$, we have constructed $U$ to be disjoint from $D_{m,n}'$. It follows that $A_{m,k} \circ A_{n,k}^{-1}(U) \subset D_{m,n}$ for all $k \gg 0$, and hence $U \cap (A_{m,k} \circ A_{n,k}^{-1}(U)) = \varnothing$ for all sufficiently large $k$. Therefore, the sets
\[
A_{n_0,k}^{-1}(U), \, A_{n_0+1,k}^{-1}(U), \, \ldots \, , \, A_{N,k}^{-1}(U)
\]
are pairwise disjoint for all $k \gg 0$. (Again, restricting to finitely many sets allows us to do this uniformly.) But then
$$\mu_k (\P^1) \geq \sum_{n=n_0}^N \mu_k\left( A_{n,k}^{-1}(U)\right)
> \sum_{n=n_0}^N \left(\nu^n(\P^1) - \eps/2^n\right) > 2 - \eps > 1,$$
contradicting the fact that $\mu_k$ is a probability measure. This completes the proof of Theorem~A.
\begin{remark}
In the case where the sequence $f_k$ lies in a meromorphic family $f_t$, the condition that $\deg \phi_n \not= o(d^n)$ is characterized in the proof of Proposition \ref{Prop: surplus equidistribution}(2), in terms of dynamics on the Berkovich $\Berk$.
\end{remark}
\bigskip
\section{1-parameter families and complex surfaces}
\label{surface def}
In this section, we carry out Step 1 in the proof of Theorem~B. To start, we consider a meromorphic family $\{f_t \ : \ t \in \mathbb{D}\}$ of rational functions of degree $d \geq 2$ and set up a geometric framework in which to talk about pullback of measures when $t=0$. Under the hypothesis of Theorem~B, the family $f_t$ defines a holomorphic disk in $\overline{\Rat}_d$ with $f_0\in \del\Rat_d$. It is convenient to package the given 1-parameter family into one map on the complex surface $X = \D\times\P^1$, as
$$F: X \dashrightarrow X,$$
defined by $F(t,x) = (t,f_t(x))$ for $t\not=0$. The map $F$ extends to a meromorphic map on the surface $X$ with a finite set of indeterminacy points in the central fiber $X_0 := \{0\}\times \P^1$. The indeterminacy points coincide with roots of the polynomial $H_{f_0}$ defined in \S\ref{space}. On any compact subset of $\P^1\setminus \{H_{f_0} = 0\}$, the functions $f_t$ converge uniformly to the reduction $\phi_{f_0}$ as $t\to 0$.
\subsection{The modified surface $Y$}
There is a unique (up to isomorphism) minimal modification $\pi: Y \to X$, so that the induced rational map
$$F: X \dashrightarrow Y$$
is nonconstant on $X_0$. In other words, either $Y=X$, or we can blow up the image surface $X$ at a unique point of $X_0$ so that $F$ has no exceptional curve. The resulting surface $Y$ may be singular. In coordinates, the existence and uniqueness of $Y$ is immediate from Lemma~\ref{nontrivial limit}: If $(t,z)$ are local coordinates on $X$, then $(t,w)$ are local coordinates on $Y$, where $z = A_t(w)$. Moreover, we see that the central fiber $Y_0$ is reduced.
\begin{figure} [ht]
\includegraphics[width=5in]{surface.pdf}
\caption{The surface map $F: X\dashrightarrow Y$ when the given reduction $\phi_{f_0}$ is constant. }
\label{surface pic}
\end{figure}
We fix a family $A_t$ of M\"obius transformations --- as guaranteed by Lemma~\ref{nontrivial limit} --- such that $A_t \circ f_t$ converges to a point $\Phi \in \Ratbar_d$ with gcd $H$ and reduction $\phi$ of degree $>0$. If the reduction of $f_0$ is nonconstant, we let $A_t$ be the identity for all $t$. Away from its points of indeterminacy, the rational map $F: X\dashrightarrow Y$ coincides with $\phi$ along the central fiber $X_0$. The central fiber $Y_0$ of $Y$ has at most two irreducible components. If $Y\not= X$, we let $E_0$ denote the exceptional curve of the projection $\pi$ and let $C_0$ be the other component of $Y_0$; see Figure~\ref{surface pic}.
\subsection{Pullback of measures from $Y_0$ to $X_0$} \label{surface pullback}
For any Borel probability measure $\mu$ on the central fiber $Y_0$ of $Y$, we can define a measure $F^*\mu$ on the central fiber $X_0$ of $X$ of total mass~$d$. We use the language of paired measures and their pullbacks defined in \S\ref{Sec: Paired measures}. If $Y=X$, we simply set
\begin{equation} \label{pullback X}
F^*\mu := \Phi^* (\mu, \mu) = \, \phi^*\mu + \sum_{x\in\P^1} s(x) \delta_x
\end{equation}
where $(\mu, \mu)$ is a paired measure on two copies of $Y_0 = X_0$. In case $Y \not=X$, recall that the projection $\pi: Y\to X$ collapses $E_0$ to a point. There is also a continuous projection $\pi_E: Y_0 \to E_0$ that collapses $C_0$ to a point. We define,
\begin{equation} \label{pullback Y}
F^*\mu := \Phi^* (\pi_* \mu, (\pi_E)_* \mu) = \, \phi^*\, (\pi_E)_* \mu \, + \sum_{x\in\P^1} s(x) \delta_x .
\end{equation}
Now suppose that $\mu_t$ is a family of probability measures on the fibers $Y_t$ on the surface $Y$. We say $\mu_0$ on $Y_0$ is a weak limit of the measures $\mu_t$ if there is a sequence $t_n \to 0$ so that
$$\int_{Y_{t_n}} \psi \, \mu_{t_n} \to \int_{Y_0} \psi \, \mu_0$$
for every continuous function $\psi$ on $Y$. If $Y=X = \D\times\P^1$, this notion of weak limit agrees with the usual notion for measures on a single $\P^1$. In case $Y\not=X$, it is not hard to see that this notion of convergence coincides with $\{A_{t_n}\}$-weak convergence of $\mu_{t_n}$ to the paired measure $(\pi_* \mu_0, (\pi_E)_*\mu_0)$ on $C_0 \cup E_0 = Y_0$.
We already know that weak limits of maximal measures satisfy a paired measure pullback formula (Theorem~\ref{paired pullback formula}). Translating into our surface framework, we immediately obtain the main result of this section:
\begin{thm} \label{complex pullback}
Any weak limit $\mu_0$ of the maximal measures $\mu_t$ on the central fiber $Y_0$ of $Y$ will satisfy the pullback formula
$$\frac{1}{d} F^*\mu_0 = \pi_* \mu_0$$
on the central fiber $X_0$ of $X$.
\end{thm}
\begin{comment}
\begin{proof}
Let $(t_k)$ be a sequence in $\mathbb{D}$ that converges to $0$ for which $\mu_{t_k} \to \mu_0$. Set $A_k = A_{t_k}$, $f_k = f_{t_k}$, and $\mu_k = \mu_{t_k}$. The proof divides into two cases depending upon the shape of $Y_0$.
\noindent \textbf{Case $Y_0 = X_0$.} Here $A_k(z) = z$ for all $k$ by construction. Evidently $\mu_k$ converges $A$-weakly to the paired measure $(\mu_0, \mu_0)$. By Theorem~\ref{paired pullback formula} and the definition of $F^*$, we find that
\[
\pi_* \mu_0 = \mu_0 = \frac{1}{d} \Phi^*(\mu_0, \mu_0) = \frac{1}{d}F^*\mu_0.
\]
\noindent \textbf{Case $Y_0 \neq X_0$.} For the pair of curves $C_0, E_0$, we take $c_ 0 = e_0 = y_s$. We obtain a paired measure $(\mu_{C_0}, \mu_{E_0})$ (as in \S\ref{Sec: Paired measures}) by setting $\mu_{C_0} := \pi_*\mu_0$, and
\[
\mu_{E_0} := \mu_0\big|_{E_0 \smallsetminus \{y_s\}} + \mu(C_0)\delta_{y_s}.
\]
Observe that $\phi^* \delta_{y_s} = \sum_{x \in X_0} m(x) \eps(x) \delta_x$. Hence,
\begin{align*}
F^* \mu_0 &= \phi^*\left(\mu_0\big|_{E_0} - \mu_s \delta_{y_s}\right) + \sum_{x \in X_0} \left\{s(x) + m(x) \eps(x) \mu_0(C_0) \right\} \delta_x \\
&= \phi^*\left(\mu_0\big|_{E_0} - \mu_s \delta_{y_s} + \mu_0(C_0)\delta_{y_s}\right) + \sum_{x \in \PP^1} s(x)\delta_x
= \Phi^*(\mu_{C_0}, \mu_{E_0}).
\end{align*}
Theorem~\ref{paired pullback formula} immediately implies the desired result if we can show that $(\mu_k)$ converges $A$-weakly to $(\mu_{C_0}, \mu_{E_0})$
Let $\psi_0$ be a continuous function on $X = \mathbb{D} \times \PP^1$ that is constant in the $t$-variable, and let $\psi = \psi_0 \circ \pi$ be the induced function on $Y$. Then weak convergence $\mu_k \to \mu_0$ on $Y$ shows that
\[
\int_{\PP^1} \psi_0(\cdot, x) \ \mu_k(x) = \int_{Y_{t_k}} \psi \ \mu_k \to \int_{Y_0} \psi \ \mu_0
= \int_{C_0} \psi_0(\cdot ,x) \pi_*\mu_0(x),
\]
so that $\mu_k \to \pi_* \mu_0$ weakly in the usual sense. Similarly, we write $\pi': Y \to X'$ for the blow down of the $(-1)$-curve $C_0$; note that $\pi'_* \mu_0 = \mu_{E_0}$. Given a continuous function $\psi'_0$ on $X'$, we set $\psi' = \psi'_0 \circ \pi'$ (a continuous function on $Y$). If $(t,z)$ are local coordinates on $X$, then $(t,z')$ are local coordinates for $X'$, where $z = A_t(z')$. It follows that $A_t^{-1}(z)$ converges to the identity on the central fiber of $X'$. By a uniform continuity argument, we find that $(A_t)_*\psi' = \psi' \circ A_t^{-1} \approx \psi_0$ near the central fiber of $X'$. Hence, for $k$ sufficiently large we have
\begin{align*}
\int_{\PP^1} \psi'_0(0,x) \ A_{k*} \mu_k(x) &\approx \int_{Y_{t_k}} (\psi' \circ A_k^{-1}) \ A_{k*} \mu_k \\
&= \int_{Y_{t_k}} \psi' \ \mu_k
\to \int_{Y_0} \psi' \ \mu_0
= \int_{E_0} \psi'_0(0,x) \ \mu_{E_0}(x).
\end{align*}
Thus $(\mu_k)$ converges $A$-weakly to $(\mu_{C_0}, \mu_{E_0})$.
\end{proof}
\end{comment}
\bigskip
\section{Dynamics and $\Gamma$-measures on the Berkovich projective line}
\label{Sec: Measures}
In this section, we quantize a dynamical system $f$ on the Berkovich projective line and describe the solutions to a system of pullback formulas, thereby completing Step~2 of our program outlined in the introduction. Throughout, we let $k$ be an algebraically closed field of characteristic zero that is complete with respect to a nontrivial non-Archimedean absolute value. Only the case where $k$ has residue characteristic zero is necessary for our application; however, with essentially no extra work, we obtain a more general result. The Berkovich projective line over $k$ will be denoted $\Berk$ for brevity.
\subsection{Vertex sets and measures}
\label{Sec: Gamma}
A \textbf{vertex set} for $\Berk$ is a finite nonempty set of type~II points, which we denote by $\Gamma$. The connected components of $\Berk \smallsetminus \Gamma$ will be referred to as \textbf{$\Gamma$-domains}. When a $\Gamma$-domain has only one boundary point, we call it a \textbf{$\Gamma$-disk}. Write $\SS(\Gamma)$ for the partition of $\Berk$ consisting of the elements of $\Gamma$ and all of its $\Gamma$-domains.
Let $(\Berk, \Gamma)$ be the measurable space structure on $\Berk$ equipped with the $\sigma$-algebra generated by $\SS(\Gamma)$. A measurable function on $(\Berk, \Gamma)$ will be called \textbf{$\Gamma$-measurable}. The space of complex measures on $(\Berk, \Gamma)$ will be denoted $M(\Gamma)$, and we call any such measure a \textbf{$\Gamma$-measure}. We write $M^\ell(\Gamma)$ for the convex subspace of $M(\Gamma)$ consisting of positive measures of volume~$\ell$.
\begin{remark}
A function $\phi: \Berk \to \CC$ is $\Gamma$-measurable if and only if it is constant on subsets of $\SS(\Gamma)$.
\end{remark}
Suppose that $\Gamma \subset \Gamma'$ are two vertex sets. If we write $\pi: \Berk \to \Berk$ for the identity morphism, then $\pi: (\Berk, \Gamma') \to (\Berk, \Gamma)$ is a measurable morphism. In particular, the projection
\begin{equation*} \label{pi_*}
\pi_* : M(\Gamma') \to M(\Gamma)
\end{equation*}
is $\CC$-linear and preserves positivity and volume of measures.
\subsection{Pulling back measures by a rational function}
\label{Sec: pullback}
Throughout this section we assume that $f: \Berk \to \Berk$ is a rational function of degree $d \geq 2$. Suppose that $\Gamma = \{\zeta\}$ is a singleton vertex set, and let $\Gamma' = \{\zeta, f(\zeta)\}$ be a second vertex set. For the applications in this article, we will only need to consider vertex sets of cardinality 1 or 2.
Now we define a pullback map $f^*: M(\Gamma') \to M(\Gamma)$. As a first step, we define certain multiplicities $m_{U,V} \in \{0, 1, \ldots, d\}$ for each $U \in \SS(\Gamma')$ and $V \in \SS(\Gamma)$. If $V = \{\zeta\}$, set $m_{U,V} = m_f(\zeta)$, the usual local degree of $f$ at $\zeta$. For a $\Gamma$-disk $V$, we may write $V = D(\vec{v})$ for some tangent vector $\vec{v} \in T\Berk_{\zeta}$. Set $\bar f (V) = D(Tf(\vec{v}))$. Write $m_f(V)$ and $s_f(V)$ for the directional and surplus multiplicities for $f$ associated to $V$. (See \cite[\S3]{Faber_Berk_RamI_2013}.) By definition, we have
\[
\#\left(f^{-1}(y) \cap V \right) = \begin{cases}
m_f(V) + s_f(V) & \text{if } y \in \bar f (V) \\
s_f(V) & \text{if } y \not\in \bar f(V).
\end{cases}
\]
Here we count each pre-image $x$ with multiplicity $m_f(x)$. Since $\bar f(V)$ is a union of elements of $\SS(\Gamma')$, the function $y \mapsto \#\left(f^{-1}(y) \cap V\right)$ is constant on elements of $\SS(\Gamma')$. For each $U \in \SS(\Gamma')$, define $m_{U,V}$ to be this constant value. The following lemma gives a compatibility relation among the multiplicities~$m_{U,V}$.
\begin{lem}
\label{Lem: Consistent Multiplicities1}
For each $U \in \SS(\Gamma')$, we have
\[
\sum_{V \in \SS(\Gamma)} m_{U,V} = \deg(f).
\]
\end{lem}
\begin{proof}
Choose a point $y \in U$. For each $V \in \SS(\Gamma)$, we have that $m_{U,V} = \#\left(f^{-1}(y) \cap V\right)$. Since $f$ is everywhere $\deg(f)$--to--1, the result follows.
\end{proof}
For a measurable function $\phi: (\Berk, \Gamma) \to \CC$, we define a $\Gamma'$-measurable function $f_*\phi$ by
\begin{equation*}
\label{Eq: pushforward2}
f_*\phi(U) = \sum_{W \in \SS(\Gamma)} m_{U,W} \cdot \phi(W) \qquad (U \in \SS(\Gamma')).
\end{equation*}
Here we have abused notation by writing $f_*\phi(U)$ for the constant value of $f_*\phi$ on $U$, and similarly for $\phi(W)$. Note that the sum defining $f_*\phi(U)$ is finite by Lemma~\ref{Lem: Consistent Multiplicities1}.
If $\phi$ is a bounded $\Gamma$-measurable function, then $\|f_*\phi\| \leq d \|\phi\|$, where we have written $\|\cdot\|$ for the sup norm. For each $\nu \in M(\Gamma')$, the linear functional $\phi \mapsto \int f_*\phi \ \nu$ is bounded, and by duality there exists a $\Gamma$-measure $f^*\nu$ satisfying $\int \phi \ f^* \nu = \int f_*\phi \ \nu$ for all bounded $\Gamma$-measurable functions $\phi$. Evidently $f^*: M(\Gamma') \to M(\Gamma)$ preserves positivity of measures, and Lemma~\ref{Lem: Consistent Multiplicities1} shows that $f^*$ carries $M^\ell(\Gamma')$ into $M^{\ell d}(\Gamma)$ for each $\ell \in \CC$. In particular, $\frac{1}{d}f^*$ maps probability measures to probability measures.
\subsection{The equilibrium and exceptional $\Gamma$-measures}
\label{Sec: distinguished}
For a given rational function $f: \Berk \to \Berk$ of degree $d \geq 2$ and $\Gamma = \{\zeta\}$, there are two distinguished $\Gamma$-measures that will play a key role in our theory.
Write $\mu_f$ for the equilibrium measure on $\Berk$ relative to $f$ \cite{Favre_Rivera-Letelier_Ergodic_2010}. (Another common name in the literature is ``canonical measure'' \cite[\S10]{Baker-Rumely_BerkBook_2010}.) It is the unique Borel probability measure $\nu$ that satisfies $f^*\nu = d \cdot \nu$ and that does not charge classical points of $\Berk$ \cite[Thm.~A]{Favre_Rivera-Letelier_Ergodic_2010}. Here $f^*$ is the usual pullback operator for Borel measures on $\Berk$ --- not the one defined in \S\ref{Sec: pullback}. For a vertex set $\Gamma$, we define the \textbf{equilibrium $\Gamma$-measure} $\omega_{f,\Gamma}$ by the formula
$$\omega_{f, \Gamma}(U) := \mu_f(U)$$
for each $U \in \SS(\Gamma)$. Note that it is supported on a countable subset of $\SS(\Gamma)$.
\begin{lem}
\label{Lem: canonical measure}
Let $f: \Berk \to \Berk$ be a rational function of degree $d \geq 2$, let $\Gamma = \{\zeta\}$ be a singleton vertex set, let $\Gamma' = \{\zeta, f(\zeta)\}$, and let $\pi_*$ and $f^*$ be the operators defined in the previous section. Then
$
\pi_* \omega_{f, \Gamma'} = \omega_{f, \Gamma}$ and $
f^* \omega_{f, \Gamma'} = d \cdot \pi_* \omega_{f, \Gamma'}.
$
\end{lem}
\begin{proof}
The statement about $\pi_*$ is immediate from the definitions.
Let $\phi: \Berk \to \CC$ be a $\Gamma$-measurable function. It is also Borel measurable on $\Berk$ since each element of $\SS(\Gamma)$ is either an open set or a point. The definitions of the multiplicities $m_{U,V}$ show that
\[
f_*\phi(y) = \sum_{f(x) = y} m_f(x) \phi(x) \qquad (y \in \Berk),
\]
which agrees with the formula for the pushforward of Borel measurable functions.
Since $f^* \mu_f = d \cdot \mu_f$ as Borel measures on $\Berk$, we find that
\[
\int \phi \ f^* \omega_{f,\Gamma'} = \int f_* \phi \ \omega_{f,\Gamma'} = \int f_*\phi \ \mu_f = \int \phi \ f^*\mu_f = d \int \phi \ \mu_f = d \int \phi \ \pi_* \omega_{f,\Gamma'}.
\]
Hence $f^*\omega_{f,\Gamma'} = d \cdot \pi_* \omega_{f,\Gamma'}$ as elements of $M(\Gamma)$.
\end{proof}
Suppose now that the rational function $f: \Berk \to \Berk$ has an exceptional orbit $\mathcal{E}$. The \textbf{exceptional $\Gamma$-measure} associated to the orbit $\mathcal{E}$ is defined to be the probability measure $\delta_\mathcal{E} \in M(\Gamma)$ given by
\[
\delta_\mathcal{E}(U) = \frac{ \#\left(\mathcal{E} \cap U\right)}{\# \mathcal{E}} .
\]
\begin{remark}
Recall that an exceptional orbit $\mathcal{E}$ is finite and $f^{-1}(\mathcal{E}) = \mathcal{E}$. Since $k$ has characteristic zero, the function $f$ admits at most two classical exceptional points and at most one exceptional point in $\Berk \smallsetminus \PP^1(k)$ (necessarily of type~II).
\end{remark}
\begin{lem}
\label{Lem: exceptional solutions}
Let $f: \Berk \to \Berk$ be a rational function of degree $d \geq 2$, let $\Gamma = \{\zeta\}$ be a singleton vertex set, and let $\Gamma' = \{\zeta, f(\zeta)\}$. Suppose that $\mathcal{E}$ is an exceptional orbit for $f$. Write $\delta_\mathcal{E}$ and $\delta_{\mathcal{E}}'$ for the associated probability measures with respect to $\Gamma$ and $\Gamma'$, respectively. Then $\pi_* \delta_\mathcal{E}' = \delta_\mathcal{E}$ and $f^* \delta_{\mathcal{E}}' = d \cdot \pi_* \delta_{\mathcal{E}}'$.
\end{lem}
\begin{proof}
Since exceptional measures count the number of exceptional points, we evidently have $\pi_* \delta_\mathcal{E}' = \delta_\mathcal{E}$. For the other equality, let $U \in \SS(\Gamma)$. Then
\[
f^*\delta_\mathcal{E}'(U)
= \sum_{\substack{V \in \SS(\Gamma') \\ V \subset f(U)}} m_{V,U}
\frac{ \#(\mathcal{E} \cap V)}{ \# \mathcal{E}}.
\]
The quantity $m_{V,U}$ is the constant value of $\#\left(f^{-1}(y) \cap U\right)$ for $y \in V$, counted with multiplicities. In particular, if $c \in \mathcal{E} \cap V$, then $m_{V,U} = 0$ or $d$, depending on whether $f^{-1}(c) \cap U$ is empty or not. Note also that $\#(\mathcal{E} \cap U) = \#(\mathcal{E} \cap f(U))$, since $\mathcal{E}$ is a totally invariant set. Hence,
\[
f^*\delta_\mathcal{E}'(U) = \sum_{\substack{V \in \SS(\Gamma') \\ V \subset f(U)}}
d \, \frac{\#(\mathcal{E} \cap V)}{\#\mathcal{E}} = d \, \frac{ \#(\mathcal{E} \cap U)
}{\#\mathcal{E}} = d \cdot \pi_* \delta_\mathcal{E}'(U). \qedhere
\]
\end{proof}
\subsection{Surplus equidistribution and surplus estimates}
We now give two technical results that will be used to prove the main result in the next section. The first is of interest in its own right: it describes how surplus multiplicities of disks behave under iteration. The second gives a lower bound for the mass of a $\Gamma$-disk in terms of its surplus multiplicity.
\begin{prop}[Surplus Equidistribution]
\label{Prop: surplus equidistribution}
Let $f: \Berk \to \Berk$ be a rational function of degree $d \geq 2$ with associated equilibrium measure $\mu_f$. Suppose that the Julia set of $f$ is not equal to $\{\zeta\}$. Let $U$ be an open Berkovich disk with boundary point $\zeta$. Then exactly one of the following is true:
\begin{enumerate}
\item
\label{Item1} The iterated surplus multiplicities of $U$ satisfy
\[
s_{f^n}(U) = \mu_f(U) \cdot d^n + o(d^n).
\]
\item
\label{Item2} The orbit $\OO_f(\zeta)$ converges along the locus of total ramification
to a classical exceptional orbit (of length~1 or~2), and
\[
s_{f^n}(U) = 0 \text{ and } \mu_f\left(f^n(U) \right) = 1 \text{ for all $n\geq 1$.}
\]
\end{enumerate}
\end{prop}
\begin{proof}
The two cases of the proposition are mutually exclusive. For if \eqref{Item2} holds, then $s_f(U) = 0$, so that $f(U) \neq \Berk$. The relation $\mu = \frac{1}{d} f^* \mu$ of Borel measures yields
\[
\mu(U) = \frac{m_f(U)}{d}\mu\left(f(U)\right) = \frac{m_f(U)}{d} > 0.
\]
But then \eqref{Item1} is contradicted.
In the remainder of the proof, let us assume that case \eqref{Item1} of the proposition does not hold. The equilibrium measure $\mu_f$ does not charge~$\zeta$ by hypothesis on the Julia set of $f$. Let $y$ be an arbitrary point of $\Berk$ that is not a classical exceptional point. Using equidistribution of iterated pre-images \cite[Thm.~A]{Favre_Rivera-Letelier_Ergodic_2010}, we find that
\[
\mu_f(U) = \lim_{n \to \infty} \frac{\#\left(f^{-n}(y) \cap U\right)}{d^n} = \lim_{n \to \infty}
\frac{\varepsilon(y, n, U) \cdot m_{f^n}(U) + s_{f^n}(U)}{d^n},
\]
where $\varepsilon(y, n, U) = 1$ if $y \in \overline{f^n}(U)$ and $0$ otherwise. We conclude that $m_{f^n}(U) \neq o(d^n)$; for otherwise, we are in case \eqref{Item1} of the proposition.
Let $\zeta_0 = \zeta$ and set $\zeta_n = f(\zeta_{n-1})$ for each $n \geq 1$. We can write $\vec{v}_n \in T\Berk_{\zeta_n}$ for the tangent vector such that $D(\vec{v}_0) = U$ and $Tf^n(\vec{v}_0) = \vec{v}_n$. Then
\[
m_{f^n}(U) = \prod_{i = 0}^{n-1} m_f(D(\vec{v}_i)).
\]
Each factor in the product is an integer in the range $1, \ldots, d$. If infinitely many of the multiplicities $m_f(D(\vec{v}_i))$ are strictly smaller than $d$, then $m_{f^n}(U) = o(d^n)$. Thus $m_f(D(\vec{v}_n)) = d$ for all $n \gg 0$. As multiplicities are upper semi-continuous, this shows $m_f(\zeta_n) = d$ for all $n$ sufficiently large, so that the orbit $\OO_f(\zeta)$ eventually lies in the locus of total ramification for~$f$.
We now show that $\OO_f(\zeta)$ converges to a classical exceptional orbit. Let $n_0$ be such that $\zeta_n \in \mathcal{R}^{\mathrm{tot}}_f$ for all $n \geq n_0$. The locus of total ramification is connected \cite[Thm.~8.2]{Faber_Berk_RamI_2013}, and any pair of points in $\mathcal{R}^{\mathrm{tot}}_f$ lie at finite hyperbolic distance to each other unless one is a classical critical point. So it suffices to prove that the hyperbolic distance $\rhoH(\zeta_{n_0}, \zeta_n)$ grows without bound. For ease of notation, let us assume that $n_0 = 0$. Since $\zeta_n, \zeta_{n+1} \in \mathcal{R}^{\mathrm{tot}}_f$, the entire segment connecting them must lie in the locus of total ramification as well. Hence $f$ maps $[\zeta_n, \zeta_{n+1}]$ injectively onto $[\zeta_{n+1}, \zeta_{n+2}]$. Moreover, $\rhoH(\zeta_{n+1}, \zeta_{n+2}) = d \cdot \rhoH(\zeta_n, \zeta_{n+1})$. By induction, we see that
\[
\rhoH(\zeta_{n + \ell}, \zeta_{n + \ell + 1}) = d^\ell \cdot \rhoH(\zeta_n, \zeta_{n+1}), \quad \ell = 0, 1, 2, \ldots,
\]
so that the locus of total ramification has infinite diameter. The locus of total ramification has at most two classical points in it; hence, some classical totally ramified point $c$ is an accumulation point of $\OO_f(\zeta)$. By (weak) continuity of $f$, we find $f(c) \in \mathcal{R}^{\mathrm{tot}}_f$. So $c$ is exceptional of period~1 or~2. The orbit $\OO_f(\zeta)$ must actually converge to the orbit of $c$ since the latter is attractive.
Since $f$ has a classical exceptional point, it is conjugate either to a polynomial or to $z \mapsto z^{-d}$. We treat the former case and leave the latter to the reader. Without loss of generality, we now assume that $f$ is a polynomial and that $f^n(\zeta)$ converges to $\infty$ along the locus of total ramification.
As $k$ has characteristic zero, the ramification locus near $\infty$ is contained in a strong tubular neighborhood of finite radius around $(\zeta_{0,R}, \infty)$ for some $R > 1$ \cite[Thm.~F]{Faber_Berk_RamII_2012}. Since hyperbolic distance is expanding on the ramification locus, we see that $f^n(\zeta)$ converges to infinity along the segment $(\zeta_{0,R}, \infty)$.
In particular, since the Julia set of $f$ is bounded away from $\infty$, and since $f$ preserves the ordering of points in $\Berk$ relative to $\infty$, we see that $\zeta$ must lie above the entire Julia set. That is, every segment from a Julia point to $\infty$ must pass through $\zeta$.
Note that if $V$ is a $\Gamma$-disk, then either $V$ does not meet infinity or it does not meet the Julia set (or both). In particular, the surplus multiplicities satisfy $s_{f^n}(U) = 0$ for all $n \geq 1$. Consequently, $U$ must meet the Julia set; else, $\mu_f(U) = 0$ and we are in case~\eqref{Item1}. Observe that $\zeta \in f^n(U)$ for each $n \geq 1$, so that the entire Julia set of $f$ is contained in $f^n(U)$. This shows $\mu_f\left(f^n(U)\right) = 1$, and we are in case~\eqref{Item2} of the proposition as desired.
\end{proof}
\begin{lem}[Surplus Estimate]
Let $f: \Berk \to \Berk$ be a rational function of degree $d \geq 2$, and let $\Gamma = \{\zeta\}$ be a singleton vertex set. Set $\Gamma' = \{\zeta, f(\zeta)\}$. (Note that $\Gamma = \Gamma'$ is allowed.) For any $\Gamma$-disk $U$ and any $\Gamma'$-measure solution $\nu$ to the equation $f^*\nu = d \cdot \pi_* \nu$, we find that
\[
\nu(U) \geq \frac{s_f(U)}{d}.
\]
\end{lem}
\begin{proof}
For ease of notation, let us write $m = m_f(U)$ and $s = s_f(U)$. We may explicitly compute the multiplicities appearing in the pullback operator to be
\[
m_{V,U} =
\begin{cases}
m+s & \text{if $V \subset \bar f(U)$} \\
s & \text{if $V \not\subset \bar f(U)$}.
\end{cases}
\]
Then for $\chi_U$ the characteristic function on the $\Gamma$-disk $U$,
\begin{equation*}
\begin{aligned}
d \cdot \pi_* \nu(U) = f^* \nu(U) &= \int_{\bar f(U)} f_*\chi_U \ \nu
+ \int_{\Berk \smallsetminus \bar f(U)} f_*\chi_U \ \nu\\
&= (m + s) \cdot \nu\left( \bar f(U) \right)
+ s \cdot \nu\left(\Berk \smallsetminus \bar f(U)\right) \\
&= s + m \cdot \nu\left( \bar f(U) \right) \geq s.
\end{aligned}
\end{equation*}
Dividing by $d$ gives the result.
\end{proof}
\subsection{Simultaneous solutions to iterated pullback formulas}
\label{Sec: Main Berkovich Theorem}
The equation $f^* \nu = d \cdot \pi_* \nu$ does not necessarily have a unique solution $\nu \in M^1(\Gamma)$ as one might expect by analogy with the standard setting. However, the solution does become essentially unique if we impose all pullback relations $(f^n)^* \nu = d^n \cdot \pi_{n*} \nu$ for $n = 1, 2, 3, \ldots$
Let $\Gamma = \{\zeta\}$ be a singleton vertex set for $\Berk$. Let $\Gamma_n = \{\zeta, f^n(\zeta)\}$ for each $n \geq 1$, and write $(f^n)^*$ and $\pi_{n*}$ for the pullback and pushforward operators relative to $\Gamma$ and $\Gamma_n$, respectively. We define a set of $\Gamma$-measures $\Delta_f \subset M^1(\Gamma)$ by
$$
\Delta_f = \bigcap_{n \geq 1} \pi_{n*} \left\{ \omega \in M^1(\Gamma_n) :
(f^n)^*\omega = d^n \cdot \pi_{n*} \omega \right\}.
$$
Each element of $\Delta_f$ is the projection of a solution to a pullback formula for each iterate of $f$, although we do not require any compatibility among these solutions. Linearity of the pullback and pushforward operators shows that $\Delta_f$ is a convex polyhedral set in the space $M^1(\Gamma)$. Note that $\Delta_f$ is nonempty: since $\omega_{f,\Gamma} = \omega_{f^n, \Gamma}$, the set $\Delta_f$ must contain the equilibrium $\Gamma$-measure $\omega_{f, \Gamma}$ (Lemma~\ref{Lem: canonical measure}).
\begin{remark}
The intersected sets that define $\Delta_f$ are typically not nested.
\end{remark}
\begin{thm}
\label{Thm: One-vertex Unique}
Let $f : \Berk \to \Berk$ be a rational function of degree $d \geq 2$, and let $\Gamma = \{\zeta\}$ be a singleton vertex set. Suppose that the Julia set of $f$ is not equal to $\{\zeta\}$. With the above notation, $\Delta_f$ is the convex hull of the equilibrium $\Gamma$-measure $\omega_{f,\Gamma}$ and at most one probability measure $\delta_{\mathcal{E}}$ supported on a classical exceptional orbit~$\mathcal{E}$. Moreover, if $\Delta_f \neq \{\omega_{f, \Gamma}\}$, then $f^n(\zeta)$ converges to an exceptional orbit along the locus of total ramification for $f$.
\end{thm}
\begin{remark}
For our application to complex dynamics, it is sufficient to restrict to countably supported measures in the definition of $\Delta_f$. But the theorem shows that this hypothesis is unnecessary: an arbitrary $\Gamma$-measure satisfying all pullback formulas is countably supported.
\end{remark}
\begin{remark}
\label{Rem: Single Vertex p}
With a little more work, one can show that this result continues to hold when $k$ has positive characteristic provided that $\OO_f(\zeta)$ does not converge to a wildly ramified exceptional orbit.
\end{remark}
\begin{cor}
With the hypotheses of Theorem \ref{Thm: One-vertex Unique}, no measure in $\Delta_f$ charges $\zeta$.
\end{cor}
\begin{proof}
The hypothesis on the Julia set guarantees that $\zeta$ is not exceptional and that $\mu_f$ does not charge $\zeta$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Thm: One-vertex Unique}]
Suppose that $f^n(\zeta)$ does not converge along the locus of total ramification to a classical exceptional periodic orbit for $f$. Let $U$ be any $\Gamma$-domain for $\Gamma = \{\zeta\}$. If $\nu \in \Delta_f$, Proposition~\ref{Prop: surplus equidistribution} and the Surplus Estimate applied to $f^n$ and $U$ show that
\[
\nu(U) \geq \frac{s_{f^n}(U)}{d^n} = \mu_f(U) + o(1).
\]
Since this is true for any $\Gamma$-disk $U$, and since $\mu_f$ is a probability measure with no support at $\zeta$, we conclude that $\nu(U) = \mu_f(U)$ for every $U \in \SS(\Gamma)$.
Now suppose that $f^n(\zeta)$ converges along the locus of total ramification to the orbit of a classical exceptional point. Without loss, we may assume that the exceptional point is fixed by replacing $f$ with $f^2$. After conjugating the exceptional fixed point to $\infty$, we may assume that $f$ is a polynomial. As in the proof of Proposition~\ref{Prop: surplus equidistribution}, we find that $f^n(\zeta)$ converges to $\infty$ along the segment $(\zeta_{0,R}, \infty)$ for some $R > 1$, and $\zeta$ lies above the entire Julia set.
Suppose that $U$ is a $\Gamma$-domain that meets the Julia set. Then $f(U)$ contains the entire Julia set, and the standard pullback formula $f^* \mu_f = d \cdot \mu_f$ on $\Berk$ shows that
\begin{equation}
\label{Eq: mu on U}
d \cdot \mu_f(U) = m_f(U) \cdot \mu_f(f(U)) = m_f(U) \quad \Rightarrow \quad \mu_f(U) = \frac{m_f(U)}{d}.
\end{equation}
In particular, only finitely many $\Gamma$-disks may meet the Julia set.
Fix any $\nu \in \Delta_f$. Write $U_\infty$ for the unique $\Gamma$-domain containing infinity; write $U_1, \ldots, U_r$ for the $\Gamma$-domains that meet the Julia set; write $U_0$ for the union of the remaining elements of $\SS(\Gamma)$. Note that since we are in case~\eqref{Item2} of Proposition~\ref{Prop: surplus equidistribution}, the surplus multiplicity satisfies $s_{f^n}(U) = 0$ for all $n \geq 1$ and $U\in\SS(\Gamma)$. Furthermore, we observe that $f(U_\infty) \subset U_\infty$ and $m_{f^n}(U_\infty) = d^n$, and that $f^n$ maps $U_0$ onto $f^n(U_0) \subset U_\infty$ in everywhere $d^n$-to-1 fashion.
First we show that $\nu(U_0) = 0$. For each $n \geq 1$, there exists $\nu_n \in M^1(\Gamma_n)$ such that $(f^n)^* \nu_n = d^n \cdot \pi_{n*} \nu_n = d^n \cdot \nu$. Then
\begin{equation}
\label{Eq: mass at infinity}
d^n \cdot \pi_{n*} \nu_n(U_\infty) = (f^n)^* \nu_n(U_\infty)
= \int (f^n)_*\chi_{U_\infty} \ \nu_n
= d^n \cdot \nu_n\left( f^n(U_\infty) \right).
\end{equation}
Thus $\nu(U_\infty) = \nu_n\left( f^n(U_\infty) \right)$ for any $n \geq 1$. Write $A$ for the annulus with boundary points $\zeta$ and $f^n(\zeta)$. By definition of the pushforward, we see that
\[
\nu(U_\infty) = \pi_{n*} \nu_n(U_\infty) = \nu_n(f^n(U_\infty)) +
\nu_n(f^n(U_0)) + \nu_n(A).
\]
Therefore, $\nu_n(A) = \nu_n(f^n(U_0)) = 0$. But the calculation \eqref{Eq: mass at infinity} applies equally well to $U_0$ to show that $\nu(U_0) = \nu_n\left( f^n(U_0) \right)$, and so we conclude that $\nu(U_0) = 0$.
Next we observe that for $i = 1, \ldots, r$, we have
\begin{equation*}
d^n \cdot \pi_{n*} \nu_n(U_i) = (f^n)^* \nu_n(U_i)
= \int (f^n)_*\chi_{U_i} \ \nu_n
= m_{f^n}(U_i)\cdot \nu_n\left( f^n(U_i) \right).
\end{equation*}
From \eqref{Eq: mu on U}, we see that
\[
\mu_f(U_i) = \frac{m_f(U_i)}{d} = \frac{m_{f^n}(U_i)}{d^n}, \quad n \geq 1, \quad i = 1, \ldots, r.
\]
Combining the last two displayed equations gives
\[
\nu(U_i) = \pi_{n*} \nu_n(U_i) = \mu_f(U_i) \nu_n\left( f^n(U_i) \right).
\]
The quantity $a := \nu_n\left( f^n(U_i) \right)$ is independent of $n$ and $i$ since $\mu_f(U_i) > 0$ for $i = 1, \ldots, r$ and $f^n(U_1) = \cdots = f^n(U_r)$. Setting $b = \nu(U_\infty)$, we have proved that
$\nu = a \cdot \omega_{f, \Gamma} + b \cdot \delta_\infty$.
\end{proof}
\bigskip
\section{A transfer principle}
\label{Sec: Transfer}
In this section, we complete the proof of Theorem~B. We explain the transfer of solutions of the pullback formula for dynamics on the our complex surfaces to $\Gamma$-measure solutions of the pullback formula on~$\Berk$, and vice versa.
\subsection{Reduction and the residual measures} \label{residual measures}
Let $X \to \mathbb{D}$ be a proper fibered surface over a complex disk with generic fiber $\PP^1_\CC$. Assume that the fiber $X_0$ over the origin is reduced. Let $\mathbb{L}$ be the completion of an algebraic closure of $\mathbb{C}(\!(t)\!)$ endowed with the natural non-Archimedean absolute value, and write $\mathbb{L}^\circ$ for its valuation ring. We claim that $X$ gives rise, canonically, to a vertex set $\Gamma \subset \Berk$. The local ring of $\mathbb{D}$ at the origin is contained inside $\mathbb{L}^\circ$, and hence so is its completion. By completing along the central fiber $X_0$ and base extending to $\mathbb{L}^\circ$, we obtain a formal scheme $\XX$ over $\mathbb{L}^\circ$ with generic fiber $\Berk = \Berk_{\mathbb{L}}$. Note that since $X_0$ is reduced, it may be identified with the special fiber $\XX_s$ as $\CC$-schemes. Let
$$\mathrm{red}_X: \Berk \to X_0$$
be the surjective reduction map \cite[2.4.4]{Berkovich_Spectral_Theory_1990}. Let $\eta_1, \ldots, \eta_r$ be the generic points of the irreducible components of the special fiber $X_0$. There exist unique type~II points $\zeta_1, \ldots, \zeta_r \in \Berk$ such that $\mathrm{red}_X(\zeta_i) = \eta_i$ for $i = 1, \ldots, r$. The desired vertex set is $\Gamma = \{\zeta_1, \ldots, \zeta_r\}$.
For each closed point $x \in X_0$, the formal fiber $\mathrm{red}_X^{-1}(x)$ is a $\Gamma$-domain, as defined in \S\ref{Sec: Gamma}. The association $x \mapsto \mathrm{red}_X^{-1}(x)$ induces a bijection between points of the scheme $X_0$ and elements of $\SS(\Gamma)$. We obtain a projection of measures,
$$\mathrm{red}_X^*: M^1(X_0) \to M^1(\Gamma),$$
where $M^1(X_0)$ is the space of Borel probability measures on $X_0(\CC)$ (with its analytic topology) and $M^1(\Gamma)$ is the space of positive $\Gamma$-measures of total mass~1 on $\Berk$, defined as follows. Given $\mu\in M^1(X_0)$, let $B = \{x \in X_0(\CC) : \mu(\{x\}) > 0\}$. The set $B$ is at most countable. Write $\eta_1, \ldots, \eta_r$ for the generic points of the irreducible components $C_1, \ldots, C_r$ of $X_0$. Define $\omega = \mathrm{red}_X^*(\mu)$ by
\[
\omega\left( \mathrm{red}_X^{-1}(x)\right) := \begin{cases}
\mu\left( x \right) & \text{if } x \in X_0(\CC) \\
\mu(C_i \smallsetminus B) & \text{if $x = \eta_i$ for some $i = 1, \ldots, r$}.
\end{cases}
\]
Evidently, $\omega(\Berk) = \mu(X_0) = 1$.
Now let $M^1(\Gamma)^{\dagger} \subset M^1(\Gamma)$ be the subset of $\Gamma$-measures that assign no mass to the elements of $\Gamma$. The reduction map $\mathrm{red}_X$ induces
$$\mathrm{red}_{X*}: M^1(\Gamma)^\dagger \to M^1(X_0)$$
as a partial inverse to $\mathrm{red}_X^*$. Explicitly, the \textbf{residual measure} $\mu = \mathrm{red}_{X*}(\omega) \in M^1(X_0)$ is defined by
$$\mu(\{x\}) := \omega(\mathrm{red}_X^{-1}(x)) \qquad \left(x \in X_0(\CC) \right). $$
For each $\omega \in M^1(\Gamma)^\dagger$, the residual measure $\mu$ is an atomic probability measure on $X_0$.
The terminology is explained by the case where $X_0$ is irreducible and $\Gamma = \{\zeta_{0,1}\}$ is the Gauss point of $\Berk$; the mass of the residual measure at a closed point $x \in X_0(\CC)$ is precisely the volume of the residue class $\mathrm{red}_X^{-1}(x) \subset \Berk$.
\subsection{Compatibility of pullbacks}
Let $f_t$ be a 1-parameter family of dynamical systems of degree $d \geq 2$ with $t$ varying holomorphically in a small punctured disk $\mathbb{D}^*$ and extending meromorphically over the puncture. As in \S\ref{surface def}, we let $X = \mathbb{D} \times \PP^1(\CC)$ and write $\pi: Y \to X$ for the minimal modification of $X$ along $X_0$ such that the induced rational map $F: X \dashrightarrow Y$ is not constant along $X_0$. The surfaces $X$ and $Y$ induce vertex sets $\Gamma = \{\zeta\}$ and $\Gamma' = \{\zeta, f(\zeta)\}$ on $\Berk = \Berk_{\mathbb{L}}$, where $\mathbb{L}$ is the completion of an algebraic closure of $\mathbb{C}(\!(t)\!)$ endowed with the natural non-Archimedean absolute value, and the family $f_t$ defines $f: \Berk \to \Berk$. The pullback $F^*$ from measures on $Y_0$ to measures on $X_0$ is given by the formula (\ref{pullback X}) or (\ref{pullback Y}), depending on whether or not $f$ fixes $\zeta$.
\begin{prop}[Transfer Principle] \label{transfer}
Let $F: X \dashrightarrow Y$, $f:\Berk\to\Berk$, $\Gamma$, and $\Gamma'$ be as above. The following conclusions hold.
\begin{enumerate}
\item If $\mu$ is a measure on the central fiber $Y_0$ such that $F^*\mu = d \cdot \pi_* \mu$, then $\omega = \mathrm{red}_Y^*\mu$ is a $\Gamma'$-measure satisfying $f^*\omega = d \cdot \pi_* \omega$.
\item If $\omega$ is a countably supported $\Gamma'$ probability measure satisfying $\omega(\Gamma') = 0$ and $f^*\omega = d \cdot \pi_* \omega$, then the residual measure $\mu = \mathrm{red}_{Y*} (\omega)$ satisfies $F^*\mu = d \cdot \pi_* \mu$.
\end{enumerate}
\end{prop}
\begin{proof}
We begin by comparing the notions of multiplicity defined for $F$ (on $X_0$) and for $f$ (on $\Berk$). Lemma~\ref{nontrivial limit} gives a meromorphic family of M\"obius transformations $A_t \in \PGL_2(\CC)$ for $t \in \mathbb{D}$, holomorphic away from $t = 0$, such that $A_t \circ f_t$ converges as $t\to 0$ to $\Phi\in \Ratbar_d$ with nonconstant reduction $\phi$. One one hand, this implies that $\phi$ describes the meromorphic map $F$ from the fiber $X_0$ onto its image component $E_0$ in $Y_0$ (or $C_0$ if $X = Y$). Evidently the local degree $m(x)$ for each point of $X_0$ may be read off algebraically as the order of vanishing of $\phi(z) - \phi(x)$ at $x$. On the other hand, we may view $A_t$ as an element $A \in \PGL_2\left(\mathbb{C}(\!(t)\!) \right)$. In particular, $A \circ f$ has nonconstant reduction as a rational function on $\Berk$, and the reduction is equal to $\phi$. If $U_x$ is a $\Gamma$-disk with reduction $x \in X_0$, Rivera-Letelier's Algebraic Reduction Formula \cite[Cor.~9.25]{Baker-Rumely_BerkBook_2010} shows that the directional multiplicity $m_f(U_x)$ is equal to the order of vanishing of $\phi(z) - \phi(x)$ at $x$, and so we conclude that
\[
m(x) = m_f(U_x).
\]
From the description of the surplus multiplicity of the map $\Phi$ in (\ref{complex m and s}) and the corresponding description of the surplus multiplicity in \cite[Lem.~3.17]{Faber_Berk_RamI_2013}, we also see that
\[
s(x) = s_f(U_x).
\]
Finally, the Algebraic Reduction Formula shows that $\deg(\phi) = m_f(\zeta)$, where $\zeta$ is the unique vertex in $\Gamma$.
Since $\omega = \mathrm{red}^*_Y (\mu)$ is supported on countably many $\Gamma'$-domains, to prove the first statement of the Transfer Principle it suffices to show that
\begin{equation}
\label{Eq: To check1}
F^* \mu = d \cdot \pi_* \mu \ \Rightarrow \ f^*\omega(U) = d \cdot \pi_* \omega(U) \text{ for every $U \in \SS(\Gamma)$}. \tag{TP1}
\end{equation}
Under the hypotheses of the second statement of the Transfer Principle, we find that $\mu$ is countably supported. Thus it suffices to show that
\begin{equation}
\label{Eq: To check2}
f^* \omega = d \cdot \pi_* \omega \ \Rightarrow \
F^*\mu(\{x\}) = d \cdot \pi_* \mu(\{x\}) \text{ for every closed point $x \in X_0$}.
\tag{TP2}
\end{equation}
We now prove these statements.
\medskip
\noindent \textbf{Case $Y = X$.}
Let $x \in X_0$ be a closed point, and write $U_x = \mathrm{red}_X^{-1}(x) \in \SS(\Gamma)$. Then
\begin{equation*}
\begin{aligned}
f^*\omega(U_x) &= \sum_{V \in S(\Gamma)} m_{V,U_x} \omega(V) \\
&= m_{\bar f(U_x), U_x} \omega(\bar f (U_x))
+ \sum_{V \in \SS(\Gamma)} s_f(U_x) \omega(V) \\
&= m(x) \mu(\{\phi(x)\}) + s(x) \\
&= F^* \mu(\{x\}),
\end{aligned}
\end{equation*}
while $d \cdot \omega(U_x) = d \cdot \mu(\{x\})$ by definition. This immediately implies \eqref{Eq: To check2}.
To verify \eqref{Eq: To check1}, it remains to consider the mass on $\Gamma = \Gamma'$. Set $B = \{x \in X_0: \mu(x) > 0\}$. If $F^*\mu = d\cdot \mu$, then $F^*\mu$ has no atoms in $X_0\smallsetminus B$. From the definition of $F^*\mu$ in (\ref{pullback X}), we see that $F^*\mu$ agrees with $\phi^*\mu$ on $X_0\smallsetminus B$. Thus,
\begin{equation*}
\begin{aligned}
d \cdot \omega(\zeta) = d\cdot \mu(X_0 \smallsetminus B) &= F^*\mu(X_0 \smallsetminus B) \\
&= \phi^* \mu(X_0 \smallsetminus B) = \deg(\phi) \cdot \mu(X_0 \smallsetminus B) = m_f(\zeta) \omega(\zeta)
= f^*\omega(\zeta).
\end{aligned}
\end{equation*}
This proves \eqref{Eq: To check1} for all $U$ in $\SS(\Gamma)$.
\bigskip
\noindent \textbf{Case $Y \neq X$.} Recall that $Y_0 = C_0 \cup E_0$, where $C_0$ is the proper transform of $X_0$, and $E_0$ is the exceptional fiber of $\pi: Y\to X$. In \S\ref{surface pullback}, to define $F^*\mu$ we introduced the (continuous) projection $\pi_E: Y_0 \to E_0$ that collapses $C_0$ to a point.
Let $U_x$ be the $\Gamma$-disk corresponding to a closed point $x \in X_0$. Recall that we set $\varepsilon(V,U_x) = 1$ or~$0$ depending on whether $\bar f(U_x) = V$ or not. We see that
\begin{equation*}
\begin{aligned}
f^*\omega(U_x) &= \sum_{V \in \SS(\Gamma')} m_{V, U_x} \omega(V) \\
&= \sum_{V \in \SS(\Gamma')} \left(s_f(U_x) + m_f(U_x)
\varepsilon(V, U_x) \right)\omega(V) \\
&= s(x) + m(x) \, (\pi_{E*}\mu) (\{\phi(x)\}) \\
&= F^*\mu\left(\{x\}\right),
\end{aligned}
\end{equation*}
while $\pi_* \omega(U_x) = \pi_* \mu\left(\{x\}\right)$ is immediate. Evidently \eqref{Eq: To check2} follows, and \eqref{Eq: To check1} holds for all $\Gamma$-disks.
To verify \eqref{Eq: To check1}, it remains to check the pullback relation for the mass on vertices. Let $B = \{y \in Y_0 : \mu(\{y\}) > 0\}$, and set $B' = \pi(B \cup E_0) \subset X_0$. Then
$$
d \cdot \pi_* \mu \left(X_0 \smallsetminus B'\right)
= d \cdot \mu \left(\pi^{-1}(X_0 \smallsetminus B')\right)
= d \cdot \mu(C_0 \smallsetminus B). $$
If $F^*\mu = d\cdot \pi_* \mu$, there are no atoms of $F^*\mu$ outside $B'$. From the definition of $F^*\mu$ in (\ref{pullback Y}), the measure $F^*\mu$ must agree with the pullback of $\pi_{E*}\mu$ by $\phi$ on the set $X_0 \smallsetminus B'$; therefore,
$$ F^*\mu(X_0 \smallsetminus B') = \phi^*(\pi_{E*}\mu) (X_0 \smallsetminus B')
= \deg(\phi) \cdot \mu(E_0 \smallsetminus B). $$
Putting these observations together yields
\begin{equation*}
f^*\omega(\zeta) = m_f(\zeta) \omega(f(\zeta))
= \deg(\phi) \mu(E_0 \smallsetminus B)
= d \cdot \mu(C_0\smallsetminus B)
= d \cdot \pi_*\omega (\zeta),
\end{equation*}
so that \eqref{Eq: To check1} is verified for all $U$ in $\SS(\Gamma)$.
\end{proof}
\subsection{Proof of Theorem~B}
\label{Sec: Main Proof}
We retain all of the notation from previous sections.
For each $n \geq 1$, let $F_n: X \dashrightarrow Y^n$ be the rational map of surfaces associated to the $1$-parameter family $f_t^n$ as constructed in \S\ref{Sec: Complex Surface}, and write $\pi_n: Y^n \to X$ for the blowing up morphism. Define
\[
\Delta_0 = \bigcap_{n \geq 1} \pi_{n*} \left\{ \mu \in M^1\left(Y^n_0\right) :
F_n^*\mu = d^n \cdot \pi_{n*} \mu \right\} \subset M^1(X_0).
\]
Write $\omega_{f, \Gamma}$ for the equilibrium $\Gamma$-measure for $f: \Berk \to \Berk$ associated to $\Gamma = \{\zeta_{0,1}\}$. Recall that $\Delta_f$ was defined in \S\ref{Sec: Main Berkovich Theorem}.
\begin{thm}
\label{Thm: Complex solutions}
Let $f_t$ be a meromorphic $1$-parameter family of rational functions of degree $d \geq 2$. Suppose that the family is not holomorphic at $t = 0$; i.e., $\deg(f_0) < d$. The reduction map induces a bijection
\[
\mathrm{red}_X^*: \Delta_0 \simarrow \Delta_f,
\]
with inverse given by the residual measure construction $\mathrm{red}_{X*}$.
\end{thm}
\begin{proof}
No measure in $\Delta_f$ charges the vertex $\zeta_{0,1} \in \Gamma$, and every measure in $\Delta_f$ is countably supported (Theorem~\ref{Thm: One-vertex Unique}). The Transfer Principle (applied to all iterates of $f_t$ and $f$) shows that the maps
\[
\mathrm{red}_X^* : \Delta_0 \to \Delta_f \qquad \text{and} \qquad \mathrm{red}_{X*} : \Delta_f \to \Delta_0
\]
are well defined. That they are inverse to one another follows from the definitions of $\mathrm{red}_X^*$ and $\mathrm{red}_{X*}$.
\end{proof}
\begin{cor}
With the setup of Theorem~\ref{Thm: Complex solutions}, $\Delta_0$ always contains the residual measure $\mathrm{red}_{X*}(\omega_{f,\Gamma})$, and $\Delta_0$ is either a point or a segment in the space of all probability measures. In the latter case, there exists a point mass $\delta_{p_0} \in \Delta_0$ and a $1$-parameter family of exceptional periodic points $p_t$ for $f_t$ such that $f_0$ is constant with value $p_0$, and $p_0$ is not an indeterminacy point for the rational map $F: X \dashrightarrow X$.
\end{cor}
\begin{proof}
Theorem~\ref{Thm: Complex solutions} allows us to transfer the statements about $\Delta_0$ to $\Delta_f$. The first statement is immediate from Theorem~\ref{Thm: One-vertex Unique}. If $\Delta_f \neq \{\omega_{f, \Gamma}\}$, then $f^n(\zeta)$ converges along the locus of total ramification to a classical exceptional orbit $\mathcal{E}$. By replacing $f$ and $f_t$ with their second iterates if necessary, we may assume that $\mathcal{E} = \{p\}$ is a single point. Now $p \in \PP^1(\mathbb{C}(\!(t)\!))$ by completeness. A priori, this gives a formal 1-parameter family $p_t$ with complex coefficients. Since $f_t(p_t) = p_t$ and $\frac{d f_t}{dz}(p_t) \equiv 0$, the implicit function theorem shows $p_t$ is a meromorphic $1$-parameter family in a small disk about $t = 0$. That is, $p = p_t$ is a 1-parameter family of exceptional fixed points for the family $f_t$. Since $f^n(\zeta)$ converges to $p$, and since $p$ is a super attracting fixed point for $f$, it follows that $f_0$ is constant with value equal to $p_0$. If $U$ is the open $\Gamma$-disk containing $p$, then $f(U) \subsetneq U$. In particular, this shows $s_f(U) = s(p_0) = 0$, so that $p_0$ is not an indeterminacy point for the rational map $F$.
\end{proof}
We are now ready to prove the second main result of the article. With the terminology we have set up in the preceding sections, our goal is to show that the family of measures of maximal entropy $\{ \mu_t : t \in \mathbb{D}^*\}$ converges weakly to the residual measure $\mathrm{red}_{X*}(\omega_{f, \Gamma})$ as $t \to 0$, where $\Gamma = \{\zeta_{0,1}\}$ is the Gauss point of $\Berk$.
\begin{proof}[Proof of Theorem~B]
Let $\mu^1$ be any weak limit of the family $\mu_t$ of maximal measures as $t \to 0$ on the surface $Y^1$.
Fix a subsequence $(t_\ell)_{\ell \geq 1}$ so that $t_\ell \to 0$ and $\mu_{t_\ell} \to \mu^1$ weakly on $Y^1$. Set $\mu_0 = \pi_{1*} \mu^1$; then $\mu_{t_\ell} \to \mu_0$ weakly on $X$.
For each $n \geq 2$, let $\mu^n$ be a weak limit of the sequence $(\mu_{t_\ell})$ on the surface $Y^n$. Note that $\mu_0 = \pi_{n*} \mu^n$ by construction. Moreover, we have $F_n^* \mu^n = d^n \cdot \pi_{n*} \mu^n$ for all $n \geq 1$ (Theorem~\ref{complex pullback}). Hence $\mu_0 \in \Delta_0$.
It remains to prove that $\mu_0 = \mathrm{red}_{X*}(\omega_{f, \Gamma})$, the residual measure associated to $\omega_f$ and the vertex set $\Gamma$. This follows immediately from the preceding corollary unless there exists a family of exceptional periodic points $p_t$ for $f_t$, the reduction of $f_0$ is equal to the constant $p_0$, and $p_0$ is not indeterminate for the rational map $F$. In that case, $\mu_0 = a \cdot \mathrm{red}_{X*}( \omega_{f, \Gamma}) + b \cdot \mathrm{red}_{X*}( \delta_\mathcal{E})$, for some $a, b \geq 0$, where $p_0 \in \supp (\mathrm{red}_{X*} (\delta_\mathcal{E}))$. We must prove that $b = 0$.
Since $p_0$ is not indeterminate, by continuity there exists a neighborhood $N$ of $p_0$ such that $f_t(N) \subset N$ for all $t$ sufficiently close to zero. Hence, $N$ is contained in the Fatou set of $f_t$, and $\mu_t$ assigns no mass to $N$. By weak continuity, $\mu_0(N) = 0$. That is, $b = 0$ and $\mu_0 = \mathrm{red}_{X*}(\omega_{f, \Gamma})$ as desired.
\end{proof}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,098
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If you wish to use a database as part of your web site one of the options we can offer is MS SQL server. SQL server offers an enterprise level server based database solution for maximum functionality and performance.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,626
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Gemini 12 var NASA:s 10:e bemannade och sista flygningen i Geminiprogrammet och 16:e bemannade färden totalt. Astronauterna Buzz Aldrin och Jim Lovell flög ombord. Färden genomfördes 11 - 15 november 1966 och varade i 74 timmar 34 minuter och 31 sekunder.
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Källor
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|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,623
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Q: Unable to access FETCH response data i've created an endpoint for a personal api (that is obscured in the code), and it works well, i can fetch correctly and i get the array in console.log
the problem is that i can't access the data to populate a table, i can't figure what is wrong.
my code :
export default class lastExtensions extends Page {
oninit(vnode){
super.oninit(vnode)
this.loading = true
}
oncreate(vnode) {
super.oncreate(vnode);
app.setTitle(app.translator.trans('justoverclock-extiverse-ext-api.forum.pagetitle'));
app.setTitleCount(0);
const settings = {
"async": true,
"crossDomain": true,
"url": "url",
"method": "GET",
"headers": {
"header1": "header1",
"header1-key": "000000000000000000000jsn213b8019cc43"
}
};
$.ajax(settings).done(function (response) {
this.ext = response.slice(0,14);
console.log(this.ext)
this.loading = false;
m.redraw();
})
}
view() {
return (
<div className="lastExtPage">
{IndexPage.prototype.hero()}
<div className="container">
<div className="sideNavContainer">
<nav className="nav IndexPage-nav sideNav">
<ul>{listItems(IndexPage.prototype.sidebarItems().toArray())}</ul>
</nav>
<div className="content sideNavOffset">
<h1 className="lastExtTitle">
{app.translator.trans('justoverclock-extiverse-ext-api.forum.pageTitle')}
</h1>
<p className="pagedescription">{app.translator.trans('justoverclock-extiverse-ext-api.forum.pageText')}</p>
<div className="containerExtensions" id="extList">
{this.ext.map((ext) =>{ // HERE THE PROBLEM
{console.log(ext)}
return (
<tr class="listaEst">
<td class="tdone">{I WANT DATA HERE}</td>
<td class="tdtwo">{I WANT DATA HERE}</td>
<td class="tdthree">{I WANT DATA HERE}</td>
</tr>
)
})}
</div>
</div>
</div>
</div>
</div>
);
}
i can't figure why i can't map and access my fetch response to populate a table...what i'm doing wrong?
this.ext console log is:
[
{
"name": " aaaaaa ",
"desc": "bbbbbbbbb.",
"totalDownload": " 0 downloads ",
"image": "https://aaa.png",
"url": "https://aaa.png"
},
{
"name": " aaaaaa ",
"desc": "bbbbbbbbb.",
"totalDownload": " 0 downloads ",
"image": "https://aaa.png",
"url": "https://aaa.png"
},
{
"name": " aaaaaa ",
"desc": "bbbbbbbbb.",
"totalDownload": " 0 downloads ",
"image": "https://aaa.png",
"url": "https://aaa.png"
}
]
A: I'd try something like this example so you know you're referencing the right this:
oninit(vnode){
...
this.ext = null;
}
setExt = (data) => {
this.ext = data;
}
$.ajax(settings).done(function (response) {
setExt(response.slice(0,14));
...
})
A: Use arrow function, so that 'this' refers to 'this' in outer scope.
$.ajax(settings).done((response) => {
this.ext = response.slice(0,14);
console.log(this.ext)
this.loading = false;
m.redraw();
})
To fix the other problem, initialize the data
oninit(vnode){
super.oninit(vnode);
this.loading = true;
this.ext = [];
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,000
|
Q: How to deny location permission forever after first deny on Android like on IOS in Flutter? I'm building an app that asks permission to access location on different screens, the problem occurs if I deny permission on screen 1 then when I get to screen 2 it still asks permission again. This problem occurs on android and i found out that android will deny forever if denied 2 or more times. How can I make it permanently denied after only 1 first denition? Please tell me the solution. Thanks
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,484
|
\section{Introduction}
\label{sec:Intro}
With no signs as yet of physics beyond the Standard Model (SM), it is essential that measurements at the Large Hadron Collider (LHC) become increasingly precise in the coming years, allowing tests of new SM effects and leading to greater sensitivity to subtle non-SM phenomena. In many cases the limiting factor is a lack of confidence in theoretical calculations, so it is particularly important to find more examples of measurable quantities that are widely agreed to have small theoretical uncertainties.
In this paper we consider production of pairs of electroweak (EW) bosons, collectively referred to as ``diboson processes'' or $pp \to V_1 V_2$, where $V_i = \gamma, W^\pm, Z$, which have by now been an object of study for almost four decades~\cite{Brown:1978mq,Brown:1979ux,Mikaelian:1979nr,Combridge:1980sx,Glover:1988rg,Smith:1989xz,Ohnemus:1990za,Mele:1990bq,Ohnemus:1991gb,Ohnemus:1991kk,Frixione:1992pj,Bailey:1992br,Ohnemus:1992jn,Frixione:1993yp,Dixon:1998py,Campbell:1999ah,Campbell:2011bn}. These processes have been measured individually by the ATLAS and CMS collaborations~\cite{Chatrchyan:2011rr,Chatrchyan:2011qt,Aad:2012tba,Aad:2013izg,ATLAS:2013gma,ATLAS:2013fma,CMS:2013qea,ATLAS:2014xea,CMS:2014xja,Khachatryan:2015kea,Khachatryan:2015sga,ATL-PHYS-PUB-2015-20}. Our goal here is to consider combinations of these measurements.
In the SM the EW bosons originate from a triplet and singlet of $SU(2) \times U(1)$, becoming massive and mixing after EW symmetry breaking. But at the high energies accessible to the LHC, the symmetry breaking effects are moderated, and one might imagine the underlying $SU(2) \times U(1)$ structure might more directly relate diboson processes to one another. It turns out that although this naive expectation is not automatically satisfied, there are nevertheless some elegant and interesting relations.
In this paper we identify numerous independent ratios of diboson measurements that are special at tree level and that offer moderate to excellent potential for both high-precision predictions and high-precision measurements. These ratios, in contrast to the differential cross sections themselves, are flat or slowly-varying as functions of \ensuremath{p_T}\xspace (and other kinematic variables), making them stable against certain experimental problems. Moreover, we expect that many of them receive controllable QCD corrections, especially at high \ensuremath{p_T}\xspace. Electroweak corrections are expected to be important at the 10--20\% level, and may be visible in these ratios, without clutter from large QCD uncertainties. Since the uncertainties on these EW corrections will be small after ongoing calculations are completed, the ratios potentially also offer sensitivity to high-energy beyond-the-Standard-Model (BSM) phenomena. These would include BSM corrections to triple-gauge-boson vertices and broad diboson resonances, though we do not investigate this issue carefully here.
To illustrate these features, we will perform a detailed study of three related ratios, each of which has a different pattern of uncertainties, though only two of the central values are independent. We will show that their special properties survive to higher order, though with an interesting array of subtleties. Specifically we will consider $d\sigma/d\ensuremath{\bar{m}_T}\xspace$ for $\gamma\gamma$, $Z\gamma$ and $ZZ$ at next-to-leading order (NLO), where \ensuremath{\bar{m}_T}\xspace is the average transverse mass of the two vector bosons:
\begin{equation}
\ensuremath{\bar{m}_T}\xspace = \frac{1}{2}\left(\sqrt{p_{T,1}^2 + m_{1}^2} + \sqrt{p_{T,2}^2 + m_{2}^2} \, \right)\,.
\end{equation}
We will discuss issues arising at NNLO, and include the $gg$-initiated loop contribution explicitly. We will give evidence that a number of uncertainties are reduced by taking the various ratios of these three processes, and also argue that experimental technicalities do not interfere with the measurements. The effect of higher-order corrections on our other observables will be studied elsewhere.
The use of ratios of measurements to reduce theoretical and experimental errors has a long history, with perhaps the most famous and successful in particle physics involving the measurements of $R_\text{had} = \sigma(e^+e^- \to \text{hadrons})/\sigma(e^+e^- \to \mu^+\mu^-)$ in the early 1970s. In the study of hadronic decay processes, ratios have long been used to reduce systematic uncertainties from higher-order and non-perturbative corrections (see ref.~\cite{Charles:2004jd} and references therein). These methods have seen continuing use at the LHC, and similar approaches have been extended to the study of Higgs decays in order to better constrain its properties~\cite{Djouadi:2012rh,Goertz:2013eka,Banerjee:2015bla}.
Ratios of production cross sections at hadron colliders have seen more limited use due to the more complex initial state. At the LHC in particular, the use of $d\sigma(\gamma + nj)$ to calibrate the process $d\sigma(Z + nj)$, an irreducible background for many BSM searches, has been investigated at leading order (LO) for $n = 1$~\cite{Ask:2011xf} and NLO for $n = 2$ and 3~\cite{Bern:2011pa,Bern:2012vx}, and implemented in an analysis by the CMS collaboration~\cite{Chatrchyan:2014lfa}. Similar studies have been carried out for ratios of $Z$ and $W^\pm$ processes~\cite{Malik:2013kba}. Moreover, data comparing $Z$ to $\gamma$ production has recently been shown to be in good agreement with theoretical predictions~\cite{Khachatryan:2015ira}, and ratios of single-boson production cross sections have been measured~\cite{Aad:2011dm,ATLAS-CONF-2015-039}, primarily to aid with fits for parton distribution functions. Searches for new colored states in ratios of multijet processes have been proposed in ref.~\cite{Becciolini:2014lya}, while the gradual ramp-up of beam energies at the LHC has also motivated looking at total cross sections of individual processes across a range of energies~\cite{Mangano:2012mh}. More recently it has been argued that a very precise measurement of the top quark Yukawa can be obtained from the ratio of $t\bar t h$ to $t\bar t Z$ production \cite{Plehn:2015cta}.
\section{Executive summary}
\label{sec:ExecSumm}
The restoration of $SU(2) \times U(1)$ well above $m_Z$, along with some happy accidents, leads to some interesting relations among the various diboson partonic differential cross sections. These are obscured once the partonic processes are convolved with parton distribution functions (PDFs), and are affected by experimental realities that impact photons, $W$s and $Z$s differently. Nevertheless, at LO we find numerous ratios of differential cross sections for LHC diboson production that have the potential to be interesting observables.
In \sec{LO} below, we investigate possible diboson variables at LO. We show that diboson processes naturally divide up into three classes:
\begin{equation}
(1)~\gamma\gamma,\, Z\gamma,\, ZZ, \qquad
(2)~W^\pm \gamma,\, W^\pm Z, \qquad
(3)~W^+ W^-.
\end{equation}
(We do not consider same-sign $W^\pm W^\pm$ processes here since extra jets must accompany them.) Each of the first two classes is self-contained, and observables can be built by taking ratios of various differential cross sections. The $W^+W^-$ process can be related to linear combinations of processes in the first two classes, but is more complicated theoretically.
Our observables involve differential cross sections for $V_1 V_2$ production binned in various kinematic variables, which we loosely denote $\sigma(V_1V_2)$ here for brevity. We are interested in symmetric and antisymmetric combinations $\sigma_{S}$ and $\sigma_A$; here the asymmetry is taken with respect to reversing the relative pseudorapidity $\Delta \eta \equiv \eta_1 - \eta_2$ of the two bosons, signed relative to their longitudinal boost direction. (That is, events are weighted by $\text{sign}(y_{12} \Delta \eta)$, where $y_{12} \approx \frac12 (\eta_1 + \eta_2)$ is the diboson rapidity. See \ssec{pdfs} for more details.)
We propose that the following ratios are of interest:\footnote{Although the central values of these observables are not all independent --- for instance $R_{1c} = R_{1b}/R_{1a}$, $R_2^+/R_2^- = C_{2b}/C_{2a}$, $A_2^+/A_2^- = D_{2b}/D_{2a}$ --- the pattern of theoretical and statistical uncertainties is different for each ratio.}
\begin{align}
\displaystyle
&\bullet~~ R_{1a}=\frac{\sigma_S(Z \gamma)}{\sigma_S(\gamma\gamma)}, \quad
R_{1b}=\frac{\sigma_S(ZZ)}{\sigma_S(\gamma\gamma)}, \quad
R_{1c}=\frac{\sigma_S(ZZ)}{\sigma_S(Z \gamma)}, \nonumber \\[5pt]
&\bullet~~ C_{2a} = \frac{\sigma_S(W^+\gamma)}{\sigma_S(W^-\gamma)}, \quad
C_{2b} = \frac{\sigma_S(W^+Z)}{\sigma_S(W^-Z)}, \quad
D_{2a} = \frac{\sigma_A(W^+\gamma)}{\sigma_A(W^-\gamma)}, \quad
D_{2b} = \frac{\sigma_A(W^+Z)}{\sigma_A(W^-Z)}, \nonumber \\[5pt]
&\hspace{7mm} R_{2}^\pm = \frac{\sigma_S(W^{\pm}Z)}{\sigma_S(W^{\pm}\gamma)}, \quad
A_2^\pm = \frac{\sigma_A(W^{\pm}Z)}{\sigma_A(W^{\pm}\gamma)}, \nonumber \\[5pt]
&\bullet~~ R_3 = \frac{\sigma_S(W^+W^-)}{\sigma_S(V_1^0V_2^0)}, \quad
A_3 = \frac{\sigma_A(W^+W^-)}{\sigma_A(WV^0)},
\label{eq:OurRatios}
\end{align}
where $V^0$ denotes $Z$ or $\gamma$, and $\sigma_A(WV^0)$ is some linear combination of $\sigma_A(W^+V^0)$ and $\sigma_A(W^-V^0)$. See \ssec{RatObs} for a more precise discussion of $R_3$ and $A_3$.
In figures~\ref{fig:rat-za-lo}--\ref{fig:ww-lo} of \ssec{RatObs}, these ratios, calculated at LO and binned in $\hat{s}$, are shown. All of the ratios are slowly varying, and each has its own special features. Observables $R_{1a}$, $R_2^\pm$, and $A_2^{\pm}$ are, to first approximation, independent of the PDFs (and hence have very small PDF uncertainties). At LO they depend only on ratios of SM couplings and charges, from which we learn $R_{1a}$ is nearly constant, $R_2^+ \approx R_2^-$, and $A_2^\pm \approx -1$. By contrast, observables $R_{1b}, R_{1c}, C_{2a},C_{2b}, D_{2a}, D_{2b}$ are dominated by the difference between up and down PDFs; all SM couplings cancel in the $C_2$ and $D_2$ ratios. Observables $R_3$ and $A_3$ are more complex.
These observables are simplest for $\sqrt{\hat{s}} \gg 2 m_Z$ or $\ensuremath{\bar{m}_T}\xspace \gg m_Z$, where the difference between the massless $\gamma$ and the massive $W,Z$ is of diminished importance. But as discussed in \ssec{StatUnc}, the low production rates for diboson processes at these high scales, and the low branching fraction for $Z \to \text{leptons}$, gives our observables relatively large statistical uncertainties, potentially negating the value of their low theoretical uncertainties. (In this paper we will only consider leptonic decays of $W$s and $Z$s, though we briefly discuss other options in \ssec{Final}.) At 300 fb$^{-1}$, the $R_{1a}$, $C_{2a}$ and $R_3$ observables can be measured in multiple bins with 5\% statistical uncertainties. This is comparable to the theoretical uncertainties that we will claim below. The variables $R_2^\pm$ and $D_{2a}$ can only be measured in a single bin, making them only marginally useful. At 3000 fb$^{-1}$, it appears all the variables are potentially useful excepting only $D_{2b}$ and $A_2^-$, and with $A_2^+$ marginal.
In \sec{beyondLO}, we study the simplest of these observables, the $R_1$ ratios, beyond LO. As described in \ssec{cuts}, we choose our cuts and our observable carefully to avoid strong jet vetoes, problematic kinematic regions with very large \ensuremath{K}\xspace factors, etc.; see \tab{bosonJetCuts} and \tab{leptonCuts} below. We also include $gg$ production, formally NNLO but numerically important. To fix its normalization, we use the fact that the dominant correction to $gg \to \gamma\gamma$ at the next order is known \cite{Bern:2002jx}. We also use this to normalize the other $gg \to V_1^0 V_2^0$ processes.\footnote{As this paper was nearing completion, a calculation for $gg\to ZZ$ analogous to ref.~\cite{Bern:2002jx} appeared in ref.~\cite{Caola:2015psa}. Our normalization estimate appears to agree with their results.}
In \ssec{NLO-QCD}, we show that many NLO QCD corrections do cancel in these ratios, except for the region where a final-state jet is collinear with a vector boson. There the photon has a collinear singularity which must be regulated with, e.g., a fragmentation function, while the $Z$ singularity is regulated by its mass. Although the ratios shift significantly in this region, we argue in \ssec{PhotonIso} that use of a ``staircase'' isolation method, as in ref.~\cite{Binoth:2010nha,Hance:2011ysa}, leaves small theoretical uncertainties. We also show in \ssec{gg} that $gg \to V_1^0 V_2^0$ causes shifts in the ratios as large as 5--20\% at low $\bar m_T$, due in part to an interesting accidental cancellation in $gg \to Z\gamma$, though these effects are reduced at high \ensuremath{\bar{m}_T}\xspace. Moreover, we argue that the uncertainties on these shifts are small. We also discuss other known NNLO effects on our ratios. Finally, we find in \ssec{pdf-scale} that certain other QCD theoretical uncertainties --- PDF uncertainties and scale uncertainties in particular --- do largely cancel, especially for $R_{1a}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\linewidth]{R1aFig1v4.pdf}\\
\includegraphics[width=0.49\linewidth]{R1bFig1v4.pdf}
\includegraphics[width=0.49\linewidth]{R1cFig1v4.pdf}
\end{center}
\caption{(Top) $R_{1a} = \sigma_S(Z\gamma)/\sigma_S(\gamma\gamma)$. (Left) $R_{1b} = \sigma_S(ZZ)/\sigma_S(\gamma\gamma)$. (Right) $R_{1c} = \sigma_S(ZZ)/\sigma_S(Z\gamma)$. The solid symbols represent our NLO (+ NNLO $gg$) theoretical prediction. Their error bars indicate the expected statistical uncertainties after 300 (3000) fb$^{-1}$ for $R_{1a}$ ($R_{1b}$ and $R_{1c}$). The shaded band around these points represents our estimate of QCD theory uncertainties; see text for important details. The corresponding LO theory prediction is given in open symbols. (By chance, higher-order corrections to $R_{1c}$ nearly cancel.) The bottom plot for each ratio shows the expected fractional correction (relative to unity) from additional non-QCD corrections: an orange solid line for the effect of $Z \to \ell\ell$ decays on the experimental measurement, a blue dashed line for an estimate of the effect of electroweak Sudakov logarithms, with a band indicating its uncertainty, and a horizontal band for the uncertainty from the undetermined choice of $\alpha_\text{QED}$.}
\label{fig:mainResult}
\end{figure}
These statements are summarized in \fig{mainResult}. To explain this figure, let us focus first on the top plot, which shows results for $R_{1a}$, the ratio of $Z\gamma$ to $\gamma\gamma$ differential cross sections with respect to \ensuremath{\bar{m}_T}\xspace, obtained for the 13 TeV LHC. The upper portion of the plot shows the ratio $R_{1a}$ as would be measured in 6 bins of 5--6\% statistical uncertainty; the last bin includes events with \ensuremath{\bar{m}_T}\xspace extending up to the kinematic limit. The open circles indicate a LO prediction, while the closed circles are our result including NLO and $gg$-initiated production. The dominant corrections are driven by the gluon PDF, and decrease with \ensuremath{\bar{m}_T}\xspace. The error bars on the closed circles indicate the expected statistical errors at 300 fb$^{-1}$. The shaded band indicates the theoretical uncertainties mentioned in the previous paragraphs, itemized in \tab{uncBudget} of \sec{Summ} and with all uncertainties combined linearly, except for PDF extraction uncertainties which are combined in quadrature with the others. This combination gives a conservative estimate of \emph{known} uncertainties.
We emphasize that we have not proven it impossible for additional \emph{unknown} sources at NNLO to shift the ratios' central values by larger amounts than our uncertainty estimates. Although we believe we identified all obvious effects that do not cancel in ratios, and have either included them or estimated our uncertainties from not including them, we cannot demonstrate this directly. Only the complete NNLO calculations, for which code is not yet public, will confirm that there are no additional subtleties.
The lower portion of the plot shows estimates of three sources of additional corrections and their uncertainties, \emph{expressed as a relative shift} of the ratio; (i.e.\ 1.05 indicates an upward shift of 5\% on the ratio.) First, as discussed in \ssec{doubleLog}, leading-log EW corrections only partially cancel in the $R_{1}$ ratios. At high \ensuremath{\bar{m}_T}\xspace Sudakov logarithmic effects will dominate and can be roughly estimated using the soft-collinear approximation, as studied in ref.~\cite{Becher:2013zua}. The effect on $R_{1a}$ arises as a difference between the $Z$ and $\gamma$ jet functions, and is of order $5$--$10\%$ at high \ensuremath{\bar{m}_T}\xspace, though this is probably an overestimate.
We show this estimate by plotting the effect on our ratios of the calculation of ref.~\cite{Becher:2013zua} as a blue dashed line, along with an estimate of its uncertainty band as a shaded blue region.
At low \ensuremath{\bar{m}_T}\xspace a finite correction, still relatively small, may make the true EW shift of $R_{1a}$ somewhat larger than indicated by our blue band --- see \cite{Bierweiler:2013dja,Denner:2015fca}, although their cuts are significantly different from ours.
Nevertheless, and more importantly, our uncertainty band is conservative.
The band correctly shows the dominant uncertainty at high \ensuremath{\bar{m}_T}\xspace, from matching the resummed and fixed-order calculations. At small \ensuremath{\bar{m}_T}\xspace the leading uncertainty, from scale variation of the EW couplings, is smaller than the band.
Second, the tan horizontal shaded bar represents an unresolved disagreement in
the community, discussed in \ssec{alphaQED}, regarding the choice of scale
$\mu$ for evaluating $\alpha_{\text{QED}}$ when an on-shell photon is emitted in a
hadronic setting. The difference between using $\mu = 0$ and $\mu = m_Z$
--- for each observable, an overall shift of all the bins by a nearly equal
amount --- is indicated by this bar. This issue is temporary; the
uncertainty will be eliminated once the controversy is settled.
Third, we have chosen to show our results in the upper portion of the figure without including effects from $Z$ decays to leptons. That is, in the figure we applied cuts on the vector bosons but ignored the finite $Z$ width and the kinematic and isolation cuts that must be imposed on the leptons. As we study in \ssec{Zdecay}, these effects, shown as an orange solid line in the lower portion of the figure, do materially change the ratios at the $\sim 5$--$15\%$ level, but with very low uncertainty.
In the other two plots of \fig{mainResult}, we show similar results for $R_{1b}$ and $R_{1c}$, but at 3000 fb$^{-1}$. The increased integrated luminosity is required in order to obtain small statistical errors, because of the small branching fraction of $ZZ$ to four leptons. Both QCD and EW corrections to $R_{1b}$ are larger because the differences between $Z$ and $\gamma$ contribute twice.
We see from \fig{mainResult} that the variables $R_{1a}$, $R_{1b}$ and $R_{1c}$ are nearly flat in \ensuremath{\bar{m}_T}\xspace, are potentially predictable at better than 5\%, and are measurable in several bins (using only leptonic $Z$ decays) at the $\sim 5$--$6\%$ level with 300, 3000 and 3000 fb$^{-1}$ respectively. Corrections to the LO prediction are moderate at low \ensuremath{\bar{m}_T}\xspace and decrease with \ensuremath{\bar{m}_T}\xspace. (In $R_{1c}$ the prediction at higher-order is nearly the same as at LO, due to an accidental cancellation between the $gg$ contribution and other corrections.) Moreover, at 3000 fb$^{-1}$ the $R_{1a}$ ratio can be measured using tens of bins (the precise number depending on \ensuremath{\bar{m}_T}\xspace resolution) with the highest bin starting above 600 GeV, nearly double what is possible at 300 fb$^{-1}$.
At this level of precision, these ratios are potentially sensitive both to interesting soft-collinear EW corrections and to BSM phenomena. We are optimistic that other variables in our list will prove comparably useful, though this remains to be shown in future work.
\section{The story at leading order}
\label{sec:LO}
We begin with a study of diboson processes at tree level, which were first computed at this order almost four decades ago~\cite{Brown:1978mq,Brown:1979ux,Mikaelian:1979nr}. In the form originally presented, the underlying broken gauge and custodial symmetries were not manifest. Making these more explicit, we identify ratios of particular interest. As we will see, each ratio has its own unique features, strengths and weaknesses, even at leading order. We will study these features first at the partonic level, where the $SU(2) \times U(1)$ structure of the rates is most clear. We then use this structure as a guide to construct our ratio observables. Finally we show and explain the behavior of these ratios in proton-proton collisions at 13 TeV. We conclude this section with a short discussion of the statistical uncertainties on these variables at 300 and 3000 fb$^{-1}$ at 13 TeV.
\subsection{High energy limit}
\label{subsec:LO-intro}
Well above the scale of EW symmetry breaking,
we may rewrite the SM EW bosons $W^\pm, Z, \gamma$ as the triplet $w^\pm, w^3$ and singlet $x$ of massless gauge bosons of $SU(2) \times U(1)$, along with the Goldstone scalars $\phi^\pm, \phi^3$. (We use lowercase letters for massless gauge bosons and capital letters for the mass eigenstates.) One basis for the massless diboson states consists, up to normalizations, of $SU(2) \times U(1)$ singlets and triplets:
\begin{align}
\label{eq:xx}
xx_{\bf 1} \equiv xx &: \quad \ket{xx}\,, \\
\label{eq:wx-3}
wx_{\bf 3} \equiv wx &: \quad \ket{w^+x}, \quad \ket{w^3x}, \quad \ket{w^-x}\,, \\
\label{eq:ww-1}
ww_{\bf1} &: \quad \ket{w^+w^-} + \ket{w^-w^+} - \ket{w^3w^3}\,, \\
\label{eq:ww-3}
ww_{\bf3}\, &: \quad \ket{w^+w^3}-\ket{w^3w^+}, \quad \ket{w^+w^-}-\ket{w^-w^+},
\quad \ket{w^3w^-}-\ket{w^-w^3}\,.
\end{align}
There are also quintet $ww$ states, such as $W^+W^+$, but they require two final-state jets at LO, whereas we will focus on production with no jets at LO. This means we only deal at LO with three $SU(2)$-singlet $q\bar{q}$ initial states
\begin{equation}
{\ket{u_R \bar u_R}\,, \quad \ket{d_R \bar d_R}}\,, \quad \ket{u_L\bar u_L} - \ket{d_L\bar d_L}\,,
\end{equation}
and the triplet of states
\begin{equation}
\left\{ \ket{u_L \bar d_L}\,, \quad \ket{u_L\bar u_L} + \ket{d_L \bar d_L}\,, \quad
\ket{d_L \bar u_L} \right\}.
\end{equation}
\begin{figure}[tb!]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,10)
\put(0,0){\includegraphics[width=0.6\linewidth]{trees-eps-converted-to.pdf}}
\end{picture}
\end{center}
\caption{At leading order, diboson processes proceed from $q\bar q$ initial states. The $t,u$ channels (left) and the $s$ channel (right) contribute only to particular amplitudes under $SU(2)\times U(1)$. }
\label{fig:LOgraphs}
\end{figure}
Production rates at LO involve $s$-, $t$-, $u$-channel Feynman diagrams; see \fig{LOgraphs}.
The $s$-channel diagram, with an $f^{abc}$ symbol, only contributes for $ww_{\bf3}$ states. Because of this, the LO production rates for $xx$, $wx$, and $ww_{\bf1}$ are proportional, differing only in the coupling constants.
This suggests that symmetries should exist among the observable cross sections of interest $\sigma(pp \to V_1 V_2)$. To determine the implications more precisely, we must take into account the production of scalars (e.g., the $\phi^3$ inside $Z$), the interference between different channels (e.g., since $W^-\gamma$ is a superposition of $wx$ and $ww_{\bf3}$), and the convolution with PDFs.
Since the quark-scalar couplings are proportional to quark masses, we can neglect scalar production in the $t$- and $u$-channel diagrams, so the scalars contribute only to triplet processes. When final-state scalars do contribute, they do so in the spin-sum of squared helicity-amplitudes, so there are no associated interference effects.
\subsection{Squared amplitudes}
\label{subsec:SquAmp}
The production of dibosons in the limit in which their masses can be neglected can be written in a simple form. We will denote the coupling-stripped LO singlet-, triplet- and scalar amplitudes by
\begin{align}
\label{eq:a1}
a_1 &\propto \mathcal{M}(xx) \propto \mathcal{M}(wx) \propto \mathcal{M}(ww_{\bf1})\,, \\
\label{eq:a3}
a_3 &\propto \mathcal{M}(ww_{\bf3})\,, \\
\label{eq:aphi}
a_\phi &\propto \mathcal{M}(\phi\phi)\,,
\end{align}
in a notation which corresponds to eqs.~\eqref{eq:xx}--\eqref{eq:ww-3}. In these schematic definitions, we leave polarizations implicit since we will always compute spin-averaged cross sections. The three amplitudes in the first line are all proportional, and this continues to hold when one includes NLO QCD corrections but not NLO EW corrections.\footnote{For instance, a virtual $w$ can attach to the final-state lines in $\mathcal{M}(ww_{\bf1})$ but not in $\mathcal{M}(xx)$.}
In the high energy limit, the partonic cross sections of interest $d\hat{\sigma}(q\bar{q} \to V_1 V_2)$ are quadratic in the $a_i$s. The products of $a_i$s that are relevant for diboson production include\footnote{These expressions can be extracted from the high-energy limit of the partonic rates in Eqs.~\eqref{eq:firstXsec}--\eqref{eq:lastXsec} below, which were computed in refs.~\cite{Brown:1978mq,Brown:1979ux,Mikaelian:1979nr}.}
\begin{align}
\label{eq:a1sq}
|a_1|^2 &= \frac{\th}{\hat{u}} + \frac{\hat{u}}{\th} \,, \\
(a_1 a_3) &= \left(\frac{\th - \hat{u}}{2\hat{s}}\right)
+ \frac14 \left(\frac{\th}{\hat{u}} - \frac{\hat{u}}{\th}\right) \,, \\
\label{eq:a3sq}
|a_3|^2 &= \frac{\th\hat{u}}{4\hat{s}^2} - \frac18
+ \frac1{32} \left(\frac{\th}{\hat{u}} + \frac{\hat{u}}{\th}\right) \,, \\
\label{eq:aLsq}
|a_\phi|^2 &= \frac{\th\hat{u}}{4\hat{s}^2}\,.
\end{align}
Here, $(a_1a_3)$ is shorthand for Re$(a_1^\star a_3)$. The $a_i$ amplitudes transform simply under $\th \leftrightarrow \hat{u}$ exchange:
\begin{equation}
\label{eq:fbprops}
a_1(\th,\hat{u}) = a_1(\hat{u},\th), \qquad a_3(\th,\hat{u}) = -a_3(\hat{u},\th), \qquad
|a_\phi(\th,\hat{u})| = |a_\phi(\hat{u},\th)|.
\end{equation}
These properties of $a_1$ and $a_3$, required by Bose statistics and by the fact that $ww_{\bf1}$ ($ww_{\bf3}$) is symmetric (antisymmetric) in the two $w$s,\footnote{Notice that NLO EW corrections break the $\th \leftrightarrow \hat{u}$ symmetry of $\mathcal{M}(wx)$ since a virtual $w$ can attach to the final-state $w$ line but not to the $x$ line.} explain why in eqs.~\eqref{eq:a1sq}--\eqref{eq:aLsq} only $(a_1a_3)$ is antisymmetric under $\th \leftrightarrow \hat{u}$.
The $\th \leftrightarrow \hat{u}$ symmetry properties of the $a_i$s play an important role in what follows. These are forward-backward symmetries, since swapping $\th \leftrightarrow \hat{u}$ in a $q\bar{q} \to V_1 V_2$ event reverses the sign of $\eta_1 - \eta_2$, with $\eta$ defined relative to the $q$'s momentum direction. In what follows, we will use $d \hat{\sigma}_S$ ($d \hat{\sigma}_A$) to denote $\th \leftrightarrow \hat{u}$ symmetrized (antisymmetrized) \emph{partonic} differential cross sections. We will discuss symmetric and antisymmetric \emph{hadronic} cross sections $\sigma_S, \sigma_A$ in \ssec{pdfs}.
One important consequence of \eq{fbprops} is that $a_3$ vanishes at $\th = \hat{u}$, that is, at center-of-mass-frame (CM) scattering angle $\theta = \pi/2$. This ``radiation zero'' has an important impact on the diboson processes.
\subsection{Partonic cross sections at high energies}
\label{subsec:partonic}
Next we write the partonic cross sections for the production of physical dibosons $V_1V_2$, ignoring mass corrections of order $m_Z^2/p_T^2$. Our formulas are written in terms of the $a_i$s given in eqs.~\eqref{eq:a1sq}--\eqref{eq:aLsq}, making various relations among the cross sections manifest and motivating the ratio observables mentioned in \sec{ExecSumm}.
The full formulas including $\ensuremath{O}(m_Z^2/p_T^2)$ terms are given in \app{MassCorr}. There we define $\mathcal{A}_i$s as straightforward generalizations of the $a_i$s including mass corrections. These corrections are subleading in the region of phase space we study in this paper compared to certain QCD corrections, and they introduce no uncertainties. We include them in our numerical results, but have no need to discuss them further. In fact a few useful relations, such as eqs.~\eqref{eq:WV-rel,f+b}--\eqref{eq:WV-rel,f-b}, are unaffected by the boson masses.
\subsubsection{\texorpdfstring{$\gamma\gamma, \,Z\gamma, \,ZZ$}{Diphoton, Z-photon, ZZ}}
\label{subsec:za}
Writing $c_W=\cos\theta_W$ and $s_W=\sin\theta_W$, we have
\begin{align}
\label{eq:A=wx}
\gamma &= c_W \, x + s_W \, w^3 \,, \\
\label{eq:Z=wx}
Z &= c_W \, w^3 - s_W \, x \,,
\end{align}
and $Z$ also contains the scalar $\phi^3$. Pairs of photons and $Z$s can be produced in $xx$, $w^3x$, and $w^3w^3$ channels. Since $w^3w^3$ is orthogonal to the $ww_{\bf3}$ states, the production rates in this sector are all proportional to $|a_1|^2$; see \eq{a1}. Inserting the appropriate coupling constants and writing $V^0=\gamma,Z$,
we have
\begin{equation}
\label{eq:za}
\diff{\hat{\sigma}}{\th}(q\bar{q} \to V^0_1 V^0_2) = \frac{C^q_{12}}{\hat{s}^2} |a_1|^2 \, ,
\end{equation}
where
\begin{align}
\label{eq:Cqaa}
C_{\gamma\gamma}^q &= \frac12 \,\frac{\pi \alpha_2^2 s_W^4}{N_c}\, 2Q^4\,, \\
C_{Z\gamma}^q &= \frac{\pi \alpha_2^2 s_W^2 c_W^2}{N_c}\, \left( L^2 Q^2 + R^2 Q^2 \right)\,, \\
\label{eq:Cqzz}
C_{ZZ}^q &= \frac12 \,\frac{\pi \alpha_2^2 c_W^4}{N_c}\, \left( L^4 + R^4 \right)\,.
\end{align}
Here, a symmetry factor of $1/2$ has been included for identical particles, $\alpha_2$ is the $SU(2)$ coupling of the SM, $Q=T_3+Y$ is the electric charge of quark $q$, and
\begin{equation}
\label{eq:LR}
L = T_3 - Y_L\, t_W^2, \qquad R = -Y_R\,t_W^2,
\end{equation}
with $t_W = s_W/c_W$. The $\ensuremath{O}(m_Z^2/p_T^2)$ corrections to \eq{za} are given in \app{MassCorr}. Each partonic rate in this sector is forward-backward symmetric, so $d\hat{\sigma}_A(V^0_1 V^0_2) = 0$ (though NLO EW corrections give a non-zero $d\hat\sigma_A(Z\gamma)$.)
\subsubsection{\texorpdfstring{$W^{\pm}\gamma, \,W^{\pm} Z$}{W-photon, WZ}}
\label{subsec:wza}
We begin this section by discussing relations among $W^+V^0$ and $W^-V^0$ rates. Since $W^+V^0$ and $W^-V^0$ production are related by $CP$, which takes $u\bar{d} \to W^+V^0$ into $d\bar{u} \to V^0 W^-$, we have (in the notation of \ssec{SquAmp})
\begin{align}
\label{eq:WV-rel,f+b}
d\hat{\sigma}_S(u\bar{d} \to W^+V^0) &= d\hat{\sigma}_S(d\bar{u} \to W^-V^0)\,, \\
\label{eq:WV-rel,f-b}
d\hat{\sigma}_A(u\bar{d} \to W^+V^0) &= -d\hat{\sigma}_A(d\bar{u} \to W^-V^0)\,.
\end{align}
Next we write down the partonic cross sections for producing $W^\pm V^0$. These arise from $w^\pm w^3$ and $w^\pm x$ and involve both $a_1$ and $a_3$, as seen from eqs.~\eqref{eq:wx-3}--\eqref{eq:ww-3} and \eqref{eq:a1}--\eqref{eq:aphi}. Scalar production $a_\phi$ also appears in $W^\pm Z$. In particular,
\begin{align}
\label{eq:wa}
\frac{d\hat{\sigma}}{d\th}(q\bar{q}' \to W^\pm\gamma)
&= \frac{\pi|V_{ud}|^2 \alpha_2^2 s_W^2}{N_c\,\hat{s}^2}
\left[\frac{Y_L^2}{2} |a_1|^2 \pm 2Y_L (a_1a_3) + 4|a_3|^2 \right]\,, \\
\label{eq:wz}
\frac{d\hat{\sigma}}{d\th}(q\bar{q}' \to W^\pm Z)
&= \frac{\pi|V_{ud}|^2 \alpha_2^2}{N_c\,\hat{s}^2} \left[\frac{s_W^2 t_W^2 Y_L^2}{2} |a_1|^2
\mp 2s_W^2 Y_L(a_1a_3) + 4c_W^2|a_3|^2 + \frac12|a_\phi|^2\right] ,
\end{align}
where $q\bar{q}'$ is $u\bar{d}$ ($d\bar{u}$) for $W^+V^0$ ($W^-V^0$). The $\ensuremath{O}(m_Z^2/p_T^2)$ terms in these rates are given in \app{MassCorr}. As seen from \eq{fbprops}, these formulas obey eqs.~\eqref{eq:WV-rel,f+b}--\eqref{eq:WV-rel,f-b}.
Next we compare $W^\pm \gamma$ to $W^\pm Z$. Notice that the forward-backward antisymmetric terms in these two rates, those proportional to $Y_L(a_1a_3)$, are equal but opposite:
\begin{equation}
\label{eq:WAZ-rel,f-b}
d\hat{\sigma}_A(W^\pm\gamma) = -d\hat{\sigma}_A(W^\pm Z)\,.
\end{equation}
These asymmetries arise from the interference between $w^\pm w^3$ and $w^\pm x$ production, a cross term that carries opposite sign for the photon versus the $Z$; see eqs.~\eqref{eq:A=wx}--\eqref{eq:Z=wx}. Alternatively, completeness requires that in the high energy limit,
\begin{equation}
d\hat{\sigma}(W^\pm \gamma) + d\hat{\sigma}(W^\pm Z)
= d\hat{\sigma}(w^\pm x) + d\hat{\sigma}(w^\pm w^3) + d\hat{\sigma}(\phi^\pm \phi^3) \ .
\end{equation}
Since the three terms on the right hand side are respectively proportional to $|a_1|^2$, $|a_3|^2$ and $|a_\phi|^2$, which are forward-backward symmetric, \eq{WAZ-rel,f-b} follows.
The forward-backward symmetric rates in this sector can be read from \eqs{wa}{wz} by omitting the $(a_1a_3)$ terms. Because of the smallness of $Y_L^2 = 1/36$ and the relative factor of $(8\,c_W)^{-1}$ suppressing $|a_\phi|^2$, the $|a_3|^2$ terms naively dominate the cross sections, leading to a ratio $d\hat{\sigma}_S(W^+\gamma)/d\hat{\sigma}_S(W^+Z)$ of $t_W^2 \approx 0.29$.
However, there is a small subtlety with this estimate. We noted earlier that $a_3$, antisymmetric under $\th \leftrightarrow \hat{u}$, has a radiation zero.\footnote{This radiation zero of $a_3$ combines with $a_1$ to give the famous tree-level $f\bar f'\to W\gamma$ radiation zero \cite{Mikaelian:1979nr}, at an angle that depends on the electric charge of $f$.} Nonetheless, the coefficients of $|a_1|^2$ and $|a_\phi|^2$ are small, so this zero is only important very close to $\theta \sim \pi/2$. Moreover, by chance, the ratio of $d\hat{\sigma}_S(W^+\gamma)$ to $d\hat{\sigma}_S(W^+Z)$ is 0.19 at $\theta=\pi/2$, protecting the naive estimate of $t_W^2$ from a large correction. We will say more about this in \ssec{RatObs}.
\subsubsection{\texorpdfstring{$W^- W^+$}{WW}}
\label{subsec:ww}
The partonic amplitude for producing transversely-polarized $W^-W^+$ is a linear combination of $a_1$ and $a_3$ in the high-energy limit. One must also include the contribution $a_\phi$ from scalars $\phi^-\phi^+$, which are produced through an $\hat{s}$-channel $w^3$ or $x$ in $q_L\bar{q}_L$-initiated processes, or through an $\hat{s}$-channel $x$ from $q_R\bar{q}_R$.
In the high energy limit, the partonic cross sections are
\begin{multline}
\label{eq:uuww}
\diff{\hat{\sigma}}{\th}(q\bar{q} \to W^-W^+)
= \frac{\pi\alpha_2^2}{N_c \hat{s}^2} \, \left\{\frac1{16} |a_1|^2 \pm \frac12(a_1a_3) + 2|a_3|^2 \right. \\
+ \left[(t_W^2\,Y_R)^2 + (t_W^2\,Y_L + T_3)^2 \right] |a_\phi|^2 \Big\}\,,
\end{multline}
where the upper (lower) sign holds for $u$-type ($d$-type) quarks. Here $T_3,Y_L,Y_R$ are the quantum numbers of quark $q$.
Note that the forward-backward symmetric rates for transversely polarized $W^-W^+$ are the same in $u\bar{u}$ and $d\bar{d}$ channels, while the forward-backward antisymmetric rates are equal and opposite; that is,
\begin{align}
\label{eq:ww,f+b}
d\hat{\sigma}_S(u\bar{u} \to W_T^- W_T^+) &= d\hat{\sigma}_S(d\bar{d} \to W_T^- W_T^+)\,, \\
\label{eq:ww,f-b}
d\hat{\sigma}_A(u\bar{u} \to W_T^- W_T^+) &= -d\hat{\sigma}_A(d\bar{d} \to W_T^- W_T^+)\,.
\end{align}
These relations are a consequence of $G$-parity (charge conjugation $C$ followed by a rotation by $\pi$ around the second isospin axis) which takes $u\bar{u} \to w^-w^+$ into $d\bar{d} \to w^+w^-$. Indeed, high energy production of $W_T^- W_T^+$ (which in our notation is equivalent to $w^-w^+$) proceeds at LO only through $SU(2)$ interactions, which respect $G$-parity. Alternatively one can derive \eqs{ww,f+b}{ww,f-b} using Clebsch-Gordan coefficients:
\begin{align}
\label{eq:ww|uu}
\mathcal{M}(u\bar{u} \to w^-w^+) &= \frac12 \mathcal{M}(q\bar{q}_{\bf3} \to ww_{\bf3})
+ \frac{1}{\sqrt{6}} \mathcal{M}(q\bar{q}_{\bf1} \to ww_{\bf1}) \,, \\
\label{eq:ww|dd}
\mathcal{M}(d\bar{d} \to w^-w^+) &= \frac12 \mathcal{M}(q\bar{q}_{\bf3} \to ww_{\bf3})
- \frac{1}{\sqrt{6}} \mathcal{M}(q\bar{q}_{\bf1} \to ww_{\bf1})\,.
\end{align}
Squaring these equations and referring to relations \eq{fbprops}, one finds that \eq{ww,f+b} must hold, with $d\hat{\sigma}_S$ given by a linear combination of $|a_1|^2$ and $|a_3|^2$. And since the cross terms have opposite signs, \eq{ww,f-b} follows, with $d\hat{\sigma}_A$ proportional to $(a_1a_3)$.
On the other hand, note that the $Y_L T_3$ terms in $d\hat{\sigma}(u\bar{u} \to \phi^-\phi^+)$ and $d\hat{\sigma}(d\bar{d} \to \phi^-\phi^+)$ are not equal even though they are forward-backward symmetric. These terms arise from an $\hat{s}$-channel $x$ boson, which interacts with the initial-state quarks with couplings that violate $G$-parity. However, these terms are numerically small.
Since $d\hat{\sigma}_A(W^-W^+) \propto (a_1a_3)$, the partonic asymmetry of $W^-W^+$ is proportional to\footnote{But note $d\hat{\sigma}_A(W^-W^+)$ arises as interference between $\mathcal{M}(ww_{\bf3})$ and $\mathcal{M}(ww_{\bf1})$, while $d\hat{\sigma}_A(W^\pm V^0)$ is an interference between $\mathcal{M}(ww_{\bf3})$ and $\mathcal{M}(wx)$. Since NLO EW corrections break the LO relation $\mathcal{M}(ww_{\bf1}) \propto \mathcal{M}(wx)$, they also violate $d\hat{\sigma}_A(W^-W^+) \propto d\hat{\sigma}_A(W^\pm V^0)$. \label{foot:WW-asym}} that of $W^\pm\gamma$ and $W^\pm Z$. Meanwhile the radiation zero of $a_3$ is quite important for $d\hat{\sigma}_S(W^-W^+)$. Later we will see that $|a_1|^2$ actually dominates the $W^-W^+$ cross section, though not overwhelmingly. This motivates comparing $d\hat{\sigma}_S(W^+W^-)$ to $d\hat{\sigma}_S(V_1^0 V_2^0) \propto |a_1|^2$, or perhaps to a linear combination of $d\hat{\sigma}_S(V_1^0 V_2^0)$ and $d\hat{\sigma}_S(WV^0)$.
\subsection{Convolution with PDFs}
\label{subsec:pdfs}
Having discussed the partonic cross sections in detail, we now turn to the observable hadronic cross sections
\begin{eqnarray}
d\sigma(pp \to V_1V_2)
&=& \sum_{q,q'} dx_1 dx_2\, f_{q}(x_1)\,f_{q'}(x_2)\, d\hat{\sigma}(qq' \to V_1 V_2)
\cr
&=& \sum_{q,q'} \frac{d\hat{s}}{s}\, dy\, f_{q}(x_1)\,f_{q'}(x_2)\, d\hat{\sigma}(qq' \to V_1 V_2)\,.
\end{eqnarray}
Here $f_i(x)$ is the PDF of parton $i$, $\hat{s} = x_1 x_2 s$ is the CM energy, and $y = \frac12 \log(x_1/x_2)$ is the rapidity of the partonic collision.
To fully specify an event, kinematic variables describing the final state must be chosen. Since our purpose is to study ratios of different diboson processes, we want variables that keep the different processes on equal footing to the extent possible. One useful variable is $m_{VV}$, the invariant mass of the two bosons; this equals $\sqrt{\hat{s}}$ at LO. Considerations at LO might also suggest the use of the transverse momentum \ensuremath{p_T}\xspace of either boson. However, the threshold value of $\hat{s}$ required to produce the $V_1 V_2$ pair with a given \ensuremath{p_T}\xspace differs among the processes:
\begin{equation}
\label{eq:whymT}
\hat{s}_\text{thresh} = \left(\sqrt{p_T^2+m_1^2} + \sqrt{p_T^2+m_2^2}\right)^2 = 4\ensuremath{\bar{m}_T}\xspace^2\,,
\end{equation}
where \ensuremath{\bar{m}_T}\xspace is the average transverse mass of the two final-state bosons. Since our ratios are simpler if partonic kinematics span the same range in numerator and denominator, the above relation suggests that \ensuremath{\bar{m}_T}\xspace is a more useful kinematic variable than \ensuremath{p_T}\xspace.
The partonic cross sections $d\hat{\sigma}/d\th$ given in \ssec{partonic} can be rewritten in terms of \ensuremath{\bar{m}_T}\xspace as
\begin{equation}
\diff{\hat\sigma}{\ensuremath{\bar{m}_T}\xspace}(qq' \to V_1 V_2)
= \left|\diff{\th}{\ensuremath{\bar{m}_T}\xspace}\right| \, \diff{\hat{\sigma}}{\th}(qq' \to V_1 V_2) \,,
\end{equation}
where, if $m_1 = m_2$ or if both $m_1$ and $m_2$ are negligible,\footnote{The Jacobian is considerably more complicated when $m_1 \neq m_2$.}
\begin{equation}
\left|\diff{\th}{\ensuremath{\bar{m}_T}\xspace}\right| = 2\ensuremath{\bar{m}_T}\xspace \left(1-\frac{4\ensuremath{\bar{m}_T}\xspace^2}{\hat{s}}\right)^{-1/2}.
\end{equation}
The corresponding observable cross section takes the form
\begin{equation}
\label{eq:obsXsec}
\sigma(pp \to V_1 V_2) = \sum_{q,q'} \int \frac{d\hat{s}}{s} \int d\ensuremath{\bar{m}_T}\xspace
\diff{\hat{\sigma}}{\ensuremath{\bar{m}_T}\xspace}(qq' \to V_1 V_2) \int dy\, f_{q}(x_1)\,f_{q'}(x_2)\,,
\end{equation}
where the domain of integration depends on the observable being computed and the kinematic cuts imposed.
The observables we propose in this paper involve the quantities $\sigma_S$ and $\sigma_A$ which we now define. We have already introduced $d\hat{\sigma}_S$ ($d\hat{\sigma}_A$) as the $\th \leftrightarrow \hat{u}$ symmetric (antisymmetric) part of the differential partonic cross section. That is, $d\hat{\sigma}_A(q\bar q \to V_1 V_2)$ weights events by $\text{sign}(\eta_1 - \eta_2)$, while $d\hat{\sigma}_S$ weights events symmetrically with $+1$. At $pp$ colliders, the $q$ direction is unobservable but is typically aligned with the longitudinal boost $y_{12}$ of the diboson system, which at LO is the same as the boost $y$ of the $q\bar{q}$ center-of-mass frame. We may thus define $\sigma_A$ at LO by assigning to events the weight $\text{sign}[y(\eta_1-\eta_2)]$, as in
\begin{equation}
\sigma_X^{\text{LO}}(pp \to V_1 V_2) = \sum_{q_i, \bar{q}_j} \int \frac{d\hat{s}}{s} \int d\ensuremath{\bar{m}_T}\xspace\,
\diff{\hat{\sigma}_X^{\text{LO}}}{\ensuremath{\bar{m}_T}\xspace} (q_i\bar{q}_j \to V_1V_2)\, \mathscr{L}^X_{q_i\bar{q}_j}\,,
\end{equation}
where $X=S,A$ and we have introduced
\begin{equation}
\mathscr{L}^{\{{S,A}\}}_{q_i\bar{q}_j}
= \int dy\, \{{1,\text{sign}(y)}\} \, 2f_{q_i}(x_1) f_{\bar q_j}(x_2)
\end{equation}
as symmetric and antisymmetric parton luminosities. The limits of integration on $y$ depend on $\hat s$ and $\bar m_T$ once cuts are imposed on the pseudorapidity of the bosons.
Triply-differential cross sections would show the relations among the diboson processes most directly, since the PDFs would be evaluated in small $x_1,x_2$ ranges. However, the statistical samples required for binning in all three variables would be far larger than are available at the LHC. To obtain measurements with small statistical errors we must integrate over two variables, namely $y$ and either $\hat{s}$ or \ensuremath{\bar{m}_T}\xspace, and bin in the third variable. Fortunately, even though this involves convolution with the PDFs, many of the good qualities of the partonic relations discussed above survive to $d\sigma/d\ensuremath{\bar{m}_T}\xspace$ and $d\sigma/d\hat{s}$.
In our study of $pp \to V^0_1 V^0_2$ beyond LO in \sec{beyondLO}, we will focus on $d\sigma/d\ensuremath{\bar{m}_T}\xspace$. However, our immediate goal in the remainder of \sec{LO} is to explain heuristically how the ratios of \eq{OurRatios} behave, and to point out their most striking features. In this regard it is most useful to work with the variable $\hat{s} = m_{VV}^2$. The \ensuremath{\bar{m}_T}\xspace and $y$ integrals split cleanly as separate functions of $\hat{s}$; see \eq{sigma,za} below. This feature makes formulas look simpler and permits simple heuristic arguments. Typically the features seen in $d\sigma/d\ensuremath{\bar{m}_T}\xspace$ are nearly the same as those seen in $d\sigma/dm_{VV}$, and moreover survive largely intact to NLO. We will see this for neutral diboson production later.
Of course the above-mentioned separation of \ensuremath{\bar{m}_T}\xspace and $y$ integrals is only formal; it ceases to hold, even at LO, when realistic kinematic cuts are included. Such cuts are always necessary when photons are involved, since production rates diverge as $\ensuremath{p_T}\xspace^\gamma \to 0$. Thus we must introduce a lower bound $(\ensuremath{\bar{m}_T}\xspace)_\text{min}$ when integrating over \ensuremath{\bar{m}_T}\xspace in \eq{obsXsec} to compute an observable rate. In \sec{beyondLO} below we bin with respect to \ensuremath{\bar{m}_T}\xspace, beginning at 200 GeV, so this requirement is automatically satisfied there. But in our heuristic LO discussion, where we bin with respect to $m_{VV}$, we achieve this goal by imposing a cut on pseudorapidity
\begin{equation}
\label{eq:etacut}
|\eta(V)| < 1.5
\end{equation}
for each final state boson $V$; this cut renders the LO cross sections finite. This will not impact our heuristic reasoning but does play a role in the plots shown.
\subsection{Ratio observables}
\label{subsec:RatObs}
We now discuss the ratio observables of \eq{OurRatios}, already mentioned in \sec{ExecSumm}. We will present precise LO results in figures, and we will use schematic or approximate equations to understand the results. In this and following sections, all results are for a 13 TeV $pp$ collider, and are obtained using \textsc{MCFM 6.8}~\cite{Campbell:1999ah,Campbell:2011bn}. The plots of our ratios are given for diboson cross sections without decays and do not include $Z$ or $W$ branching fractions to leptons.
For $V^0_1 V^0_2 = \gamma\gamma,\,Z\gamma,\,ZZ$ we found that all the partonic cross sections are forward-backward symmetric and proportional to the kinematic function $|a_1|^2$. For each of these processes, schematically,\footnote{The lower limit of integration over \ensuremath{\bar{m}_T}\xspace depends on the pseudorapidity cut imposed at $\eta_\text{cut}=1.5$\,. In the $m_Z \to 0$ limit, $(\ensuremath{\bar{m}_T}\xspace)_\text{min} = \sqrt{\hat{s}}/(2\cosh\eta_\text{cut})$. The limits of integration over $y$ in $\mathscr{L}^S_{q\bar{q}}$ also depend on \ensuremath{\bar{m}_T}\xspace, a point we can ignore for the heuristic arguments presented here.}
\begin{equation}
\label{eq:sigma,za}
\diff{\sigma_S}{\hat{s}}(pp \to V^0_1V^0_2)
\sim \frac {\sum_q C^q_{12} \mathscr{L}^S_{q\bar{q}}(\hat{s})}{s\,\hat{s}^2}
\int^{\sqrt{\hat{s}}/2} d\ensuremath{\bar{m}_T}\xspace \left|\diff{\th}{\ensuremath{\bar{m}_T}\xspace}\right| \, |a_1|^2\,,
\end{equation}
where the $C^q_{12}$s were defined in eqs.~\eqref{eq:Cqaa}--\eqref{eq:Cqzz}. Note the numerator of the prefactor is a weighted parton luminosity, with the PDFs weighted by process-dependent couplings and charges. Our observable $R_{1a}$ then satisfies
\begin{equation}
\label{eq:R1a}
R_{1a}(\hat{s})
\equiv \left[\frac{\sigma_S(pp \to Z\gamma)}{\sigma_S(pp \to \gamma\gamma)}\right]_{\hat{s}}
\sim \frac{\sum_q C^q_{Z\gamma} \mathscr{L}^S_{q\bar{q}}(\hat{s})}
{\sum_q C^q_{\gamma\gamma} \mathscr{L}^S_{q\bar{q}}(\hat{s})} \,,
\end{equation}
with similar relations for $R_{1b} = \sigma_S(ZZ)/\sigma_S(\gamma\gamma)$ and $R_{1c} = \sigma_S(ZZ)/\sigma_S(Z\gamma)$.
\begin{table}[tb!]
\begin{center}
\begin{tabular}{|c|c|c|}\hline
$V_1^0V_2^0$ & $C^u_{12} \cdot 10^5$ & $C^d_{12} \cdot 10^5$ \\
\hline\hline
$\gamma\gamma$ & 1.2 & 0.07 \\ \hline
$Z\gamma$ & 2.2 & 0.7 \\ \hline
$ZZ$ & 1.6 & 3.3 \\ \hline
\end{tabular}
\end{center}
\caption{The values of $C^q_{12}$ relevant for the $R_1$ ratios.}
\label{tab:Cq12}
\end{table}
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.5\linewidth]{R1.pdf}
\end{center}
\caption{The $R_1$ ratios of $V_1^0V_2^0$ cross sections at LO, computed in MCFM at a $pp$ collider with $\sqrt{s}=13\text{ TeV}$. A pseudorapidity cut of $|\eta(V)|<1.5$ is imposed. These curves are determined almost entirely by ratios of parton luminosities, weighted by SM couplings.}
\label{fig:rat-za-lo}
\end{figure}
One can then get a rough estimate for the $R_1$ ratios by using \tab{Cq12} and applying the very crude relation $\mathscr{L}^S_{u\bar{u}} \sim 2\mathscr{L}^S_{d\bar{d}}$. The small values of $C^d_{\gamma\gamma}, C^d_{Z\gamma}$ imply that $u\bar{u}$ initial states matter most for $R_{1a}$, and the parton luminosities largely cancel. We may therefore estimate $R_{1a} \sim C^u_{Z\gamma}/C^u_{\gamma\gamma} \sim 1.8$. Including $C^d_{12}$ and the crude relation among parton luminosities, the estimate increases to 2.1. This estimate is very good, as we can see by looking at the actual LO $R_{1a}$ ratio in \fig{rat-za-lo}. For $ZZ$, however, both $u\bar{u}$ and $d\bar{d}$ initial states are important. Although the similarly crude estimates $R_{1b} \sim 2.6$ and $R_{1c} \sim 1.3$ work quite well in the 1--2 TeV range, they are somewhat too small at low $\hat{s}$ because\footnote{Effects from the $Z$ mass, neglected in these estimates, are indeed small, reaching only 3--6\% for $\sqrt{\hat s} \sim 500$ GeV.} $\mathscr{L}^S_{u\bar{u}} < 2\mathscr{L}^S_{d\bar{d}}$ for $\sqrt{\hat{s}} \ll 1\text{ TeV}$. We will see later that NLO QCD makes only minor corrections to these ratios, especially at high energy.
Next, we turn to the observables relating $W^+V^0$ and $W^-V^0$. We know from \eq{WV-rel,f+b} that the partonic cross sections $d\hat{\sigma}_S(W^+V^0)$ and $d\hat{\sigma}_S(W^-V^0)$ are identical. This leads to the following formula for the observable ``charge asymmetry'',
\begin{equation}
C_{2a}(\hat{s})
\equiv \left[\frac{\sigma_S(W^+\gamma)}{\sigma_S(W^-\gamma)}\right]_{\hat{s}}
\sim \frac{\sum_{q_u,q_d} |V_{q_u q_d}|^2 \mathscr{L}^S_{q_u\bar{q}_d}}
{\sum_{q_u,q_d} |V_{q_u q_d}|^2 \mathscr{L}^S_{q_d\bar{q}_u}}\,,
\end{equation}
written as a ratio of weighted parton luminosities, with $V_{ij}$ the CKM matrix. The same result holds for $C_{2b} = \sigma_S(W^+Z)/\sigma_S(W^-Z)$. To derive an expectation for the magnitude and slope of these $C_2$ observables, we use the fact that $W^+V^0$ and $W^-V^0$ are produced predominantly at LO by $u\bar{d}$ and $d\bar{u}$, respectively. Then we have roughly that $C_2 \sim \mathscr{L}^S_{u\bar{d}}/\mathscr{L}^S_{d\bar{u}} \sim f_u/f_d$, which has a magnitude of order 2, grows with energy, and is identical for $W\gamma$ and $WZ$ with negligible mass corrections. These expectations are confirmed in \fig{Qrat-wza-lo}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.48\linewidth]{C2.pdf}
\includegraphics[width=0.48\linewidth]{D2.pdf}
\end{center}
\caption{(Left) The $C_2$ charge ratios at LO, which go roughly like $f_u/f_d$ and are identical for $W\gamma$ and $WZ$. (Right) The $D_2$ variables, also identical for $W\gamma,WZ$. These forward-backward asymmetric charge ratios have a similar dependence on the PDFs, complicated by $\text{sign}(y)$ in the asymmetric parton luminosity which results in $|D_2|>C_2$. }
\label{fig:Qrat-wza-lo}
\end{figure}
Similarly, because $d\hat\sigma_A(W^+V^0)$ and $d\hat \sigma_A(W^-V^0)$ are equal in magnitude and opposite in sign (see \eq{WV-rel,f-b}), we define
\begin{align}
\label{eq:D2}
D_{2a}(\hat s) ~&\equiv~
\left[\sigma_A(W^+\gamma) \over \sigma_A(W^-\gamma)\right]_{\hat s}
~\sim~ {\sum_{q_u,\,q_d} \, |V_{q_uq_d}|^2 {\mathscr L}^A_{q_u\bar q_d}\,
\over - \sum_{q_u,\,q_d} \, |V_{q_uq_d}|^2 {\mathscr L}^A_{q_d\bar q_u}\, }
~\sim~- {{\mathscr L}^A_{u\bar d}\over {\mathscr L}^A_{d\bar u}}\,.
\end{align}
An identical result, with negligible mass corrections, holds for the $WZ$ processes in $D_{2b}$.
As we can see in \fig{Qrat-wza-lo}, $D_2$ has a similar shape to $C_2$, but with opposite sign and somewhat larger magnitude. This can be understood by recalling ${\mathscr L}^A_{q\bar q} = \int dy \ \text{sign}(y) \,2\,f_q(x_1) f_{\bar q}(x_2)$. If the $y<0$ portion of the integral were zero, then we would have $|D_2|= C_2$. Instead, this portion is small, negative, and nearly identical for ${\mathscr L}^A_{u\bar d}$ and ${\mathscr L}^A_{d\bar u}$. The fact that $|D_2|$ is fractionally larger than $C_2$ is merely a consequence of the inequality $(a-\epsilon)/(b-\epsilon)>a/b$ for $a>b>\epsilon>0$.
Now we consider the observables that compare $W^\pm\gamma$ to $W^\pm Z$.
Both $\sigma_A(W^+\gamma)$ and $\sigma_A(W^+Z)$ depend on the same weighted parton luminosity, which appears as the numerator of \eq{D2}. The antisymmetric partonic cross sections are equal in magnitude, opposite in sign, and proportional to $(a_1a_3)$. Everything thus cancels out of their ratio, leaving
\begin{equation}
A_2^+(\hat s) ~\equiv~ \left[{\sigma_A(W^+\gamma) \over \sigma_A(W^+Z)}\right]_{\hat s}
\approx - 1 \ .
\end{equation}
As seen in \fig{Arat-wza-lo}, this ratio differs from $-1$ at low $\hat s$ due to few-percent $m_Z^2/\hat s$ effects.\footnote{In addition to the mass corrections to $(a_1a_3)$ given in \app{MassCorr}, the Jacobian $|d\hat t/d\bar m_T^2|$ and the limits of integration also have mass dependence that differs in numerator and denominator.} The same holds for the $W^-V^0$ processes in $A_2^-$. Since the PDFs are absent, these ratio observables can be computed with relatively low theoretical uncertainty. It is most unfortunate that these ratios have the largest statistical errors, as we will see in \ssec{StatUnc}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.48\linewidth]{R2.pdf}
\includegraphics[width=0.48\linewidth]{A2.pdf}
\end{center}
\caption{
(Left) The $R_2$ ratios, identical for $W^+,W^-$. These ratios are nearly $\tan^2(\theta_W)$, due to the coefficients of $|a_3|^2$ in the partonic rates.
(Right) The $A_2$ ratios at LO, also identical for $W^+$ and $W^-$. These equal $-1$ because partonic forward-backward asymmetries are equal and opposite for $W\gamma$ and $WZ$, which depend on the same PDFs. }
\label{fig:Arat-wza-lo}
\end{figure}
As we discussed at the end of \ssec{wza}, we naively expect
\begin{equation}
R_2^\pm(\hat s) \equiv \left[{\sigma_S(W^\pm\gamma) \over \sigma_S(W^\pm Z)} \right]_{\hat s}
\sim \tan^2 \theta_W \approx 0.29\,.
\end{equation}
The one subtlety is the radiation zero in $a_3$ at $\theta=\pi/2$, which is potentially important because this is the region of phase space where $d\hat\sigma_S/d\bar m_T$ peaks (due to the Jacobian $|dt/d\bar m_T|$). However, as seen in \fig{Arat-wza-lo}, the above estimate is a good one. The reason is a combination of two pieces of good fortune. The first is that the ratio of the partonic amplitudes everywhere lies between 0.29 and 0.19\,. Since $|a_1|^2=2$ and $|a_\phi|^2=1/16$ at $\theta=\pi/2$, we see from \eqs{wa}{wz} that
\begin{align}
{d\hat\sigma_S}(u \bar d \to W^+\gamma) \propto {s_W^2\,Y_L^2}\,, \qquad
{d\hat\sigma_S}(u\bar d \to W^+Z) \propto s_W^2\,t_W^2\,Y_L^2 + {1 \over 32}\,,
\end{align}
which means $d\hat\sigma_S(W^+\gamma)/d\hat\sigma_S(W^+Z)\to 0.19$ there. The second is that the coefficients of $|a_1|^2$ and $|a_\phi|^2$ are so small that $|a_3|^2$ is numerically very important despite its radiation zero.
This last statement is not true for $W^-W^+$; from \eq{uuww}, the relative coefficient of $|a_1|^2$ is 1/32, {\it vs.} $Y_L^2/8$ in $W\gamma$. Consequently $d \sigma_S(W^-W^+)$ is dominated by the singlet term, making it roughly proportional to $d\sigma_S(V^0_1V^0_2)\sim |a_1|^2$. This leads us to consider ratios such as
\begin{equation}
R_{3a}(\hat s) \equiv \left[{\sigma_S(W^-W^+) \over \sigma_S(\gamma\gamma)} \right]_{\hat s}\,,
\end{equation}
and similarly $R_{3b} = \sigma_S(W^-W^+)/\sigma_S(Z\gamma)$ and $R_{3c} = \sigma_S(W^-W^+)/\sigma_S(ZZ)$. These possibilities are displayed in \fig{ww-lo}. We can estimate their magnitudes just as we did for the $R_{1}$ ratios above. Comparing the coefficients of $|a_1|^2$ in \eqs{za}{uuww}, referring to \tab{Cq12}, and using the crude relation $\mathscr L^S_{u\bar u} \sim 2\,\mathscr L^S_{d \bar d}$, we get an estimate
\begin{equation}
R_{3a} \sim {{1 \over 16}\left(\mathscr L^S_{u\bar u} + \mathscr L^S_{d\bar d}\right) \over s_W^4\,Q_u^4\,\mathscr L^S_{u\bar u}} \sim 10\,.
\end{equation}
Similar estimates for $R_{3b}$ and $R_{3c}$ then follow from the $R_1$ ratios in \fig{rat-za-lo}.
\begin{figure}[tb!]
\begin{center}
\includegraphics[width=0.48\linewidth]{R3.pdf}~~
\includegraphics[width=0.48\linewidth]{A3.pdf}
\end{center}
\caption{(Left) Possibly useful $R_3$ variables involving $W^-W^+$. Ratios are taken with $V_1^0V_2^0$ processes because $W^-W^+$ is dominantly produced as an $SU(2)$-singlet at LO. (Right) Possible $A_3$ variables involving forward-backward asymmetric $W^-W^+$ production. A property of the PDFs explains the flatness of the lower curve.}
\label{fig:ww-lo}
\end{figure}
Although these estimates are not wildly off, they do come up somewhat short, even after allowing for $\mathscr L^S_{u\bar u} < 2 \, \mathscr L^S_{d \bar d}$ at low $\hat s$. This is because we cannot actually ignore the $|a_3|^2$ contribution to $\sigma(W^-W^+)$, which makes up about 20\% of the total cross section. Because of this, one may be led to include some admixture of $\sigma_S(W^\pm V^0)$ in the denominators of the $R_3$ ratios. We leave it to further study to decide which admixture would have the most desirable properties at NLO.
Finally, we turn to ratios involving $\sigma_A(W^-W^+)$. As we saw earlier, the leading order partonic asymmetry in $W^-W^+$ is proportional to $(a_1a_3)$, as was the case for $W^\pm\gamma$ and $W^\pm Z$ (but see footnote \ref{foot:WW-asym}). We therefore expect that a ratio of $d\sigma_A(W^-W^+)$ to any linear combination of the $d\sigma_A(W^\pm V^0)$ is given by a ratio of parton luminosities weighted by SM coefficients. The asymmetries in $W^\pm Z$ suffer from low statistics, so we consider linear combinations of $d\sigma_A(W^+\gamma)$ and $d\sigma_A(W^-\gamma)$:
\begin{equation}
\label{eq:A3}
A_3(\hat s) \equiv \left[{\sigma_A(W^-W^+) \over a\,\sigma_A(W^+\gamma) + b\,\sigma_A(W^-\gamma)} \right]_{\hat s} \sim {\mathscr L^A_{u\bar u} - \mathscr L^A_{d\bar d} \over 4\,|V_{ud}|^2\,s_W^2\,Y_L\,\left(a\,\mathscr L^A_{u\bar d} - b\,\mathscr L^A_{d\bar u}\right)}\,.
\end{equation}
It is an interesting non-obvious feature of the PDFs that, as functions of $\hat s$ in the kinematic region of interest,
\begin{equation}
\mathscr L^A_{u\bar u} ~\propto~ \mathscr L^A_{d\bar d} ~\propto~ \mathscr L^A_{u\bar d}+ \mathscr L^A_{d\bar u}\,;
\end{equation}
the first (second) relation holds at the 2\% (15\%) level. This suggests the use of $a=-b=1$, which has the further advantage of minimizing the relative statistical uncertainty in the denominator of \eq{A3}. Whether this is the ideal choice after NLO corrections are included remains to be seen. We can see in \fig{ww-lo} that, at LO, this choice leads to a much flatter and smaller ratio than the choice $a=b=1$.
\subsection{Limitations of finite statistics}
\label{subsec:StatUnc}
Attractive as these ratios are, the reality of low cross sections means that many of these observables are not useful in the near term. In \tab{numEvents} we show a rough estimate of the number of high-energy events ($m_{VV} > 400$ GeV) expected for each process. We assume 300 fb$^{-1}$ at $\sqrt{s}=13$ TeV and account for leptonic branching fractions of the $Z$ and $W$. We computed these numbers imposing a pseudorapidity cut $|\eta(V)|<1.5$ on the bosons (as in \tab{bosonJetCuts} below), and have separated events into ``Forward'' and ``Backward'' by the sign of $y(\eta_1-\eta_2$) as described in \ssec{pdfs}.
Any one of our ratios becomes interesting as a precision observable once its statistical uncertainty becomes of order 5--10\%, so that its exceptionally low theoretical errors become experimentally relevant. If such small uncertainties are possible for a particular ratio only by combining all events together into a single bin, e.g.~using ratios of total cross sections with $\ensuremath{\bar{m}_T}\xspace > 200\text{ GeV}$, then this measurement is likely to be useful only for testing methods for SM predictions, since it will be sensitive mainly to physics only up to the $\sqrt{\hat s} \sim 400$ GeV range. However, more can be done once the events can be divided into multiple bins of varying width, each with statistical uncertainty of order 5--10\%, as in \fig{mainResult}. In this case the lower bins serve as a test of the predictive techniques, while the higher ones are useful for other purposes, including searches for BSM phenomena and tests of important EW corrections that grow with energy and do not entirely cancel in these ratios.
\begin{table}
\begin{center}
\begin{tabular}{| c | c | c |}\hline
$V_1V_2$ & $N_f+N_b$ & $N_f-N_b$ \\ \hline\hline
$\gamma\gamma$ & 12\,000 & 0 \\
$Z\gamma$ & 2000 & 0 \\
$ZZ$ & 220 & 0 \\
$W^+\gamma$ & 3300 & $-500$ \\
$W^-\gamma$ & 2100 & 220 \\
$W^+Z$ & 790 & 33 \\
$W^-Z$ & 520 & $-16$ \\
$W^-W^+$ & 9500 & $-430$ \\\hline
\end{tabular}
\end{center}
\caption{At LO, the number of events with $\sqrt{\hat s} = m_{VV} > (400\text{ GeV})^2$ assuming 300 fb$^{-1}$ and leptonic decays of $W$ and $Z$. $N_f$ and $N_b$ indicate forward- and backward events. These numbers increase by a factor of order 1.5--2 at NLO, but are reduced by a comparable amount when using the variable $\bar m_T$ instead of $m_{VV}$.}
\label{tab:numEvents}
\end{table}
As we saw in \fig{mainResult} of \sec{ExecSumm}, the ratio $R_{1a}$ permits 6 bins at 300 fb$^{-1}$ with 6\% statistical uncertainties. At this integrated luminosity, the other variables that allow multiple bins with $\sim$5\% uncertainties are $C_{2a}$ and $R_3$, as one can see using \tab{numEvents}. Meanwhile $R_2^\pm$, $C_{2b}$, and $D_{2a}$ allow for a single bin.
The situation will improve at 3000 fb$^{-1}$, though the high pileup environment may lead to some loss of statistics. If we simply assume the total rate increases by a factor of 10 without significant losses, we find that in addition to the above six variables, the variables $R_{1b}$, $R_{1c}$ and $A_3$ also permit multiple bins. The $A_2^+$ ratio can be used in a single bin. The two variables $A_2^-$ and $D_{2b}$ involving $\sigma_A(W^-Z)$ are too small to measure.
It may prove useful to improve statistics slightly by combining observables predicted to be equal within the SM. For instance, one could replace $R_2^+$ and $R_2^-$ with
\begin{equation}
R_2^{0}=
\frac{\sigma_S(W^+Z)+\sigma_S(W^-Z)}{\sigma_S(W^+\gamma)+\sigma_S(W^-\gamma)}.
\end{equation}
Similar combinations would assist with $A_2^+$ and $A_2^-$ (see \fig{Arat-wza-lo}), $C_{2a}$ and $C_{2b}$, and $D_{2a}$ and $D_{2b}$ (see \fig{Qrat-wza-lo}).
\section{Beyond leading order for \texorpdfstring{$\gamma\gamma$, $Z\gamma$, $ZZ$}{diphoton, Z-photon, ZZ}}
\label{sec:beyondLO}
In \ssec{partonic} we saw that the differential LO partonic cross sections for $V_1^0V_2^0 = \gamma\gamma$, $Z\gamma$, $ZZ$ are all proportional to the same function $|a_1|^2$, up to $m_Z^2/\ensuremath{p_T}\xspace^2$ effects (provided in \app{MassCorr}). Consequently, at high energy, the ratios of these partonic cross sections are given by constants of the SM. Since the up quark PDF dominates $\gamma\gamma$ and largely dominates $Z\gamma$, the hadronic ratio $R_{1a}$ is approximately constant and equal to a simple partonic ratio. Although the PDFs have a greater effect on the hadronic cross sections for $R_{1b}$ and $R_{1c}$, these two observables still vary rather slowly with $\sqrt{\hat{s}}=m_{VV}$, with easily understandable values, as we saw in \fig{rat-za-lo}.
Beyond LO, we will study the $R_{1}$ ratios differentially with respect to $\bar{m}_T$, the average $m_T$ of the two vector bosons, \eq{whymT}. The LO ratios in this variable are given in \fig{mtRatio}.
Comparing with \fig{rat-za-lo}, one can see that the LO ratios as functions of $\bar m_T$ and as functions of $m_{VV}/2$ are quite similar. This is because the hadronic cross section for a given $\bar m_T$ is dominated by the region with $\sqrt{\hat s}\sim 2\bar m_T$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\linewidth]{R1mt.pdf}
\end{center}
\caption{The $R_1$ ratios at LO binned in $\bar m_T$.}
\label{fig:mtRatio}
\end{figure}
The fully-differential cross-sections for the diboson processes have been known for quite some time~\cite{Smith:1989xz,Ohnemus:1990za,Mele:1990bq,Ohnemus:1991gb,Ohnemus:1991kk,Frixione:1992pj,Bailey:1992br,Ohnemus:1992jn,Frixione:1993yp}. In this and following sections, all calculations are carried out using \textsc{MCFM 6.8}~\cite{Dixon:1998py,Campbell:1999ah,Campbell:2011bn}, except for an NNLO real emission study which used \textsc{MadGraph 2.3.0}~\cite{Alwall:2014hca}. Renormalization and factorization scales $\mu_R,\mu_F$ are chosen at $m_{VV}$ except when otherwise specified. We use MSTW 2008 NLO [LO] PDFs~\cite{Martin:2009iq} for all NLO [LO] calculations and for $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ [$\ensuremath{O}(\ensuremath{\alpha_S}\xspace^2)$] $gg\to V_1^0V_2^0$ calculations. Our cuts on the bosons are presented in \tab{bosonJetCuts}.
See \sec{Practical} for cuts on their decay products.
\subsection{Choices of observable and of cuts}
\label{subsec:cuts}
We begin with a discussion of our cuts and our observable. It is important to choose these carefully in order to avoid large NLO and NNLO corrections to our ratios, and associated large uncertainties.
We will discuss certain experimental realities in \sec{Practical}, but for now we neglect $Z$ decay and impose cuts on the vector bosons and on any jets,\footnote{We will refer to all final-state colored partons, for brevity only, as ``jets''. We do not include showering and hadronization in our study, but we expect these to have small effects, since we impose cuts on our observables to avoid regions where resummation plays an important role.} as in \tab{bosonJetCuts}. (Our cuts on leptons in $Z \to \ell^+\ell^-$ are given in \tab{leptonCuts} of \ssec{Zdecay}.) In our discussion we will have at most one jet and so for us $H_T$ is simply the \ensuremath{p_T}\xspace of that jet, but it is important that $H_T$ be the variable used at higher jet multiplicity, not maximum jet \ensuremath{p_T}\xspace. This cut ensures that multiple jets with \ensuremath{p_T}\xspace just below our cuts cannot combine together on one side of the event and force the two bosons to be close in angle, or allow one boson to be soft relative to the QCD activity. Either of these effects would allow events that are far in phase space from the LO kinematics to enter the measurement, and potentially cause large corrections and failures of cancellations in our ratios. Note also that we choose identical kinematic cuts for $Z$ and $\gamma$, which we supplement in \sec{Practical} when being more experimentally realistic. We discuss angular isolation of the bosons in \ssec{NLO-QCD} and \ssec{PhotonIso}.
{\renewcommand{\arraystretch}{1.5}
\begin{table}
\begin{center}
\begin{tabular}{| c |}\hline
Kinematic Cuts \\ \hline \hline
$|\eta(V_i)| < 1.5$ \\ \hline
$p_T(V_2) > \frac12 \,p_T(V_1)$ \\ \hline
$H_T = \sum_{\text{jets}} |p_T^{j}| < \frac12 \, p_T(V_2)$ \\ \hline
\end{tabular}
\end{center}
\caption{ Kinematic cuts imposed on vector bosons $V_i$ and on the jets $j$ from real emission at NLO. In our calculations we work only to single real emission so $H_T$ is simply $\ensuremath{p_T}\xspace^j$, but the use of an $H_T$ cut is important at higher orders. We define $V_1,V_2$ by $p_T(V_1) > p_T(V_2)$. Isolation requirements and cuts on decay products are described in \sec{Practical}.}
\label{tab:bosonJetCuts}
\end{table}
}
A variety of problems can arise that can invalidate or destabilize fixed-order calculations. Our cuts, which allow the vector bosons to have unequal \ensuremath{p_T}\xspace, but require both bosons have substantially higher \ensuremath{p_T}\xspace than any jet from real emission, are chosen to avoid them. Note also that our cuts generally scale with the overall average \ensuremath{p_T}\xspace, and roughly with our observable \ensuremath{\bar{m}_T}\xspace.
One issue we must avoid is large logarithms. The fairly loose cut on additional hadronic activity, $H_T < \frac12\, p_T(V_2)$, means that logarithms of $p_T(V)/H_{T,\text{min}}$ never become so large as to require jet veto resummation~\cite{Banfi:2012yh,Banfi:2012jm,Tackmann:2012bt}. But because our cuts scale with the average \ensuremath{p_T}\xspace, we also avoid large logarithms of $p_T(V_1)/p_T(V_2)$, which (in combination with a large $qg$ parton luminosity) could have led to very large corrections \cite{Rubin:2010xp}. Simultaneously,\footnote{We thank Z.~Bern for alerting us to possible subtleties with these cuts and specifically to ref.~\cite{Frixione:1997ks}.} asymmetric cuts on the bosons avoid logarithms of $p_T(V)/\Delta$, where $\Delta = \ensuremath{p_T}\xspace^\text{cut}(V_1) - \ensuremath{p_T}\xspace^\text{cut}(V_2)$; these logarithms, which arise from soft gluon emission, were first identified in ref.~\cite{Frixione:1997ks} and resummed in ref.~\cite{Banfi:2003jj}.
Meanwhile our observable itself, $\ensuremath{\bar{m}_T}\xspace$, does not appear in large logarithms and requires no resummation.
Other effects can enhance the size of fixed-order terms relative to naive expectations. For instance, if radiative corrections are allowed to populate phase space at lower $\sqrt{\hat{s}}$ than is accessible at tree level, the formally NLO calculation carries \emph{de facto} LO scale uncertainties. This does not happen with our cuts and observable; all bins in \ensuremath{\bar{m}_T}\xspace are dominated by the LO contribution.
Another common issue with $q\bar q$ processes is the opening of new channels with large parton luminosities at higher orders. At NLO, we have the new channel $qg\to qV_1^0V_2^0$, but our cuts mitigate the \ensuremath{K}\xspace factors, making them of order 1.5. Moreover, these \ensuremath{K}\xspace factors are nearly process independent and largely cancel in our ratios. At NNLO, we have the new channel $gg\to V_1^0V_2^0$, which is substantial and process dependent; we include it in our calculation. Also at NNLO is the new channel $qq\to qqV_1^0V_2^0$, which is process dependent and potentially large for valence quarks. We estimate that with our cuts, (which avoid any large logarithmic enhancements,) this process is subleading; we do not evaluate it but include it in our uncertainty estimates.
We must also avoid situations where higher-order matrix elements (at a particular jet multiplicity) are enhanced relative to LO matrix elements dressed with soft and collinear factors (at the same multiplicity). One way this can happen is if an additional jet emission can make a threshold or resonance accessible that was inaccessible at lower order. This can occur in QCD corrections to the $Z\gamma$ process, via radiative $Z$ decays $Z \to \ell^+\ell^- \gamma$. Simply because we take \ensuremath{\bar{m}_T}\xspace $>$ 200 GeV, this is irrelevant at LO, and our $H_T$ cut assures this does not arise at any order in \ensuremath{\alpha_S}\xspace.
A further potential problem can appear if a radiative emission can significantly decrease an internal propagator's virtuality compared to the analogous propagator in the LO process, thus enhancing the amplitude. (Strictly speaking, this way of stating things is not gauge invariant,
but the enhancement itself clearly is.) With our cuts and observable, this too does not occur.
\subsection{NLO QCD corrections}
\label{subsec:NLO-QCD}
For our observables and with our choice of cuts, virtual and real QCD corrections to $d\hat\sigma(q\bar q\to V_1^0V_2^0)$ are largely proportional to the LO values. Consequently the $R_1$ ratios receive only small NLO QCD corrections in most regions of phase space. The exception is in the region where a final-state quark is nearly collinear with a vector boson; this region is enhanced for photons by large logs from collinear emission, whereas for $Z$s the logarithmic enhancement is cut off by $m_Z$. More specifically, for $Z$ emission the quark propagator in \fig{CollFeyn} is bounded from above by $1/m_Z^2$, while the photon's collinear singularity at low $m_{q\gamma}$ must be absorbed into a non-perturbative fragmentation function, or evaded through an angle-dependent energy isolation cut that avoids generating soft divergences at higher order. This fundamental difference between $Z$ and $\gamma$ cannot be removed experimentally, and gives a significant NLO shift to the $R_1$ ratios at low $\bar m_T$.
\begin{figure}
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(100,25)
\put(20,0){\includegraphics[width=0.4\linewidth]{collinear-eps-converted-to.pdf}}
\end{picture}
\end{center}
\caption{The regime in which $V$ and $q$ are nearly collinear in the final state, the source of a significant difference between photon- and $Z$-rates.}
\label{fig:CollFeyn}
\end{figure}
The collinear-$\gamma q$ singularity can be dealt with using the smooth-cone isolation method of Frixione \cite{Frixione:1998jh}. (While theoretically elegant, this method is not practical; we will employ a more experimentally realistic version of Frixione isolation, and discuss the uncertainties inherent in its use, in \ssec{PhotonIso}.) In this method, one chooses two parameters $\delta,\epsilon$ and requires that in any cone of radius $R<\delta$, the hadronic activity is bounded by a function that goes smoothly to zero as $R \to 0$; in particular\footnote{Frixione included a third parameter $n$ as an exponent on the trigonometric function here; we have chosen $n=1$.}
\begin{eqnarray}
\label{eq:frix}
\sum_{h \in R} \ensuremath{p_T}\xspace^h < p_T(V)\ {\mathcal I}(R;\epsilon,\delta) \quad \text{for all $R < \delta$} \ ,\\
\label{eq:frixfunc}
{\mathcal I}(R;\epsilon,\delta) = \epsilon \left({1-\cos R \over 1-\cos\delta}\right) .
\end{eqnarray}
Here the sum is over all hadrons $h$ within a cone of radius $R$ around the boson.
That the $R_1$ ratios remain unchanged outside the collinear regime may be seen by applying the Frixione method with extreme parameters $(\delta,\epsilon)=(1.2,0.2)$. This choice largely removes the collinear region. Here (but see below) we apply isolation {\it both} to photons and $Z$s, to maintain as much congruence as possible. At left in \fig{dead-cone}, we see that the \ensuremath{K}\xspace factors are then almost identical for the three $V^0_1V^0_2$ processes, and so the $R_1$ ratios at NLO are the same as at LO.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\linewidth]{removeDeadcone.pdf}~~
\includegraphics[width=0.48\linewidth]{seeDeadcone.pdf}
\end{center}
\caption{The $V_1^0V_2^0$ cross section at NLO, shown relative to the LO rates. At left, the collinear region was removed by a very strict smooth-cone isolation cut $(\delta,\epsilon)=(1.2,0.2)$ applied to both $\gamma$s and $Z$s. All 3 processes receive identical NLO corrections, thus leaving the ratios invariant. At right, with a reasonable isolation cut $(\delta,\epsilon)=(0.4,0.5)$ the NLO corrections differ significantly among the processes at low energies.}
\label{fig:dead-cone}
\end{figure}
However, as seen at right in the same figure, when the collinear region is restored by using more reasonable smooth-cone parameters $(\delta,\epsilon)=(0.4,0.5)$, there is a significant splitting in the \ensuremath{K}\xspace factors at low $\bar m_T$, where the $Z$ mass is particularly relevant, and thus a shift in the $R_1$ ratios away from their LO values. Note that the splitting of $\gamma\gamma$ from $ZZ$ is roughly double that of $\gamma\gamma$ from $Z\gamma$, so the effect of the collinear regime is largest on $R_{1b}$.
In all results beyond this point we use $(\delta,\epsilon) = (0.4,0.5)$, with appropriate practical modifications discussed in \ssec{PhotonIso}. For this choice, and for the range of \ensuremath{\bar{m}_T}\xspace that is relevant for the LHC, we find it unnecessary to impose isolation on $Z$s, for the following reasons. At low \ensuremath{\bar{m}_T}\xspace the Frixione cut removes a region where the amplitude for $Z$ emission is not enhanced. Meanwhile at larger \ensuremath{\bar{m}_T}\xspace the falling $qg$ parton luminosity makes the collinear region less important even for photons, an effect seen at right in \fig{dead-cone}, and also tends to favor the region of low $p_T^q/p_T^Z$, which is not removed by the Frixione cut. Altogether this reduces the impact of Frixione isolation on $Z$s to the percent level, relative to the total differential cross section.
Therefore, {\it in what follows below and in our final results, we impose isolation only on photons, not on $Z$s}, and believe it is safe for the LHC experiments to do the same without negatively impacting the ratios. At a higher-energy collider this would need to be revisited.
With these Frixione parameters, our lowest $\bar m_T$ bin sees a downward shift of $R_{1a} = \sigma(Z\gamma)/\sigma(\gamma\gamma)$ by 15\%, of $R_{1b}=\sigma(ZZ)/\sigma(\gamma\gamma)$ by 25\%, and of $R_{1c}=\sigma(ZZ)/\sigma(Z\gamma)$ by 12\% relative to the LO values. In higher bins, the effect of the collinear regime is muted as the $gq$ parton luminosity falls and the difference between photon and $Z$ amplitudes decreases.
It is instructive to understand why the NLO corrections to the $R_1$ ratios are so small outside of the collinear region. The point is that most logarithmically-enhanced corrections are themselves proportional to the LO process, for reasons that even extend to many regions of phase space that are not log-enhanced. For instance, in the NLO process $q\bar q\to V_1^0V_2^0 g$, our cuts are inclusive in the initial state radiation (ISR) region of phase space, where the final-state gluon is collinear with the initial partons. Consequently a fixed-order calculation is a reliable guide, and the NLO diagrams that appear are the same for all three processes. Thus no large process-dependent corrections arise, and the $R_1$ ratios are hardly affected. Meanwhile emissions of hard gluons are suppressed by our jet cuts.
Similarly, for the ISR region of $qg \to V_1^0V_2^0 q$, the ratios are little changed, for two reasons. First, the partonic cross section near this singular region displays a factorization into the tree-level cross section and a universal factor that is absorbed into the definition of the PDFs. Second, the replacement of an anti-quark PDF with a gluon PDF has a small impact, because $\bar u$ and $\bar d$ PDFs are similar. We may see this heuristically by writing
$f_{\bar q}= \frac12(f_{\bar u}+f_{\bar d})$ and $\bar\delta= \frac12(f_{\bar u}-f_{\bar d})$, and noting the $qg$ integrand is roughly proportional to
\begin{equation}
\label{eq:qgpdfs}
\left[f_{u}(x_1) d\hat\sigma^{LO}_{u\bar u} +
f_{d}(x_1) d\hat\sigma^{LO}_{d\bar d}\right]\left[f_{g}(x_2/z) P_{q\leftarrow g}(z) \right]
\end{equation}
while the tree-level process has integrand
\begin{equation}
\label{eq:qqpdfs}
\left[f_{u}(x_1) d\hat\sigma^{LO}_{u\bar u} +
f_{d}(x_1) d\hat\sigma^{LO}_{d\bar d}\right]f_{\bar q}(x_2) + \ \ensuremath{O}(\bar{\delta}) \ .
\end{equation}
Here $P_{q\leftarrow g}$ is the gluon-to-quark splitting function, and we have ignored small contributions from subdominant initial states. Since $\bar \delta\ll f_{\bar q}$ in the relevant $x$ range, these integrands are proportional, so no large correction to the LO ratios is expected from the ISR region.
\subsection{NNLO QCD corrections}
\label{subsec:gg}
Although NNLO calculations of diboson processes have been carried out for all processes except $WZ$~\cite{Catani:2011qz,Grazzini:2013bna,Cascioli:2014yka,Gehrmann:2014fva,Grazzini:2015nwa,Grazzini:2015hta}, most of these are not yet accessible in public code. This limits our ability to refine our NLO results or to estimate the theoretical uncertainties from which they suffer. In this context, we take the following approach. On the one hand, we study in detail the largest known NNLO correction to our ratios, namely $gg\to V_1^0V_2^0$, which is large enough that it must be included, but fortunately is available publicly. On the other hand, we search for additional NNLO corrections that should affect our ratios, and make rough estimates of their size to see if they are important; if so we include them as a theoretical uncertainty.
We saw in \fig{dead-cone} and eqs.~\eqref{eq:qgpdfs}--\eqref{eq:qqpdfs} that many NLO corrections are common to all three $V^0_1V^0_2$ processes and cancel in the $R_1$ ratios. Similar logic would suggest that many NNLO QCD corrections are also common to the three processes and that, away from the collinear-$qV$ regions, new real contributions like $q\bar{q} \to V_1^0V_2^0 gg$, or $qg \to V_1^0V_2^0 qg$ are likely to cancel. But by looking carefully at the physical origin of various effects, we can also see where such cancellations will fail.
Before we do so, let us forestall an obvious question. Below, we will assume that many NNLO corrections cancel in ratios, and that the largest one that does not cancel comes from the $gg\to V_1^0V_2^0$ loop graph (as suggested in ref.~\cite{Bern:2002jx}), which we will include explicitly below.
One might question this assumption based on the existing NNLO and near-NNLO literature, which suggests potentially large $\ensuremath{K}\xspace_\text{NNLO/NLO}$ factors ($1.3 - 1.6$), substantial process-dependence in these \ensuremath{K}\xspace factors, and effects that can be much larger than the $gg\to V_1^0V_2^0$ loop graph. How, then, can we possibly claim that NNLO corrections to our ratios could be brought under control, and further assume that even higher-order effects can be ignored?
Here one needs to look carefully at the details, which we do in \app{NNLO}. The large $\ensuremath{K}\xspace_\text{NNLO/NLO}$ arise only in situations where the cuts on the bosons and jets are very different from our own, causing even the $\ensuremath{K}\xspace_\text{NLO/LO}$ factor to be much larger than the $\sim 1.5$ that we found above in \fig{dead-cone}. The process-dependent differences among the $\ensuremath{K}\xspace_\text{NNLO/NLO}$ factors also appear much smaller when one restricts to kinematic regions and observables similar to the ones we are considering. In those regions there is no clear indication that the $gg$ loop is not the main process-dependent effect. Thus there is no clear evidence against our assumptions, and even some mild (though hardly decisive) evidence in their favour. Let us note again that our choice of observable and of cuts appears to be crucial in this regard; many other observables and cuts would have larger NNLO corrections in ratios.
With that issue set aside, we now consider obvious sources of NNLO corrections that will not cancel in our ratios. Since the dominant NLO correction to the ratios, shown in the right-hand plot of \fig{dead-cone}, was from the collinear-$qV$ region, corrections to that region of phase space will not cancel. NNLO real and virtual corrections to this single-collinear effect will impact the ratios. However, we expect these to give an order $\alpha_s$ adjustment to the splitting shown in \fig{dead-cone}, which puts them below 2\%.
Another important contribution could come from the double-collinear region in $q\bar q, gg \to V_1^0V_2^0 q \bar q$. This too is very small, despite the large NLO single-collinear correction. To see this, note the following. The reason that $qg\to V_1^0V_2^0q$ is so important is that ${\mathscr L}_{qg}\gg {\mathscr L}_{q\bar q}$, partially canceling the extra $\alpha_s$ at NLO. There is no corresponding enhancement for two independent collinear emissions. The double-collinear region at the next order should be thought of predominantly as $q\bar q\to q\bar q$, $gg\to q\bar q$, with double emission $q\to q V_1$ and $\bar q\to \bar q V_2$. (Our cuts remove the region where both $V_1$ and $V_2$ radiate off a single quark.) For $q\bar q\to q\bar q$ the parton luminosity is the same as that arising at LO, so the $q\bar q$-initiated process is indeed suppressed by $\ensuremath{O}(\alpha_s^2)\sim 1\%$ compared to LO. Meanwhile, ${\mathscr L}_{gg}$ is comparable to or smaller than ${\mathscr L}_{qg}$ at the relevant energies; and furthermore $gg \to q\bar{q}$, which lacks a $t$-channel gluon, has a smaller partonic cross section than $qq\to qq$ and $q\bar{q} \to q\bar{q}$. Altogether it appears the double-collinear regime shifts the ratios at the percent level or below.
A qualitatively new source of non-canceling corrections is from the opening of a new channel at NNLO, namely the (dominantly valence-quark) process $qq \to qqV_1^0V_2^0$. When each of the two fermion lines emits one vector boson, the resulting contribution is generally no longer proportional to the LO $q\bar{q} \to V_1^0V_2^0$ process. Still, we estimate that the $qq$-initiated processes at NNLO correct the ratios by just a few percent. Our argument proceeds as follows. The process $qq \to qqV_1^0V_2^0$ has a collinear divergence near the beampipe and can only be defined by requiring both jets to have \ensuremath{p_T}\xspace greater than some minimum $\ensuremath{p_T}\xspace^{j,\text{min}}$. However, the divergence is proportional to the LO $q\bar q\to V_1^0V_2^0$ process, and largely cancels in the ratios. Calculating the effect on the ratios for different values of $p^j_{T,\text{min}}$ between 5 and 30 GeV, and extrapolating $p^j_{T,\text{min}}\to 0$ by fitting to a falling exponential, we find shifts for $R_{1a}$ ($R_{1b}$) [$R_{1c}$] of 3\% (3.5\%) [2.5\%] or less. Consequently, although our estimates are crude and this source of NNLO corrections may well be one of the largest on the $R_1$ ratios, it does not seem to present issues that exceed our fiducial benchmark of 5--6\% theoretical uncertainties.
Finally, the largest known NNLO correction to the $R_1$ ratios is from $g g\to V_1^0V_2^0$. Fortunately, much is already known about this correction, which is separately gauge-invariant and finite. It has been known for some time \cite{Combridge:1980sx,Glover:1988rg} and can consistently be combined with the NLO calculation on its own. As it gives the largest source of NNLO corrections in most regions of phase space and has a different dependence on EW quantum numbers than does the tree-level process, it has an important effect on our ratio observables.
Because $u$- and $d$-type quarks contribute coherently in the loop, the formulas for $gg\to V_1^0V_2^0$ are not proportional to the tree-level $q\bar q\to V_1^0V_2^0$ formulas. In fact $gg\to w^3 x$ is zero by $SU(2)$ conservation, and so $gg\to Z\gamma$ is relatively small compared to $gg\to ZZ, \gamma\gamma$. In \fig{kgg} the $gg$ contributions to the cross sections are shown relative to the corresponding NLO
differential cross sections; they represent a 13\% (5\%) [20\%] correction for $\gamma\gamma$ $(Z\gamma)$ $[ZZ]$ at low $\bar m_T$,
though less at higher energies where the gluon PDFs are smaller.
Partial cancellations still take place in our ratios.
The observable $R_{1a}$ is shifted downward by as much as 7\% from its NLO value at the lowest values of $\bar m_T$ we consider; however, this $gg$-shift is reduced at higher $\bar m_T$, quickly becoming of order $3\%$. Meanwhile $R_{1b}$ $(R_{1c})$ shifts up 7\% (14\%) at low $\bar m_T$; this $gg$-shift remains at the 6\% (9\%) level for moderate $\bar m_T$ before shrinking more rapidly to 3\% (3\%) at high $\bar m_T$.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\linewidth]{ggOVERnlo.pdf}~~
\includegraphics[width=0.48\linewidth]{R1gg.pdf}
\end{center}
\caption{(Left) Contribution from $gg\to V_1^0V_2^0$ to the $V_1^0V_2^0$ cross sections, expressed relative to the corresponding NLO
cross section. We used $gg \to \gamma\gamma$ at $\ensuremath{O}(\alpha_S^3)$ to estimate $gg \to Z\gamma,ZZ$ at this order. (Right) The $R_1$ ratios, including the NLO and $gg\to V_1^0V_2^0$ contributions.}
\label{fig:kgg}
\end{figure}
\Fig{kgg} displays the $R_1$ ratios including the $gg\to V_1^0V_2^0$ channel along with the NLO
contributions. This plot should be compared with \fig{mtRatio}, which shows the LO ratios. Notice that $R_{1a}$ is accidentally flatter than at LO, as a result of the above-mentioned corrections.
This plot of course depends on a choice of renormalization and factorization scales $\mu_R$ and $\mu_F$ used for the $gg\to V_1^0V_2^0$ computation. For $gg\to\gamma\gamma$ the scale dependence can be reduced because the dominant\footnote{In \app{gg} we argue that the terms neglected in ref.~\cite{Bern:2002jx} are indeed subleading. For $gg \to ZZ$ a similar calculation appeared very recently \cite{Caola:2015psa}, as this paper was nearing completion.} part of the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ correction is known \cite{Bern:2002jx}. For $gg\to Z\gamma, ZZ$, we can use the fact that at NLO all three processes have a nearly universal $\mu_R,\mu_F$ dependence for $\hat s\gg m_Z^2$. This is because (i) the three processes have the same $\ensuremath{\alpha_S}\xspace$-dependence and involve the same PDFs, (ii) the SM is anomaly free and so no new non-universal diagrams appear at $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$, and (iii) the contribution of longitudinal $Z$s to $gg \to ZZ$ is rather small \cite{Glover:1988rg}, of order 10--15\%. Thus for reasonable values of $\mu_R$ and $\mu_F$,
\begin{equation}
\label{eq:Kgg}
K_{gg} \equiv \frac{d\sigma_{(3)}(gg \to \gamma\gamma)}
{d\sigma_{(2)}(gg \to \gamma\gamma)}
\approx
\frac{d\sigma_{(3)}(gg \to Z\gamma)}
{d\sigma_{(2)}(gg \to Z\gamma)}
\approx
\frac{d\sigma_{(3)}(gg \to ZZ)}
{d\sigma_{(2)}(gg \to ZZ)},
\end{equation}
where $d\sigma_{(n)}$ marks the cross section calculated at order $\ensuremath{\alpha_S}\xspace^n$. We can then use MCFM to compute the known $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^2)$ and $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ cross sections for $gg\to \gamma\gamma$, thereby determining the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ cross sections for the other processes to a fairly good approximation. For our central values we choose scales $\mu_R = \mu_F = m_{\gamma\gamma}$ everywhere in \eq{Kgg}.\footnote{We have observed, by direct comparison across our \ensuremath{\bar{m}_T}\xspace range, that the procedure just outlined is essentially identical to calculating the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^2)$ cross sections for the three processes with scales $\mu_R \sim 0.34\,m_{VV}$ and $\mu_F\sim 0.20\, m_{VV}$. The fact that these are reasonable scales serves as a sanity check of our method.}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\linewidth]{Kgg-crf.pdf}~~
\includegraphics[width=0.48\linewidth]{ggScaleDep.pdf}
\end{center}
\caption{(Left) The size of $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ corrections $K_{gg}$ to $gg \to \gamma\gamma$ as a function of $\bar{m}_T$, with scales set to $\mu_R=\mu_F=m_{\gamma\gamma}$ in numerator and denominator. This function allows us to estimate the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ cross section for $gg \to Z\gamma$ and $ZZ$. (Right) As a function of scale $\mu_R=\mu_F=\mu$, the $gg \to \gamma\gamma$ rate in the kinematic region $\bar m_T > 200$ GeV, shown at $\ensuremath{O}(\alpha_s^2)$ and (with the partial calculation implemented in MCFM) at $\ensuremath{O}(\alpha_s^3)$. The cross sections are normalized with respect to $\sigma_0\equiv \sigma_{(3)}(gg \to \gamma\gamma, \mu=m_{VV})$.}
\label{fig:ggCorr}
\end{figure}
We show the values of $K_{gg}$ in left panel of \fig{ggCorr}. Since the values of $K_{gg}$ are large, one might wonder whether, as in $gg\to h$, the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^4)$ correction to $gg \to V_1^0V_2^0$ could itself be quite large. However, unlike $gg\to h$, where the NLO prediction exceeds the LO substantially at all $\mu$, the situation is milder here. As can be seen in the right panel of \fig{ggCorr}, which shows $gg \to \gamma\gamma$ at $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^2)$ and $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ with a variety of scale choices, the higher-order prediction turns over at small $\mu$, and above the turnover varies only slowly. We therefore expect $\ensuremath{O}(\alpha_s)\sim 10-20\%$ uncertainties on $gg\to\gamma\gamma$, and $\sim 1-2\%$ uncertainties on the $R_{1}$ variables, from the unknown $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^4)$ terms. We will estimate uncertainties from this source in \ssec{pdf-scale} and find them consistent with this expectation.
\subsection{Partial cancellation of PDF and scale uncertainties}
\label{subsec:pdf-scale}
Now we turn to standard sources for potential theoretical uncertainties: the PDFs and the choices of renormalization and factorization scales in QCD corrections. These show significant cancellations and become subleading compared to other uncertainties that we have already discussed.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.48\linewidth]{unc_pdf_chs_amt.pdf}~~
\includegraphics[width=0.48\linewidth]{unc_pdf_rat_amt.pdf}
\end{center}
\caption{The relative PDF uncertainty bands for the individual $V^0_1V^0_2$ cross sections (left) and the $R_1$ ratios (right). PDF variations of $gg \to V^0_1V^0_2$ are included. See text for more details. }
\label{fig:pdfunc}
\end{figure}
The PDF uncertainties for the individual channels, and their reduced values for the ratios, are shown in \fig{pdfunc}. For the $R_{1a}$ ratio the uncertainties are of order $1\%$ and can be essentially ignored; as we saw in \ssec{RatObs}, the parton luminosity ${\mathscr{L}}^S_{u\bar u}$ dominates both numerator and denominator, so that PDF variations nearly cancel.
For the others, the uncertainties are still significantly reduced, rising only to about $2\%$ even up to $\bar m_T\sim 1$ TeV.
These uncertainties were determined using \textsc{MCFM 6.8}. The $pp\to V_1^0V_2^0$ cross sections are evaluated for the central ($S_0$) and all 20 pairs of error sets ($S_i^\pm$) of the MSTW 2008 PDF set \cite{Martin:2009iq}.
With the cross sections $d\sigma(S_i) $, we use the prescription of ref.~\cite{Martin:2009iq} to determine the PDF uncertainties on individual channels. The upper edge of the uncertainty band is calculated with
\begin{equation}
\Delta_+(d\sigma) =
\sqrt{ \sum_i \left(\max \left[0,d\sigma(S_i^+)-d\sigma(S_0),d\sigma(S_i^-)-d\sigma(S_0)\right]\right)^2},
\end{equation}
while the lower edge is the same with ``max'' replaced with ``min''.\footnote{We actually carry this out with the 90\% confidence-level NLO MSTW 2008 PDF sets, and then rescale the result, formally a $2\sigma$ variation, by 1.645 to obtain a formally $1\sigma$ variation. This is almost the same as using the 68\%-level confidence sets, but because of non-Gaussian tails gives a slightly more conservative estimate of uncertainties.}
Because the error sets of MSTW 2008 are eigenvectors of the covariance matrix, the PDF uncertainties for the ratios can then be obtained in a similar fashion.\footnote{
For example:
\begin{equation} \nonumber
\Delta_+\left(R_{1a}\right) =
\sqrt{ \sum_i \left(\max \left[0,\frac{d\sigma(Z\gamma,S_i^+)}{d\sigma(\gamma\gamma,S_i^+)}-\frac{d\sigma(Z\gamma,S_0)}{d\sigma(\gamma\gamma,S_0)},\frac{d\sigma(Z\gamma,S_i^-)}{d\sigma(\gamma\gamma,S_i^-)}-\frac{d\sigma(Z\gamma,S_0)}{d\sigma(\gamma\gamma,S_0)}\right]\right)^2}.
\end{equation}}
All this is straightforward except for one subtlety.
Since we do not have access to the $\ensuremath{O}(\ensuremath{\alpha_S}\xspace^3)$ calculation for $gg\to Z\gamma$ and $gg\to ZZ$, we obtain them by rearranging \eq{Kgg} as
\begin{equation}
d\sigma_{(3)} (gg \to Z\gamma,\text{pdf}_1) \approx d\sigma_{(3)}(gg \to\gamma\gamma,\text{pdf}_1)\frac{d\sigma_{(2)} (gg \to Z\gamma,\text{pdf}_2)}{d\sigma_{(2)} (gg \to \gamma\gamma,\text{pdf}_2)} ,
\label{eq:ggpdf}
\end{equation}
where $d\sigma(\ldots;\text{pdf}_i)$ is the cross section evaluated for PDF set $S_i$. A similar expression holds for $gg \to ZZ$. Inaccuracies in this procedure will be subleading in our uncertainties since $gg\to V_1^0V_2^0$ is itself sufficiently small.
Now we turn to uncertainties in our NLO calculation from renormalization and factorization scales $\mu_R,\mu_F$. Typically the cancellation of correlated scale variations in ratios of various processes should be viewed as accidental, since the actual structure of higher-order corrections in differing processes is uncorrelated. We wish to argue that this is not the case here. The renormalization scale is sensitive to the ultraviolet region of higher-order corrections, where EW symmetry is restored (up to longitudinal polarizations, which first appear at NNLO in $gg \to \phi^3\phi^3)$, and where we expect higher-order corrections in general to take a nearly identical form for all $V^0_1V^0_2$ processes. Meanwhile, factorization scale sensitivity primarily comes from divergences associated with emissions off the initial state. While this is not directly affected by the restoration of EW symmetry, it is sensitive to the color structure of the processes order-by-order in the perturbative expansion of QCD, which is also identical for the three $V^0_1V^0_2$ processes. For these reasons the cancellation of scale dependence we observe in our ratios is physical, since the scale choices really are probing correlated higher-order effects.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.48\linewidth]{sigmaScaleEnvFlat.pdf}~~
\includegraphics[width=0.48\linewidth]{ratioScaleEnvFlat.pdf}
\end{center}
\caption{The relative uncertainty band on the $V^0_1V^0_2$ cross sections (left) and $R_1$ ratios (right) found by varying the renormalization and factorization scales $\mu_R,\mu_F$ up and down by a factor of 2. Here the scales appearing in the $gg\to V_1^0V_2^0$ process are {\it not} varied; see \fig{scalevarGG} below.}
\label{fig:scalevar}
\end{figure}
As shown in \fig{scalevar}, scale-dependence is reduced from several percent in the cross sections to 1--2\% in the ratios, where the cancellation is significant for all three ratios and works best at high energy. Here we have varied the scales $(\mu_R,\mu_F)$ independently from $\frac12 \,m_{VV}$ to $2\,m_{VV}$ and plotted the envelope of the relative variation in each quantity.
However, in \fig{scalevar} we have held the scales in the $gg\to V_1^0V_2^0$ processes {\it fixed}. The calculation to NLO of $q\bar{q} \to V_1^0V_2^0$ begins at $\ensuremath{O}(\alpha_s^0)$, while the calculation of $gg\to V_1^0V_2^0$ begins at $\ensuremath{O}(\alpha_s^2$). To the order we are working there are no terms in the former calculation which are at the same order as terms in the latter, and thus there is no sense in which the perturbative expansion of the one can affect that of the other. Correspondingly there is no sense in which these two calculations must or should be evaluated with the same value of $\mu_R$, and so their $\mu_R$ dependence must be computed separately. While in principle there could be correlation in the $\mu_F$-dependence through the pdfs, it turns out that $gg\to V_1^0V_2^0$ depends much more strongly on $\mu_R$, and so any such correlation is unimportant.
Based on this reasoning, we have also computed the effects of scale variations on the $gg \to V_1^0V_2^0$ component of the cross sections, holding all other components fixed. Lacking the $\ensuremath{O}(\alpha_s^3)$ differential cross sections for $gg\to Z\gamma$ and $gg \to ZZ$, we again rely on another incarnation of \eq{Kgg}:
\begin{equation}
d\sigma_{(3)} (gg \to Z\gamma,\{\mu_1\}) \approx d\sigma_{(3)}(gg \to\gamma\gamma,\{\mu_1\}) \frac{d\sigma_{(2)} (gg \to Z\gamma,\{\mu_2\})}{d\sigma_{(2)} (gg \to \gamma\gamma,\{\mu_2\})},
\label{eq:ggscal}
\end{equation}
where $\{\mu_i\}$ stands for a choice of $\mu_R$ and $\mu_F$.
The resulting uncertainties due to scale variation of the $gg\to V_1^0V_2^0$ processes are shown in \fig{scalevarGG}; these are consistent with our estimate from \ssec{gg}. Although small for each individual channel compared to the scale variation in the left-hand plot of \fig{scalevar}, cancellations are not as significant as for the NLO scale variations. Consequently the two classes of scale variation turn out to be quite similar in size and shape for the $R_1$ observables, as can be seen in the right-hand plots of \fig{scalevar} and \fig{scalevarGG}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.48\linewidth]{sigmaScaleEnvFlatGG.pdf}~~
\includegraphics[width=0.48\linewidth]{ratioScaleEnvFlatGG.pdf}
\end{center}
\caption{The relative error band on the $V^0_1V^0_2$ cross sections (left) and $R_1$ ratios (right) found by varying $\mu_R,\mu_F$ up and down by a factor of 2. Here {\it only} the scales appearing in $gg\to V_1^0V_2^0$ are varied.}
\label{fig:scalevarGG}
\end{figure}
Overall, we can see that while the PDF and scale uncertainties form a significant portion of the theoretical error budget for individual cross sections, these uncertainties are substantially reduced in ratios (in particular in $R_{1a}$) and become subleading. This presumably reflects true symmetry-related cancellations in the many NNLO corrections that are common to the three neutral diboson processes.
\subsection{EW corrections}
\label{subsec:EW-corr}
\subsubsection{Sudakov enhancements}
\label{subsec:doubleLog}
For the level of precision we pursue, higher-order EW corrections to our
ratio observables are important. Complete calculations of NLO EW effects
for $\gamma\gamma$, $Z\gamma$, and $ZZ$ exist,
though public code is not yet at our disposal and the results have been
presented with different cuts from our own. As an approximation of the EW corrections,
and to estimate the magnitude of their uncertainties, we employ a leading-log calculation
in the threshold limit. Comparison of our results below with the full NLO calculations of
refs.~\cite{Bierweiler:2013dja,Denner:2015fca} reassures us that our estimates are reasonable.
Because of various sources of $SU(2) \times U(1)$ breaking, large EW logarithms do not
entirely cancel even in fairly inclusive observables such as $d\sigma/d\ensuremath{\bar{m}_T}\xspace$. At very large \ensuremath{\bar{m}_T}\xspace,
ignoring finite NLO EW corrections and resumming the leading Sudakov logarithms,
of the form $\alpha^n \log^{2n}([\ensuremath{\bar{m}_T}\xspace/m_{W,Z}]^2)$, is justified and should give a good approximation of the dominant effects.
An estimate of the Sudakov logarithm-enhanced corrections can be obtained from a
calculation at threshold, where all the energy of the initial state goes into
production of the electroweak states. The threshold limit corresponds to a strict veto
on the real emission of EW bosons, so at high \ensuremath{\bar{m}_T}\xspace it overestimates the true EW correction.
Since we do not have such a strict veto in our observables, the large virtual corrections
above are reduced by our partial inclusion of the real radiation of gauge bosons.
For instance, soft $W$ and $Z$ bosons are partially included: a soft $Z$ or $W$ that decays
hadronically typically produces soft daughters at wide angles to the hard boson,
and thus its daughter jets will neither fail our jet cuts nor ruin isolation of the
boson or its daughter leptons. Leptonic decays of the soft bosons are potentially more subtle,
depending on how the extra leptons are treated experimentally. Our less extreme veto of
soft-collinear bosons should lead to some reduction of the soft-collinear corrections.
Conversely, finite NLO corrections that we ignore in our estimates should increase the size of the
EW correction. For moderate values of \ensuremath{\bar{m}_T}\xspace, this effect may partially compensate
the above-mentioned reduction.
Our estimates below are therefore rough guides, and the issue deserves further study.
This threshold regime was studied in the context of boson + jet
production~\cite{Becher:2013zua}.\footnote{We thank T.~Becher for extensive discussions
and Xavier Garcia i Tormo for providing detailed results of their calculation.}
It was found that the EW corrections reduce the photon + jet cross section by
$\Delta\sigma_\text{EW} = -6\%^{+3\%}_{-2\%}$ ($-11\%^{+3\%}_{-2\%}$) at
$\ensuremath{p_T}\xspace^\gamma = 500$ (1000) GeV, while reduction of the $Z$ + jet cross section is
roughly double this, $\Delta\sigma_\text{EW} = -13^{+4\%}_{-1\%}$ ($-22^{+4\%}_{-1\%}$).
The difference between $Z$ and $\gamma$ arises mainly from loops involving $W$ bosons.
As these effects are primarily associated with the phase space collinear to the hard boson,
we anticipate the effect on $\gamma\gamma$ to be roughly the square of the effect on
$\gamma$ + jet, leading to a 12--21\% reduction in $\sigma(\gamma\gamma)$ for
$500\text{ GeV} < \ensuremath{p_T}\xspace^\gamma < 1000\text{ GeV}$. Similarly, we expect reductions in
$\sigma(Z\gamma)$ $[\sigma(ZZ)]$ by 18--31\% [24--39\%]. But these effects partly
cancel in the $R_1$ ratios, reducing $R_{1a}$ $(R_{1b})$ $[R_{1c}]$ by just
7--12\% (14--23\%) [7--12\%] in this \ensuremath{p_T}\xspace range. At high enough \ensuremath{p_T}\xspace, EW effects
become the leading correction to our ratios, dominating over QCD effects.
Importantly, the uncertainties on these EW corrections are not large and are further
reduced in our ratios. There are several scale choices which appear in the calculation
of ref.~\cite{Becher:2013zua}, but the scale dependence of photons and $Z$s is correlated,
as can be seen in figure 3 of that paper. This correlation reduces the uncertainty in the
EW corrections to our ratios. We estimate that the NLO EW uncertainty from scale choices
that propagates into our ratios $R_{1a}$ $(R_{1b})$ $[R_{1c}]$ is no more than
${}^{+2\%}_{-1\%}$ $\left({}^{+3\%}_{-1\%}\right)$ $\left[{}^{+2\%}_{-1\%}\right]$
for $\ensuremath{p_T}\xspace \sim 500$--$1000 \text{ GeV}$. These uncertainties are comparable in size to the
uncertainties from PDFs and unknown QCD corrections.
At lower values of \ensuremath{\bar{m}_T}\xspace,
the finite NLO EW corrections become important,
but our resummation approximation still serves as a rough guide to their magnitudes.
For $\sigma(\gamma\gamma)$ and $\sigma(ZZ)$, ref.~\cite{Bierweiler:2013dja} has calculated these
corrections as functions of $p_T$.
The EW correction is dominated by a logarithmically growing component over
much of the \ensuremath{p_T}\xspace range relevant for our ratios,
suggesting that our approximation remains applicable in this region.
Moreover, comparison of ref.~\cite{Bierweiler:2013dja} to an
earlier calculation of the $\alpha \log^2([p_T/m_{W,Z}]^2)$ term alone~\cite{Accomando:2004de},
corresponding to truncation of the resummed calculation to first nontrivial order,
found agreement at the several percent level. For similar cuts to ours,
ref.~\cite{Bierweiler:2013dja} claims reductions in $\sigma(\gamma\gamma)$ $[\sigma(ZZ)]$
by 13--21\% [39--60\%] over the range $p_T \sim 500$--1000 GeV. These reductions are somewhat larger than the ones we obtained, and resummation is undoubtedly an important part of the discrepancy.
At somewhat lower $p_T$, only $\sigma(ZZ)$ shows a clear subleading $p_T$-independent correction,
which will certainly shift the EW corrections to $R_{1b},R_{1c}$ away from our leading-log predictions.
NLO EW results for $\sigma(Z\gamma)$ are given in ref.~\cite{Denner:2015fca},
but only with a fixed and low cut
on $p_{T,Z}$.
This makes comparison with our estimates impossible,
because large logarithms of $p_{T,\gamma}/p_{T,Z}^\text{cut}$ arise and are
indistinguishable from inclusive EW Sudakov
logarithms.
Still, we have no reason to suspect
that the behavior of the finite EW corrections should be qualitatively different
from those of $\gamma\gamma$ and $ZZ$.
Most importantly for our purposes, when finite pieces numerically
dominate the NLO EW correction, its uncertainty arises mainly from
scale variation in the EW couplings. Our earlier estimate of
the uncertainty using ref.~\cite{Becher:2013zua} is therefore an
overestimate at small \ensuremath{\bar{m}_T}\xspace.
We have summarized these statements in \fig{mainResult} of \sec{ExecSumm} by
indicating the expected fractional shifts in the ratios due to the source of
EW corrections derived in ref.~\cite{Becher:2013zua}, along with an estimate of
their uncertainties. This shows that these EW effects might be observable in our
ratios in the highest bins, where they dominate QCD effects.
Furthermore, EW effects are under sufficient control that there will still
be substantial sensitivity to other, non-SM contributions at high \ensuremath{\bar{m}_T}\xspace.
\subsubsection{Proper choice of EW scales for on-shell external photons}
\label{subsec:alphaQED}
Another EW issue concerns the correct choice of electromagnetic coupling
corresponding to emission of a photon.\footnote{We thank Z.~Bern for
pointing out the issue, and for conversations.} In the literature one
finds preference for evaluating $\alpha(\mu_\text{QED})$ both at $\mu_\text{QED} = 0$
and at $\mu_\text{QED}= {\text {min}}(m_Z,\sqrt{\hat{s}})$ (or some fraction
thereof). Since the QED coupling runs by 7\% from 0 to $m_Z$, this
difference affects $R_{1a}$ and $R_{1c}$ by 7\% and $R_{1b}$ by 14\%.
Typical QCD calculations may seem to suggest using $\mu_\text{QED} \sim
\sqrt{\hat{s}}$.
But in contrast to a quark or gluon, we can experimentally require that a
photon is on-shell and does not shower, i.e., does not form an
electromagnetic jet of leptons and hadrons with a finite mass. For abelian
gauge bosons, the leading effect of requiring an \emph{on-shell} photon,
rather than a photon that could be off-shell by as much as $q^2\sim \hat
s$, is given by running the coupling down from $\mu_\text{QED} = \sqrt{\hat{s}}$ to
$\mu_\text{QED} = 0$. (Importantly this is not true for nonabelian gauge bosons.)
This choice removes photons that, for instance, split to a $\mu^+\mu^-$
pair or mix with the $\rho$. We find this argument reliable in a pure
color-singlet situation, such as Higgs decay to two photons.
Subtleties could arise, however, in a colored environment: soft ISR gluons
are present in $pp$ collisions and can be radiated into the photon
isolation cone. On the one hand, we still want to forbid $\gamma^*\to
\mu^+\mu^-$ since this would be experimentally rejected; this tends to
suggest $\mu_\text{QED}<2 m_\mu$. On the other hand, we should include photons
with nearby soft gluons that lie below the isolation cut
$p_{T,\text{min}}^{\text{had}}$, which could suggest\footnote{Suggested to
us by T.~Becher following ref.~\cite{Becher:2013zua}. A related suggestion was
made by M.~Schwartz.} $\mu_\text{QED} \sim p_{T,\text{min}}^{\text{had}}$.
Faced with a lack of consensus, we have chosen not to directly address this
issue in this paper. Instead we use MCFM 6.8
``out of the box'', for which $\mu_\text{QED}=m_Z$ throughout. In
\fig{mainResult} of \sec{ExecSumm}, we have indicated the potential shift
from switching to $\mu_\text{QED}=0$ as an overall 7\% or 14\% error band that
is essentially flat and fully correlated across all bins. (Even if this
dispute were not resolved theoretically, the measurement of the average
ratio of the lowest bins would largely fix the value of $\mu_\text{QED}$.) In no
sense should this be thought of as a Gaussian error band, since no
probability extends beyond the band. For now readers may adjust our results
according to their individual opinions, but clearly it is important that
consensus on the matter be reached in the near future.
\section{Additional practical considerations}
\label{sec:Practical}
\subsection{Photon isolation}
\label{subsec:PhotonIso}
In \sec{beyondLO} we used the smooth-cone photon-isolation method of Frixione, \eq{frix}, but this is experimentally impractical.
More traditional is hard-cone isolation, simply requiring that the energy in a cone of size $R_h$ around the photon be less than $\epsilon_h \, \ensuremath{p_T}\xspace^\gamma$.
But if $\epsilon_h$ is small, a hard cone produces large logarithms due to the incomplete cancellation of virtual and soft gluon effects. Meanwhile if $\epsilon_h$ is not small, the hard cone introduces large sensitivity to the fragmentation function $D_{q \to \gamma}(z)$ at $z\to 1$, which is dangerous to a precision calculation since
$D_{q\to\gamma}(z\to 1)$ has substantial associated uncertainties. The Frixione algorithm avoids these issues by removing the divergent regions of phase space that require the introduction of a fragmentation function in the first place. The isolation parameters can then be set so that no large perturbatively calculable logarithms appear.
However, the smooth cone cannot be implemented experimentally since it requires the energy in a small cone around the photon to go literally to zero as that cone decreases in size. This difficulty may be evaded by using a discretized or ``staircase'' version of the smooth cone~\cite{Binoth:2010nha,Hance:2011ysa}. Although sensitivity to the photon fragmentation function is thereby reintroduced, this sensitivity can be maintained small while keeping the associated logarithms of manageable size, so as to not call the accuracy of the fixed-order calculation into question.
Our staircase isolation approximates the smooth cone of \eq{frix}, which has parameters $(\delta,\epsilon)=(0.4,0.5)$. We choose four nested cones ($n=1,2,3,4$) with radii $R_h^{(n)}=0.1 \times n$, and approximate the function ${\mathcal I}(R;\epsilon,\delta)$ of \eq{frixfunc} by a piecewise constant function
\begin{equation}
\label{eq:staircasefunc}
\hat{\mathcal I}(R;\epsilon, \delta) = \epsilon
\left[\frac{ 1 - \cos\left( \frac{1}{2} [R_h^{(n)}+R_h^{(n-1)}] \right) }{1-\cos\delta}
\right] \equiv \epsilon_h^{(n)}, \quad \text{for} \quad R_h^{(n-1)}< R < R_h^{(n)},
\end{equation}
where we define $R_h^{(0)}\equiv 0$. The constants $\epsilon_h^{(n)}$ are shown in \tab{stairs}; the functions ${\mathcal I}$ and
$\hat{\mathcal I}$ are plotted at left in \fig{staircase}. Then our staircase isolation criterion requires
\begin{equation}
\label{eq:staircase}
\sum_{h \in R^{(n)}} \ensuremath{p_T}\xspace^h <
\text{max}\left\{\epsilon_h^{(n)}\,\ensuremath{p_T}\xspace^\gamma\,,
E_\text{min}^{(n)}\right\},
\end{equation}
where the energies $E_\text{min}^{(n)}$, given in \tab{stairs}, are chosen so that they lie at or above the expected level of pile-up (up to an average of 60 $pp$ collisions per crossing) over Run 2 and 3 of the LHC.
Since event-by-event pile-up subtraction techniques will remove a significant fraction of the energy deposited in the isolation cone, this choice will assure that our technique will not suffer from large efficiency losses due to pile-up.
\begin{table}
\begin{center}
\begin{tabular}{| c | c | c |}\hline
$R$ & $\epsilon_h$ & $E_\text{min}$ \\ \hline \hline
0.1 & 0.01 & 5 GeV \\ \hline
0.2 & 0.07 & 10 GeV \\ \hline
0.3 & 0.20 & 23 GeV \\ \hline
0.4 & 0.38 & 40 GeV \\ \hline
\end{tabular}
\end{center}
\caption{Four concentric hard cones used to approximate smooth-cone isolation. $R$ is the cone angle, $\epsilon_h$ is the energy fraction, and $E_\text{min}$ is a threshold below which we do not reject events, regardless of hadronic energy fraction in the cone. Note that the value $\epsilon_h^{(1)}$ is so small that, in our kinematic regime, isolation in the innermost cone is always controlled by the energy cutoff $E_\text{min}$.}
\label{tab:stairs}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=0.475\linewidth]{stairCurve.pdf}~~
\includegraphics[width=0.5\linewidth]{stairVSfrix.pdf}
\end{center}
\caption{Comparing staircase isolation to the Frixione algorithm. (Left) The smooth curve is $\mathcal I(R;\epsilon,\delta)$ of \eq{frixfunc}, while the piecewise-constant curve is $\hat{\mathcal I}(R;\epsilon,\delta)$ of \eq{staircasefunc}. (Right) The effect, on $\sigma(\gamma\gamma)$ at NLO, of changing the isolation procedure.
Here, $\sigma$(smooth) corresponds to pure Frixione isolation, \eq{frix}, while $\sigma$(stair) is computed using \eq{staircase}. At high energies, staircase isolation is indistinguishable from the Frixione algorithm; even at low energies, the difference is slight.}
\label{fig:staircase}
\end{figure}
At right in \fig{staircase}, we compare our staircase isolation with the Frixione algorithm, by computing $\sigma(\gamma\gamma)$ with each isolation method and taking the relative difference of the results. The two methods differ by at most 4\% [2\%] in $\sigma(\gamma\gamma)$ $[\sigma(Z\gamma)$], and the difference decreases with energy. Staircase isolation thus shifts the central value of $R_{1a}$ $(R_{1b})$ $[R_{1c}]$ up by at most 2\% (4\%) [2\%] from the values computed in \sec{beyondLO} with smooth-cone isolation.
Now, having seen that the two photon-isolation procedures are not substantially different for our ratios, let us discuss the uncertainties associated with the staircase method. One source of uncertainties stems from the experimental extraction of the fragmentation function. We use the leading-order $q \to \gamma$ fragmentation function,
since our NLO calculations involve working only to leading order in $q\to q\gamma$ splitting. The photon fragmentation function for a quark parent has been measured most precisely at ALEPH~\cite{Buskulic:1995au}, in $Z\to \gamma$ + hadrons, in which the final state is dominated by $Z \to q\bar{q}\gamma$ and the fragmentation function contributes to the region where a quark or antiquark becomes collinear with the photon. The function extracted at leading order by ALEPH, based on a QCD analysis proposed in ref.~\cite{Glover:1993xc}, is
\begin{align}
D^\text{LO}_{\gamma\leftarrow q}(z,\mu_0) &= \frac{\alpha Q_q^2}{2\pi}
\left(P^{(0)}_{\gamma\leftarrow q} \log \frac{\mu_F^2}{\mu_0^2(1-z)^2} +C \right),\\
\label{eq:muBigUnc}
\mu_0 &= 0.22^{+1.3}_{- 0.19} \text{ GeV}\ , \\
\label{eq:cBigUnc}
C &= -12.1 \pm 4.3 \ ,
\end{align}
where $P^{(0)}_{\gamma \leftarrow q}$ is the tree-level perturbative splitting function.
Uncertainties on the two parameters appear large at first glance, but the parameters are highly correlated. ALEPH suggested that one should take the relation
\begin{equation}
\label{eq:correlatedparams}
C = \left.-1-\log\left({s \over 2\mu_0^2}\right)\right|_{s=m_Z^2}\,,
\end{equation}
and found
\begin{equation}
\mu_0 = 0.14^{+0.21+0.22}_{-0.08-0.04}\text{ GeV}, \quad C=-13.26\,.
\end{equation}
This uncertainty in $\mu_0$ propagates into a minute (per-mil) uncertainty in our ratios. But since the correlation in \eq{correlatedparams} is not assigned an uncertainty, this approach is slightly over-optimistic. On the other hand we can obtain an overly-conservative estimate if we ignore the correlation and vary both parameters independently by the uncertainties listed in \eqs{muBigUnc}{cBigUnc}. In this case we find uncertainties of about 1\% on our ratio $R_{1a}$. As this is surely a considerable over-estimate, we believe that this source of uncertainty is unimportant.
Several other sources might inflate the uncertainties of the isolation contribution if not handled correctly. First is the fact that, working to NLO in $V_1^0V_2^0$ production, we have done only a leading-order calculation for quark-photon splitting (and used the corresponding LO fragmentation function). However, since the sensitivity to the fragmentation function is minimized by the staircase method, we do not think the next order correction will affect our ratios in a material way. At the same time, since none of the currently available fragmentation function fits perform any resummation of the logarithms of $\log^2(1-z)$ that appear in the perturbative fragmentation contribution, one must be careful to implement isolation in such a way that one does not weight the $z \to 1$ region of phase space too strongly. The staircase isolation that we advocate here does precisely this, in contrast to hard-cone isolation with a small radius.
\subsection{\texorpdfstring{$Z$}{Z} decay and lepton isolation}
\label{subsec:Zdecay}
Up to this point we have treated the $Z$ as though it does not decay, and imposed the same cuts on $\gamma$ and $Z$ as shown in \tab{bosonJetCuts}. But the decay of the $Z$ forces us to impose kinematic and isolation cuts on its daughter leptons and to account for the $Z$ peak's width in defining what we mean by a $Z$. (We consider non-leptonic decays of the $Z$ briefly in \ssec{Final}.) This has a significant though highly predictable effect on the measurements.
To a good approximation we find that the effects of these three experimental realities factorize, meaning that the overall acceptance $\zeta$ of these three effects can be written as the product of separate acceptance factors:
\begin{equation}
\zeta = \zeta_{\Gamma,\Delta} \times \zeta_{\text{kin}} \times \zeta_{\text{iso}},
\end{equation}
where the $\zeta_i$ are defined as a relative change to the cross section due to a particular effect: $\zeta_{\Gamma,\Delta}$ is the acceptance after requiring the dilepton mass be within $\Delta$ of the $Z$ pole, $\zeta_{\text{kin}}$ is the acceptance of our lepton kinematic cuts, and $\zeta_{\text{iso}}$ is the acceptance of our lepton isolation cuts. Let us now discuss each of them in turn.
{\renewcommand{\arraystretch}{1.5}
\begin{table}
\begin{center}
\begin{tabular}{| c |}\hline Kinematic Cuts \\
\hline\hline
$| m_{\ell\ell}-m_Z| <$ 25 GeV \\ \hline
$\ensuremath{p_T}\xspace^{\ell_1} > 20$ GeV \\ \hline
$\ensuremath{p_T}\xspace^{\ell_{2,3,4}} > 7$ GeV \\ \hline
$|\eta(\ell)| < 2.5$ \\ \hline
\end{tabular}
\hspace{0.25in}
\begin{tabular}{| c |}\hline Isolation Cuts \\
\hline\hline
$\Delta R_{\ell \gamma} > 0.2$ \\ \hline
$\Delta R_{\ell^+\ell^-} \geq 0.0$ \\ \hline
$p_{T}(j) < 0.2 \times p_{T}(\ell)$ \\
if $\Delta R_{\ell j} < 0.4$ \\ \hline
\end{tabular}
\end{center}
\caption{Kinematic and isolation cuts imposed on daughter leptons. The leptons are \ensuremath{p_T}\xspace-ordered such that $\ensuremath{p_T}\xspace^{\ell_1} > \ensuremath{p_T}\xspace^{\ell_2} > \ensuremath{p_T}\xspace^{\ell_3} > \ensuremath{p_T}\xspace^{\ell_4}$.}
\label{tab:leptonCuts}
\end{table}
}
The $Z$'s finite width and $Z$--$\gamma^*$ interference require that we define what we mean by a $Z$ boson. We take a $Z$ to be an opposite-sign same-flavor dilepton pair whose mass $m_{\ell\ell}$ falls within $\Delta = 25$ GeV of $m_Z=91.2$ GeV. To quantify effects of this mass window, we define $\zeta_{\Gamma,\Delta}$ as the ratio of a finite-width cross section with mass window $\Delta$ divided by the (fictitious) zero-width cross section that we have used up to now. Note $\zeta_{\Gamma,\Delta}$ can exceed 1 if the window $\Delta$ is taken sufficiently wide.
Also at this stage, to remove a divergence in the $pp\to Z\gamma$ cross section, we apply an isolation cut between leptons and photons by requiring that $\Delta R_{\ell\gamma} > 0.2$ for any lepton-photon pair in the event. We find that if we change $\Delta R_{\gamma\ell}>0.2$ to $\Delta R_{\gamma \ell}>0.4$, the cross section changes by less than 0.5\%. This is unsurprising since the kinematic cuts of \tab{bosonJetCuts} force the $Z$ and $\gamma$ to be well-separated. The small effect of this isolation cut is included in $\zeta_{\Gamma,\Delta}$.
We next impose realistic kinematic cuts on individual leptons, as shown in the kinematic cuts section of \tab{leptonCuts}. In order to quantify the effects of kinematic cuts on individual leptons, we define $\zeta_\text{kin}$ as the ratio of cross sections with and without these kinematic cuts.
Although the $Z$ is not itself observed, we retain the cut on $Z$s shown in \tab{bosonJetCuts}; that is, we reject $Z$s with $|\eta|>1.5$ even if the leptons pass the cuts in \tab{leptonCuts}. This choice is somewhat arbitrary and may not be necessary, but making different $\eta$ cuts on $Z$s and $\gamma$s might inflate PDF uncertainties, which otherwise have substantial cancellations.
Finally, we impose isolation cuts between the daughter leptons of a $Z$ and all the jets in the final state. Specifically, we require that $p_{T}(j) < 0.2 \times p_{T}(\ell)$ for any jet-lepton pair with $\Delta R_{\ell j} < 0.4$. The effect of these isolation cuts is described by $\zeta_{\text{iso}}$. We define $\zeta_{\text{iso}}$ as the ratio of cross sections with and without the jet-lepton isolation cuts.
Our $Z$ bosons are often boosted. To avoid unnecessary acceptance losses, we do not require any lepton to be isolated from another lepton of the same flavor and opposite sign. We do not expect this to cause a large $Z$ fake rate at the relevant $\ensuremath{\bar{m}_T}\xspace$.
As shown in \fig{Zaccept}, these cuts lower the $Z$ acceptance; for $Z\gamma$ ($ZZ$) events, acceptance drops to 94\% (85\%) for the lowest values of $\bar{m}_T$, rising toward 100\% (90\%) at higher values. Losses are small at high $\bar{m}_T$ because the leptons have large \ensuremath{p_T}\xspace and have similar $\eta$ to the parent $Z$. Losses would be much greater (78\% and 58\% acceptance for $Z\gamma$ and $ZZ$) if all four leptons were required to have $\ensuremath{p_T}\xspace > 20$ GeV.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\linewidth]{ZA_leptoncuts.pdf}~~
\includegraphics[width=0.45\linewidth]{ZZ_leptoncuts.pdf}
\end{center}
\caption{Effects of finite $Z$ width and lepton cuts on $Z\gamma$ (left) and $ZZ$ cross sections (right). Each plot is normalized by the respective cross section with no lepton cuts and $\Gamma_Z=0$. The red circles show the effect of the $Z$ width.
The green triangles combine the width with the kinematic cuts of \tab{leptonCuts} and photon-lepton isolation cuts. The blue squares combine these with the other lepton isolation cuts of \tab{leptonCuts}. See text for more details and notation. }
\label{fig:Zaccept}
\end{figure}
These effects thus change our ratios, reducing cross sections by 5--8\% for each $Z$. However these effects are calculable and do not increase uncertainties. There should be no problem to include them in the theoretical predictions or unfold them from the experimental measurements.
\section{Discussion and summary}
\label{sec:Summ}
\subsection{Uncertainty budget}
\label{subsec:UncBudget}
{\renewcommand{\arraystretch}{1.4}
\begin{table}[h!]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}\hline
Effect
& $R_{1a}
& $R_{1b}
& $R_{1c}
& Comments \\
& ($Z\gamma/\gamma\gamma$) & ($ZZ/\gamma\gamma$)
& ($ZZ/Z\gamma$) & \\
\hline\hline
$qq\to VVqq$ & 2--3\% & 3--3.5\% & 1.5--2.5\%
& extrapolating $p_{T,\text{min}}^{j} \to 0$ (\ssec{NLO-QCD})\\ \hline
$\mu_R,\mu_F$ ($gg$) & 0.5--1\% & 1\% & 1--2\%
& uses NLO $gg\to\gamma\gamma$ (\ssec{pdf-scale})\\ \hline
$\mu_R,\mu_F$ (NLO) & 0.5--1\% & 1.5--2.5\% & 1--1.5\%
& varied independently (\ssec{pdf-scale})\\ \hline
PDF & 0.5\% & 1--1.5\% & 0.5--1\%
& MSTW 2008 using MCFM (\ssec{pdf-scale})\\ \hline
\hline
EW (LL) & $^{+2\%}_{-1\%}$ & $^{+3\%}_{-1\%}$ & $^{+2\%}_{-1\%}$
& EFT scale uncertainty (\ssec{doubleLog}) \\ \hline
$\alpha_\text{QED}$ & 7\% & 14\% & 7\%
& Fully correlated (\ssec{alphaQED})\\ \hline
\end{tabular}
\end{center}
\caption{Summary of overall uncertainty budget. The first three entries are not independent sources of uncertainty, and combining them assuming no correlation provides a conservative estimate.}
\label{tab:uncBudget}
\end{table}
}
In \sec{ExecSumm}, we presented our claim that the three $R_1$ ratios (whose central values are related but which have different cancellations among their uncertainties) are under exceptional theoretical control. Here we present a detailed breakdown of what we include in our estimate of known theory uncertainties, as shown in \tab{uncBudget}, and justify our confidence in the small size of further higher-order effects. We now review the table line by line.
The first three lines of \tab{uncBudget} are not truly independent, as they are all striving to capture aspects of the uncertainty associated with higher-order corrections to our calculations of the ratios. Our goal in isolating them was to try to identify any particularly large effects, ones that would not show up in overall NLO scale variations, that we have not already included and would not cancel in our ratios. Although the separation we have made is both scheme and scale dependent and thus unphysical, our methods are probably sufficient to estimate the rough magnitude of the higher-order corrections that we did not include. We have also been quite conservative in our estimates and in how we combined uncertainties.
Once NNLO calculations of all diboson processes become publicly
accessible, the uncertainties from all sources should be subsumed in the
scale variation of the analogous NNLO calculations, with the exception of
the $gg$ initial state, which only first appears at NNLO. For this last
part, two-loop results for $\gamma\gamma$ and $ZZ$ already exist \cite{Bern:2002jx,Caola:2015psa}, as do
most components of the $Z\gamma$ calculation \cite{Gehrmann:2013vga}, allowing for a more robust
characterization of the associated uncertainties than the estimates we have
performed here.
As we noted in \ssec{NLO-QCD} above, many NNLO corrections are expected to cancel in the $R_1$ ratios. Valence quark scattering $qq\to V_1^0V_2^0 qq$, which has terms that are not proportional to the LO cross sections, gives one of the largest non-canceling terms that we cannot currently compute. Our method for obtaining these estimates was described in \ssec{gg}.
We obtained estimates of the $\ensuremath{O}(\alpha_s^4)$-uncertainty in $gg\to \gamma\gamma$ production by varying the scales $\mu_R, \mu_F$ in $gg\to \gamma\gamma$, computed to $\ensuremath{O}(\alpha_s^3)$, up and down by a factor of two. Because of $SU(2)\times U(1)$ relations, we assumed that nearly the same relative uncertainty applies to $gg\to Z\gamma$ and $gg\to ZZ$. See \ssec{pdf-scale} for more details.
Although it is possible that there are other large non-cancelling NNLO effects, we have not been able to identify them. In particular, although collinear effects make a large contribution at NLO, their contribution at NNLO appears to be much smaller. Moreover, there are no other new channels or new regions of phase space that open up at this order. Consequently we naively expect other NNLO shifts to the ratios to largely cancel.
We estimated these effects in \ssec{pdf-scale} by seeing how varying the renormalization and factorization scales for the strictly NLO calculations independently affect the ratios; the $\sim 5\%$ corrections in each channel have very substantial correlations, and largely cancel in the ratios.
PDF uncertainties are extracted from the calculations that led to \fig{pdfunc}. We find that they are very small for $R_{1a}$, and even for the other ratios are significantly smaller in percentage terms than for the individual diboson processes.
Now we turn to the EW uncertainties. The leading-log EW uncertainties, dominated by the choice of matching scales, were extracted from the threshold resummation calculation of ref.~\cite{Becher:2013zua} as described in \ssec{EW-corr}.
We also account for the differing views of how to set the scale for $\alpha(\mu_\text{QED})$ by varying $\mu_\text{QED}$ between $0$ and $m_Z$.
Note this is a window and not in any sense a $1\sigma$ Gaussian uncertainty.
One item for which we do not have an error estimation is photon isolation. An essential part of our proposal involves the use of staircase isolation, an experimentally practical approximation to the Frixone smooth-cone method, discussed in \ssec{PhotonIso}. The use of a hard cone for isolation would introduce a substantial shift to our result and significant sensitivity to $q \to \gamma$ fragmentation.\footnote{Note that uncertainties in the fragmentation function might be reducible. The ALEPH measurement could be repeated at the LHC, using $W$ decays arising in $t\bar t$ events. Selecting events with a lepton, $\slashed{E}_T$, two $b$ tags and a loose photon, one could then reconstruct the tops and extract the probability that $W\to\gamma$ + hadrons.} Staircase isolation minimizes these effects. We saw in \ssec{PhotonIso} that the difference between smooth and staircase cone, most important at low $\bar m_T$, is at most $2 \%$ for $R_{1a}$ and $R_{1c}$, and double this for $R_{1b}$. The effect of experimental uncertainties on the fragmentation function, which we estimated by varying the parameters in ALEPH's fit, appears to be negligible.
\subsection{Final comments}
\label{subsec:Final}
\subsubsection{General reflections on our methods}
We have proposed a wide variety of ratios using LO reasoning about the $SU(2)\times U(1)$ structure of the SM. Interestingly, the structure of $SU(2)\times U(1)$ and the radiation zero in the amplitude $a_3$ means that the naive guess for custodial-$SU(2)$ relationships among $W$ and $Z$ do not hold.
The only interesting relation between $ZZ$, $W^\pm Z$ and $W^-W^+$ production is an imperfect (and somewhat impractical) relation between $W^-W^+$ and $ZZ$, which follows only because $|a_3|^2$ is subdominant in $W^-W^+$ production. This could be generalized to a relation between $W^-W^+$, $W^\pm Z$ and $ZZ$, but the relation is complicated as well as impractical. We also saw no interesting relation between $W^\pm \gamma$ and $Z\gamma$. That said, charge ratios of $W^+Z$ to $W^-Z$, and of $W^+\gamma$ to $W^-\gamma$, are important tools. Although we focused on diboson ratios at high $\bar m_T$, the nice properties of these charge ratios do not require $SU(2)\times U(1)$, and would remain interesting even down to low $\bar m_T$.
An issue that we have not addressed is the experimental systematic uncertainty from fake photons. We have seen in the Higgs boson search that the experiments have fairly large contributions to their $\gamma\gamma$ searches from $\gamma + \text{jet}$. Although these decrease at moderate \ensuremath{\bar{m}_T}\xspace, partially cancel in our ratios, and are typically smaller for photons in the barrel of ATLAS and CMS, they are by no means negligible, as can be seen in ref.~\cite{Khachatryan:2015qba}. We have implicitly assumed that the systematic uncertainties from these fakes will be under very good control for $\ensuremath{p_T}\xspace^\gamma \geq 150\text{ GeV}$ by the time 300 fb$^{-1}$ has been accumulated. If this is not true it could make the $R_1$ ratios, and others we have proposed, somewhat less useful.
We have limited our detailed study to NLO QCD effects, for practical reasons. Many NNLO QCD and higher order EW corrections have been performed already, so it should soon be possible to improve upon our results and, most importantly, check our uncertainty estimates. At NNLO, with two jets accompanying the two bosons, one would encounter many new issues, including vector boson scattering and potential sensitivity to new phenomena therein, as well as $SU(2)$-quintet amplitudes, including same-sign $WW$ production. However, few of these issues may be essential in the ratios we propose, since the low rates for diboson production mean that theoretical predictions more precise than a few percent may often not be needed at the LHC.
Our results for the $R_1$ ratios involved many arbitrary choices including specific kinematic cuts, isolation requirements, binning, etc. Although we have carefully considered these choices, we have not in any sense optimized them, and further consideration, both theoretical and experimental, should be given to them.
Finally, in our results we have imposed an isolation cone around photons but not around $Z$s. This appears to be a sub-percent effect (for each $Z$) with the isolation criteria that we selected. However, at a higher energy collider this must be revisited, since at sufficiently high energy the $Z$ and photon will have to be treated on equal footing and a photon-like isolation on the reconstructed $Z$ will have to be applied; otherwise large EW logarithms will afflict our ratios.
\subsubsection{Other $Z$ decays}
The $R_1$ ratios, especially the two involving $ZZ$, suffer from low statistics, due to the small $Z\to \ell^+\ell^-$ branching fraction. One might wonder whether one can gain by looking at $Z\gamma$ events in which the $Z$ decays to neutrinos, and especially at $ZZ$ events by looking for $\ell^+\ell^-$ plus missing transverse momentum ($\slashed{E}_T$). We have not explored this, but in the ATLAS measurement of the $ZZ$ production cross section~\cite{Aad:2012awa}, such signal events are incorporated.
An obvious downside to this approach would be an inability to put the same $\eta$ cuts on $Z$s and $\gamma$s. Since these $SU(2)$-singlet processes are generated in the $t$ and $u$ channel, they are particularly sensitive to the $\eta$ cut, so having different cuts for $\gamma$ and $Z$ could potentially cause large NLO corrections due to imperfect cancellations. Also, the excellent cancellation of PDF uncertainties in $R_{1a}$ could potentially fail. There would also be backgrounds from $W\gamma$ and $WZ$ events where the $W$ decays to a hadronic tau or a soft lepton, and is mistaken for an invisibly decaying $Z$. The ideal balance between smaller statistical uncertainties and larger theoretical uncertainties will be time-dependent, and requires study by the experimental LHC groups at the time of the measurement.
Nevertheless, there might be a practical strategy using $Z\to\nu\bar\nu$ events. One could measure $\gamma\gamma$ differentially, and use the $R_{1a}$ and $R_{1b}$ ratios to predict, in the central region, the $Z\gamma$ and $ZZ$ distributions. Then, assuming the SM, the full $Z\gamma$ and $ZZ$ distributions extending to larger $\eta$ could be predicted with lower uncertainties, and this prediction could then be checked against events with $\slashed{E}_T$ and a single $\gamma$ or leptonic $Z$.\footnote{A similar method is presumably needed for any ratios involving $W^+W^-$ production, where both $\eta$ and especially $\ensuremath{\bar{m}_T}\xspace$ are somewhat uncertain in each event.}
The option of using hadronic decays of the $Z$ seems daunting. The backgrounds from $Z$- or $\gamma$-plus-jet events, where a QCD jet fakes a boosted $W$ or $Z$, are not small, and will leak into the diboson measurement. Moreover, mass- and charge resolution on hadronically decaying vector bosons is poor, so one cannot distinguish $Z\gamma$ from $W^\pm\gamma$, processes with completely different differential distributions even at LO.
A further tool that we have not explored is the use of $Z$ polarization, potentially of interest due to the parity violation in the SM. BSM physics might alter polarization ratios.
\subsubsection{Applications of the $R_1$ ratios}
The $R_1$ ratios should be useful in several ways even within the SM. First, they allow high-precision tests of SM calculations, including Monte Carlo methods. Second, they may serve as a place to explore higher-order EW effects. As we described in \ssec{EW-corr}, EW corrections partly cancel in these ratios, but can reach the 10--20\% level, above the level of theoretical uncertainties. The fact that QCD corrections cancel rather completely, especially at high \ensuremath{\bar{m}_T}\xspace, means these ratios may serve as a particularly clean place to examine logarithmically-enhanced EW effects.
In this paper we have not addressed the question of how sensitive these variables would be to BSM phenomena. An obvious potential use of these variables is in searching for BSM interactions of the SM gauge bosons, e.g.~through anomalous triple and quartic gauge couplings (aTGCs, aQCSs). In exploring this, one should use an $SU(2)\times U(1)$ invariant classification of the various operators. We leave this for future work. Wide resonances decaying to EW bosons might also alter the ratios without being observable in some simpler way. Most other phenomena would introduce additional hard jets, which would often be vetoed by our cuts, or large amounts of $\slashed{E}_T$, which would not impact the $R_1$ ratios but could affect ratios involving leptonic $W$s, as well as any measurements that try to use $Z \to \nu\bar{\nu}$ as we outlined above. Strategies for events with additional jets and/or $\slashed{E}_T$ are worth further study, but it is far from clear whether our methods can be suitably generalized to such cases.
At higher collision energy, such as a 30 or 100 TeV $pp$ collider, these observables are probably still useful, but will perhaps be more complicated. On the one hand, finite mass effects will be completely negligible, and the fragmentation contribution for $W$s and $Z$s will become similar to that for photons. EW corrections become quite large and could easily be observed in these ratios, as would any TeV-scale new physics effects. However, other issues, such as the non-negligible rate for a hard lepton to radiate a real EW boson, as well as the general challenges of resolving and identifying gauge bosons at ultra-high boosts, will begin to have a practical impact on precise diboson measurements. A dedicated study of this question is needed.
\subsubsection{Prospects for the other ratios}
What can we expect for the other ratios of \eq{OurRatios} at NLO? The $R_1$ ratios are somewhat special. First, the $\gamma\gamma$, $Z\gamma$ and $ZZ$ processes are fully reconstructible, though at the cost of $Z\to \ell^+\ell^-$ branching fractions. By contrast, $W\gamma$ and $WZ$ events with a single neutrino are only reconstructible up to a two-fold ambiguity, and leptonic $WW$ events cannot be reconstructed event-by-event.
Second, $\gamma\gamma$, $Z\gamma$ and $ZZ$ cross sections are all proportional to the singlet amplitude-squared $|a_1|^2$ at LO, and many of their NLO corrections are identical at high energy. This is not true for the other processes. A particular complication is the fact that the $SU(2)$-triplet amplitude-squared $|a_3|^2$ vanishes at scattering angle $\pi/2$, or in other words at $\hat s=4 \bar m_T^2$. The falling PDFs assure this is a kinematic region of particular importance for production rates at hadron colliders. Differential cross sections for $W^\pm\gamma$, $W^\pm Z$ and $W^-W^+$ are suppressed at LO by this ``radiation zero'', but to different degrees, and what remains behind is different in each case. The radiation zero is removed at NLO, and consequently some ratios, particularly certain asymmetries which are quite small at LO, may end up with large NLO corrections and NNLO uncertainties. Indeed it is already well-known that the $\ensuremath{K}\xspace_\text{NLO/LO}$ factor for $W\gamma$ is much larger than that for $Z\gamma$ \cite{Ohnemus:1992jn,Grazzini:2015nwa}.
Despite these challenges, there are enough variables in our list that some may evade these concerns. We are optimistic that a few of the remaining variables will be as precisely predictable as the $R_1$ ratios, and we plan to explore this possibility further. In the meantime, we hope that our methods will inspire invention of other precision observables, perhaps more sophisticated and less obvious, for the LHC and for hadron colliders of the future.
\acknowledgments
We are grateful for conversations with T.~Becher, Z.~Bern, J.~Campbell, C.~Rogan, G.~Salam, M.~Schwartz, K.~Tackmann, and C.~Williams. The authors were supported in part by US Department of Energy grant DE-SC0013607, and National Science Foundation grants PHY-1258729, PHY-0855591, and PHY-1216270. The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
M.J.S.~thanks Harvard University for support and hospitality during this project, as well as the Simons Foundation for support. For hospitality and support during this research, M.F. and M.J.S. thank the Galileo Galilei Institute, C.F. thanks the Perimeter Institute, J.S. and M.J.S. thank the CERN theory group, J.S. thanks the Munich Institute for Astro- and Particle Physics, and M.F. thanks the Aspen Center for Physics (via NSF grant PHY-1066293) and the Center for Future High Energy Physics, Beijing.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Unconventional superconducting states are often defined by the breaking of an additional symmetry beyond global gauge invariance \cite{AnnettReview,AnnettRev,SigristUeda}. Typically this means a reduction of the point group symmetry, but could also include breaking time reversal or, for triplet superconductors, spin rotation symmetries \cite{Vollhardt&Wolfle}.
Starting from the SO(3)$\times$SO(3)$\times$U(1) symmetry of superfluid $^3$He \cite{Vollhardt&Wolfle}, the discussion of unconventional superconductivity has focused on superconductivity in high symmetry environments. In this context, unconventional superconducting states can be characterised by the irreducible representations of the point group of the crystal \cite{AnnettReview,AnnettRev,SigristUeda}.
The symmetry of the superconducting order parameter provides important clues about the mechanism of superconductivity. As such the determination of the exact form of the superconducting gap is of significant importance. In practice, such a determination is not straightforward. The Josephson interference experiments responsible for unambiguously identifying the `d$_{x^2-y^2}$-wave' symmetry of the cuprates \cite{Wollman,Tsuei} have not been possible in many materials. The interpretation of other experimental results can be ambiguous. In particular, the limiting low temperature behaviours of many experimental probes (e.g., heat capacity, nuclear magnetic relaxation rate, or penetration depth) can, in principle, distinguish between a fully gapped state, line nodes, and point nodes. However, these results are often controversial. And, even in principle, such experiments cannot differentiate between different gap symmetries with the same class of nodes (point or line). This has led to the study of directional probes, such as thermal conductivity \cite{IzawaOrganic,IzawaSRO}.
However, superconductivity is observed in many materials with rather low point group symmetries. Non-centrosymmetric materials are a prominent example. Here spin-orbit coupling can mix singlet and triplet superconducting states \cite{Sigrist}. Many organic superconductors, e.g., those based on the BEDT-TTF, Pd(dmit)$_2$, TMTSF, or TMTTF molecules, form monoclinic or orthorhombic crystals \cite{Ishiguro}. This means that superconducting symmetries that are distinct on the square lattice such as s-wave, d$_{xy}$, or d$_{x^2-y^2}$ often belong to the same irreducible representation \cite{BenGroupTh,BenGroupTh2}.
Similarly, a number of transition metal oxides with orthorhombic crystal structures superconduct \cite{Dynes,Ono,YBCOPG,RotSymLSCO}. In some cuprates, chemical doping results in a distortion of the lattice, reducing the rotational symmetry to C$_2$ (i.e., orthorhombic as opposed to tetragonal). This distortion is on the order of $<10\%$ of the lattice spacing \cite{YBCOPG,RotSymLSCO}.
Emergent physics can also lower the symmetry of a material, for example, via electronic `nematicity'. Indeed, in some cuprates, even if the crystal lattice is constrained to reduce this distortion, evidence of electronic nematicity has been observed in transport properties \cite{RotSymLSCO}, while nematic phases (with reduced rotational symmetry) have been theorised, resulting from spin or charge density wave order \cite{NemCDWSDW}, and evidence of such phases, and their connection to the pseudogap phase, has been observed in some cuprate materials from magnetic torque measurements \cite{NPhys_NemThermo}. Additionally, nematic phases arise in iron-based superconductors \cite{Kasahara12,FernandesMillis13} (in fact, as temperature is lowered, FeSe undergoes a structural transition to an orthorhombic state well above the superconducting critical temeprature \cite{FeSeOrth}) and strong anisotropy has been observed in resistivity measurements of the heavy-fermion superconductor CeRhIn$_5$ \cite{NemInCeRhIn5}, indicating the presence of some nematic order.
Superconductivity in materials such as the cuprate, organic, heavy fermion and iron-based families of unconventional superconductors, is widely believed to arise from electronic correlations. These unconventional superconductors share many similar properties, including complex phase diagrams with multiple phases and (spin singlet) superconductivity in particular proximity to some magnetically ordered phase. While the order parameter is believed to be anisotropic in the majority of these materials, disagreement remains over the exact form of the gap function in many materials. For example, in the organic superconductors, specific heat measurements have been taken to indicate nodeless (`s-wave') superconductivity \cite{Wosnitza00,Muller02,Wosnitza03} while others indicate the presence of nodes of the gap function \cite{Nakazawa97,Taylor07,Malone10}, and similarly penetration depth measurements were inconclusive until recently \cite{BrounkBr}. This has led both theorists \cite{Schmalian,BenGroupTh,BenRVB2,Guterding16,PowellPRL17} and experimentalists \cite{Dion,Guterding16PRL} to discuss the possibility of accidental nodes in organic superconductors. In the iron-based superconductors both `d-wave' (nodal) gap structures and nodeless `s$_\pm$-wave' structures, with band-dependent magnitudes, have been proposed for various materials \cite{ChenFeSe,Hanaguri_spm}, while in some heavy fermion superconductors a band-dependent gap symmetry has been discussed \cite{Broun2,Tanatar_HFgap} (i.e. with nodes present on some bands and isotropic gap magnitude on others).
There are important differences between accidental nodes and those required by symmetry. In the latter case, the location of the nodes is restricted to satisfy a symmetry constraint (for example, a reflection through the plane of the node line) and therefore the gap function transforms as a non-trivial representation of the point group. In the case of accidental nodes, the positioning of the nodes is unrestricted by symmetry requirements, and the gap function transforms as the trivial representation of the point group, the same representation to which an isotropic gap function belongs. This also allows the possibility of a mixed symmetry `s+d-wave' state \cite{Guterding16,Guterding16PRL}. For example, the `d$_{xy}$-wave' and `d$_{x^2-y^2}$-wave' gaps belong respectively to the B$_{2g}$ and A$_{1g}$ (trivial) representation of the D$_{2h}$ point group, which captures the orthorhombic symmetry of a square lattice with a rectangular distortion \cite{BenGroupTh,groupfoot}. As many models find d$_{x^2-y^2}$ superconductivity on the square lattice one expects that, at least for small distortions, this will also be the dominant superconducting channel for similar models with D$_{2h}$ point group symmetry. However, generically a real material with this symmetry will be able to lower its energy by producing an admixture of isotropic (s-wave) superconductivity, e.g., via sub-dominant interactions. If this admixture is small it will not remove the nodes, but will move them (note that an admixture with a complex phase breaks time reversal symmetry, and so will not be considered here).
Below we consider the nuclear spin-lattice relaxation rate $1/T_1$ in superconductors with accidental nodes. We show that in clean non-interacting models there is a logarithmic divergence $1/T_1T$ as $T\rightarrow T_c$ even if there is no isotropic component of the gap, $\Delta_{\bm k}$, i.e. $\int d^3{\bm k}\Delta_{\bm k}=0$, where $T$ is the temperature and $T_c$ is the superconducting critical temperature. We show, numerically in a D$_{2h}$ symmetric model -- similar to those discussed above -- that this divergence is controlled but not removed entirely by either disorder or electron-electron interactions, giving rise to a Hebel-Slichter-like peak. However, it shows some subtle differences from the true Hebel-Slichter peak both in its microscopic origin and in that it is not controlled by gap anisotropy.
\section{Nuclear magnetic resonance and the relaxation rate $1/T_1T$}
The spins of atomic nuclei relax by exchanging energy with their environment. In the case of a metal or superconductor this means the conduction electrons.
Thus the relaxation rate of nuclei in an electronic environment is related to the transverse dynamic susceptibility of the quasiparticles, $\chi_{+-}\left(\bm{q},\omega\right)=\chi'_{+-}\left(\bm{q},\omega\right)+i\chi''_{+-}\left(\bm{q},\omega\right)$, via \cite{Slichter,Tinkham}
\begin{eqnarray}
\frac{1}{T_1T}&=& \lim\limits_{\omega\rightarrow 0} \frac{2k_B}{\gamma_e^2\hbar^4}\sum\limits_{\bm{q}}\left|A_H\left(\bm{q}\right)\right|^2\frac{\chi''_{+-}\left(\bm{q},\omega\right)}{\omega},\label{T1T_Gen}
\end{eqnarray}
where $\gamma_e$ is the (electron) gyromagnetic ratio and $A_H\left(\bm{q}\right)$ is the hyperfine coupling, which we approximate by a point contact interaction [$A_H\left(\bm{q}\right)=A_H$] for simplicity below. Neglecting vertex corrections, the dynamic susceptibility can be expressed in terms of the spectral density function, $A_{\bm{k}}\left(E\right)$, \cite{Mahan,Coleman,Eddy_K}
\begin{widetext}
\begin{eqnarray}
\chi''_{+-}\left(\bm{q},\omega\right)&=&\sum\limits_{\bm{k}}\int\limits_{-\infty}^{\infty} \frac{dE_1dE_2}{4\pi^2}\left\lbrace \frac{1}{2}\left[1+\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[f\left( E_{2}\right)-f\left(E_{1}\right)\right]\delta\left[\omega -\left(E_{2} - E_{1}\right)\right]A_{\bm{k}}\left(E_{1}\right)A_{\bm{k}+\bm{q}}\left( E_{2}\right)\right. \nonumber\\
&&+\frac{1}{4}\left[1-\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[\bar{f}\left( E_{2}\right)-f\left(E_{1}\right)\right]\delta\left[\omega +\left(E_{2} + E_{1}\right)\right]A_{\bm{k}}\left(E_{1}\right)A_{\bm{k}+\bm{q}}\left( E_{2}\right)
\nonumber\\
&&\left. +\frac{1}{4}\left[1-\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[f\left( E_{2}\right)-\bar{f}\left(E_{1}\right)\right]\delta\left[\omega -\left(E_{2} + E_{1}\right)\right]A_{\bm{k}}\left(E_{1}\right)A_{\bm{k}+\bm{q}}\left( E_{2}\right)\right\rbrace , \label{GeneralChi1}
\end{eqnarray}
\end{widetext}
where $\xi_{\bm{k}}$ is the single particle dispersion, measured from the chemical potential,
$E_{\bm{k}}=\sqrt{\xi_{\bm{k}}^2+\left|\Delta_{\bm{k}}\right|^2}$, $f\left(E\right)$ is the Fermi-Dirac distribution function, and $\bar{f}\left(E\right)=1-f\left(E\right)$. The coherence factors in the above expression arise naturally from the trace over Nambu spinors.
From Eq. (\ref{GeneralChi1}) we consider three approximations: (i) The clean, BCS limit, where interactions are limited to those giving rise to superconductivity and no disorder is present.
(ii) Uncorrelated disorder, where electron-impurity interactions result in a broadening of the peak in the spectral function (i.e. giving rise to a finite quasiparticle lifetime). The strong disorder regime is irrelevant as strong disorder suppresses unconventional superconductivity \cite{Mineev&Samokhin}. And (iii) disorder and electron-electron interactions, with the latter taken into account via the random phase approximation (RPA). In the following analysis we will treat the clean BCS limit analytically, then extend our results to account for disorder and electron-electron interactions numerically.
\begin{figure}
\includegraphics[trim = 40mm 75mm 50mm 75mm, clip, width=0.4\textwidth]{Figure_contours-v1.pdf}
\caption{Contours of constant energy for an isotropic gap (black dashed lines), symmetry required nodes (blue dot dashed line) and accidental nodes (red dotted line), sketched for an elliptical Fermi surface. The black dotted line indicates the position of the accidental node, while the symmetry required nodes reside on the axes (with the contour around the $k_x$-axis highlighted).}
\label{contourFig}
\end{figure}
\section{The clean limit}
Inserting Eq. (\ref{GeneralChi1}) for the dynamic susceptibility into the expression for the relaxation rate, Eq. (\ref{T1T_Gen}), we have
\begin{eqnarray}
\frac{1}{T_1T} &\propto &\frac{1}{4\pi^2}\int\limits_{-\infty}^{\infty} dE\left[-\frac{df}{dE}\right]
\sum_{n=1}^3
\left[\mathcal{K}_n\left(E\right)\right]^2 ,\label{GeneralKernel}
\end{eqnarray}
where
\begin{subequations}
\begin{eqnarray}
\mathcal{K}_1\left(E\right)&=&\sum_{\bm{k}}A_{\bm{k}}\left(E\right),\\
\mathcal{K}_2\left(E\right)&=&\sum_{\bm{k}}\frac{\xi_{\bm{k}}}{E_{\bm{k}}}A_{\bm{k}}\left(E\right),\\
\mathcal{K}_3\left(E\right)&=&\sum_{\bm{k}}\frac{\Delta_{\bm{k}}}{E_{\bm{k}}}A_{\bm{k}}\left(E\right).
\end{eqnarray}
\label{eq:Kdef}
\end{subequations}
Each term in Eq. (\ref{GeneralKernel}) represents the average of a function over a contour of approximately constant energy $E_{\bm{k}}$, due to the peak in $A_{\bm k}(E)$ at $E\simeq E_{\bm k}$. These averages are then integrated over energy, with the integral restricted by the derivative of the Fermi function to a range of order $k_BT$. For an s-wave superconductor, this contour will wrap around the entire Fermi surface, while for a gap with nodes, the contour will form closed surfaces around the nodes (for energies smaller than the maximum gap). Examples of these contours are shown in Fig. \ref{contourFig}.
\begin{figure}
\includegraphics[trim = 50mm 105mm 45mm 45mm, clip, width=0.4\textwidth]{Figure_anglecombined-v1.pdf}
\caption{The geometry of a system with accidental nodes. Arrows denote the gradients of the normal dispersion ($v_F$) and the gap function ($v_\Delta$), in the local co-ordinate system defined at the node. The solid line denotes the Fermi surface and the dotted line the node location, while $k_\parallel$ and $k_\perp$ respectively denote the coordinates parallel and perpendicular to the Fermi surface.}
\label{angleFig}
\end{figure}
\subsection{Anisotropic gap with accidental nodes}\label{accidental}
In the clean limit, the spectral functions are given by Dirac delta functions,
\begin{eqnarray}
A_{\bm{k}}\left(E\right)&=&\pi\delta\left(E_{\bm{k}}-E\right), \label{Spectral}
\end{eqnarray}
each of which can be decomposed \cite{Arfken,G&V} into delta functions acting on $k_\perp$, the component of $\bm{k}$ perpendicular to the Fermi surface,
\begin{eqnarray}
\delta\left(E_{\bm{k}}-E\right)&=&\sum\limits_{\bm k_\perp\left(E\right)}\left|\frac{\partial E_{\bm{k}}}{\partial \bm k_\perp}\right|^{-1}\delta\left[\bm k_{\perp}-\bm k_{\perp}\left(E\right)\right].
\end{eqnarray}
These delta functions then constrain $\bm k_\perp$ to the value corresponding to the constant energy surface, $\bm k_\perp\left(E\right)$. The gradient of the gap function is then, in terms of
$\bm{k}_\perp$ and a local $\left(D-1\right)$-dimensional manifold parallel to the Fermi surface, $\{\bm{k}_i\}$, given by
\begin{eqnarray}
\bm{v}_{\Delta}\equiv\boldsymbol{\nabla}_{\bm{k}}\Delta_{\bm{k}}&=&\sum\limits_{i=1}^{D-1}\hat{\bm{k}}_{i}\left(v_{\Delta}\right)_i +\hat{\bm{k}}_{\perp}v_{\perp},
\end{eqnarray}
where $\left(v_{\Delta}\right)_i=\bm{v}_{\Delta}\cdot\bm k_i$ is the projection of $\boldsymbol{\nabla}_{\bm{k}}\Delta_{\bm{k}}$ onto $\bm{k}_i$, the $i$th dimension on the manifold parallel to the Fermi surface, $v_{\perp}=\bm{v}_{\Delta}\cdot\bm k_\perp$ is the projection of $\boldsymbol{\nabla}_{\bm{k}}\Delta_{\bm{k}}$ onto $\bm{k}_\perp$, $\bm k_i=\hat{\bm k}_ik_i$, and $\bm k_\perp=\hat{\bm k}_\perp k_\perp$. In general, the $(v_{\Delta})_i$ and $v_{\perp}$ are functions of the position on the manifold. In the simplest, two dimensional, case (with $\bm{k}_i=\bm{k}_\parallel$), this gives
\begin{eqnarray}
\bm{v}_{\Delta}\equiv\boldsymbol{\nabla}_{\bm{k}}\Delta_{\bm{k}}&=&\hat{\bm{k}}_{\parallel} v_{\Delta} \cos\theta
+\hat{\bm{k}}_{\perp}v_{\Delta} \sin\theta ,
\end{eqnarray}
where $\theta $ is the angle between the nodal line and the normal to the Fermi surface (see Fig. \ref{angleFig}).
In general, the gap function will be independent of the coordinate parallel to the node line. Importantly, in the accidental case this is not required to be normal to the Fermi surface, which results in a nonvanishing average of the gap over the Fermi surface. The Hebel-Slichter peak present in $1/T_1T$ in an isotropic superconductor is a probe of this average gap \cite{AnnettRev}. The angle $\theta$ parametrises the existence of such a nonvanishing average gap, and can be defined via the overlap between the gradients of the gap and dispersion, as the dispersion varies solely in the direction normal to the Fermi surface,
\begin{eqnarray}
\sin\theta =\frac{\bm{v}_F\cdot \bm{v}_{\Delta}}{\left|\bm{v}_F\right|\left|\bm{v}_{\Delta}\right|},\label{sinth_def}
\end{eqnarray}
where $\bm{v}_{F}=\boldsymbol{\nabla}_{\bm{k}}\xi_{\bm{k}}$ is the Fermi velocity and $\theta=\phi+\pi/2$, where $\phi$ is the angle between $\bm{v}_{\Delta}$ and $\bm{v}_F$. Near the node the energy is then $E_{\bm{k}} \sim \sqrt{\left[v_Fk_{\perp}\right]^2+v_\Delta^2\left[k_\parallel \cos\theta + k_{\perp}\sin\theta \right]^2}$.
The delta function in Eq. (\ref{Spectral}) {then constrains the component perpendicular to the Fermi surface, allowing the simplification,
\begin{widetext}
\begin{eqnarray}
\delta\left[E_{\bm{k}} - E\right]&=&\sum\limits_{k_\perp\left(E\right)}\left|\frac{E}{v_F\sqrt{E^2-\left|\Delta_{\bm{k}}\right|^2}+\Delta_{\bm{k}}v_{\Delta} \sin\theta }\right|\delta\left[k_\perp-k_\perp\left(E\right)\right].\label{myDeltaExp}
\end{eqnarray}
Previous analyses \cite{Tinkham,Samokhin} have primarily focused on the symmetry required case. A symmetry required node must reside on an axis of symmetry, to which the Fermi surface must be perpendicular; thus $\theta=0$. In fact, we find that the vanishing of this angle is responsible for the lack of analytical divergences encountered in the symmetry required case, see Section \ref{sect:req}.
As a first approximation, we perform a binomial expansion in $\Delta_{\bm{k}}v_{\Delta}\sin\theta/v_F\sqrt{E^2-\left|\Delta_{\bm{k}}\right|^2}$ in the denominators in Eqs. (\ref{eq:Kdef}). Such an approximation is valid for $T\sim T_c$, where $\Delta_{\bm{k}}\rightarrow 0$ provided $v_\Delta/v_F$ is not too large; i.e., away from van Hove singularities, where $v_F$ vanishes. Additionally, for sufficiently small $\sin\theta $, such an expansion will be reasonable at all temperatures given the same caveat.
Performing the expansion gives
\begin{subequations}
\begin{eqnarray}
\mathcal{K}_1\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{E}{v_F\sqrt{E^2-\Delta_{\bm{k}}^2}+\Delta_{\bm{k}}v_{\Delta} \sin\theta }\approx \int_{E}d\bm{k}_\parallel \left[\frac{E}{v_F\sqrt{E^2-\Delta_{\bm{k}}^2}}-\frac{E\Delta_{\bm{k}}v_{\Delta} \sin\theta}{v_F^2\left(E^2-\Delta_{\bm{k}}^2\right)}\right]\label{Terms1}\\
\mathcal{K}_2\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{\sqrt{E^2-\Delta_{\bm{k}}^2}}{v_F\sqrt{E^2-\Delta_{\bm{k}}^2}+\Delta_{\bm{k}}v_{\Delta} \sin\theta }\approx \int_{E}d\bm{k}_\parallel \left[\frac{1}{v_F}-\frac{\Delta_{\bm{k}}v_{\Delta} \sin\theta }{v_F^2\sqrt{E^2-\Delta_{\bm{k}}^2}}\right]\label{Terms2}\\
\mathcal{K}_3\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{\Delta_{\bm{k}}}{v_F\sqrt{E^2-\Delta_{\bm{k}}^2}+\Delta_{\bm{k}}v_{\Delta} \sin\theta } \approx \int_{E}d\bm{k}_\parallel\left[ \frac{\Delta_{\bm{k}}}{v_F\sqrt{E^2-\Delta_{\bm{k}}^2}}-\frac{\Delta_{\bm{k}}^2v_{\Delta} \sin\theta }{v_F^2\left(E^2-\Delta_{\bm{k}}^2\right)}\right],\label{Terms3}
\end{eqnarray}
\label{allTerms}
\end{subequations}
\end{widetext}
where $\int_{E}d\bm{k}_\parallel$ denotes the integral over the $\left(D-1\right)$-dimensional surface in momentum space at energy $E$.
In the terms depending on the gap, we approximate the gap function by a Taylor series in the momentum components parallel to the Fermi surface, near the node, in $D$-dimensions this is given by
\begin{eqnarray}
\Delta_{\bm{k}}&=\sum\limits_{i=1}^{D-1}\left(v_{\Delta}\right)_i \left(k^i_\parallel-k^{i\left(0\right)}_\parallel\right)+\mathcal{O}\left(k^i_\parallel-k^{i\left(0\right)}_\parallel\right)^2, \label{NearNodegen}
\end{eqnarray}
where $\bm{k}_\parallel^{\left(0\right)}=\hat{\bm{k}}_\parallel^{\left(0\right)} k_\parallel^{\left(0\right)}$ denotes the position of the node. In the case of $D=2$ this gives
\begin{eqnarray}
\Delta_{\bm{k}}&=&v_{\Delta}\cos\theta \left(k_\parallel-k_\parallel^{\left(0\right)}\right)+\mathcal{O}\left(k_\parallel-k_\parallel^{\left(0\right)}\right)^2. \label{NearNode2D}
\end{eqnarray}
Under this approximation for the gap, we arrive at
\begin{subequations}
\begin{eqnarray}
\mathcal{K}_1\left(E\right)&=& \lim\limits_{\delta\rightarrow 0}\frac{E}{2}\left\langle\frac{\text{sgn}\left(\Delta\right)}{v_F\cos\theta}\left[\frac{\pi }{v_{\Delta}} +\frac{\sin \theta }{v_F}\ln \left(\delta\right)\right]\right\rangle_{E}\label{K1acc}\hspace{0.7cm}\\
\mathcal{K}_2\left(E\right)&=& \left\langle v_F^{-1}\right\rangle_{E}\label{K2acc}\\
\mathcal{K}_3\left(E\right)&=& \lim\limits_{\delta\rightarrow 0}\frac{E}{2}\left\langle\frac{\tan \theta }{v_F^2}\right\rangle_{E}\ln \left(\delta\right),
\label{K3acc}
\end{eqnarray}
\end{subequations}
where $\left\langle \ldots \right\rangle_{E}$ denotes the average over the contour(s) of energy $E$.
Note that $\mathcal{K}_1\left(E\right)$ depends on the difference between the averages taken on the segments of the energy contour with positive and negative superconducting gap, while $\mathcal{K}_2$ and $\mathcal{K}_3$ depend only on averages over the entire energy contour. The logarithmically divergent contributions arise due to the vanishing denominators in the expansion terms of Eqs. (\ref{Terms1}) and (\ref{Terms3}), while the terms with square roots in the denominator give a convergent contribution.
In the general case of accidental node placement, the velocity magnitudes $v_{\Delta}$ and $v_{F}$ may vary freely across the energy surface, but as an demonstrative example, in the simplest case, $v_F$ and $v_{\Delta}$ are constant near the node, giving
\begin{eqnarray}
\mathcal{K}_1\left(E\right)&=&0\label{Ccdef}\\
\mathcal{K}_3\left(E\right)&=&\frac{E}{2}\lim\limits_{\delta\rightarrow 0}\left\langle\frac{\tan \theta }{v_F^2}\right\rangle_{E}\ln \left(\delta\right)\label{Bcdef}.
\end{eqnarray}
Both the linear correction to Eq. (\ref{Terms2}) and the zero order contribution to Eq. (\ref{Terms3}) vanish as the gap function is odd with respect to the position of the node, but higher order corrections will diverge, similar to the divergence encountered in Eq. (\ref{K3acc}). In a more realistic model, the velocities $v_F$ and $v_\Delta$, as well as the angle $\theta$ will depend on $\bm{k}$, and so $\mathcal{K}_1\left(E\right)$ may also be nonvanishing. In general, the node in an anisotropic gap function is not required to be near a portion of the gap function where a linear expansion in $k_\parallel$ is valid, in particular the presence of an isotropic component of the gap will shift the node position. Including higher order terms in either the Taylor series for the gap or the expansions of the $\mathcal{K}_i$ is, however, insufficient to remove these divergences.
\subsection{Anisotropic gap with symmetry required nodes}\label{sect:req}
If $\Delta_{\bm{k}}$ transforms as a non-trivial representation of the point group, the nodes are required by symmetry. This implies that $\theta $ vanishes, as the gap function near the node is independent of the direction perpendicular to the Fermi surface. Further, as the node in this case is required to reside on a symmetry axis for the material, $\mathcal{K}_3$ must vanish, given that an equal length of the contour is on either side of the node where the gap changes sign. In this case, $\mathcal{K}_2$ [Eq. (\ref{Terms2})] again gives a non-divergent contribution with the form of the density of states, as the surface integral over the energy contour; and Eq. (\ref{Terms1}) reduces to $\mathcal{K}_1\left(E\right)=\pi E/\left\langle v_{\Delta} v_F\right\rangle$. In this way, we recover the well known result \cite{Coleman,Ketterson&Song} that no Hebel-Slichter peak is observed.
\subsection{Isotropic gap}
In a purely isotropic gap superconductor, the gap function is independent of momentum, so $\bm{v}_\Delta=0$ and $\Delta_{\bm{k}}=\Delta_0$, and the momentum sums in the relaxation rate give the constant energy surface integral $S\left(E\right)=\sum_{\bm{k}_\parallel ,k_\perp\left(E\right)}\frac{1}{v_F\left(\bm{k}\right)}\delta\left[k_\perp -k_\perp\left(E\right)\right]$,
\begin{eqnarray}
\mathcal{K}_1\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{E}{v_F\sqrt{E^2-\Delta_{0}^2}}=\frac{E}{\sqrt{E^2-\Delta_{0}^2}}S\left(E\right)\\
\mathcal{K}_2\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{\sqrt{E^2-\Delta_{0}^2}}{v_F\sqrt{E^2-\Delta_{0}^2} }=S\left(E\right)\\
\mathcal{K}_3\left(E\right) &=& \int_{E}d\bm{k}_\parallel \frac{\Delta_{0}}{v_F\sqrt{E^2-\Delta_{0}^2} }=\frac{\Delta_0}{\sqrt{E^2-\Delta_{0}^2}}S\left(E\right).\hspace*{0.6cm}
\end{eqnarray}
Thus, $\mathcal{K}_2$ is once more non-divergent.
The energy integrals over $\mathcal{K}_1^2$ and $\mathcal{K}_3^2$ result in a logarithmic divergence in the energy domain when $E\sim T \rightarrow\Delta$, giving rise to the Hebel-Slichter peak.
The divergence is in general controlled by one or more of gap anisotropy, the presence of impurities or strong coupling effects \cite{Tinkham, Ketterson&Song, Samokhin}.
This is in marked contrast to the accidental node case, where the peak results from a divergence in the momentum integral, independently of the energy value. The peak observed in the isotropic case arises due to a divergence at a particular energy.
\section{Disorder and Electron-Electron Interactions}
The classical s-wave Hebel-Slicter peak is controlled by disorder, electron-electron interactions, and gap anisotropy. The latter is necessarily present in the case of accidental nodes. Therefore, it is important to understand the affects of the former two effects in the current context.
If we evaluate the energy integrals in Eq. (\ref{GeneralChi1}), first using the delta function to constrain $E_2$ before using the strongly peaked nature of the spectral function to evaluate $E_1\approx E_{\bm{k}}$, we have
\begin{widetext}
\begin{eqnarray}
\chi''_{+-}\left(\bm{q},\omega\right)&=&\sum\limits_{\bm{k}}\left\lbrace \frac{1}{2}\left[1+\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[f\left(E_{\bm{k}}+\omega\right)-f\left(E_{\bm{k}}\right)\right]A_{\bm{k}+\bm{q}}\left( E_{\bm{k}}+\omega\right)\right. \nonumber\\
&&+\frac{1}{4}\left[1-\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[f\left(E_{\bm{k}}+\omega\right)-f\left(E_{\bm{k}}\right)\right]A_{\bm{k}+\bm{q}}\left( -E_{\bm{k}}-\omega\right)
\nonumber\\
&&\left. +\frac{1}{4}\left[1-\frac{\xi_{\bm{k}}\xi_{\bm{k}+\bm{q}}+\Delta_{\bm{k}}\Delta_{\bm{k}+\bm{q}}}{E_{\bm{k}}E_{\bm{k}+\bm{q}}}\right]\left[f\left( \omega -E_{\bm{k}}\right)-\bar{f}\left(E_{\bm{k}}\right)\right]A_{\bm{k}+\bm{q}}\left( \omega - E_{\bm{k}}\right)\right\rbrace , \label{NumChi2}
\end{eqnarray}
\end{widetext}
where the remaining spectral function is evaluated numerically, by introducing a finite Lorentzian broadening, of width $\eta$. This is equivalent to introducing a finite lifetime to the familiar BCS susceptibility \cite{Coleman,Ketterson&Song,Scalapino92}
To evaluate the relaxation rate, Eq. (\ref{T1T_Gen}), each of the nested momentum integrals (over $\bm{k}$ and $\bm{q}$) are performed numerically on a discrete grid, with a small finite frequency, which is then reduced until further variations no longer affect the result, allowing the limit $\omega\rightarrow 0$ to be approximated numerically.
\subsection{Orthorhombic model}\label{analytical}
In order to investigate this behaviour numerically, we consider a simple tight-binding model on an orthorhombic lattice, with nearest neighbour couplings ($t_x$ and $t_y$) allowed to vary independently. The dispersion relation is thus
\begin{eqnarray}
\varepsilon_{\bm{k}} &=& t_x\cos k_x+t_y\cos k_y, \label{dispersion}
\end{eqnarray}
where we have set $a_x=a_y=1$ (where $a_x$ and $a_y$ are the lattice constants in the $x$ and $y$ directions, respectively).
We consider two different symmetry states, a d$_{x^2-y^2}$ gap, with nodes located at $k_y=\pm k_x$, given by
\begin{eqnarray}
\Delta^{\left(x^2-y^2\right)}_{\bm{k}} &=& \frac{\Delta_0}{2}\left(\cos k_x-\cos k_y \right),\label{dgap}
\end{eqnarray}
and a d$_{xy}$ gap, with nodes located on the axes ($k_x=0$ and $k_y=0$)
\begin{eqnarray}
\Delta^{\left(xy\right)}_{\bm{k}} &=& \Delta_0\sin k_x \sin k_y.\label{xygap}
\end{eqnarray}
In both cases, the maximum magnitude of the gap is $|\Delta_0|$. The presence or absence of a divergent peak in the $1/T_1T$ relaxation rate is dependent on the angle between the quasiparticle group velocity and the `gap velocity' at the position of the node on the Fermi surface. This angle gives an approximate measure of the value of the superconducting gap averaged over the Fermi surface, which in turn controls the presence of a Hebel-Slichter divergence. In a material with symmetry required nodes, the angle vanishes, $\theta^{(xy)}=0$ for this model, as does the average of the gap, resulting in an absence of the Hebel-Slichter peak.
Near the $k_x=k_y$ node of the d$_{x^2-y^2}$ symmetry gap, we find
\begin{eqnarray}
\sin\theta^{(x^2-y^2)}&=& \frac{\left(t_x - t_y\right)}{\sqrt{2}\sqrt{t_x^2+t_y^2}}.
\label{theta_x2y2}
\end{eqnarray}
The above calculations estimate the parameters relevant to the divergence for specific `d-wave' gap symmetries, as these are the focus in many families of unconventional superconductor, especially the cuprate and organic superconductors. If the nodes of the gap are accidental, by definition there is no preferred node placement on the Fermi surface and the above case is fine tuned. To explore the possibilities of other node locations, we include a finite isotropic component into the gap function. This results in a shift of the node position on the Fermi surface, while retaining the symmetry properties of the fully anisotropic gap. Additionally, such an isotropic component will alter the magnitude of the average gap on the Fermi surface, unless such effects are negated by the shifted node position. As an example, we consider a d$_{x^2-y^2}$-wave gap with an isotropic component parametrised by a real coefficient, $\alpha$, given by
\begin{eqnarray}
\Delta_{\bm{k}}\left(\alpha\right) &=& \Delta_0\left[\alpha +\left(1-\left|\alpha\right|\right)\frac{\cos k_x-\cos k_y}{2}\right],\nonumber\\\label{s+dwave}
\end{eqnarray}
where $\alpha = \pm1$ corresponds to the conventional isotropic gap, $\alpha = 0$ corresponds to the situations described in the previous section, and the absolute value of $\alpha$ is taken in the prefactor to the second term so that the magnitude of the maximum gap remains constant. We do not consider complex $\alpha$ as this would break time reversal symmetry and is hence detectable by other methods \cite{SigristUeda,BenGroupTh}. Hence,
\begin{widetext}
\begin{eqnarray}
\sin\left[\theta(\alpha , k_y)\right]&=& \frac{\left(t_x - t_y\right)\sin^2 k_y-\frac{4 t_x\alpha}{\left|\alpha\right| -1}\left[\frac{\alpha}{\left|\alpha\right| -1}-\cos k_y\right]}
{\sqrt{2}\sqrt{\left[\left(t^2_x + t_y^2\right)\sin^2 k_y
-\frac{4 t_x \alpha}{\left|\alpha\right|-1}\left(\frac{\alpha}{\left|\alpha\right|-1}-\cos k_y\right)\right]\left[\sin^2 k_y -\frac{2\alpha}{\left|\alpha\right|-1}\left(\frac{\alpha}{\left|\alpha\right|-1}-\cos k_y \right)\right]}}.\label{theta_s+dwave}
\end{eqnarray}
\end{widetext}
Notably, this expression is now explicitly dependent on $k_y$, and therefore the shape and size of the Fermi surface, unlike in the $\alpha=0$ case considered previously. Additionally, it can be seen that, while the isotropic component may enhance the peak in the accidental node case, it can also potentially reduce the peak, depending on the relative magnitudes of the anisotropy $t_y/t_x$, the isotropic component $\alpha$ and the size and position of the Fermi surface, via $k_y$.
Eq. (\ref{theta_s+dwave}) also indicates that, even in the limit of vanishing anisotropy in the hopping parameters ($t_x\rightarrow t_y$), there arises a divergence in the relaxation rate due to the second term in the numerator for finite $\alpha$. This is entirely expected, as the gap has a non-zero average value over the Fermi surface for non-zero $\alpha$. In terms of the angle $\theta$, this can be interpreted as the isotropic component altering the nodal structure of the gap in the Brillouin zone, deforming the surface upon which nodes exist.
We take the temperature dependence of the gap to be given by given by the strong coupling BCS form:
\begin{eqnarray}
\Delta_0\left(T\right)=\frac{\Delta_0}{2}\tanh\left(3\sqrt{\frac{T_c}{T}-1}\right)
\end{eqnarray} with $\Delta_0/2=2.5k_BT_c=0.25t$, typical of a number of unconventional superconductors \cite{BrounkBr,DHS}.
\subsection{Robustness of the Hebel-Slichter-like peak}
\begin{figure
\includegraphics[trim = 5mm 70mm 5mm 80mm, clip, width=0.45\textwidth]{T1T_N300_ortho_tpt4_markers+linesv4.pdf}%
\\
\includegraphics[trim = 5mm 70mm 5mm 80mm, clip, width=0.45\textwidth]{T1T_N300_ortho_tpt4_markers+linesv4_zoom4.pdf}%
\caption{Peak structure in the presence of disorder. The divergence observed in the clean limit, Eq. (\ref{K1acc}-\ref{K3acc}), is controlled by the introduction of disorder, but a clear peak remains even in the limit of large disorder. Top: Orthorhombic model, Eq. (\ref{dispersion}), with $t_y= 0.4t_x$. The relaxation rates in both the isotropic s-wave ($\alpha=1$) and symmetry required ($d_{xy}$) gap cases match conventional expectations with a Hebel-Slichter peak and its absence, respectively. In the accidental node case, we see a peak present at $\alpha=0$, which grows smoothly to the s-wave magnitude with increasing isotropic component. Furthermore, the variation of the peak is also smooth for $\alpha <0$. Interestingly this decreases the peak magnitude, as the angle is decreased in this case, cf. Eq. (\ref{theta_s+dwave}). Bottom: The same data, close to $T_c$, highlighting the peak structure. For these plots, frequency $\omega=5\times 10^{-3}t$, Lorentzian broadening $\eta=10^{-3}t$ (corresponding to a residual resistivity of order $\sim 10~\Omega$\,cm for $a_x,a_y\sim 3$\,\r{A}, relevant to cuprates and other transition metal oxides, up to $\sim 100~\Omega$\,cm for organic materials, with $a_x,a_y\sim 10$\,\r{A}, well above measured values in irradiated crystals \cite{Analytis}), number of grid points $N=300^4$ ($300$ per dimension in the $\bm{q}$ and $\bm{k}$ integrals) and $\left\langle n \right\rangle=0.5$ (quarter filling).}
\label{orthopt4}
\end{figure}
In Fig. \ref{orthopt4}, we show the results of the numerical calculations for the above model with $t_y=0.4t_x$, for various gap symmetries at quarter filling.
In the symmetry required ($\theta=0$) case, $1/T_1T$ decreases immediately below T$_c$, never increasing above the Fermi liquid value, as expected.
For the isotropic gap ($\alpha=1$) and gaps with accidental nodes ($-1/2\leq\alpha\leq1/2$) the logarithmic divergence found in the pure case is controlled by the introduction of disorder for all gaps studied.
Nevertheless, we find clear Hebel-Slichter-like peaks for all of the gaps with accidental nodes studied, indicating that the essential physics of this effect survives even quite strong disorder.
It is interesting to note that the size of the peak varies smoothly with $\alpha$, cf. Eq. (\ref{s+dwave}). In particular the case $\alpha=0$, where there is no isotropic component in the gap, is not special. Indeed the peak is smaller for $\alpha<0$ than it is for $\alpha=0$. This is a straightforward consequence of the anisotropy of the Fermi surface. For $\alpha=-1/4$ the average gap over Fermi surface is less than the average for $\alpha=0$. As $\alpha$ is further decreased this average must vanish and then increase again with the peak for $\alpha=-1$ being identical to that for $\alpha=1$.
\begin{figure
\includegraphics[trim = 10mm 75mm 10mm 60mm, clip, width=0.45\textwidth]{T1T_N300_ortho_tcomp_markers+lines.pdf}%
\caption{Effect of the band structure anisotropy on the relaxation rate $1/T_1T$. Here we plot the calculated $1/T_1T$ for the orthorhombic model, Eq. (\ref{dispersion}), for various values of $t_y/t_x$ for the case of accidental nodes with no isotropic component ($\alpha=0$). The magnitude of the peak initially grows with increasing anisotropy, reaching a maximum value for $t_y=0.4t_x$, before decreasing again. The initial growth arises from the increase in $\theta^{(x^2-y^2)}$, cf. Eq. (\ref{theta_x2y2}). The suppression of the Hebel-Slichter-like peak for $t_y<0.4t_x$ is caused by the proximity to a van Hove singularity when the Fermi surface crosses the Brillouin zone boundary.
Notably, this behaviour is only seen close to $T_c$, where contours with energy $\sim k_BT$ wrap around a significant portion of the Fermi surface. At sufficiently low temperatures $1/T_1T$ also increases monotonically with increasing anisotropy (decreasing $t_y/t_x$). Note that in the normal state $1/T_1T$ depends on the hopping anisotropy, as visible from the spread of the data above $T_c$.
Parameters: $\omega=5\times 10^{-3}t$, $\eta=10^{-3}t$, $N=300^4$ and $\left\langle n \right\rangle=0.5$. }
\label{orthotcomp}
\end{figure}
To better understand the dependence of the peak magnitude on the Fermi surface anisotropy we show the $\alpha=0$ accidental node case for varying hopping anisotropy in Fig. \ref{orthotcomp}. These numerical results should be compared to the analytical prediction that $\sin\theta \propto t_x-t_y$, Eq. (\ref{theta_x2y2}).
At low temperatures increasing the anisotropy always increases $1/T_1T$, consistent with the changes in $\theta$. For weak anisotropies the peak grows, consistent with this prediction. However, a maximum is reached at $t_y=0.4t_x$, further increasing the anisotropy (decreasing $t_y$) decreases the peak immediately below $T_c$. This behaviour is not explained by the variation of $\theta$.
The supression of the Hebel-Slichter-like peak for $t_y<0.4t_x$ is due to the presence of a van Hove singularity in the density of states which approaches the the Fermi energy at quarter filling as $t_y$ is reduced.
Close to $T_c$ the gap is small, $k_BT\gtrsim \Delta_0\left(T\right)$, and contours with energy $\sim\mu\pm k_BT$ wrap around a large segment of the Fermi surface. As a result, such contours include the region of the Fermi surface where the van Hove singularity is relevant, enhancing the spectral weight (density of states) in this region. This, in turn, affects the average of the gap within $\sim k_BT$ of the Fermi surface. In the example considered here, the superconducting gap in the vicinity of the van Hove singularity is of the minority sign of the gap, and thus the van Hove singularity reduces the average gap value over the Fermi surface. In the orthorhombic model, the van Hove singularity arises as the Fermi surface crosses the Brillouin zone boundary ($k_y=\pm\pi$), enhancing the contribution for $\Delta_{\bm{k}} <0$. As the accidental nodes are on the $k_x=\pm k_y$ diagonals, the average of the gap within $\sim k_BT$ of the Fermi surface $\left\langle \Delta_{\bm{k}}\right\rangle_{\mu\pm k_BT
>0$ is then reduced by the contribution due to the van Hove singularity.
Thus, for temperatures close to $T_c$, the enhancement of the spectral density at the van Hove point becomes significant, while it is less relevant at lower temperatures where the contours of energy $\sim k_BT$ are further from the van Hove point.
Such behaviour is not apparent from variation of $\theta$ (see Sec. \ref{accidental}), as the binomial expansion in the derivation of Eqs. (\ref{allTerms}) fails due to the divergence of $1/v_F$ near the van Hove point. The importance of such singularities are, however, apparent from Eqs. (\ref{eq:Kdef}).
In the low temperature regime, where the gap is maximal, the relevant contours are restricted to be near the nodes, well away from the van Hove singularity, and thus the relaxation rate increases smoothly as a function of decreasing $t_y$. In the regime of smaller anisotropy ($t_y\ge0.4t_x$), the effects of the van Hove singularity are not strong enough to overwhelm the effects due to the variation of $\theta$, and the peak size increases smoothly with decreasing $t_y$.
For all levels of anisotropy ($t_y<t_x$), we find the qualitative features observed in the $t_y=0.4t_x$ case largely unchanged, though at very low anisotropy ($t_y\gtrsim 0.95t_x$) the $\alpha = 0$ peak is strongly suppressed and not clearly resolved in the numerics.
\begin{figure}
\includegraphics[trim = 10mm 75mm 10mm 60mm, clip, width=0.45\textwidth]{T1T_N300_ortho_tpt4_RPAv4.pdf}%
\caption{Robustness of the accidental node peak to electron-electron interactions. Orthorhombic model, Eq. (\ref{dispersion}), with $t_y= 0.4t_x$, and $U=2t$ [Eq. (\ref{RPA})]. It is apparent here that the inclusion of electron-electron interactions via the RPA susceptibility does not alter the qualitative features of the previous figures. A clear Hebel-Slichter-like peak is still apparent for all values of $\alpha$ in the accidental node case, though the width of said peaks is reduced, even in the s-wave case ($\alpha=1$). The Fermi liquid relaxation rate also acquires a much stronger temperature dependence. Parameters: $\omega=5\times 10^{-3}t$, $\eta=10^{-3}t$, $N=300^4$ and $\left\langle n \right\rangle=0.5$. }
\label{orthoRPA}
\end{figure}
Finally, to investigate the effects of including electron-electron interactions, we present results for the random phase approximation. The RPA for the magnetic susceptibility is the sum over ladder diagrams \cite{Doniach&Sondheimer}, therefore this treatment includes the vertex corrections that we have neglected above. Explicitly, we replace the magnetic susceptibility by
\begin{eqnarray}
\chi_{RPA}\left(\bm{q},\omega\right)&=&\frac{\chi_{+-}\left(\bm{q},\omega\right)}{1-U\chi_{+-}\left(\bm{q},\omega\right)},\label{RPA}
\end{eqnarray}
where $\chi_{+-}\left(\bm{q},\omega\right)$ is the magnetic susceptibility (in either the superconducting or normal state, as appropriate) in the absence of electron-electron interactions. For simplicity we limit our treatment to a Hubbard-like model with a contact interaction, $U$.
As shown in Fig. \ref{orthoRPA}, the qualitative features of the relaxation rate survive the inclusion of vertex corrections via the RPA susceptibility. Nevertheless it is important to note that the RPA treatment predicts that electron-electron interactions tend to suppress the Hebel-Slichter-like peak.
Beyond vertex corrections electron-electron interactions lead to a temperature dependence for the quasiparticle lifetime. Including such effects, for example via the phenomenological form described in \cite{Jacko}, does not lead to significant changes in $1/T_1T$.
\section{Conclusions}
We have shown that there is a logarithmic divergence in $1/T_1T$ in superconductors with accidental nodes as $T\rightarrow T_c$ from below. The microscopic origin of this divergence is distinct from that of the Hebel-Slichter peak familiar from s-wave superconductors. One signature of this is that the anisotropy in the gap, necessary for accidental nodes, does not control the divergence, as it does in the Hebel-Slichter case. We have confirmed that both impurities and electron-electron interactions can control the divergence, but for reasonable values these effects no not completely suppress the effect.
Thus, we predict a Hebel-Slichter-like peak should be observed in superconductors with accidental nodes. This provides an important test for theories of superconductivity in low symmetry materials that predict the presence of accidental nodes.
\section*{Acknowledgements}
This work was supported by the Australian Research Council (FT13010016 and DP160100060).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,807
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* by the given pattern.
+ * by the given pattern, ignoring the scheme.
* pattern and the given pattern.
* The match pattern string represented by this pattern.
* Returns true if any sub-pattern subsumes the given pattern.
+ * ignoring any of the schemes in the patterns.
* Returns true if any sub-pattern overlaps the given pattern.
+ // formatPermissionStrings ignores any scheme, so only look at the domain.
+ // changes, update the comparePermissions method as needed.
// combinations of host permissions.
+// Tests that an update with less permissions has no warning.
+// result in additional permission warnings.
+// an existing permission does not result in additional permission warnings.
+ // (no new warning) Unchanged permission from old extension.
+ // (no new warning) Different schemes, host should match "*.b" wildcard.
+ // (expect warning) Wildcard was added.
+ // (no new warning) file:-scheme, but host "f" is same as "http://f/".
+ // (expect warning) New permission was added.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,575
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Q: Navbar is not scrolling to sections on page anymore? I added an li to my navbar and all of a sudden whenever I click the menu item, it doesn't scroll anymore. I need to make my navbar scroll to my section with Javascript.
Here is my navbar created with Javascript
const navMenu = document.querySelectorAll("section");
const navList = document.getElementById("navbar__list");
const items = ["Section 1", "Section 2", "Section 3", "Section 4"];
//Build the nav
items.forEach((item, i) => {
const el = document.createElement("a");
el.innerText = item;
el.classList.add("menu-items");
el.setAttribute("id", `menu-${i + 1}`);
el.href = `#section${i + 1}`;
navList.appendChild(el);
const li = document.createElement("li");
li.classList.add("menu-list");
li.appendChild(el);
// Append the list item to the list
navList.appendChild(li);
});
//Make Nav Active when Clicked and scrolls down to section
document.addEventListener("click", function (event) {
let active = document.querySelector(".menu-list.active");
if (active) active.classList.remove("active");
if (event.target.classList.contains("menu-list")) {
event.target.classList.add("active");
}
});
Before I only had the a tag added and I targeted .menu-items instead of .menu-list in the addEventListener, but once I added the li tag to my navbar, the class for li doesn't work. I'm not sure what to edit or change
A: idea of system is that give an id to menu item and using id to scrolling here u forgot to give id to element . I put it then in your click event get id and href to that section with using this id.
items.forEach((item, i) => {
const el = document.createElement("a");
el.innerText = item;
el.classList.add("menu-items");
el.setAttribute("id", `menu-${i + 1}`);
el.href = `#section${i + 1}`;
navList.appendChild(el);
const li = document.createElement("li");
li.classList.add("menu-list");
li.appendChild(el);
li.setAttribute("id", `${i + 1}`);
// Append the list item to the list
navList.appendChild(li);
});
//Make Nav Active when Clicked and scrolls down to section
document.addEventListener("click", function (event) {
let active = document.querySelector(".menu-list.active");
if (active) active.classList.remove("active");
if (event.target.classList.contains("menu-list")) {
event.target.classList.add("active");
console.log(event.target.id);
window.location.href="#section"+event.target.id
}
});
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,760
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{"url":"https:\/\/hal.archives-ouvertes.fr\/hal-01248760","text":"# On tensor products of CSS Codes\n\n2 GRACE - Geometry, arithmetic, algorithms, codes and encryption\nLIX - Laboratoire d'informatique de l'\u00c9cole polytechnique [Palaiseau], Inria Saclay - Ile de France\nAbstract : CSS codes are in one-to-one correspondence with length 3 chain complexes. The latter are naturally endowed with a tensor product which induces a similar operation on the former. We investigate this operation, and in particular its behavior with regard to minimum distances. Given a CSS code $C$, we give a criterion which provides a lower bound on the minimum distance of $C \\otimes D$ for every CSS code $D$. We apply this result to study the behaviour of iterated tensor powers of codes. Such sequences of codes are logarithmically LDPC and we prove in particular that their minimum distances tend generically to infinity. Different known results are reinterpretated in terms of tensor products. Three new families of CSS codes are defined, and their iterated tensor powers produce LDPC sequences of codes with length n, row weight in O(logn) and minimum distances larger than $n^{\\alpha \/2}$ for any $\\alpha<1$. One family produces sequences with dimensions larger than $n^{\\beta}$ for any $\\beta<1$.\nKeywords :\nDocument type :\nJournal articles\nDomain :\n\nCited literature [27 references]\n\nhttps:\/\/hal.archives-ouvertes.fr\/hal-01248760\nContributor : Benjamin Audoux Connect in order to contact the contributor\nSubmitted on : Thursday, October 4, 2018 - 4:17:21 PM\nLast modification on : Wednesday, November 3, 2021 - 9:43:22 AM\nLong-term archiving on: : Saturday, January 5, 2019 - 5:00:40 PM\n\n### File\n\nArticle_On tensor product of C...\nFiles produced by the author(s)\n\n### Citation\n\nBenjamin Audoux, Alain Couvreur. On tensor products of CSS Codes. Annales de l\u2019Institut Henri Poincar\u00e9 (D) Combinatorics, Physics and their Interactions, European Mathematical Society, 2019, 6 (2), pp.239-287. \u27e810.4171\/AIHPD\/71\u27e9. \u27e8hal-01248760v2\u27e9\n\n### Metrics\n\nLes m\u00e9triques sont temporairement indisponibles","date":"2022-01-19 07:59:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.17067638039588928, \"perplexity\": 1695.4503042344686}, \"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-2022-05\/segments\/1642320301264.36\/warc\/CC-MAIN-20220119064554-20220119094554-00433.warc.gz\"}"}
| null | null |
Bitcoin – the potential Pandora's Box of the coin world – has never been short of contention. Whether it be helping the black market or scamming users out of millions, bitcoin is no alien to the front page.
Still, the jury is out on the legality and practicality of bitcoin – leaving it in a current grey area. Nevertheless, there have been several unauthorized bitcoin scams that have become notorious – but, what are the top 7 bitcoin scams? And how can you avoid them?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,015
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This volume provides a comprehensive overview of the contemporary Maghreb. Made up of contributions from leading academics in the field, it highlights specific issues of importance, including international and security affairs.
With profiles of individual countries and regional issues, such as migration, gender, integration, economics, and war in Western Sahara, as well as a section dealing with international relations and the Maghreb, including US and EU foreign policy and security issues, North Africa: Politics, Region, and the Limits of Transformation is a major resource for all students of Middle Eastern Studies and North African Politics.
Policy Reforms in Algeria: Genuine Change or Adjustments?
Emigration, Immigration, and Transit in the Maghreb: Externalization of EU Policy?
The United States, Islamism, Terrorism, and Democracy in the Maghreb: The Predominance of Security?
France and the Maghreb: The End of the Special Relationship?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,482
|
<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="com.nostra13.universalimageloader"
android:versionCode="38"
android:versionName="1.9.3" >
</manifest>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 789
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Q: Click button with spacebar I have a button with an anchor, that I would like to trigger with the spacebar for accessibility reasons. Instead, clicking the spacebar jumps the page down when button is in focus.
<a href="stackoverflow.com">Go to Stack Overflow</a>
I have tried eating the spacebar key:
window.onkeydown = function(e) {
return !(e.keyCode == 32);
};
but of course this is not what I want. I'm not sure if mapping the spacebar key to the enter key is a smart solution, or if its possible. How can I trigger a button with the spacebar using pure JS?
A: You might want to look into a prevent default solution:
window.onkeydown = function(event){
if(event.keyCode === 32) {
event.preventDefault();
document.querySelector('a').click(); //This will trigger a click on the first <a> element.
}
};
That will stop the space bar from performing the default action (to send a space) and then you can add your scroll to command below that inside the function.
A: Give your a link an id and try this:
var link = document.getElementById("link");
document.onkeydown = function (e) {
if (e.keyCode == 32) {
link.click();
}
};
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,532
|
El tiburón sedoso (Carcharhinus falciformis) es una especie de elasmobranquio carcarriniforme de la familia Carcharhinidae que habita latitudes intertropicales.
Características
Carcharhinus faciformis es gris oscuro con reflejos bronceados en el dorso y blanco en el vientre. Las puntas de las aletas dorsales son más oscuras que el resto del cuerpo.
Hábitat
Aunque fundamentalmente pelágico, el tiburón sedoso no se limita a mar abierto y se han registrado casos de avistamientos en profundidades de 18 metros. Se trata de un tiburón muy activo y rápido que prefiere las aguas cálidas (23 °C). Es frecuente encontrarlos cerca de los bordes de plataformas continentales y en los arrecifes de aguas profundas, abundante fuente de alimento. Normalmente, nada hasta profundidades de 500 m, pero se han recogido datos de ejemplares a más profundidad. Este tiburón presenta segregación sexual, que se refleja en el hábito de viajar con congéneres del mismo tamaño.
Distribución geográfica
El tiburón sedoso sudoroso es un tiburón común de las zonas tropicales, subterráneas y subtropicales del océano Atlántico, Pacífico e Índico. En el Atlántico occidental, se distribuye desde Massachusetts a Brasil (incluyendo el Golfo de México y Mar Caribe) y de España a Angola en el Atlántico oriental. En el Océano Índico occidental se puede encontrar en el Mar Rojo y desde Tanzania a Mozambique, incluyendo Madagascar y Comoras, y en el Índico oriental desde las Maldivas y Sri Lanka a Australia Occidental. Se encuentra también de China a Nueva Zelanda en el Pacífico occidental (incluyendo las islas hawaianas), y desde California hasta Chile en el Pacífico Oriental. Es muy común en la Isla Malpelo(Colombia), pero muchas poblaciones se ven amenazadas por el aleteo de tiburones.
Véase también
Anexo:Taxonomía de los tiburones
Referencias
Enciclopedia de tiburones
Enlaces externos
falciformis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,009
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• Preparing test samples and setting-up testing equipment as per MEC testing procedures to ensure accuracy of test results in a timely manner.
• Performing materials, components, and product level "fit for purpose" lab testing and analyzing test data results in a consistent and accurate manner.
• Documenting test results in a clear and concise manner. Data can be tracked for reference in future considerations.
• Ensuring the laboratory and equipment remain in good operating order.
• Assisting in identifying and implementing improvements for test lab processes and fixtures.
• Assisting in the research and development of new testing requirements and best practices to meet the evolving needs of the MEC Brand product.
• Engaging as part of the MEC Brand Design and Development Department as a solution provider to help ensure MEC is able to meet our product level functional and quality commitments.
• Experience with continuous quality improvement systems such as Six Sigma.
• Strong communication skills, both oral and written. The ability to present information well, both verbally and in written form.
• Strong attention to detail, a high level of accuracy, exceptional time management skills.
• Patient, meticulous and methodical approach.
• Curious, engaged and willing to learn.
PLEASE NOTE: Interested candidates are asked to apply on our website only. Please visit www.mec.ca/jobs to set up a profile, apply for this position or browse through current job postings at MEC. Only applications received through our website will be considered. We thank all applicants for their time and interest in MEC, but will only contact those selected for an interview. Deadline to apply is December 15, 2012.
Mountain Equipment Co-op (mec.ca) is Canada's leading retailer of clothing, gear, and services for active lifestyles, including cycling, running, hiking, camping, climbing, canoeing and kayaking, yoga and fitness. Established in 1971, MEC has close to 3.8 million members throughout Canada whom it serves through 16 stores in six provinces as well as online at mec.ca. Widely recognized for its commitment to sustainability, MEC is a member of 1% For The Planet and supports various outdoor recreation and environmental initiatives through its community grants program.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,837
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Mass. fare hikes targeted in Statehouse protest
Apr 5, 2012 at 1:15 PM Apr 5, 2012 at 9:00 PM
An offshoot of the Occupy Boston movement has begun what it says will be a ten-day protest against MBTA fare hikes and service cuts.
Occupy the MBTA established what it called Camp Charlie on the Statehouse steps shortly after the T's board voted Wednesday to raise fares an average 23 percent for holders of automated Charlie Cards on July 1.
Advocates for riders have called on the Legislature to find a long-term solution to the T's money woes.
Senate President Therese Murray, in a speech to business leaders on Thursday, said any long-term fix must involve not only the T but the state's entire transportation network — including roads, bridges and regional transit systems.
She called on the board overseeing the Massachusetts Department to Transportation to develop such a plan.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,537
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The 1946 United States Senate election in Maryland was held on November 5, 1946.
Incumbent Democratic Senator George L. P. Radcliffe ran for a third consecutive term in office, but lost the Democratic primary to Governor of Maryland Herbert O'Conor. O'Conor narrowly defeated Republican D. John Markey to win the open seat.
O'Conor's general election victory and the subsequent recount by a federal Senate Subcommittee were controversial, with each party claiming partisan manipulation by the other.
Democratic primary
Candidates
John Emerson LaVeck
Herbert O'Conor, Governor of Maryland
George L. P. Radcliffe, incumbent Senator since 1935
Results
Republican primary
Candidates
D. John Markey, businessman, U.S. Army veteran, and former Maryland Agricultural College football coach
Roscoe F. Walter
Joseph Allison Wilmer
Results
General election
Results
After the vote, both candidates claimed victory, before the official count declared O'Conor the winner by a margin of 2,232 out of more than 470,000 votes cast. On December 10, 1946, Markey requested the U.S. Senate Special Committee to Investigate Senatorial Campaign Expenditures (now controlled by Republicans after their landslide victories in the 1946 elections) conduct a recount in Baltimore City and Montgomery County, which had used electronic voting machines. He also alleged the O'Conor campaign had committed financing violations. The committee agreed because Maryland was unable to conduct its own official recount and found a variation of about 400 votes. The committee then sought to survey five additional counties that were likely to have irregularities. Markey requested a full recount of the entire state.
In the meantime, O'Conor was sworn into the Senate seat on January 4, 1947, after a slight delay. Throughout the recounts, Markey implored the process be done quickly, and implied that the election evidence could go missing at any moment. In May 1947, upon completion of the recount of the five additional counties, O'Conor still maintained a margin of 1,465 votes.
In the aftermath, Markey complained of the O'Conor administration's control of the state government, the Democratic Party's control of the state since 1864, and law enforcement's failure to prevent polling abuses. By contrast, Democratic Maryland senator Millard Tydings alleged partisan bias on the part of the Republican-led investigating subcommittee. The committee completed its full recount of the state in January 1948, and concluded that O'Conor had secured a 1,624-vote majority.
Results by county
Counties that flipped from Democrat to Republican
Allegany
Anne Arundel
Baltimore (County)
Carroll
Frederick
Somerset
Talbot
Washington
Counties that flipped from Republican to Democrat
Cecil
Harford
St. Mary's
See also
1946 United States Senate elections
1946 United States elections
References
Notes
1946
Maryland
United States Senate
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,929
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